# Euclid

### book elements books propositions pappus edition proclus time title straight

EUCLID. Of the lives of the Greek mathematicians generally very little is known, and among the number Euclid is no exception ; we are ignorant not only of the dates of his birth and death, but also of his parentage, his teachers, and the residence of his early years. In some of the editions of his works, as will be seen, he is called iffegarensis, as if he had been born at Megara in Greece, a mistake which arose from confounding him with another Euclid, a disciple of Socrates. Proclus, the Neoplatonist (412-485 A.D.), is the authority for most of our information regarding Euclid, which is contained in his commentary on the first book of the Elements. He there states that Euclid lived in the time of Ptolemy I., king of Egypt, who reigned from 323 to 285 B.C., that he was younger titan the associates of Plato, but older than Eratosthenes (276-196 n.c.) and Archimedes (287-212 n.c.) Euclid is said to have founded the mathematical school of Alexandria, which was at that time becoming a centre, not only of commerce, but of learning and research, and for this service to the cause of exact science he would have deserved commemoration, even if his writings had not secured him a worthier title to fame. Proclus preserves a reply made by Euclid. to King Ptolemy, who asked whether he could not learn geometry more easily than by studying the Elements" There is no royal road to geometry." Pappus of Alexandria, whose date is rather uncertain, but is probably a century earlier than that of Proclus, says that Euclid was a man of mild and inoffensive temperament, unpretending, and kind to all genuine students of mathematics. This being all that is known of the life and character of Euclid, it only remains therefore to speak of his works.

