No More Learning

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.


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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.


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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.


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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


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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? We then understand why pulse was given by them. Commentaries, and
he placed his six postulates in the prominent posi- even original writings, followed ; but so few of
tion which they occupy, and we can find no fault these are known among us, that it is only from
with his tacit admission of many others, the neces- the Saracen writings on astronomy (a science which
sity of which had perhaps never been questioned. always carries its own history along with it) that
But if we are to consider him as meaning to be we can form a good idea of the very striking pro-
what his commentators have taken him to be, a gress which the Mohammedans made under their
model of the most scrupulous formal rigour, we can Greek teachers. Some writers speak slightingly of
then deny that he has altogether succeeded, though this progress, the results of which they are too apt
we may admit that he has made the nearest ap- to compare with those of our own time: they
proach.
ought rather to place the Saracens by the side of
The literary history of the writings of Euclid their own Gothic ancestors, and, making some al-
would contain that of the rise and progress of geo-lowance for the more advantageous circumstances
metry in every Christian and Mohammedan na- under which the first started, they should view
tion : our notice, therefore, must be but slight, and the second systematically dispersing the remains of
various points of it will be confirmed by the biblio- Greek civilization, while the first were concentrat-
graphical account which will follow.
ing the geometry of Alexandria, the arithmetic
in Greece, including Asia Minor, Alexandria, and algebra of India, and the astronomy of both,
and the Italian colonies, the Elements soon became to form a nucleus for the present state of science.
the universal study of geometers. Commentators The Elements of Euclid were restored to Europe
were not wanting ; Proclus mentions Heron and by translation from the Arabic. In connection
Pappus, and Aeneas of Hierapolis, who made an with this restoration four Eastern editors may be


