one, would have seemed to shew great want of The history of
geometry
before the time of
judgment.
judgment.
William Smith - 1844 - Dictionary of Greek and Roman Antiquities - b
1458;
but which were afterwards given to real persons. Burmann, Anthol. Lat. v. 133, or n. 385, ed.
[CHEIRISOPHUS. ) 1. Eucheir, a relation of Dae- Meyer. )
(W. R. )
dales, and the inventor of painting in Greece, ac- ÈUCHE'RIUS, bishop of Lyons, was born,
cording to Aristotle, is no doubt only a mythical during the latter half of the fourth century, of an
personage. (Plin. vii. 56. )
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. Eu-
bly be considered also a mythical personage, desig- cherius married Gallia, a lady not inferior to him-
nating 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, Corsurtia and
## p. 63 (#79) ##############################################
EUCHERIUS.
63
EUCLEIDES.
165.
At all
Es gises
papir
-acter of
2. ) He
C5th
HARTAS
prolabir
t see
II. &
anus.
Atters a
nes, in cis
3. TH. 14.
m the
eference ta
[P. S. ]
R, No. 3]
f Coera
He took
(Pansi
is called a
ere are two
E. (Apk
. (LS. )
steen elegaze
e indication
Iths witor-
- moši ausard
nich are to be
in compariga
the piece mai
lines
mus amantes?
ne sequenti
ula damnae,
de resemblance
o the Dirse of
Tullia. About the year a. D. 410, while still in the separate tracts are carefully enumerated by
the vigour of his age, he determined to retire from Schönemann, and the greater number of them will
the world, and accordingly betook himself, with be found in the “Chronologia S. insulae Lerinen-
his 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 Lng. Episc. doctiss. Lucubrationes
St. Margaret, where he lived the life of a hermit, cura Joannis Alexandri Brassicani," Basil. fol.
devoting 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. Put. Mar. Lugdun.
During his retirement in this secluded spot, he ac- fol. 1077, vol. vi. p. 822. (Gennad. de Viris. 10.
quired so high a reputation for learning and sanc- c. 63; Schoenemann, Bill. Patrum. Lat. ii. $ 36. )
tity, that he was chosen bishop of Lyons about This Eucherius must not be confounded witb
A. D. 431, a dignity enjoyed by him until his another Gaulish prelate of the same name whe
death, which is believed to have happened in 450, flourished during the carly part of the sixth cen-
under the emperors Valentinianus III. and Marci- tury, and was a member of ccclesiastical councils
Veranius was appointed his successor in held in Gaul during the years a. D. 527, 527, 529.
the episcopal chair, while Salonius became the head The latter, although a bishop, was certainly not
of the church at Geneva.
bishop of Lyons. See Jos. Antelmius, asscrtio pro
The following works bear the name of this pre- unico S. Eucherio Lugdunensi episcopo, Paris, 410.
late : I. De luule Eremi, written about the year 1726.
A. D. 428, in the form of an epistle to Ililarius of There is yet another Eucherius who was bishop
Arles. It would appear that Eucherius, in his of Orleans in the eighth century. [W. R. )
passion for a solitary life, had at one time formed EUCLEIA (Eurheia), a divinity who was wor-
the project of visiting Egypt, that he might profit shipped at Athens, and to whom a sanctuary wns
by the bright example of the anchorets who dedicated there out of the spoils which the Athe-
thronged the deserts near the Nile. He requested nians had taken in the battle of Marathon. (Paus.
information from Cassianus (CASSIANUS), who re i. 14. & 4. ) The goddess was only a personification
plied by addressing to him some of those collationes of the glory which the Athenians liad reaped in
in which are painted in such lively colours the the day of that memorable battle. (Comp. Böckh,
habits and rules pursued by the monks and ere- Corp. Inscript. n. 258. ) Eucleia was also used at
mites of the Thebaid. The enthusiasm excited by Athens as a surname of Artemis, and her sanctuary
these details called forth the letter bearing thic was of an earlier date, for Euchidas died in it.
above title.
(Plut. Arist. 20; Euchidas. ) Plutarch remarks,
2. Epistola paracnetica ad Vulerianum cognatum that many took Eucleia for Artemis, and thus
de contemtu mundi et secularis philosophiae, composed made her the same as Artemis Eucleia, but that
abont A. D. 432, in which the author endeavours others described her as a daughter of Heracles and
to detach his wealthy and magnificent kinsman Myrto, a daughter of Menoetius; and he adds that
from the pomps and vanities of the world. An this Eucleia died as a maiden, and was worshipped
edition with scholia was published by Erasmus at in Boeotia and Locris, where she had an altar and
Basle in 1520.
a statue in every market-place, on which persons on
3. Liber formularum spiritalis intelligentiae ad the point of marrying used to offer sacrifices to her.
Veranium filium, or, as the title sometimes appears, Whether and what connexion there existed be-
De forma spiritalis intellectus, divided into eleven tween the Attic and Boeotian Eucleia is unknown,
chapters, containing an exposition of many phrases though it is probable that the Attic divinity was,
and texts in Scripture upon allegorical, typical, as is remarked above, a mere personification, and
and mystical principles.
consequently quite independent of Eucleia, the
4. Instructionum Libri II. ad Salonium filium. daughter of Heracles. Artemis Eucleia had also a
The first book treats" De Quaestionibus difficilio. temple at Thebes. (Paus. ix. 17. § 1. ) (L. S. ]
ribus Veteris et Novi Testamenti," the second EUCLEIDES (Evrheidns) of ALEXANDRRIA.
contains “ Explicationes nominum Hebraicorum. ” The length of this article will not be blamed by
5. Homiliwe. Those, namely, published by Li- any one who considers that, the sacred writers
vineius at the end of the “Sermones Catechetici excepted, no Greek has been so much read or so
Theodori Studitae," Antverp. , 8vo. 1602.
variously translated as Euclid. To this it may be
The authenticity of the following is very doubtful. added, that there is hardly any book in our lan-
6. Historia Passionis S. Mauritü et Sociurum guage in which the young scholar or the young
Martyrum Legionis Felicis Thebueae Agaunensium. mathematician can find all the information about
7. Exhortatio ad Monachos, the first of three this name which its celebrity would make him
printed by Holstenius in his “ Codex Regularum,” desire to have.
Rom. 1661, p. 89.
Euclid has almost given his own name to the
8. Epitome Operum Cassiani.
science of geometry, in every country in which his
The following are certainly spurious : 1. Com- writings are studied; and yet all we know of his
mentarius in Genesiin, 2. Commentariorum in private history amounts to very little. He lived,
libros Regum Libri IV. 3. Epistola ad Faustinum. according to Proclus (Comm. in Eucl. ii. 4), in the
4. Epistola ad Philonem. 5. Reyula duplex ad time of the first Ptolemy, B. C. 323—283. The
Monachos. 6. Homiliarum Collectio, ascribed in forty years of Ptolemy's reign are probably those
some of the larger collections of the Fathers to of Euclid's age, not of his youth ; for had he been
Eusebius of Emesa, in others to Gallicanus. Eu trained in the school of Alexandria formed by
cherius is, however, known to have composed many Ptolemny, who invited thither men of note, Proclus
homilies; but, with the exception of those men would probably have given us the name of his
tioned above (5), they are believed to have perished. teacher : but tradition rather makes Euciid the
No complete collection of the works of Eucherius founder of the Alexandrian mathematical school
has ever been published. The various editions of Ithan its pupil
. This point is very material to the
wooer is als
t clearly under
reptation of the
rfs who, acord
nans and Gaal,
stock indisside
ultirated from
uction here and
ne fact that most
es were found ia
cherius was con
and sirth cette
the period sbe
the land of ber
idence which caa
Tee of contdence.
ol. iii. p. Luc. and
v. pt. iii. p. 1458;
3, or n. 385, ed
(IF. R)
Lyons, was bers,
urth century, of an
. Valerianus is o
anus who about this
efectus Galliae, and
ܬܐܐ
nperor Avitus. Et
not inferior to his
ad two sons, Salata
ghters, Corsuttia and
## p. 64 (#80) ##############################################
64
EUCLEIDES.
EUCLEIDES.
formation of a just opinion of Euclid's writings ; he | Harless thinks that Eudorus should be read for
was, we see, a younger contemporary of Aristotle Euclid in the passage of Valerius.
(B. C. 384-322) if we suppose him to have been of In the frontispiece to Whiston's translation of
mature age when Ptolemy began to patronise litera- Tacquet's Euclid there is a bust, which is said to
ture: and on this supposition it is not likely that be taken from a brass coin in the possession of
Aristotle's writings, and his logic in particular, Christina of Sweden ; but no such coin appears in
Bhould have been read by Euclid in his youth, the published collection of those in the cabinet of
if at all. To us it seems almost certain, from the the queen of Sweden. Sidonius Apollinaris sors
structure of Euclid's writings, that he had not (Epist. xi. 9) that it was the custom to paint Euclid
read Aristotle : on this supposition, we pass over, with the fingers extended (laxatis), as if in the
as perfectly natural, things which, on the contrary act of measurement.
one, would have seemed to shew great want of The history of geometry before the time of
judgment.