Among those which have come down to us the most remarkable is the Elements (f,rotxcia). They consist of thirteen books ; two more are frequently added, but there is reason to believe that they are the work of a later mathematician, Hypsicles of Alexandria. At the outset of the first book occur the definitions or explanations of the meanings of the terms employed ; the postulates, which limit the instruments to be used in the constructions to the ruler and the compasses ; and the axioms or common notions, the fundamental principles from which mathematical truths are deduced. The propositions, which consist of both theorems and problems, deal with rectilineal figures, principally the triangle and the parallelogram, and the book concludes with the celebrated Pythagorean theorem and its converse. The second book is occupied with the consideration of the rectangular parallelograms contained by the segments of straight lines, and their relation to certain squares. It contains only two problems, the one to divide a straight line in medial section (" the divine section," as it was afterwards called), and the other which shows how to effect the quadrature of any rectilineal area. The third book, prefaced with a few definitions, discusses the properties of circles. The fourth book contains no theorems. The problems are on the inscription in, and circumscription about, circles of triangles, squares, and certain regular polygons, and on the inscription of circles in, and the circumscription of circles about, some of these figures. Though, in the definitions preliminary to this book, Euclid explains when a rectilineal figure is inscribed in end circumscribed about another rectilineal figure, he has given no proposition showing how in any case such inscription or circumscription may be effected. The equilateral triangle, the square, the regular pentagon, and such regular polygons as can be derived from these, were the only regular figures known to be inscriptible io circle by means of elementary geometry, till Gauss dis- covered, in 1796, that the circumference of a circle could be divided into 17 equal parts. In his Disquisitiones Arithmetiece, published in 1801, it is proved that there can be inscribed in a circle any regular polygon, the number of whose sides is prime, and is denoted by 2"+ 1. Euclid's second book presupposes, that is, depends to some extent upon, the first ; the third presupposes both the first and second ; the fourth presupposes the first three ; arid all four are largely concerned with the discussion of the absolute equality or inequality of certain magnitudes. The fifth book stands alone, depending upon none of the preceding books, and contains the theory of proportion, with respect not merely to geometrical magnitudes, such as lines, angles, areas, solids, but to any magnitudes of which multiples can be formed. The diagrams consist of straight lines, probably for convenience of construction, but the enunciations of the propositions and the reasoning are perfectly general. With the exception of his treatment of parallels, Euclid's doctrine of proportion has been the subject of more discussion than any other part of the Elements. The foundation of the doctrine is the criterion of proportionality laid down in the famous fifth definition. The necessity or the appropriateness of such a criterion can be seen only when the distinction between number and magnitude has been clearly apprehended, or, what amounts to the same thing, when an adequate conception has been formed of incommensurables. The ordinary arithmetical test of proportionality will then be found to suit only certain cases which occur - those, namely-, where the magnitudes considered are commensurable, and if the theory of proportion is to be rigorous and complete (as Euclid's is), it must be extended to incommensurables by the notions of continuity and limits. The difficulty therefore which :s felt by readers of the fifth book in grasping Euclid's doctrine is due mainly to the nature of the subject., and no very material simplification of the full treatment of proportion is possible. The sixth book contains the application of the theory of proportion, mostly to rectilineal figures. In the last proposition, the second part of which is due to Theon, it is noteworthy that the restricted definition of an angle, given in the first book, and adhered to throughout, is tacitly abandoned. The seventh, eighth, and ninth books are arithmetical, that is, treat of the properties of numbers. The definitions relating to them occur at the beginning of the seventh book, and same of these show perhaps the tendency of the Greeks, natural enough to a scientific people with a defective numerical notation, to consider quantity from a geometrical point of view. A. number composed of two factors was called a plane number, one composed of three a solid number, and the factors themselves were called sides. The test by which numbers are recognized to be proportionals is different from that given in the fifth book, for here it requires to be applied only to quantities which are commensurable, namely, integers. The last proposition of the ninth book shows how to construct a perfect number, that is, a number which is equal to the sum of all its divisors ; for example, 6=1 + 2 + 3, 28 =1 + 2 + 4 + 7 +14, &c. The tenth book is the longest of the Elements. It is occupied with the consideration of commensurable and incommensurable magnitudes, and ends with the proposition that the diagonal and the side of a square are incommensurable. With regard to straight lines, Euclid distinguishes between those which are commensurable or incommensurable in length, and those which are so in power, the latter being defined to be straight lines the squares on which have or have not a common measure. There are three sets of definitions to this book, the second set inserted before the forty-ninth proposition, and the third before the eighty-sixth. The eleventh, twelfth, and thirteenth books treat mainly of solid geometry. In the eleventh are given the definitions which serve for the three books, the principal properties of straight lines and planes, of solid angles, and of parallelepipeds. The twelfth book begins with two theorems of plane geometry, and then discusses chiefly the properties of pyramids, cones, and cylinders. The last two propositions relate to spheres, the last being to prove that spheres have to one another the triplicate ratio of their diameters. In this book is exemplified the method of Exhaustions, which is founded on the principle that by taking away from a magnitude more than its half, from the remainder more than its half, and so on, a remainder is at length reached which is less than any assignable magnitude (book x. prop. 1). Other applications of this method, the nearest approach made by the ancients to the differential calculus, are to be found in the works of Archimedes (see his Measurement of the Circle, Conoids and Spheroids, Sphere and Cylinder). The thirteenth book treats of lines divided in extreme and mean ratio, of some regular figures inscribed in circles, and of the five regular solids, the last proposition being to exhibit the edges of these five solids, and to compare them with one another.