## p. 69 (#85) ##############################################

DUCLEIDES
69
EUCLEIDES.
EUCLEIDES.
pomer.
the whole. Theon the younger (od
lived a little before Proclus (who died
85). The latter has made his fehlen
on the first book valuable bp its tr
ation, and was something of a lie
ore dark than his own. But Time
f another sort, and his parze ba
Ecuous and singular part in the bis
writings. He gave a new edities
some slight additions and aliez
215 60 himself
, and uses the Ford
ed to his own edition, in his re
my. He also informs to that the
to the sectors in the last post
book is his own addition: 2!
the manuscripts following the
rith which Euclid always enes
sicnsis ( Comments in prors
entions as the fourth of the
ich is the fifth in all star-
everal manuscripts the whole
TWY Oí wvos owoway, Me
what this led: but not 18
clus does not mention Thera
nce both were Platonists re
and Proclus had probably
Tiger dars, we must either
seen the two, or, Shich is
sume thai Theon's alterz-
ametry left by BOETIIS
nciations and diren
Euclid. The asserbe?
only arranged, and that
ation were the work of
I to the notions about
Until the restorative
jon from the Arabic
the only Europea
is known.
Euclid began to be
un al Raschid and
very name of Er
the l'est. Bet
Fears follosed the
medans before the
nowledge of the
ks of the ratione
and a great in
mentaries, ed
but so fer of
It is only fras
science lid
with it) that
striking pm
under the
ightingh
mentioned. Honein ben Ishak (died A. D. 873) had got " as far as the 32nd proposition of the first
published an edition which was afterwards cor- book” before he was detected, the exaggerators
rected by Thabet ben Corrah, a well-known astro- (for much exaggerated this very circumstance shews
After him, according to D'Herbclot, the truth must have been) not having the slightest
Othman of Damascus (of uncertain date, but before idea that a new invented system could proceed in
the thirteenth century) saw at Rome a Greek ma- any other order than thut of Euclid.
nuscript containing many more propositions than The vernacular translations of the Elements date
he had been accustomed to find : he had been used from the middle of the sixteenth century, from which
to 190 dingrams, and the manuscript contained 40 time the history of mathematical science divides
more. If these numbers be correct, Honein could itself into that of the several countries where it
only have had the first six books; and the new fourished. By slow steps, the continent of Europe
translation which Othman immediately made must has almost entirely abandoned the ancient Ele-
have been afterwards augmented. A little after ments, and substituted systems of geometry more
A. D. 1260, the astronomer Nasiruddin gave an- in accordance with the tastes which algebra has
other edition, which is now accessible, having been introduced : but in England, down to the present
printed in Arabic at Ronie in 1594. It is tolera- time, Euclid has held his ground. There is not in
bly complete, but yet it is not the edition from our country any system of geometry twenty years
. which the enrliest European translation was made, old, which has pretensions to anything like cur-
as Peyrard found by comparing the same proposi- rency, but it is either Euclid, or something so
tion in the two.
fashioned upon Euclid that the resemblance is as
The first European who found Euclid in Arabic, close as that of some of his professed editors. We
and translated the Elements into Latin, was Athe-cannot here go into the reasons of our opinion; but
lard or Adelard, of Bath, who was certainly alive we have no doubt that the love of accuracy in ma-
in 1130. (See “ Adelard," in the Biogr. Dict. of thematical reasoning has declined wherover Euclid
the Soc. D. U. K. ) This writer probably obtained has been abandoned. We are not so much of the
his original in Spain: and his translation is the old opinion as to say that this must necessarily have
one which became current in Europe, and is the happened ; but, feeling quite sure that all the al-
first which was printed, though under the name of terations have had their origin in the desire for