Euclid is given by Proclus, in a manner which
Euclid, says Proclus, was younger than Plato, shews that he is merely making a summary of well
and older than Eratosthenes and Archimedes, the known or at least generally received facts. He
latter of whom mentions him. He was of the begins with the absurd stories so often repeated,
Platonic sect, and well read in its doctrines. He that the Aegyptians were obliged to inrent geo-
collected the Elements, put into order much of metry in order to recover the landmarks which
what Eudoxis had done, completed many things the Nile destroyed year by year, and that the
of Theaetetus, and was the first who reduced Phoenicians were equally obliged to invent arith-
to unobjectionable demonstration the imperfect retic for the wants of their commerce. Thales, he
attempts of his predecessors. It was his an- goes on to say, brought this knowledge into Greece,
swer to Ptolemy, who asked if geometry could and added many things, attempting some in a
not be made easier, that there was no royal road general manner (KaOGAIKUT epox) and some in a
(μη είναι βασιλικήν άτραπον προς γεωμετρίαν). perceptive or sensible manner (αισθητικώτερον).
This piece of wit has had many imitators ; “ Quel Proclus clearly refers to physical discovery in geo-
diable” said a French nobleman to Rohault, his metry, by measurement of instances. Next is
teacher of geometry,
pourrait entendre cela ? ” mentioned Ameristus, the brother of Stesichorns
to which the answer was “ Ce serait un diable qui the poet. Then Pythagoras changed it into the
aurait de la patience. ” A story similar to that of form of a liberal science (maideias en evdépov), took
Euclid is related by Seneca (Ep. 91, cited by Au- higher views of the subject, and investigated his
gust) of Alexander.
theorems immaterially and intellectually (dû Aws
Pappus (lib. vii. in praef. ) states that Euclid was kal voepôs): he also wrote on incommensurable
distinguished by the fairness and kindness of his quantities (arbywv), and on the mundane figures
disposition, particularly towards those who could (the five regular solids).
do anything to advance the mathematical sciences: Barocius, whose Latin edition of Proclus has
but as he is here evidently making a contrast to been generally followed, singularly enough trans-
Apollonius, of whom he more than insinuates a lates doya by quae non explicari possunt, and
directly contrary character, and as he lived more Taylor follows him with " such things as cannot
than four centuries after both, it is difficult to give be explained. ” It is strange that two really learned
credence to his means of knowing so much about editors of Euclid's commentator should have been
either. At the same time we are to remember ignorant of one of Euclid's technical terms. Then
that he had access to many records which are now come Anaxagoras of Clazomenae, and a little after
lost. On the same principle, perhaps, the account him Oenopides of Chios; then Hippocrates of
of Nasir-eddin and other Easterns is not to be Chios, who squared the lunule, and then Theodorus
entirely rejected, who state that Euclid was sprung of Cyrene. Hippocrates is the first writer of ele-
of Greek parents, settled at Tyre; that he lived, at ments who is recorded. Plato then did much for
one time, at Damascus ; that his father's name was geometry by the mathematical character of his
Naucrates, and grandfather's Zenarchus. (August, writings; then Leodamos of Thasus, Archytas of
who cites Gartz, De Interpr. Euc. Arab. ) It is Tarentum, and Theaetetus of Athens, gave a more
against this account that Eutocius of Ascalon never scientific basis (TOTNUOVIKWTépay cústaðWV) to va-
hints at it.
rious theorems ; Neocleides and his disciple Leon
At one time Euclid was universally confounded came after the preceding, the latter of whom increas-
with Euclid of Megara, who lived near a century ed both the extent and utility of the science, in par-
before him, and heard Socrates. Valerius Maximus ticular by finding a test (diopro móv) of whether the
has a story (viii. 12) that those who came to Plato thing proposed be possible or impossible. Eudoxus
about the construction of the celebrated Delian of Cnidus, a little younger than Leon, and the
altar were referred by him to Euclid the geometer. companion of those about Plato (Eudoxus), in-
This story, which must needs be false, since Euclid creased the number of general theorems, added
of Megara, the contemporary of Plato, was not a three proportions to the three already existing, and
geometer, is probably the crigin of the confusion. in the things which concern the section (of the
cone, no doubt) which was started by Plato him-
• This celebrated anecdote breaks off in the self, much increased their number, aud employed
middle of the sentence in the Basle edition of analyses upon them. Amyclas Heracleotes, the
Proclus. Barocius, who had better manuscripts, companion of Plato, Menaechmus, the disciple of
supplies the Latin of it ; and Sir Henry Savile, Eudoxus and of Plato, and his brother Deinostratus,
who had manuscripts of all kinds in his own li- made geometry more perfect. Theudius of Magnesia
brary, quotes it as above, with only éad for apos.
August, in his edition of Euclid, has given this * We cannot well understand whether by Oura-
chapter of Proclus in Greek, but without saying róv Proclus means geometrically soluble, or possible
from whence he has taken it.
in the common sense of the word.
## p. 65 (#81) ##############################################
EUCLEIDES.
65
EUCLEIDES.
generalized many particular propositions. Cyzici- | script supports him: bow, then, did he know?
nus of Athens was his contemporary ; they took He saw that there owht to have been such a deti-
different sides on many common inquiries. Hermo- nition, and he concluded that, therefore, there hud
timus of Colophon added to what had been done been one. Now we by no means uphold Euclid
by Eudoxus and Theaetetus, discovered elementary as an all-sufficient guide to geometry, though we
propositions, and wrote something on loci. Philip feel that it is to himself that we owe the power of
(ó Metalos, others read Meduaios, Barocius reads amending his writings ; and we hope we may pro-
Mendaeus), the follower of Plato, made many ma- test against the assumption that he could not have
thematicnl inquiries connected with his master's crred, whether by omission or commission.
philosophy. Those who write on the history of Some of the characteristics of the Elements are
geo. netry bring the completion of this science thus briefly as follows:-
far. Here Proclus expressly refers to written his First. There is a total absence of distinction
tory, and in another place he particularly mentions between the various ways in which we know the
the history of Eudemus the Peripatetic.
meaning of terms : certainty, and nothing morc, is
This history of Proclus has been much kept in the thing sought. The definition of straightness,
the background, we should almost say discredited, an idea which it is impossible to put into simpler
by editors, who seem to wish it should be thought words, and which is therefore described by a more
that a finished and unassailable system sprung at difficult circumlocution, comes under the same
once from the brain of Euclid ; an armed Minerva heading as the explanation of the word " parallel. ”
from the head of a Jupiter. But Proclus, as much Hence disputes about the correctness or incorrect-
a worshipper as any of them, must have had the ness of many of the definitions.
same bias, and is therefore particularly worthy of Secondly. There is no distinction between pro-
confidence when he cites written history as to positions which require demonstration, and those
what was not done by Euclid. Make the most we which a logician would see to be nothing but
can of his preliminaries, still the thirteen books of different modes of stating a preceding proposition.
the Elements must have been a tremendous advance, When Euclid bas proved that everything which
probably even greater than that contained in the is not A is not B, he does not hold himself entitled
Principia of Newton. But still, to bring the state to infer that every B is A, though the two propo-
of our opinion of this progress down to something sitions are identically the same. Thus, having
short of painful wonder, we are told that demon- shewn that every point of a circle which is not the
stration had been given, that something had been centre is not one from which three equal straighit
written on proportion, something on incommensu- lines can be drawn, he cannot infer that any point
rables, something on loci, something on solids ; from which three equal straight lines are drawn is
that analysis had been applied, that the conic sec- the centre, but has need of a new demonstration.
tions had been thought of, that the Elements had Thus, long before he wants to use book i. prop. 6,
been distinguished from the rest and written on. he has proved it again, and independently.