The question has often been mooted, to what extent Euclid, in his Elements, is a discoverer or a compiler. To this question no entirely satisfactory answer can be given, for scarcely any of the writings of earlier geometers have come down to our times. We are dependent on Pappus and Proclus for the scanty notices we have of Euclid's predecessors, and of the problems which engaged their attention ; for the solution of problems, and not the discovery of theorems, would seem to have been their principal object. From these authors we learn that the property of the right-angled triangle had been found out, the principles of geometrical analysis laid down, the restriction of constructions in plane geometry to the straight line and the circle agreed upon, the doctrine of proportion, as well as loci, plane and solid, and sonic of the properties of the conic sections investigated, the five regular solids (often called the Platonic bodies) and the relation between the volume of a cone or pyramid and that of its circumscribed cylinder or prism discovered. Elementary works had been written, and the famous problem of the duplication of the cube reduced to the determination of two mean proportionals between two given straight lines. Notwithstanding this amount of discovery, and all that it implied, Euclid must have made a great advance beyond his predecessors (we are told that "he arranged the discoveries of Eudoxus, perfected those of Theretetus, and reduced to invincible demonstration many things that had previously been more loosely proved "), for his' Elements supplanted all similar treatises, and, as Apollonius received the title of "the great geometer," so Euclid has come down to later ages as " the elementator," The first six and, less frequently, the eleventh and twelfth books are the only parts of the Elements which are now read in the schools or universities of the United Kingdom; and, within recent years, strenuous endeavours have been made by the Association for the Improvement of Geometrical Teaching to supersede even these. On the Continent, Euclid has for many years been abandoned, and his place supplied by numerous treatises, certainly not models of geometrical rigour and arrangement. The fact that for twenty centuries the Elements, or parts of them, have held their ground as an introduction to geometry is a proof that they are, at any rate, not unsuitable for such a purpose. They are, speaking generally, not too difficult for novices in the science ; the demonstrations are rigorous, ingenious, and often elegant; the mixture of problems and theorems gives perhaps some variety, and makes their study less monotonous; and, if regard be had merely to the metrical properties of space as distinguished from the graphical, hardly any cardinal geometrical truths are omitted. With these excellences are combined a good many defects, sonic of them inevitable to a system based on a very few axioms and postulates. Thus the arrangement of his propositions seems arbitrary ; associated theorems and problems are not grouped together; the classification, in short, is imperfect. That is the main objection to the retention of Euclid as a school-book. Other objections, not to mention minor blemishes, are the prolixity of his style, arising partly from a defective nomenclature, his tre-atment of parallels depending on an axiom which is not axiomatic, and his sparing use of superposition as a method of proof. A text-book of geometry which shall be free from Euclid's faults, and not contain others of a graver character, and which shall at the same time be better adapted to purposes of elementary instruction, is much to be desired, and remains yet to be written.

Of the thirty-three ancient books subservient to geometrieal analysis, Pappus enumerates first the Data (.1E6o/26,a) of Euclid. He says it contained 90 propositions, the scope of which he describes ; it now consists of 95. It is not easy to explain this discrepancy, unless we suppose that some of the propositions, as they existed in the time of Pappus, have since been split into two, or that what were once scholia have since been erected into propositions. The object of the Patois to show that when certain things - lines, angles, spaces, ratios, cf7c. - are given by hypothesis, certain other things are given, that is, are determinable. The hook, as we are expressly told, and as we may gather from its contents, was intended for the investigation of problems ; and it has been conjectured that Euclid must have extended the method of the Data to the investigation of theorems. What prompts this conjecture is the similarity between the analysis of a theorem and the method, common enough in the Elements, of redoctio ad absurdam, - the one setting out from the supposition that the theorem is true, the other from the supposition that it is false, thence in both cases deducing a chain of consequences which ends in a conclusion previously known to be true or false.

The Introdoct irm to Harmony Mo-aray;.1 `Aptcoyue0, and the Section of the Site (Kararotcil Kaeoves), treat of music. There is good reason for believing that one at any rate, and probably both, of these books are not by Euclid. No mention is made of them by any writer previous to Ptolemy (140 A.D.), or by Ptolemy himself, and in no ancient codex are they ascribed to Euclid.

The PAcenomena (cDaudacva) contains an exposition of the appearances produced by the motion attributed to the celestial sphere. Pappus, in the few remarks prefatory to his sixth book, complains of the faults, both of omission and commission, of writers on astronomy, and cites as an example of the former the second theorem of Euclid's Plo.enomena, whence, and from the interpolation of other proofs, Gregory infers that this treatise is corrupt.

The Optics and Cato) trics ('_rruai, KaT0777pCKCA) are ascribed to Euclid by Proclus, and by Marinus in his preface to the Data, but no mention is made of them by Pappus. This latter circumstance, taken in connexion with the fact that two of the propositions in the sixth book of the Mathematical Collection prove the same things as three in time Optics, is one of the reasons given by Gregory for deeming that work spurious. Several other reasons will be found in Gregory's preface to his edition of Euclid's works.

In some editions of Euclid's works there is given a book on the Divisions of 8v/27.71f:it's, which consists of a few propositions, showing how a straight line may be drawn to divide in a given ratio triangles, quadrilaterals, and pentagons. This was supposed by John Dee of London, who transcribed or translated it, and entrusted it for publication to his friend Federic Commandine of Urbino, to be the treatise of Euclid referred to by Proclus as TO 7rEpi StacpicrEwv /3E/3A/ov. Dec mentions that, in the copy from which he wrote, the book was ascribed to Machomet of Bagdad, and adduces two or three reasons for thinking it to be Euclid's. This opinion, however, he does not seem to have held very strongly, nor does it appear that it was adopted by Commandine. The book does not exist in Greek.