Campanus. Till very lately, Campanus was supposed more facility than could be obtained by rigorous
to have been the translator. Tiraboschi takes it to deduction from postulates both true and evident,
have been Adelard, as a matter of course ; Libri we see what has happened, and why, without be-
pronounces the same opinion after inquiry; and ing at all inclined to dispute that a disposition to
Scheibel states that in his copy of Campanus the depart from the letter, carrying off the spirit, would
authorship of Adelard was asserted in a band have been attended with very different results. Of
writing as old as the work itself. (4. D. 1482. ) the two best foreign books of geometry which we
Some of the manuscripts which bear the name of know, and which are not Euclidean, one demands
Adelard have that of Campanus attached to the a right to “imagine" a thing which the writer
commentary. There are several of these manu- himself knew perfectly well was not true ; and the
scripts in existence; and a comparison of any one other is content to shew that the theorems are so
of them with the printed book which was attributed nearly true that their error, if any, is imperceptible
to Canıpanus would settle the question.
to the senses. It must be adınitted that both these
The seed thus brought by Adelard into Europe absurdities are committed to avoid the fifth book,
was sown with good effect. In the next century and that English teachers have, of late years, been
Roger Bacon quotes Euclid, and when he cites Boe much inclined to do something of the same sort,
thius, it is not for his geometry. Up to the time of less openly. But here, at least, writers have left
printing, there was at least as much dispersion of the it to teachers to shirk truth, if they like, without
Elements as of any other book : after this period, being wilful accomplices before the fact. In an
Euclid was, as we shall see, an early and frequent English translation of one of the preceding works,
product of the press. Where science flourished, the means of correcting the error were given : and
Euclid was found; and wherever he was found, the original work of most note, not Euclidean,
science flourished more or less according as more which has appeared of late years, does not attempt
or less attention was paid to his Elements. As to to get over the difficulty by any false assumption.
writing another work on geometry, the middle ages
At the time of the invention of printing, two
would as soon have thought of composing another errors were current with respect to Euclid person-
New Testament: not only did Euclid preserve his ally. The first was that he was Euclid of Megara,
right to the title of kúpios oroixewtńs down to the a totally different person. This confusion has been
end of the seventeenth century, and that in so ab- said to take its rise from a passage in Plutarch,
solute a manner, that then, as sometimes now, the but we cannot find the reference. Boëthius per-
young beginner imagined the name of the man to petuated it. The second was that Theon was the
be a synonyme for the science; but his order of demonstrator of all the propositions, and that Euclid
demonstration was thought to be necessary, and only left the definitions, postulates, &c. , with the
founded in the nature of our minds. Tartaglia,
whose bias we might suppose would have been * We must not be understood as objecting to
shaken by his knowledge of Indian arithmetic and the teacher's right to make his pupil assume any-
algebra, calls Euclid solo introduttore delle scientie thing he likes, provided only that the latter
mathemulice: and algebra was not at that time con- knows what he is about. Our contemptuous
sidered as entitled to the name of a science by expression (for such we mean it to be) is directed
those who had been formed on the Greek model; against those who substitute assumption for de
“urte maggiore” was its designation. The story monstration, or the particular for the general, and
about Pascal's discovery of geometry in his boy- leave the student in ignorance of what has been
bood (A. D. 1635) contains the statement that he done.
are too and
me : Le
be side of
some al-
ITSLID
ald rier
mains of
hcete
empe