From what Hippocrates had done, we know that Thirdly. He has not the smallest notion of
the important property of the right-angled triangle admitting any generalized use of a word, or of part-
was known ; we rely much more on the lunules ing with any ordinary notion attached to it.
than on the story about Pythagoras. The dispute Setting out with the conception of an angle rather
about the famous Delian problem had arisen, and as the sharp corner made by the meeting of two
some conventional limit to the instruments of geo- lines than as the magnitude which he afterwards
metry must have been adopted; for on keeping shews how to measure, he never gets rid of that
within them, the difficulty of this problem depends. corner, never admits two right angles to make
It will be convenient to speak separately of the one angle, and still less is able to arrive at the
Elements of Euclid, as to their contents; and after idea of an angle greater than two right angles.
wards to mention them bibliographically, among And when, in the last proposition of the sixth
the other writings The book which passes under book, his definition of proportion absolutely requires
this name, as given by Robert Simson, unexcep that he should reason on angles of even more than
tionable as Elements of Geoinetry, is not calculated four right angles, he takes no notice of this neces-
to give the scholar a proper idea of the elements of sity, and no one can tell whether it was an over-
Euclid ; but it is admirably adapted to confuse, in sight, whether Euclid thought the extension one
the mind of the young student, all those notions of which the student could make for himself, or
sound criticism which his other instructors are whether (which has sometimes struck us as not
endeavouring to instil. The idea that Euclid must unlikely) the elements were his last work, and he
be perfect had got possession of the geometrical did not live to revise them.
world ; accordingly each editor, when he made In one solitary case, Euclid seems to have made
what he took to be an alteration for the better, an omission implying that he recognized that
assumed that he was restoring, not amending, the natural extension of language by which unity is
original. If the books of Livy were to be re considered as a number, and Simson has thought it
written upon the basis of Niebuhr, and the result necessary to supply the omission (see his book v.
declared to be the real text, then Livy would no prop. A), and has shewn himself more Euclid than
more than share the fate of Euclid ; the only dir. Euclid upon the point of all others in which
ference being, that the former would undergo a Euclid's philosophy is defective.
larger quantity of alteration than editors hare seen Fourthly. There is none of that attention to
fit to inflict upon the latter. This is no caricature; the forms of accumcy with which translators bare
2. 9. , Euclid, says Robert Simson, gave, without endeavoured to invest the Elements, thereby giv-
doubt, a definition of compound ratio at the being them that appearance which has made many
ginning of the fifth book, and accordingly he there teachers think it meritorious to insist upon their
inserts, not merely a definition, but, he assures us, pupils remembering the very words of Simson.
the very one which Fuclid gave. Not a single manu- Theorems are found among the definitions : assump-
VOL. II.
## p. 66 (#82) ##############################################
66
EUCLEIDES.
EUCLEIDES.
tions are made which are not formally set down as an assumption, not as to its truth), and that
among the postulates. Things which really ought two straight lines cannot inclose a space. Lastly,
to have been proved are sometimes passed over, under the name of common notions (koival érvola)
and whether this is by mistake, or by intention of are given, either as common to all men or to all
supposing them self-evident, cannot now be known : sciences, such assertions as that things equal to the
for Euclid never refers to previous propositions by same are equal to one another—the whole is greater
name or number, but only by simple re-assertion than its part—&c. Modern editors have put the
without reference; except that occasionally, and last three postulates at the end of the common
chiefly when a negative proposition is referred to, notions, and applied the term ariom (which was
such words as “it has been demonstrated" are not used till after Euclid) to them all. The in-
employed, without further specification.
tention of Euclid seems to have been, to distin-
Fifthly. Euclid never condescends to hint at guish between that which his reader must grant,
the reason why he finds himself obliged to adopt or seek another system, whatever may be his opi-
any particular course. Be the difficulty ever so nion as to the propriety of the assumption, and
great, be removes it without mention of its exist that which there is no question every one will
ence. Accordingly, in many places, the unassisted grant. The modern editor merely distinguishes
student can only see that much trouble is taken, the assumed problem (or construction) from the
without being able to guess why.
assumed theorem. Now there is no such distino
What, then, it may be asked, is the peculiar tion in Euclid as that of problem and theorem ;
merit of the Elements which has caused them to the common term apótagis, translated proposition,
retain their ground to this day? The answer is, includes both, and is the only one used. An im-
that the preceding objections refer to matters mense preponderance of manuscripts, the testi-
which can be easily mended, without any alter-mony of Proclus, the Arabic translations, the
ation of the main parts of the work, and that no summary of Boethius, place the assumptions about
one has ever given 80 easy and natural a chain of right angles and parallels (and most of them, that
geometrical consequences. There is a never erring about two straight lines) among the postulates ;
truth in the results; and, though there may be and this seems most reasonable, for it is certain
here and there a self-evident assumption used in that the first two assumptions can have no claim
demonstration, but not formally noted, there is to rank among common notions or to be placed in
never any the smallest departure from the limit the same list with “ the whole is greater than its
ations of construction which geometers had, from part. "
the time of Plato, imposed upon themselves. The Without describing minutely the contents of
strong inclination of editors, already mentioned, to the first book of the Elements, we may observe
consider Euclid as perfect, and all negligences as that there is an arrangement of the propositions,
the work of unskilful commentators or interpo- which will enable any teacher to divide it into
lators, is in itself a proof of the approximate truth sections. Thus propp. 1–3 extend the power of
of the character they give the work ; to which it construction to the drawing of a circle with any
may be added that editors in general prefer Euclid centre and any radius; 4-8 are the basis of the
as he stands to the alterations of other editors. theory of equal triangles ; 9-12 increase the
The Elements consist of thirteen books written power of construction ; 13—15 are solely on rela-
by Euclid, and two of which it is supposed that tions of angles; 16–21 examine the relations of
Hypsicles is the author. The first four and the parts of one triangle ; 22—23 are additional con-
sixth are on plane geometry; the fifth is on the strictions ; 23—26 augment the doctrine of equal
theory of proportion, and applies to magnitude in triangles; 27—31 contain the theory of parallels;
general ; the seventh, eighth, and ninth, are on 32 stands alone, and gives the relation between
arithmetic; the tenth is on the arithmetical cha- the angles of a triangle; 33-34 give the first
racteristics of the divisions of a straight line; the properties of a parallelogram ; 35–41 consider
eleventh and twelfth are on the elements of solid parallelograms and triangles of equal areas, but
geometry; the thirteenth (and also the fourteenth different forms ; 42–46 apply what precedes to
and fifteenth) are on the regular solids, which augmenting power of construction; 47–48 give
were so much studied among the Platonists as to the celebrated property of a right angled triangle
bear the name of Platonic, and which, according to and its converse. The other books are all capable
Proclus, were the objects on which the Elements of a similar species of subdivision.
were really meant to be written.
The second book shews those properties of the
At the commencement of the first book, under rectangles contained by the parts of divided
the name of definitions (pou), are contained the straight lines, which are so closely connected with
assumption of such notions as the point, line, &c. , the common arithmetical operations of multipli-
and a number of verbal explanations. Then fol. cation and division, that a student or a teacher
low, under the name of postulates or demands who is not fully alive to the existence and diffi-
(aithuata), all that it is thought necessary to culty of incommensurables is apt to think that
state as assumed in geometry. There are six common arithmetic would be as rigorous as geo-
postulates, three of which restrict the amount of metry. Euclid knew better.
construction granted to the joining two points The third book is devoted to the consideration
by a straight line, the indefinite lengthening of a of the properties of the circle, and is much cramped
terminated straight line, and the drawing of a in several places by the imperfect idea already al-
circle with a given centre, and a given distance luded to, which Euclid took of an angle. There
measured from that centre as a radius; the other are some places in which he clearly drew upon
three assume the equality of all right angles, the experimental knowledge of the form of a circle,
much disputed property of two lines, which meet
a third at angles less than two right angles (we * See Penny Cyclopaedia, art. “ Paralleis, " for
mean, of course, much disputed as to its propriety some account of this well-worn subject.
## p. 67 (#83) ##############################################
EIDES.
67
EUCLEIDES.
EUCLEIDES.
us to its truto), and that
1 inclose a space. Lastly,
on notions (kaival éram
mon to all men or to ai
s that—things equal to the
ther—the whole is greater
em editors hare put the
the end of the con
term ariom (which was
1) to them all The in
to have been, to dira
his reader must grans
,
hatever may be his opi
of the assumption, and
juestion every one 2
or merely distinguisbas
a
construction) from the
Jere is no such distiso
problem and theren;
translated properties
,
inly one used. An i
manuscripts, the testi
abic translations, the
ibe assumptions about
ind most of therm, that
umong the postulates ;
nable, for it is certain
ons can have no claim
ons or to be placed i
lole is greater than is
tely the contents of
nts, we may eberte
t of the propositions
jer to divide it in
extend the power
of a circle with any
are the basis of the
2-12 increase the
5 are solely on reis
ine the relations of
are additional cost
ve doctrine of equal
Geory of parallels;'
e relation between
-34 give the Ert
35—41 consda
equal aress, but
what precedes to
on; 47-48 gire
it angled triangle
ks are all capable
properties of the
arts of divided
connected with
ons of multipl
ent of a teacher
stence and as
ot to think that
rigorous as gan
ne considerati
s much cramped
idea already a
1 angle. There
urly dres ypas
orm of a circle
** Paralleis, in
ject
and made tacit assumptions of a kind which are count of it in the Penny Cyclopaedia, article, “ Ir-
rarely met with in his writings.