The fragment, in Latin, De _Levi of Ponderosa which is of no value, and was printed at the end of Gregory's edition only in order that nothing might be left out, is mentioned neither by Pappus nor Proclus, and occurs first in Zamberti's edition of 1537. There is no reason for supposing it to be genuine.

The following works attributed to Euclid are not now extant :- Three books on Porisms (11Epi, Ti;;!) Hoptcri,ccirwv) are mentioned both by Pappus and Proclus, and the former gives an abstract of them, with the lemmas assumed. A porism, according to Pappus, was neither a theorem nor a problem, but something of an intermediate form, which yet could be enunciated as a theorem or as a problem. Later geometers, he says, defined it to be a local theorem wanting part of the hypothesis, but this definition he censures as imperfect. After the publication of Commandine's translation of Pappus (1588), many attempts were made to extract from this unsatisfactory description a clear idea of what a porism was, and, with the help of the lemmas, to restore Euclid's books. The mystery, which baffled the penetration even of Edmund llallev, was not resolved till the time of Simson, who, in 1722, gained some insight into the subject, and whose posthumous treatise De Porismatibue appeared in 1776. Simson's views have been objected to by recent French writers, such as M. Paul Breton, and M. Michel Chasles ; but for a discussion of the subject recourse must be had to the article Pomsv. Here it will be sufficient to state Simson's definition, which is, - "A porism is a proposition in which it is proposed to demonstrate that one or more things are given, between which and every one of innumerable other things, not given but assumed according to a given law, a certain relation, described in the proposition, is to be shown to take place ;" and to refer to Simson's Opera Reliqua ; Mayfair's paper OIL the Origin and Investigation of Porisms; Trail's Life of Dr Simson; Breton's Recherches Nouvelles 6111' les Porismes d'Eaclide, and his Question des Purisnies; Vineent's Considerations sur les Porismes; and Charles's Les Trois Livres de Porismes d'Eoclide.

Two books are mentioned, named Tcle-oa, 7pUs firCOVE4G, which is rendered 1.0'207'11»?, ad Supo:ticieni by Commandine and subsequent geometers. These books were subservient to the analysis of loci, but the four lemmas Which refer to them, and which occur at the end of the seventh book of the Nothematical Collection, throw very little light on their contents. Simson's opinion was that they treated of curves of double curvature, and he intended at one time to write a treatise on the subject. (See Trail's /Ale, pp. 60-62, 100-105).

l'appus says that Euclid wrote four books on the Conic Sections (1343AL T 6ro-apa KwycKipv), whi elm A poll oni us amplified, and to which he added other four. It is known that, in the time of Euclid, the parabola was considered as the section of a right-angled cone, the ellipse that of an acute-angled cone, the hyperbola that of an obtuse-angled .one, and that Apolionius was the first who showed that the three sections could be obtained from any cone. There is good ground therefore for supposing that the first four books of Apollonius's Comics, which are still extant, resemble Euclid's Conics even less than Euclid's Elements do those of Eudoxus and Thextetus.

This notice of Euclid would be incomplete without some account of the earliest and the most important editions of his works. Passing over the commentators of the Alexandrian school, the first European translator of any part of Euclid is Boetius (500 A.D.), author of the De Consolatione Philosophic:. His Enclidis Megarensis Geometrice libri duo contain nearly all the definitions of the first three hooks of the Elements, the postulates, and most of the axioms. The enunciations, with diagrams but no proofs, are given of most of the propositions in the first, second, and fourth books, and a few from the third.

Some centuries afterwards, Euclid was translated into Arabic, but the only printed version in that language is the one made of the thirteen books of the Elements by Nasir Al-Tasf (13th century), which appeared at Rome in 1594. Judging from the unusual number of diagrams in this edition, the translation of Euclid's text is probably rather free.

The first printed edition of Euclid was a translation of the fifteen books of the Elements from the Arabic, made, it is supposed, by Adelard of Bath (12th century), with the comments of Campanus of Novara. It appeared at Venice in 1482, printed by Erhardus Ratdolt, and dedicated to the doge Giovanni Mocenigo. This edition represents Euclid very inadequately ; the comments are often foolish, propositions are sometimes omitted, sometimes joined together, useless cases are interpolated, and now and then Euclid's order changed.