## p. 70 (#86) ##############################################

70
EUCLEIDES.
EUCLEIDES.
enunciations in their present order. So completely. The preceding works are in existence; the folo
was this notion received, that editions of Euclid, lowing are either lost, or do not remain in the
60 called, contained only enunciations ; all that original Greek.
contained demonstrations were said to be Euclid 8. Περί Διαιρέσεων βιβλίων, On Divisions. Pro-
with the commentary of Theon, Campanus, Zam- clus (L. c. ) There is a translation from the Arabic,
bertus, or some other. Also, when the enunciations with the name of Mohammed of Bagdad attached,
were given in Greek and Latin, and the demon- which has been suspected of being a translation of
strations in Latin only, this was said to constitute the book of Euclid: of this we shall see more.
an edition of Euclid in the original Greek, which 9. Kwvixwv Bibila , Four books on Conic Seo-
has occasioned a host of bibliographical errors. We tions. Pappus (lib. vii. pruef. ) affirms that Euclid
have already seen that Theon did edit Euclid, and wrote four books on conics, which Apollonius en-
that manuscripts have described this editorship larged, adding four others. Archimedes refers to
in a manner calculated to lead to the mistake the elements of conic sections in a manner which
but Proclus, who not only describes Euclid as ta shews that he could not be mentioning the new
μαλακώτερον δεικνύμενα τοις έμπροσθεν εις ανε- work of his contemporary Apollonius (which it is
λέγκτους αποδείξεις αναγαγών, and comments on most likely he never saw). Euclid may possibly
the very demonstrations which we now have, as have written on conic sections ; but it is impossible
on those Euclid, is an unanswerable witness; that the first four books of APOLLONIUS (see his
the order of the propositions themselves, connected life) can have been those of Euclid.
as it is with the mode of demonstration, is another ; 10. Πορισμάτων βιβλία γ', Tire books of Porisms.
and finally, Theon himself, in stating, as before These are mentioned by Proclus and by Pappus
noted, that a particular part of a certain demonstra- (l. c. ), the latter of whom gives a description which
tion is his own, states as distinctly that the rest is is so corrupt as to be unintelligible.
not. Sir Henry Savile (the founder of the Savilian 11. Τόπων Έπιπέδων βιβλία β', Two books on
chairs at Oxford), in the lectures on Euclid with Plane Loci. Pappus mentions these, but not Eu-
which he opened his own chair of geometry before tocius, as Fabricius affirms. (Comment. in Apulla
he resigned it to Briggs (who is said to have taken lib. i. lemm. )
up the course where his founder left off, at book i. 12. Τόπων προς Επιφάνειων βιβλία β, men-
prop. 9), notes that much discussion had taken tioned by Pappus. What these Tónou após 'Ero-
place on the subject, and gives three opinions. pávelar, or Loci ad Superfuiem, were, neither
The first, that of quidam stulti et perridiculi, above Pappus nor Eutocius inform us; the latter says
discussed: the second, that of Peter Ramus, who they derive their name from their own ideórns,
held the whole to be absolutely due to Theon, which there is no reason to doubt. We suspect
propositions as well as demonstrations, fulse, quis that the books and the meaning of the title were
negat? the third, that of Buteo of Dauphiny, a as much lost in the time of Eutocius as now.
geometer of merit, who attributes the whole to 13. Περί Ψευδαρίων, On Fullacies. On this
Euclid, quae opinio aut vera est, aut veritati certe work Proclus says, “ He gave methods of clear
proxima. It is not useless to remind the classical judgment (diopatikas opovnOews) the possession of
student of these things: the middle ages may be which enables us to exercise those who are begin-
called the “ages of faith” in their views of criticism. ning geometry in the detection of false reasonings,
Whatever was written was received without exa- and to keep them free from delusion. And the
mination; and the endorsement of an obscure scho- book which gives us this preparation is called
liast, which was perhaps the mere whim of a tran- Yevdapiwv, in wbich he enumerates the species of
scriber, was allowed to rank with the clearest as- fallacies, and exercises the mental faculty on each
sertions of the commentators and scholars who had species by all manner of theorems. He places
before them more works, now lost, written by the truth side by side with falsehood, and connects
contemporaries of the author in question, than the confutation of falsehood with experience. " It
there were letters in the stupid sentence which thus appears that Euclid did not intend his Ele-
was allowed to overbalance their testimony. From ments to be studied without any preparation, but
such practices we are now, it may well be hoped, that he had himself prepared a treatise on fallacious
finally delivered : but the time is not yet come reasoning, to precede, or at least to accompany, the
when refutation of the scholiast ” may be safely Elements. The loss of this book is much to be
abandoned.
regretted, particularly on account of the explana-
All the works that have been attributed to tions of the course adopted in the Elements which
Euclid are as follows:
it cannot but have contained.
1. Stoixeia, the Elements, in 13 books, with a
We now proceed to some bibliographical account
14th and 15th added by HYPSICLES.
of the writings of Euclid. In every case in which
2. Aedouéva, the Data, which has a preface by we do not mention the source of information, it is
Marinus of Naples
to be presumed that we take it from the edition
3. Eloayam 'Apuovingh, a Treatise on Music; itself.
and 4. Katatout) Kavóvos, the Dirision of the Scale : The first, or editio princeps, of the Elements is
one of these works, most likely the former, must that printed by Erhard Raidolt at Venice in 1482,
be rejected. Proclus says that Euclid wrote kata black letter, folio. It is the Latin of the fifteen
μουσικών στοιχειώσεις.
books of the Elements, from Adelard, with the
5. Dawóueva, the Appearances (of the heavens). commentary of Campanus following the demon-
Pappus mentions them.
strations. It has no title, but, after a short intro-
6. 'Ontiká, on Optics ; and 7. Katottpiké, on duction by the printer, opens thus : “ Preclarissimus
Catoptrics. Proclus mentions both.
liber elementorum Euclidis Perspicacissimi : in
artem geometrie incipit quā foelicissime: Punctus
* Praelectiones tresdecim in principium elementorum est cujus ps nñ est,” &c. Ratdolt states in the
Euclidis; Oxonü habitac n. dc. xx. Oxoniae, 1621. I introduction that the difficulty of printing diagrams