rational Quantities. ” Euclid has evidently in his
The fourth book treats of regular figures. Eu- mind the intention of classifying incommensurable
clid's original postulates of construction give him, quantities : perhaps the circumference of the circle,
by this time, the power of drawing them of 3, 4, 5, which we know had been an object of inquiry,
and 15 sides or of double, quadruple, &c. , any of was suspected of being incommensurable with its
these numbers, as 6, 12, 24, &c. , 8, 16, &c. &c. diameter ; and hopes were perhaps entertained
The fifth book is on the theory of proportion. that a searching attempt to arrange the incommen-
It refers to all kinds of magnitude, and is wholly surables which ordinary geometry presents might
independent of those which precede. The exist- enable the geometer to say finally to which of them,
ence of incoinmensurable quantities obliges him to if any, the circle belongs. However this may be,
introduce a definition of proportion which seems Euclid investigates, by isolated methods, and in a
at first not only difficult, but uncouth and inele- manner which, unless he had a concealed algebra,
gant; those who have examined other definitions is more astonishing to us than anything in the
know that all which are not defective are but Elements, every possible variety of lines which can
various readings of that of Euclid. The reasons be represented by v (VatVb), a and 6 repre.
for this difficult definition are not alluded to, ac- senting two commensurable lines. He divides lines
cording to his custom ; few students therefore un- which can be represented by this formula into 25
derstand the fifth book at first, and many teachers species, and he succeeds in detecting every possible
decidedly object to make it a part of the species. He shews that every individual of every
course. A distinction should be drawn between species is incommensurable with all the individuals
Euclid's definition and his manner of applying it of every other species ; and also that no line of any
Every one who understands it must see that it is species can belong to that species in two different
an application of arithmetic, and that the defective ways, or for two different sets of values of a and I.
and unwieldy forms of arithmetical expression He shews how to form other classes of incommen-
which never were banished from Greek science, surables, in number how many soever, no one of
need not be the necessary accompaniments of the which can contain an individual line which is com-
modern use of the fifth book. For ourselves, we mensurable with an individual of any other class ;
are satisfied that the only sigorous road to propor- and he demonstrates the incommensurability of a
tion is either through the fifth book, or else square and its diagonal. This book has á com-
through something much more difficult than the pleteness which none of the others (not even the
fifth book need be.
fifth) can boast of: and we could almost suspect
The sixth book applies the theory of propor- that'Euclid, having arranged his materials in his
tion, and adds to the first four books the proposi- own mind, and having completely elaborated the
tions which, for want of it, they could not contain. tenth book, wrote the preceding books after it, and
It discusses the theory of figures of the same form, did not live to revise them thoroughly.
technically called similar. To give an idea of the The eleventh and twelfth books contain the
advance which it makes, we may state that the elements of solid geometry, as to prisms, pyramids,
first book has for its highest point of constructive &c. The duplicate ratio of the diameters is
power the formation of a rectangle upon a given shewn to be that of two circles, the triplicate ratio
base, equal to a given rectilinear figure; that the that of two spheres. Instances occur of the method
second book enables us to turn this rectangle into Of exhaustions, as it has been called, which in the
a square ; but the sixth book empowers us to hands of Archimedes became an instrument of dis-
make a figure of any given rectilinear shape equal covery, producing results which are now usually
to a rectilinear figure of given size, or briefly, to referred to the differential calculus : while in those
construct a figure of the form of one given figure of Euclid it was only the mode of proving proposi-
and of the size of another. It also supplies the tions which must have been seen and believed be-
geometrical form of the solution of a quadratic fore they were proved. The method of these books
equation.
is clear and elegant, with some striking in perfec-
The seventh, eighth, and ninth books cannot tions, which have caused many to abandon them,
have their subjects usefully separated. They treat even among those who allow no substitute for the
of arithmetic, that is, of the fundamental properties first six books. The thirteenth, fourteenth, and
of numbers, on which the rules of arithmetic must fifteenth books are on the five regular solids : and
be founded. But Euclid goes further than is ne- even had they all been written by Euclid (the last
cessary merely to construct a system of computa- two are attributed to Hypsicles), they would but
tion, about which the Greeks had little anxiety. ill bear out the assertion of Proclus, that the regu-
He is able to succeed in shewing that numbers lar solids were the objects with a view to which
which are prime to one another are the least in the Elements were written : unless indeed we are
their ratio, to prove that the number of primes is to suppose that Euclid died before he could com-
infinite, and to point out the rule for constructing plete his intended structure. Proclus was an en-
what are called perfect numbers. When the mo- thusiastic Platonist: Euclid was of that school ;
dem systems began to prevail
, these books of Eu- and the former accordingly attributes to the latter
clid were abandoned to the antiquary: our elemen- a particular regard for what were sometimes called
tary books of arithmetic, which till lately were all, the Platonic bodies. But we think that the author
and now are mostly, systems of mechanical rules, himself of the Elements could hardly have considered
tell us what would have become of geometry if the them as a mere introduction to a favourite specula-
earlier books had shared the same fate.
tion : if he were so blind, we have every reason to
The tenth book is the development of all the suppose that his own contemporaries could have set
power of the preceding ones, geometrical and arith him right. From various indications, it can be col
metical. It is one of the most curious of the Greek lected that the fame of the Elements was almost
speculations : the reader will find a synoptical ac- coeval with their publication ; and by the time of
## p. 68 (#84) ##############################################
68
EUCLEIDES.
EUCLEIDES.
Marinus we learn from that writer that Euclid | epitome of the whole. Theon the younger (of
was called κύριος στοιχειωτής.
Alexandria) lived a little before Proclus (who died
The Data of Euclid should be mentioned in con- about A. D. 485). The latter has made his feeble
nection with the Elements. This is a book contain- commentary on the first book valuable by its his-
ing a hundred propositions of a peculiar and limited torical information, and was something of a lumi-
intent. Some writers have professed to see in it a nary in ages more dark than his own. But Tbeon
key to the geometrical analysis of the ancients, in was a light of another sort, and his name has
which they have greatly the advantage of us. played a conspicuous and singular part in the his-
When there is a problem to solve, it is undoubtediy tory of Euclid's writings. lle gave a new edition
advantageous to have a rapid perception of the steps of Euclid, with some slight additions and altera-
which will reach the result, if they can be succes- tions : he tells us so himself, and uses the word
sively made. Given A, B, and C, to find D: one éxdoois, as applied to his own edition, in his com-
person may be completely at a loss how to proceed; mentary on Ptolemy. He also informs us that the
another may see almost intuitively that when A, part which relates to the sectors in the last proper
B, and C are given, E can be found; from which sition of the sixth book is his own addition: and
it may be that the first person, had he perceived it, it is found in all the manuscripts following the
would have immediately found D. The formation rep deu deixar with which Euclid always ends.
of data consequential, as our ancestors would per- Alexander Aphrodisiensis ( Comment, in priora
haps have called them, things not absolutely given, Analyt. Aristot. ) mentions as the fourth of the
but the gift of which is implied in, and necessarily tenth book that which is the fifth in all manu-
follows from, that which is given, is the object of scripts. Again, in several manuscripts the whole
the hundred propositions above mentioned. Thus, work is beaded as åk Two Obwros ouvovoiav. We
when a straight line of given length is intercepted shall presently see to what this led: but now we
between two given parallels, one of these proposi- must remark that Proclus does not mention Theon
tions shews that the angle it makes with the pa- at all; from which, since both were Platonists re-
rallels is given in magnitude. There is not much siding at Alexandria, and Proclus had probably
more in this book of Data than an intelligent stu- seen Theon in his younger days, we must either
dent picks up from the Elements themselves; on infer some quarrel between the two, or, which is
which account we cannot consider it as a great step perhaps more likely, presume that Theon's altera-
in geometrical analysis. The operations of thought tions were very slight.
which it requires are indispensable, but they are The two books of Geometry left by Boethits
contained elsewhere. At the same time we cannot contain nothing but enunciations and diagrams
deny that the Data might have fixed in the mind from the first four books of Euclid. The assertion
of a Greek, with greater strength than the Ele- of Boethius that Euclid only arranged, and that
ments themselves, notions upon consequential data the discovery and demonstration were the work of
which the moderns acquire from the application of others, probably contributed to the notions about
arithmetic and algebra : perhaps it was the percep-Theon presently described. Until the restoration
tion of this which dictated the opinion about the of the Elements by translation from the Arabic,
value of the book of Data in analysis.
this work of Boethius was the only European
While on this subject, it may be useful to re-treatise on geometry, as far as is known.
mind the reader how difficult it is to judge of the The Arabic translations of Euclid began to be
character of Euclid's writings, as far as his own made under the caliphs Haroun al Raschid and
merits are concerned, ignorant as we are of the Al Mamun; by their time, the very name of Eu-
precise purpose with which any one was written. clid had almost disappeared from the West. But
For instance : was he merely shewing his contem-nearly one hundred and fifty years followed the
poraries that a connected system of demonstration capture of Egypt by the Mohammedans before the
might be made without taking more than a certain laiter began to profit by the knowledge of the
number of postulates out of a collection, the neces- Greeks. After this time, the works of the geome
sity of each of which had been advocated by some ters were sedulously translated, and a great im-
and denied by others?
but which were afterwards given to real persons. Burmann, Anthol. Lat. v. 133, or n. 385, ed.
