History. 
— 
Euclid's 
Elements of 
Geometry. 
188 | 
plane figures ; the seventh, eighth, ninth, and tenth 
relate to arithmetic, and the doctrine of incommensu- 
rables ; the eleventh and twelfth contain the elements 
of the geometry of solids ; and the thirteenth treats of 
the five regular solids, or Platonic bodies, so called be- 
cause they were studied in that celebrated school : two 
books more, viz. the fourteenth and fifteenth, on regu- 
lar solids, have been attributed to Euclid, but these 
rather appear to have been written by Hypsicles of 
Alexandria. 
It is only the first six, and the eleventh and twelfth 
books, that are now commonly taught in the schools ; 
for the books on arithmetic have been superseded by 
the modern theories of algebra, and the regular solids 
have long ceased to be particularly interesting: they 
may be compared to mines which have been abandoned 
because the produce was not equal to the expence of 
working them. Euclid’s Elements have had a number 
of commentators ; the earliest was Theon of Alexan- 
dria, who lived about the middle of the fourth century. 
Proclus also has given a commentary on the first book, 
which is only valuable on account of the information it 
contains respecting the history and metaphysics of geo- 
metry. After the revival of learning, the Elements of 
Euclid were first known in Europe, through the me- 
dium of an Arabic translation ; from this it was deci- 
phered and translated into Latin, by Athelard in Eng- 
land, and Campanus in Italy, about the same time, in 
the 12th or 13th centuries. Athelard’s translation ex- 
ists only in manuscript, in some libraries ; that of Cam- 
panus seryed as the basis of the greater part of the La- 
tin translations, made about the end of the 15th and 
the beginning of the 16th centuries. The editio prin- 
ceps is'that which Ratdolt of Augsburg, a celebrated 
printer, gave in 1482, at Venice, in folio; the Greek 
text did not appear until 1533, when it was printed at 
Basle, by J. Hervage, under the care of J. Gryneus. 
* The earliest English edition is that of Bilingsley, in 
Euclid’s 
Data, 
1570: But the history of the various editions of this 
work, either in whole or in part, that have been pub- 
lished in all countries, in which science has been culti- 
vated, is far too extensive to find a place here. The 
curious reader may find a copious list in the second 
volume of the Bibliotheca Mathematica, by Murhard. 
At present, the edition of Euclid most esteemed in this 
country, is that of the late Dr Simson of Glasgow, 
which contains the first six and the eleventh and twelfth 
books, and the book of Euclid’s Data. We have lately 
seen the first volume of an edition in the original Greek, 
accompanied with a Latin and French translation by 
Peyrard, a French professor of mathematics, and au- 
thor of a French translation of Archimedes ; it gives 
the original text as exhibited in a great number of 
manuscripts, and on this account it must be extremely 
valuable. 
Besides the Elements, the only other entire geometri- 
cal work of Euclid that has come down to the present 
times, is his Data. This is the first in order of the 
books written by the ancient geometers to facilitate the 
method of resolution or analysis. In general, a thing 
is said to be given, which is actually exhibited, or can 
be found ; and the propositions in the book of Euclid’s 
Data, shew what things can be found from those which 
a scams are already known. 
_ We learn from Pappus of Alexandria, that'there’ex- 
isted four books by Euclid on Conic Sections, and two 
concerning Loci ad Superficiem ; these were curves of 
double curvature. But his most profound work, and 
that of which the loss is most regretted, was his three 
GEOMETRY. 
books on Porisms, which Pappus says were a most art~ 
ful collection of many things that relate to the analysis 
of thé more difficult and general problems. We- 
explain this subject under the word Porrsm.  Proclus © 
cites another work of Euclid’s, which he entitles, De 
Divisionibus. This probably treated of the division of 
figures. These are all the known etrical writings 
of Euclid:—his other works do not belong to this 
place. See Evctip. brid Morbi at 
In the order of time, Archimedes is the next of the Arch 
and in particular geometry. The most difficult part of 
the science is that which relates to the areas 
lines, and to curve surfaces. Archimedes applied his 
fine genius to this subject, and he laid the foundation , 
of all the‘ subsequent discoveries relating to it. His 
writings on geometry are numerous. e have, in 
the first place, two books on the sphere and cylin« 
der ; these contain the beautiful discovery, that the 
sphere is two-thirds of the circumscribing cylinder, 
whether we compare their surfaces, or their solidities, 
observing that the two ends of the cylinder are con- 
sidered as forming a part of its surface. He likewise 
shews, that the curve surface of any segment of the 
cylinder between two planes perpendicular to its axis, 
is equal to the curve surface of the corresponding seg- 
ment of the sphere. Archimedes was so much pleased 
with these discoveries, that he requested after his death 
that his tomb might be inscribed with a sphere and 
cylinder. ; 
His book on the Measure of the Circle, is a kind of 
supplement to those on the sphere and cylinder. In 
this, he demonstrates that any circle is to a trian, 
having its base equal to the circumference, and its 
height equal to the radius; and he proves, that -if the 
diameter of a circle be reckoned unity, the circumfe- 
rence will be between 342 and 349. The «principles 
laid down by Archimedes were sufficient to carry the 
approximation to any degree of nearness; but he 
pears to have aimed at nothing more than a sim r @ 
rule, sufficiently accurate for the common concerns of 
life. vats 
His treatise on Conoids and Spheroids relates to the 
solids generated by the conic sections revolving about 
their axes: those prodtced by the rotation of the para~ 
bola and hyperbola, he called Conoids ; and such as are i 
generated by the revolution of the ellipse about either 
axis are his Spheroids. Here he com the area of 
an ellipse with that of a circle ; he also proves that the 
sections of conoids and spheroids are conic sections,’ 
and he treats of their tangent planes. He proves, for 
the first time, that a parabolic conoid is equal to three’ 
times the half of a cone of the same base and altitude, 
and he also:shews what is the ratio of any segment of a) 
hyperbolic conoid, or of a spheroid to a cone of the : 
same base and altitude. His reasoning is a model of ~~ 
accuracy; and it exhibits the true spirit of the ancient e 
synthetic method; it is however exceedingly prolix 
and difficult, insomuch that few will have patience to: 
follow the steps of the venerable mathematician, more 
especially as the same conclusions may be found ‘with . 
equal certainty by the modern analysis; at an infinitely’ " 
less expence of thought. Ce ae 
His treatise on Spirals treats of a curve which was 
the invention of his friend Conon, who it seems had Cone 
found its properties, but died before he had time to in= 
vestigate their demonstrations: these Archimedes has’ 
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