The first printed translation from the Greek is that of Bartholomew Zamberti, which appeared at Venice in 1505. Its contents will be seen from the title : Euclidis megaresis philosophi platonici Mathentaticarzqdisciplinarzi lanitoris : Hutment in hoc volumine quiciNN ad mathematicc7, substantia aspirat clem-etorunt libros xiii car expositione Theonis insignis mathematici Quibu .. adjuncta. Deputa.tum scilicet Euclidi volume xiiii cat expositioe Hypsi. Alex. Itidef:a Phaeno. LSPecet. Perspe. cum expositione Theonzs. ac mirandus ille liter Datorunt cum expositioe Pappi Mechanici mama ea Marini dialectici protheoria. Bar. '/,caber. Vene. Interine.

The first printed Greek text was published at Basel, in 1533, with the title EZKA.aoe MA? LE 1K TWV OLV03 Lrovovo-ciiw. It was edited by Simon Grynmus from two MSS. sent to him, the one from Venice by Lazarus Bayfius, and the other from Paris by John Ruellius. The four books of Proclus's commentary are given at the end from an Oxford MS. supplied by John Cla.ymundus.

The English edition, the only one which contains all the extant works attributed to Euclid, is that of Dr David Gregory, published at Oxford in 1703, with the title, 'ANAE1601) Td o-o.Vikera. guce supersunt The text is that of the Basel edition, corrected from the MSS. bequeathed by Sir Henry Savile, and from Savile's annotations on his own copy. The Latin translation, which accompanies the Greek on the same page, is for the most part that of Commandine.

peered in 1814, the second in 1816, and the third in 1818. It contains the Elements and the Data, which are, says the editor, certainly the only works which remain to us of this ever-celebrated geometer. The texts of the Basel and Oxford editions were collated with 23 MSS., one of which belonged to the library of the Vatican, but had been sent to Paris by the Comte de Peluso (Monge). The Vatican MS. was supposed to date from the 9th century; and to its readings Peyrard gave the greatest weight.

What may be called the German edition has the title EincXE(Bov tixooXEre. Euclidis Elementa ex optimis libris in USUM Tironum Greece edita ab Ernesto _Ferdinand° Augnst. It was published at Berlin in two parts, the first of which appeared in 1826, and the second in 1829. All the above-mentioned texts were collated with three other MSS.

Of translations of the Elements into modern languages the number is very large. The first English translation, published at London in 1570, has the title, The Elements of Geometric of the most auncient Philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toting, by 11. Billingsley, Citizen of London. Whereunto are annexed certaine &holies, Annotations, and Inventions, of the best Mathematiciens, both of time past, and in this our age. The first French translation of the whole of the Elements has the title, Les Quinze Livres des Elements d'Euclide. Traduicts de Latin en Francois. Par D. Henrion, Mathematicien. The first edition of it was printed in 1614, and a second, corrected and augmented, was published at Paris in 1623. An Italian translation, with the title, Euclide Megarense acntissimo philosopho solo introduttore Belle Scientie Mathematice. Diligentenzente rassettato, et alla integrita ridotto, per il degno professore di tal Scientie 1Vicolo Tartcdea Brisciano, was published at Venice in 1569 ; a Spanish version, Los Seis Libros primeros de la geometria de Euclides. Traduzidos eve legua Espanola pox Rodrigo camorano, Astrologo y Mathematic°, at Seville in 1576 ; mimic! a Turkish one at Bulak in 1825. Dr Robert Situson's editions of the first six and the eleventh and twelfth books of the Elements, and of the Data, which form the basis of all the modern school texts of Euclid, are so common that it is not considered necessary to describe them.

Authorities. - The authors and editions above referred to ; Fabricii .7? ibliothesq Graxa, vol. iv. ; Murhard'sLitteratur der Mathonalischen Wissenschaften ; Heilbronn el.'s 'Estonia Mathescos Uitiversa: ; Do Morgan's article " Eueleides " in Smith's Dictionary of IlEography and Mythology. (J. S. Al.)

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