[CHEIRISOPHUS. ) 1. Eucheir, a relation of Dae- Meyer. )
(W. R. )
dales, and the inventor of painting in Greece, ac- ÈUCHE'RIUS, bishop of Lyons, was born,
cording to Aristotle, is no doubt only a mythical during the latter half of the fourth century, of an
personage. (Plin. vii. 56. )
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. Eu-
bly be considered also a mythical personage, desig- cherius married Gallia, a lady not inferior to him-
nating 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, Corsurtia and
## p. 63 (#79) ##############################################
EUCHERIUS.
63
EUCLEIDES.
165.
At all
Es gises
papir
-acter of
2. ) He
C5th
HARTAS
prolabir
t see
II. &
anus.
Atters a
nes, in cis
3. TH. 14.
m the
eference ta
[P. S. ]
R, No. 3]
f Coera
He took
(Pansi
is called a
ere are two
E. (Apk
. (LS. )
steen elegaze
e indication
Iths witor-
- moši ausard
nich are to be
in compariga
the piece mai
lines
mus amantes?
ne sequenti
ula damnae,
de resemblance
o the Dirse of
Tullia. About the year a. D. 410, while still in the separate tracts are carefully enumerated by
the vigour of his age, he determined to retire from Schönemann, and the greater number of them will
the world, and accordingly betook himself, with be found in the “Chronologia S. insulae Lerinen-
his 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 Lng. Episc. doctiss. Lucubrationes
St. Margaret, where he lived the life of a hermit, cura Joannis Alexandri Brassicani," Basil. fol.
devoting 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. Put. Mar. Lugdun.
During his retirement in this secluded spot, he ac- fol. 1077, vol. vi. p. 822. (Gennad. de Viris. 10.
quired so high a reputation for learning and sanc- c. 63; Schoenemann, Bill. Patrum. Lat. ii. $ 36. )
tity, that he was chosen bishop of Lyons about This Eucherius must not be confounded witb
A. D. 431, a dignity enjoyed by him until his another Gaulish prelate of the same name whe
death, which is believed to have happened in 450, flourished during the carly part of the sixth cen-
under the emperors Valentinianus III. and Marci- tury, and was a member of ccclesiastical councils
Veranius was appointed his successor in held in Gaul during the years a. D. 527, 527, 529.
the episcopal chair, while Salonius became the head The latter, although a bishop, was certainly not
of the church at Geneva.
bishop of Lyons. See Jos. Antelmius, asscrtio pro
The following works bear the name of this pre- unico S. Eucherio Lugdunensi episcopo, Paris, 410.
late : I. De luule Eremi, written about the year 1726.
A. D. 428, in the form of an epistle to Ililarius of There is yet another Eucherius who was bishop
Arles. It would appear that Eucherius, in his of Orleans in the eighth century. [W. R. )
passion for a solitary life, had at one time formed EUCLEIA (Eurheia), a divinity who was wor-
the project of visiting Egypt, that he might profit shipped at Athens, and to whom a sanctuary wns
by the bright example of the anchorets who dedicated there out of the spoils which the Athe-
thronged the deserts near the Nile. He requested nians had taken in the battle of Marathon. (Paus.
information from Cassianus (CASSIANUS), who re i. 14. & 4. ) The goddess was only a personification
plied by addressing to him some of those collationes of the glory which the Athenians liad reaped in
in which are painted in such lively colours the the day of that memorable battle. (Comp. Böckh,
habits and rules pursued by the monks and ere- Corp. Inscript. n. 258. ) Eucleia was also used at
mites of the Thebaid. The enthusiasm excited by Athens as a surname of Artemis, and her sanctuary
these details called forth the letter bearing thic was of an earlier date, for Euchidas died in it.
above title.
(Plut. Arist. 20; Euchidas. ) Plutarch remarks,
2. Epistola paracnetica ad Vulerianum cognatum that many took Eucleia for Artemis, and thus
de contemtu mundi et secularis philosophiae, composed made her the same as Artemis Eucleia, but that
abont A. D. 432, in which the author endeavours others described her as a daughter of Heracles and
to detach his wealthy and magnificent kinsman Myrto, a daughter of Menoetius; and he adds that
from the pomps and vanities of the world. An this Eucleia died as a maiden, and was worshipped
edition with scholia was published by Erasmus at in Boeotia and Locris, where she had an altar and
Basle in 1520.
a statue in every market-place, on which persons on
3. Liber formularum spiritalis intelligentiae ad the point of marrying used to offer sacrifices to her.
Veranium filium, or, as the title sometimes appears, Whether and what connexion there existed be-
De forma spiritalis intellectus, divided into eleven tween the Attic and Boeotian Eucleia is unknown,
chapters, containing an exposition of many phrases though it is probable that the Attic divinity was,
and texts in Scripture upon allegorical, typical, as is remarked above, a mere personification, and
and mystical principles.
consequently quite independent of Eucleia, the
4. Instructionum Libri II. ad Salonium filium. daughter of Heracles. Artemis Eucleia had also a
The first book treats" De Quaestionibus difficilio. temple at Thebes. (Paus. ix. 17. § 1. ) (L. S. ]
ribus Veteris et Novi Testamenti," the second EUCLEIDES (Evrheidns) of ALEXANDRRIA.
contains “ Explicationes nominum Hebraicorum. ” The length of this article will not be blamed by
5. Homiliwe. Those, namely, published by Li- any one who considers that, the sacred writers
vineius at the end of the “Sermones Catechetici excepted, no Greek has been so much read or so
Theodori Studitae," Antverp. , 8vo. 1602.
variously translated as Euclid. To this it may be
The authenticity of the following is very doubtful. added, that there is hardly any book in our lan-
6. Historia Passionis S. Mauritü et Sociurum guage in which the young scholar or the young
Martyrum Legionis Felicis Thebueae Agaunensium. mathematician can find all the information about
7. Exhortatio ad Monachos, the first of three this name which its celebrity would make him
printed by Holstenius in his “ Codex Regularum,” desire to have.
Rom. 1661, p. 89.
Euclid has almost given his own name to the
8. Epitome Operum Cassiani.
science of geometry, in every country in which his
The following are certainly spurious : 1. Com- writings are studied; and yet all we know of his
mentarius in Genesiin, 2. Commentariorum in private history amounts to very little. He lived,
libros Regum Libri IV. 3. Epistola ad Faustinum. according to Proclus (Comm. in Eucl. ii. 4), in the
4. Epistola ad Philonem. 5. Reyula duplex ad time of the first Ptolemy, B. C. 323—283. The
Monachos. 6. Homiliarum Collectio, ascribed in forty years of Ptolemy's reign are probably those
some of the larger collections of the Fathers to of Euclid's age, not of his youth ; for had he been
Eusebius of Emesa, in others to Gallicanus. Eu trained in the school of Alexandria formed by
cherius is, however, known to have composed many Ptolemny, who invited thither men of note, Proclus
homilies; but, with the exception of those men would probably have given us the name of his
tioned above (5), they are believed to have perished. teacher : but tradition rather makes Euciid the
No complete collection of the works of Eucherius founder of the Alexandrian mathematical school
has ever been published. The various editions of Ithan its pupil
. This point is very material to the
wooer is als
t clearly under
reptation of the
rfs who, acord
nans and Gaal,
stock indisside
ultirated from
uction here and
ne fact that most
es were found ia
cherius was con
and sirth cette
the period sbe
the land of ber
idence which caa
Tee of contdence.
ol. iii. p. Luc. and
v. pt. iii. p. 1458;
3, or n. 385, ed
(IF. R)
Lyons, was bers,
urth century, of an
. Valerianus is o
anus who about this
efectus Galliae, and
ܬܐܐ
nperor Avitus. Et
not inferior to his
ad two sons, Salata
ghters, Corsuttia and
## p. 64 (#80) ##############################################
64
EUCLEIDES.
EUCLEIDES.
formation of a just opinion of Euclid's writings ; he | Harless thinks that Eudorus should be read for
was, we see, a younger contemporary of Aristotle Euclid in the passage of Valerius.
(B. C. 384-322) if we suppose him to have been of In the frontispiece to Whiston's translation of
mature age when Ptolemy began to patronise litera- Tacquet's Euclid there is a bust, which is said to
ture: and on this supposition it is not likely that be taken from a brass coin in the possession of
Aristotle's writings, and his logic in particular, Christina of Sweden ; but no such coin appears in
Bhould have been read by Euclid in his youth, the published collection of those in the cabinet of
if at all. To us it seems almost certain, from the the queen of Sweden. Sidonius Apollinaris sors
structure of Euclid's writings, that he had not (Epist. xi. 9) that it was the custom to paint Euclid
read Aristotle : on this supposition, we pass over, with the fingers extended (laxatis), as if in the
as perfectly natural, things which, on the contrary act of measurement.
one, would have seemed to shew great want of The history of geometry before the time of
judgment.