## p. 71 (#87) ##############################################

Seo
Lucid
ft. sia
ne mes
Lich it is
possib's
possible
5 (see li
of Porissa
by Pappes
etion which
Ero looks on
but not Es
ent. in Apel
Erla , men
To spas 'Es:
were, neither
the latter eers
ir own idées
. We suspect
of the title were
us as now,
lacies. On this
methods of clea:
) the possession de
se who are begis
of false reasoning
elusion. And the
Teparation is called
serates the special of
ental faculty on each
heorems. He placas
sehood, and connects
with experience. "
EUCLEIDES.
EUCLEIDES.
71
had prevented books of geometry from going through principis opera, &c. At the end, Venetiis impressum
the press, but that he had so completely overcome per. . . Paganinum de Paganinis. . . anno. . . MDVIII . . .
it, by great pains, that “ qua facilitate litterarum Paciolus adopts the Latin of Adelard, and occa-
elementa imprimuntur, ea etiam geometrice figure sionally quotes the comment of Campanus, intro-
conficerentur. " These diagrams are printed on the ducing his own additional comments with the bead
margin, and though at first sight they seem to be “ Castigator. ” He opens the fifth book with the
woodcuts, yet a closer inspection makes it probable account of a lecture which he gave on that book in
that they are produced from metal lines. The a church at Venice, August 11, 1508, giving the
number of propositions in Euclid (15 books) is 485, names of those present, and some subsequent lau-
of which 18 are wanting here, and 30 appear which datory correspondence. This edition is less loaded
are not in Euclid; so that there are 497 proposi- with comment than either of those which precede.
tions. The preface to the 14th book, by wbich it It is extremely scarce, and is beautifully printed :
is made almost certain that Euclid did not write it the letter is a curious intermediate step between
(for Euclid's books have no prefaces) is omitted. the old thick black letter and that of the Roman
İts Arabic origin is visible in the words helmuaym type, and makes the derivation of the latter from
and helmuariphe, which are used for a rhombus and the former very clear.
a trapezium. This edition is not very scarce in The fifth edition (Elements, Latin, Roman letter,
England; we have seen at least four copies for folio), edited by Jacobus Faber, and printed by
sale in the last ten years.
Henry Stephens at Paris in 1516, has the title
The second edition bears “ Vincentiae 1491,"Contenta followed by heads of the contents.
Roman letter, folio, and was printed “per magis There are the fifteen books of Euclid, by which
trum Leonardum de Basilea et Gulielmum de are meant the Enunciations (see the preceding re-
Papia socios. " It is entirely a reprint, with the marks on this subject); the Comment of Campanus,
introduction omitted (unless indeed it be torn out meaning the demonstrations in Adelard's Latin ;
in the only copy we ever saw), and is but a poor the Comment of Theon as given by Zambertus,
specimen, both as to letter-press and diagrams, meaning the demonstration in the Latin of Zam-
when compared with the first edition, than which bertus ; and the Comment of Hypsicles as given by
it is very much scarcer. Both these editions call Zambertus upon the last two books, meaning the
Euclid Megarensis.
demonstrations of those two books. This edition
The third edition (also Latin, Roman letter, is fairly printed, and is moderately scarce. From
folio,) containing the Elements, the Phaenomena, it we date the time when a list of enunciations
the two Optics (under the names of Specularia and merely was universally called the complete work of
Perspectiva), and the Data with the preface of Euclid.
Marinus, being the editio princeps of all but the With these editions the ancient series, as we
Elements, has the title Euclidis Megarensis philo may call it, terminates, meaning the complete La-
sophici Platonici, mathematicarum disciplinarü tin editions which preceded the publication of the
janitoris : habent in hoc volumine quicüque ad ma- Greek text. Thus we see five folio editions of the
thematică substantiã aspirāt : elemêtorum libros, Elements produced in thirty-four years.
&c. &c. Zamberto Veneto Interprete. At the end The first Greek text was published by Simon
is Impressum Venetis, &c. in edibus Joannis Ta- Gryne, or Grynoeus, Basle, 1533, folio : * contain-
cuini, &c. , M. D. V. Vili. Klendas Novēbris ing, ék tw Oewvos ouvovoiây (the title-page has
that is, 1505, often read 1508 by an obvious this statement), the fifteen books of the Elements,
mistake. Zambertus has given a long preface and the commentary of Proclus added at the end,
and a life of Euclid: he professes to have trans- so far as it remains; all Greek, without Latin.
lated from a Greek text, and this a very little On Grynoeus and his reverendt care of manuscripts,
inspection will shew he must have done ; but he see Anthony Wood. (Athen. Oxon. in verb. ) The
does not give any information upon his manu- Oxford editor is studiously silent about this Basle
scripts.