A Sketch of the Hiftory 
profpeéts into the vaft field, which the 
menturation of curvilinear figures offered 
to his refearches. He demonftrated, that 
the furface of the fphere is two thirds of 
the whole furface of the circumf{cribed cy- 
linder, and that the folidities of thofe bo- 
dies hold the fame proportion. ‘This 
difcovery. fo much pleafed Archimedes, 
that he defired the figures to be engraven 
onhis tomb. That great man feund out 
110 
the exaét quadrature of the parabola, and- 
determined the limits of the proportion 
between the circumference of the circle 
and the radius, by fhewing that, the radius 
being unity, the circumference is lefs than 
342, and greater than 3495 fo that. we 
may approach very nearly to the iwalue of 
zhe circumference by tripling the diameter, 
and adding 4 of the fame: to that triple. 
the fervice of his country a great number of 
engines which he had invented. The Ro- 
mans having laid fiege to Syracufe, the inha- 
bitants were fo intimidated by the-progrefs of 
their arms, that they neglected the prepara- 
tions necefiary for a vigorous defence; but 
Archimedes revived their courage. He dif- 
concerted the defigns of the enemy—burnt 
their fleet with reflecting mirrors, and 
obliged Marcellus to convert the fiege into a 
blockade. But the fecurity>of the Syra- 
cufans afforded the Roman general an oppor- 
tunity of furprifing the town. During the 
feaft of Diana, they forfook the ramparts. 
The Romans, boldly fcaling the walls, pene-~ 
trated into the town, which they facked, 
Marcellus ordered them to fave Archimedes 5 
but his command was badly executed, and 
the unfortunate mathematician fell by the 
hand of a foldier, 212 years before Chrift, 
Marcellus, penetrated with grief, delivered 
the body of the great man to his relations, 
along with hjs property, to be applied to the 
ere€tion of a monument to his memory. 
Cicero, when queftor of Sicily, difcovered 
that monument, furrounded with briars and 
brambles, and he knew it to be the monu- 
ment of Archimedes, by the {phere infcribed 
in a cylinder, which was carved upon it. 
Note by ihe Trunflator.—The aftonifhing 
philojophico-military exploit of Archimedes 
tempts me to digrefs a little from pure into 
mixed mathematics. That exploit is record- 
ed by Diodorus Siculus, Lucian, Dion, Zona- 
ras, Galen, Anthemius, Tzctzes, and-other 
ancient writers. he account of Tzetzes is 
fo particular, that it fuggefted to Father 
Kircher the fpecific method by which Ar- 
chimedes probably eftected his purpofe. 
«< Archimedes,” fays that author, ‘¢ fet fire 
to the fleet of Marcellus by a burning-glafs, 
compofed of fmall fquare mirrors, moving 
every way upon hinges; and which, when 
placed in the fun’s rays, refieéted them on 
the Roman fieet, fo as to reduce it to afhes 
~~ 
~ 
of Pure Mathematics. [March 1, 
28. A wonderful depth and: fagacity 
runs through all the works of Archi- 
medes. But the way which he opened te 
difeover the properties of conoids, {phe- 
roids, {pirals, &c.* is fo difficult to be 
followed, that he deferves our admiratien 
for having firtt, found it, and for having 
Dever deviated from it. skp” 
__ 29. About the time when Archimedes 
‘finifhed his career, Apollonius,} furnamed 
the 
at the diftance of a bow-fhot.” This ac= 
count gained additional probability by the ef- 
fe& which Zonaras afcribes to the burning- 
mirror of Proclus, by which he affirms, that 
the fleet of Vitellius, when befieging By- 
zantium (now Conftantinople) was utterly 
confumed, But perhaps no hiftorical tefti- 
mony could have gained beHef to fuch extra- 
ordinary fats, if fimilar ones had not 
-been feen in modern times. In the Memoirs. 
of the French Academy of Sciences for 1726, 
p- 172, we read of a plain mirror, one foot 
{fquare, refleting the fun’s rays to a concave 
mirror 1@ inches in diameter, in the focus of 
which bodies were burnt at the diftance of 
600 paces (whether geometrical paces, each ¢ 
feet, or common paces, each 24 feet, we are 
not told.) Speaking of this mirror, Father 
‘Regnault afks, (in his ‘Phyfics, yol. 3. 
difc.. 10.) ‘© What would be the effect of a 
number of plain mirrors, placed in 4 holiow 
truncated pyramid, and direéting the fun’s 
rays to the fame point? Throw the focus, 
faid he, a littte farther, and you re-difcover 
or verify the fecret of Archimedes.” This 
M. de Buffon actually effected. In the year 
1747, he read to the Academy an account of 
a mirror, which he had compofed of an af- 
femblage of plain mirrors, which made the 
fun’s rays converge to a point at a great dif- 
tance; but what that exact diftance was, I 
cannot find. For an account of this and fe- 
veral other burning-glafies, both reflectors _ 
and refra€ters, fee the Encyclopedia Britan~ 
nica, 3d edition, article Burzing-glafs. 
* The fpiral was invented by a geometri- 
cian of the name of Conon; but it bears the 
mame of Archimedes, becaufe he firft difco- 
vered its properties. 
Note by the Tranflator.—What our author 
favs of the difficulty of following Archi- 
medes,in his refined demonftrations, was juf- 
tified by the experience of M. Bulliald, 
© Though I have twice or thrice,” fays that 
eminent French mathematician, ‘* read over 
Archimedes’s Treatife of Spirals, with the 
utmoft attention, in order to compreherid the 
art employed in his fubtile demonftrations 
relating to the tangents of thofe curves, yet 
could I never rife from him witheut fome 
fufpicion that I had not apprehended the 
whole force of his reafonings.” Sullia‘dus ds 
Lineis Spiralibus, Pref. gis! 
+ Apollonius was of Perga, in ces: 
€. 