Euclid is given by Proclus, in a manner which
Euclid, says Proclus, was younger than Plato, shews that he is merely making a summary of well
and older than Eratosthenes and Archimedes, the known or at least generally received facts. He
latter of whom mentions him. He was of the begins with the absurd stories so often repeated,
Platonic sect, and well read in its doctrines. He that the Aegyptians were obliged to inrent geo-
collected the Elements, put into order much of metry in order to recover the landmarks which
what Eudoxis had done, completed many things the Nile destroyed year by year, and that the
of Theaetetus, and was the first who reduced Phoenicians were equally obliged to invent arith-
to unobjectionable demonstration the imperfect retic for the wants of their commerce. Thales, he
attempts of his predecessors. It was his an- goes on to say, brought this knowledge into Greece,
swer to Ptolemy, who asked if geometry could and added many things, attempting some in a
not be made easier, that there was no royal road general manner (KaOGAIKUT epox) and some in a
(μη είναι βασιλικήν άτραπον προς γεωμετρίαν). perceptive or sensible manner (αισθητικώτερον).
This piece of wit has had many imitators ; “ Quel Proclus clearly refers to physical discovery in geo-
diable” said a French nobleman to Rohault, his metry, by measurement of instances. Next is
teacher of geometry,
pourrait entendre cela ? ” mentioned Ameristus, the brother of Stesichorns
to which the answer was “ Ce serait un diable qui the poet. Then Pythagoras changed it into the
aurait de la patience. ” A story similar to that of form of a liberal science (maideias en evdépov), took
Euclid is related by Seneca (Ep. 91, cited by Au- higher views of the subject, and investigated his
gust) of Alexander.
theorems immaterially and intellectually (dû Aws
Pappus (lib. vii. in praef. ) states that Euclid was kal voepôs): he also wrote on incommensurable
distinguished by the fairness and kindness of his quantities (arbywv), and on the mundane figures
disposition, particularly towards those who could (the five regular solids).
do anything to advance the mathematical sciences: Barocius, whose Latin edition of Proclus has
but as he is here evidently making a contrast to been generally followed, singularly enough trans-
Apollonius, of whom he more than insinuates a lates doya by quae non explicari possunt, and
directly contrary character, and as he lived more Taylor follows him with " such things as cannot
than four centuries after both, it is difficult to give be explained. ” It is strange that two really learned
credence to his means of knowing so much about editors of Euclid's commentator should have been
either. At the same time we are to remember ignorant of one of Euclid's technical terms. Then
that he had access to many records which are now come Anaxagoras of Clazomenae, and a little after
lost. On the same principle, perhaps, the account him Oenopides of Chios; then Hippocrates of
of Nasir-eddin and other Easterns is not to be Chios, who squared the lunule, and then Theodorus
entirely rejected, who state that Euclid was sprung of Cyrene. Hippocrates is the first writer of ele-
of Greek parents, settled at Tyre; that he lived, at ments who is recorded. Plato then did much for
one time, at Damascus ; that his father's name was geometry by the mathematical character of his
Naucrates, and grandfather's Zenarchus. (August, writings; then Leodamos of Thasus, Archytas of
who cites Gartz, De Interpr. Euc. Arab. ) It is Tarentum, and Theaetetus of Athens, gave a more
against this account that Eutocius of Ascalon never scientific basis (TOTNUOVIKWTépay cústaðWV) to va-
hints at it.
rious theorems ; Neocleides and his disciple Leon
At one time Euclid was universally confounded came after the preceding, the latter of whom increas-
with Euclid of Megara, who lived near a century ed both the extent and utility of the science, in par-
before him, and heard Socrates. Valerius Maximus ticular by finding a test (diopro móv) of whether the
has a story (viii. 12) that those who came to Plato thing proposed be possible or impossible. Eudoxus
about the construction of the celebrated Delian of Cnidus, a little younger than Leon, and the
altar were referred by him to Euclid the geometer. companion of those about Plato (Eudoxus), in-
This story, which must needs be false, since Euclid creased the number of general theorems, added
of Megara, the contemporary of Plato, was not a three proportions to the three already existing, and
geometer, is probably the crigin of the confusion. in the things which concern the section (of the
cone, no doubt) which was started by Plato him-
• This celebrated anecdote breaks off in the self, much increased their number, aud employed
middle of the sentence in the Basle edition of analyses upon them. Amyclas Heracleotes, the
Proclus. Barocius, who had better manuscripts, companion of Plato, Menaechmus, the disciple of
supplies the Latin of it ; and Sir Henry Savile, Eudoxus and of Plato, and his brother Deinostratus,
who had manuscripts of all kinds in his own li- made geometry more perfect. Theudius of Magnesia
brary, quotes it as above, with only éad for apos.
August, in his edition of Euclid, has given this * We cannot well understand whether by Oura-
chapter of Proclus in Greek, but without saying róv Proclus means geometrically soluble, or possible
from whence he has taken it.
in the common sense of the word.
## p. 65 (#81) ##############################################
EUCLEIDES.
65
EUCLEIDES.
generalized many particular propositions. Cyzici- | script supports him: bow, then, did he know?
nus of Athens was his contemporary ; they took He saw that there owht to have been such a deti-
different sides on many common inquiries. Hermo- nition, and he concluded that, therefore, there hud
timus of Colophon added to what had been done been one. Now we by no means uphold Euclid
by Eudoxus and Theaetetus, discovered elementary as an all-sufficient guide to geometry, though we
propositions, and wrote something on loci. Philip feel that it is to himself that we owe the power of
(ó Metalos, others read Meduaios, Barocius reads amending his writings ; and we hope we may pro-
Mendaeus), the follower of Plato, made many ma- test against the assumption that he could not have
thematicnl inquiries connected with his master's crred, whether by omission or commission.
philosophy. Those who write on the history of Some of the characteristics of the Elements are
geo. netry bring the completion of this science thus briefly as follows:-
far. Here Proclus expressly refers to written his First. There is a total absence of distinction
tory, and in another place he particularly mentions between the various ways in which we know the
the history of Eudemus the Peripatetic.
meaning of terms : certainty, and nothing morc, is
This history of Proclus has been much kept in the thing sought. The definition of straightness,
the background, we should almost say discredited, an idea which it is impossible to put into simpler
by editors, who seem to wish it should be thought words, and which is therefore described by a more
that a finished and unassailable system sprung at difficult circumlocution, comes under the same
once from the brain of Euclid ; an armed Minerva heading as the explanation of the word " parallel. ”
from the head of a Jupiter. But Proclus, as much Hence disputes about the correctness or incorrect-
a worshipper as any of them, must have had the ness of many of the definitions.
same bias, and is therefore particularly worthy of Secondly. There is no distinction between pro-
confidence when he cites written history as to positions which require demonstration, and those
what was not done by Euclid. Make the most we which a logician would see to be nothing but
can of his preliminaries, still the thirteen books of different modes of stating a preceding proposition.
the Elements must have been a tremendous advance, When Euclid bas proved that everything which
probably even greater than that contained in the is not A is not B, he does not hold himself entitled
Principia of Newton. But still, to bring the state to infer that every B is A, though the two propo-
of our opinion of this progress down to something sitions are identically the same. Thus, having
short of painful wonder, we are told that demon- shewn that every point of a circle which is not the
stration had been given, that something had been centre is not one from which three equal straighit
written on proportion, something on incommensu- lines can be drawn, he cannot infer that any point
rables, something on loci, something on solids ; from which three equal straight lines are drawn is
that analysis had been applied, that the conic sec- the centre, but has need of a new demonstration.
tions had been thought of, that the Elements had Thus, long before he wants to use book i. prop. 6,
been distinguished from the rest and written on. he has proved it again, and independently.
From what Hippocrates had done, we know that Thirdly. He has not the smallest notion of
the important property of the right-angled triangle admitting any generalized use of a word, or of part-
was known ; we rely much more on the lunules ing with any ordinary notion attached to it.
than on the story about Pythagoras. The dispute Setting out with the conception of an angle rather
about the famous Delian problem had arisen, and as the sharp corner made by the meeting of two
some conventional limit to the instruments of geo- lines than as the magnitude which he afterwards
metry must have been adopted; for on keeping shews how to measure, he never gets rid of that
within them, the difficulty of this problem depends. corner, never admits two right angles to make
It will be convenient to speak separately of the one angle, and still less is able to arrive at the
Elements of Euclid, as to their contents; and after idea of an angle greater than two right angles.
wards to mention them bibliographically, among And when, in the last proposition of the sixth
the other writings The book which passes under book, his definition of proportion absolutely requires
this name, as given by Robert Simson, unexcep that he should reason on angles of even more than
tionable as Elements of Geoinetry, is not calculated four right angles, he takes no notice of this neces-
to give the scholar a proper idea of the elements of sity, and no one can tell whether it was an over-
Euclid ; but it is admirably adapted to confuse, in sight, whether Euclid thought the extension one
the mind of the young student, all those notions of which the student could make for himself, or
sound criticism which his other instructors are whether (which has sometimes struck us as not
endeavouring to instil. The idea that Euclid must unlikely) the elements were his last work, and he
be perfect had got possession of the geometrical did not live to revise them.
world ; accordingly each editor, when he made In one solitary case, Euclid seems to have made
what he took to be an alteration for the better, an omission implying that he recognized that
assumed that he was restoring, not amending, the natural extension of language by which unity is
original. If the books of Livy were to be re considered as a number, and Simson has thought it
written upon the basis of Niebuhr, and the result necessary to supply the omission (see his book v.
declared to be the real text, then Livy would no prop. A), and has shewn himself more Euclid than
more than share the fate of Euclid ; the only dir. Euclid upon the point of all others in which
ference being, that the former would undergo a Euclid's philosophy is defective.
larger quantity of alteration than editors hare seen Fourthly. There is none of that attention to
fit to inflict upon the latter. This is no caricature; the forms of accumcy with which translators bare
2. 9. , Euclid, says Robert Simson, gave, without endeavoured to invest the Elements, thereby giv-
doubt, a definition of compound ratio at the being them that appearance which has made many
ginning of the fifth book, and accordingly he there teachers think it meritorious to insist upon their
inserts, not merely a definition, but, he assures us, pupils remembering the very words of Simson.
the very one which Fuclid gave. Not a single manu- Theorems are found among the definitions : assump-
VOL. II.
## p. 66 (#82) ##############################################
66
EUCLEIDES.
EUCLEIDES.
tions are made which are not formally set down as an assumption, not as to its truth), and that
among the postulates. Things which really ought two straight lines cannot inclose a space. Lastly,
to have been proved are sometimes passed over, under the name of common notions (koival érvola)
and whether this is by mistake, or by intention of are given, either as common to all men or to all
supposing them self-evident, cannot now be known : sciences, such assertions as that things equal to the
for Euclid never refers to previous propositions by same are equal to one another—the whole is greater
name or number, but only by simple re-assertion than its part—&c. Modern editors have put the
without reference; except that occasionally, and last three postulates at the end of the common
chiefly when a negative proposition is referred to, notions, and applied the term ariom (which was
such words as “it has been demonstrated" are not used till after Euclid) to them all. The in-
employed, without further specification.
tention of Euclid seems to have been, to distin-
Fifthly. Euclid never condescends to hint at guish between that which his reader must grant,
the reason why he finds himself obliged to adopt or seek another system, whatever may be his opi-
any particular course. Be the difficulty ever so nion as to the propriety of the assumption, and
great, be removes it without mention of its exist that which there is no question every one will
ence. Accordingly, in many places, the unassisted grant. The modern editor merely distinguishes
student can only see that much trouble is taken, the assumed problem (or construction) from the
without being able to guess why.
assumed theorem. Now there is no such distino
What, then, it may be asked, is the peculiar tion in Euclid as that of problem and theorem ;
merit of the Elements which has caused them to the common term apótagis, translated proposition,
retain their ground to this day? The answer is, includes both, and is the only one used. An im-
that the preceding objections refer to matters mense preponderance of manuscripts, the testi-
which can be easily mended, without any alter-mony of Proclus, the Arabic translations, the
ation of the main parts of the work, and that no summary of Boethius, place the assumptions about
one has ever given 80 easy and natural a chain of right angles and parallels (and most of them, that
geometrical consequences. There is a never erring about two straight lines) among the postulates ;
truth in the results; and, though there may be and this seems most reasonable, for it is certain
here and there a self-evident assumption used in that the first two assumptions can have no claim
demonstration, but not formally noted, there is to rank among common notions or to be placed in
never any the smallest departure from the limit the same list with “ the whole is greater than its
ations of construction which geometers had, from part. "
the time of Plato, imposed upon themselves. The Without describing minutely the contents of
strong inclination of editors, already mentioned, to the first book of the Elements, we may observe
consider Euclid as perfect, and all negligences as that there is an arrangement of the propositions,
the work of unskilful commentators or interpo- which will enable any teacher to divide it into
lators, is in itself a proof of the approximate truth sections. Thus propp. 1–3 extend the power of
of the character they give the work ; to which it construction to the drawing of a circle with any
may be added that editors in general prefer Euclid centre and any radius; 4-8 are the basis of the
as he stands to the alterations of other editors. theory of equal triangles ; 9-12 increase the
The Elements consist of thirteen books written power of construction ; 13—15 are solely on rela-
by Euclid, and two of which it is supposed that tions of angles; 16–21 examine the relations of
Hypsicles is the author. The first four and the parts of one triangle ; 22—23 are additional con-
sixth are on plane geometry; the fifth is on the strictions ; 23—26 augment the doctrine of equal
theory of proportion, and applies to magnitude in triangles; 27—31 contain the theory of parallels;
general ; the seventh, eighth, and ninth, are on 32 stands alone, and gives the relation between
arithmetic; the tenth is on the arithmetical cha- the angles of a triangle; 33-34 give the first
racteristics of the divisions of a straight line; the properties of a parallelogram ; 35–41 consider
eleventh and twelfth are on the elements of solid parallelograms and triangles of equal areas, but
geometry; the thirteenth (and also the fourteenth different forms ; 42–46 apply what precedes to
and fifteenth) are on the regular solids, which augmenting power of construction; 47–48 give
were so much studied among the Platonists as to the celebrated property of a right angled triangle
bear the name of Platonic, and which, according to and its converse. The other books are all capable
Proclus, were the objects on which the Elements of a similar species of subdivision.
were really meant to be written.
The second book shews those properties of the
At the commencement of the first book, under rectangles contained by the parts of divided
the name of definitions (pou), are contained the straight lines, which are so closely connected with
assumption of such notions as the point, line, &c. , the common arithmetical operations of multipli-
and a number of verbal explanations. Then fol. cation and division, that a student or a teacher
low, under the name of postulates or demands who is not fully alive to the existence and diffi-
(aithuata), all that it is thought necessary to culty of incommensurables is apt to think that
state as assumed in geometry. There are six common arithmetic would be as rigorous as geo-
postulates, three of which restrict the amount of metry. Euclid knew better.
construction granted to the joining two points The third book is devoted to the consideration
by a straight line, the indefinite lengthening of a of the properties of the circle, and is much cramped
terminated straight line, and the drawing of a in several places by the imperfect idea already al-
circle with a given centre, and a given distance luded to, which Euclid took of an angle. There
measured from that centre as a radius; the other are some places in which he clearly drew upon
three assume the equality of all right angles, the experimental knowledge of the form of a circle,
much disputed property of two lines, which meet
a third at angles less than two right angles (we * See Penny Cyclopaedia, art. “ Paralleis, " for
mean, of course, much disputed as to its propriety some account of this well-worn subject.
## p. 67 (#83) ##############################################
EIDES.
67
EUCLEIDES.
EUCLEIDES.
us to its truto), and that
1 inclose a space. Lastly,
on notions (kaival éram
mon to all men or to ai
s that—things equal to the
ther—the whole is greater
em editors hare put the
the end of the con
term ariom (which was
1) to them all The in
to have been, to dira
his reader must grans
,
hatever may be his opi
of the assumption, and
juestion every one 2
or merely distinguisbas
a
construction) from the
Jere is no such distiso
problem and theren;
translated properties
,
inly one used. An i
manuscripts, the testi
abic translations, the
ibe assumptions about
ind most of therm, that
umong the postulates ;
nable, for it is certain
ons can have no claim
ons or to be placed i
lole is greater than is
tely the contents of
nts, we may eberte
t of the propositions
jer to divide it in
extend the power
of a circle with any
are the basis of the
2-12 increase the
5 are solely on reis
ine the relations of
are additional cost
ve doctrine of equal
Geory of parallels;'
e relation between
-34 give the Ert
35—41 consda
equal aress, but
what precedes to
on; 47-48 gire
it angled triangle
ks are all capable
properties of the
arts of divided
connected with
ons of multipl
ent of a teacher
stence and as
ot to think that
rigorous as gan
ne considerati
s much cramped
idea already a
1 angle. There
urly dres ypas
orm of a circle
** Paralleis, in
ject
and made tacit assumptions of a kind which are count of it in the Penny Cyclopaedia, article, “ Ir-
rarely met with in his writings.
rational Quantities. ” Euclid has evidently in his
The fourth book treats of regular figures. Eu- mind the intention of classifying incommensurable
clid's original postulates of construction give him, quantities : perhaps the circumference of the circle,
by this time, the power of drawing them of 3, 4, 5, which we know had been an object of inquiry,
and 15 sides or of double, quadruple, &c. , any of was suspected of being incommensurable with its
these numbers, as 6, 12, 24, &c. , 8, 16, &c. &c. diameter ; and hopes were perhaps entertained
The fifth book is on the theory of proportion. that a searching attempt to arrange the incommen-
It refers to all kinds of magnitude, and is wholly surables which ordinary geometry presents might
independent of those which precede. The exist- enable the geometer to say finally to which of them,
ence of incoinmensurable quantities obliges him to if any, the circle belongs. However this may be,
introduce a definition of proportion which seems Euclid investigates, by isolated methods, and in a
at first not only difficult, but uncouth and inele- manner which, unless he had a concealed algebra,
gant; those who have examined other definitions is more astonishing to us than anything in the
know that all which are not defective are but Elements, every possible variety of lines which can
various readings of that of Euclid. The reasons be represented by v (VatVb), a and 6 repre.
for this difficult definition are not alluded to, ac- senting two commensurable lines. He divides lines
cording to his custom ; few students therefore un- which can be represented by this formula into 25
derstand the fifth book at first, and many teachers species, and he succeeds in detecting every possible
decidedly object to make it a part of the species. He shews that every individual of every
course. A distinction should be drawn between species is incommensurable with all the individuals
Euclid's definition and his manner of applying it of every other species ; and also that no line of any
Every one who understands it must see that it is species can belong to that species in two different
an application of arithmetic, and that the defective ways, or for two different sets of values of a and I.
and unwieldy forms of arithmetical expression He shews how to form other classes of incommen-
which never were banished from Greek science, surables, in number how many soever, no one of
need not be the necessary accompaniments of the which can contain an individual line which is com-
modern use of the fifth book. For ourselves, we mensurable with an individual of any other class ;
are satisfied that the only sigorous road to propor- and he demonstrates the incommensurability of a
tion is either through the fifth book, or else square and its diagonal. This book has á com-
through something much more difficult than the pleteness which none of the others (not even the
fifth book need be.
fifth) can boast of: and we could almost suspect
The sixth book applies the theory of propor- that'Euclid, having arranged his materials in his
tion, and adds to the first four books the proposi- own mind, and having completely elaborated the
tions which, for want of it, they could not contain. tenth book, wrote the preceding books after it, and
It discusses the theory of figures of the same form, did not live to revise them thoroughly.
technically called similar. To give an idea of the The eleventh and twelfth books contain the
advance which it makes, we may state that the elements of solid geometry, as to prisms, pyramids,
first book has for its highest point of constructive &c. The duplicate ratio of the diameters is
power the formation of a rectangle upon a given shewn to be that of two circles, the triplicate ratio
base, equal to a given rectilinear figure; that the that of two spheres. Instances occur of the method
second book enables us to turn this rectangle into Of exhaustions, as it has been called, which in the
a square ; but the sixth book empowers us to hands of Archimedes became an instrument of dis-
make a figure of any given rectilinear shape equal covery, producing results which are now usually
to a rectilinear figure of given size, or briefly, to referred to the differential calculus : while in those
construct a figure of the form of one given figure of Euclid it was only the mode of proving proposi-
and of the size of another. It also supplies the tions which must have been seen and believed be-
geometrical form of the solution of a quadratic fore they were proved. The method of these books
equation.
is clear and elegant, with some striking in perfec-
The seventh, eighth, and ninth books cannot tions, which have caused many to abandon them,
have their subjects usefully separated. They treat even among those who allow no substitute for the
of arithmetic, that is, of the fundamental properties first six books. The thirteenth, fourteenth, and
of numbers, on which the rules of arithmetic must fifteenth books are on the five regular solids : and
be founded. But Euclid goes further than is ne- even had they all been written by Euclid (the last
cessary merely to construct a system of computa- two are attributed to Hypsicles), they would but
tion, about which the Greeks had little anxiety. ill bear out the assertion of Proclus, that the regu-
He is able to succeed in shewing that numbers lar solids were the objects with a view to which
which are prime to one another are the least in the Elements were written : unless indeed we are
their ratio, to prove that the number of primes is to suppose that Euclid died before he could com-
infinite, and to point out the rule for constructing plete his intended structure. Proclus was an en-
what are called perfect numbers. When the mo- thusiastic Platonist: Euclid was of that school ;
dem systems began to prevail
, these books of Eu- and the former accordingly attributes to the latter
clid were abandoned to the antiquary: our elemen- a particular regard for what were sometimes called
tary books of arithmetic, which till lately were all, the Platonic bodies. But we think that the author
and now are mostly, systems of mechanical rules, himself of the Elements could hardly have considered
tell us what would have become of geometry if the them as a mere introduction to a favourite specula-
earlier books had shared the same fate.
tion : if he were so blind, we have every reason to
The tenth book is the development of all the suppose that his own contemporaries could have set
power of the preceding ones, geometrical and arith him right. From various indications, it can be col
metical. It is one of the most curious of the Greek lected that the fame of the Elements was almost
speculations : the reader will find a synoptical ac- coeval with their publication ; and by the time of
## p. 68 (#84) ##############################################
68
EUCLEIDES.
EUCLEIDES.
Marinus we learn from that writer that Euclid | epitome of the whole. Theon the younger (of
was called κύριος στοιχειωτής.
Alexandria) lived a little before Proclus (who died
The Data of Euclid should be mentioned in con- about A. D. 485). The latter has made his feeble
nection with the Elements. This is a book contain- commentary on the first book valuable by its his-
ing a hundred propositions of a peculiar and limited torical information, and was something of a lumi-
intent. Some writers have professed to see in it a nary in ages more dark than his own. But Tbeon
key to the geometrical analysis of the ancients, in was a light of another sort, and his name has
which they have greatly the advantage of us. played a conspicuous and singular part in the his-
When there is a problem to solve, it is undoubtediy tory of Euclid's writings. lle gave a new edition
advantageous to have a rapid perception of the steps of Euclid, with some slight additions and altera-
which will reach the result, if they can be succes- tions : he tells us so himself, and uses the word
sively made. Given A, B, and C, to find D: one éxdoois, as applied to his own edition, in his com-
person may be completely at a loss how to proceed; mentary on Ptolemy. He also informs us that the
another may see almost intuitively that when A, part which relates to the sectors in the last proper
B, and C are given, E can be found; from which sition of the sixth book is his own addition: and
it may be that the first person, had he perceived it, it is found in all the manuscripts following the
would have immediately found D. The formation rep deu deixar with which Euclid always ends.
of data consequential, as our ancestors would per- Alexander Aphrodisiensis ( Comment, in priora
haps have called them, things not absolutely given, Analyt. Aristot. ) mentions as the fourth of the
but the gift of which is implied in, and necessarily tenth book that which is the fifth in all manu-
follows from, that which is given, is the object of scripts. Again, in several manuscripts the whole
the hundred propositions above mentioned. Thus, work is beaded as åk Two Obwros ouvovoiav. We
when a straight line of given length is intercepted shall presently see to what this led: but now we
between two given parallels, one of these proposi- must remark that Proclus does not mention Theon
tions shews that the angle it makes with the pa- at all; from which, since both were Platonists re-
rallels is given in magnitude. There is not much siding at Alexandria, and Proclus had probably
more in this book of Data than an intelligent stu- seen Theon in his younger days, we must either
dent picks up from the Elements themselves; on infer some quarrel between the two, or, which is
which account we cannot consider it as a great step perhaps more likely, presume that Theon's altera-
in geometrical analysis. The operations of thought tions were very slight.
which it requires are indispensable, but they are The two books of Geometry left by Boethits
contained elsewhere. At the same time we cannot contain nothing but enunciations and diagrams
deny that the Data might have fixed in the mind from the first four books of Euclid. The assertion
of a Greek, with greater strength than the Ele- of Boethius that Euclid only arranged, and that
ments themselves, notions upon consequential data the discovery and demonstration were the work of
which the moderns acquire from the application of others, probably contributed to the notions about
arithmetic and algebra : perhaps it was the percep-Theon presently described. Until the restoration
tion of this which dictated the opinion about the of the Elements by translation from the Arabic,
value of the book of Data in analysis.
this work of Boethius was the only European
While on this subject, it may be useful to re-treatise on geometry, as far as is known.
mind the reader how difficult it is to judge of the The Arabic translations of Euclid began to be
character of Euclid's writings, as far as his own made under the caliphs Haroun al Raschid and
merits are concerned, ignorant as we are of the Al Mamun; by their time, the very name of Eu-
precise purpose with which any one was written. clid had almost disappeared from the West. But
For instance : was he merely shewing his contem-nearly one hundred and fifty years followed the
poraries that a connected system of demonstration capture of Egypt by the Mohammedans before the
might be made without taking more than a certain laiter began to profit by the knowledge of the
number of postulates out of a collection, the neces- Greeks. After this time, the works of the geome
sity of each of which had been advocated by some ters were sedulously translated, and a great im-
and denied by others?
