62 A R C H I 
took care of his funeral ; and made his name a protection 
and honour to thofe who could claim a relutionfhip to him. 
His death happened about the i42d or 143d olympiad, or 
210 years before the birth of Ohrid. 
When Cicero was queftor for Sicily, he difcovered the 
tomb of Archimedes, all overgrown with buflies and bram¬ 
bles ; which he caufed to be cleared, and the place fet in 
order. There was a i'phere and cylinder cut upon it, with 
an infcription, but the latter part of the verfes quite worn 
out. Many of the works of this great man are dill extant, 
though the greared part of them are lod. The pieces re¬ 
maining are as follow : 1. Two books on the Sphere and 
Cylinder. 2. The Dimehfion of the Circle, or proportion 
between the diameter and the circumference. 3. Of Spiral 
Lines. 4. Of Conoids and Spheroids. 5. Of Equiponde¬ 
rants, or Centres of Gravity. 6. The Quadrature of the 
Parabola. 7. Of Bodies floating on Fluids. 8. Lemmata. 
9. Of the Number of the Sand. Among the works of 
Archimedes which are loir, may be reckoned the defcrip- 
tions of the following inventions, which may be gathered 
from himfelf and other ancient authors. 1. His account 
of the method which he employed to difcoverthe mixture 
of gold and filver in the crown, mentioned by Vitruvius. 
2. His defcription of the Cochleon, or engine to draw wa¬ 
ter out of places where it is llagnated, (fill in ufe under 
the name of Archimedes’s Screw. Athemeus, fpeaking 
of tire prodigious (hip built by the order of Hiero, fays, 
that Archimedes invented the cochleon, by means of which 
the hold, notwithftanding its depth, could be drained by 
one man. And Diodorus Siculus fays, that he contrived 
this machine to drain Egypt, and that by a wonderful 
mechanifm it would exhaud the water from any depth. 
3. The Helix, by means of which, Athenaeus informs us, 
he launched I-Iiero’s great fliip. 4. The Trifpafton, which, 
according to Tzetzes and Oribafius, could draw the mod 
ftupendous weights. 5. The machines, which, according 
Jo Polybius, Livy, and Plutarch, he ufed in the defence 
of Syracufe againft Marcellus, confiding of Tormenta, Ba- 
lidte, Catapults, Sagittarii, Scorpions, Cranes, &c. 6. His 
Burning-Glades, with which he fet fire to the Roman gal- 
lies. 7. His Pneumatic and Hydrodatic Engines, concern¬ 
ing which fttbjeCts he wrote fome books, according to 
Tzetzes, Pappus, and Tertullian. 8. His Sphere, which 
exhibited the celedial motions: and probably many others. 
He was the fird who fquared a curvilineal fpace; unlefs 
Hippocrates mud be excepted on account of his lunes. In 
his time the conic feCtions were admitted into geometry, 
and he applied himfelf clofely to the meafuring of them, 
as well as other figures. Accordingly, he determined the 
relations of fpheres, fpheroids, and conoids, to cylinders 
and cones; and the relations of parabolas to rectilineal 
planes whofe quadratures had long before been determined 
by Euclid. He has left us alio his attempts upon the circle: 
he proved that a circle is equal to a right-angled triangle, 
whofe bafe is equal to the circumference, and its altitude 
equal to the radius; and, confequently, that it's area is 
equal to the rectangle of half the diameter and half the 
circumference; thus reducing the quadrature of the circle 
to the determination of the ratio between the diameter and 
the circumference; which determination, however, has 
never yet been done. Being difappointed of the exad qua¬ 
drature of the circle, for want of the rectification of its 
circumference, which all his methods would not effeCt, he 
proceeded to affign an ufeful approximation to it : this he 
effected by the numeral calculation of the perimeters of 
the infcribed and circumfcribed polygons: from which 
calculation it appears that the perimeter of the circum- 
Jfcribed regular polygon of 192 (ides, is to the diameter, in 
a lei's ratio than that of 3A or 3^2 to 1 ; and that the peri¬ 
meter of the infcribed polygon of 96 (ides, is to the dia¬ 
meter, in a greater ratio than that of 3J-2 to 1 ; and confe¬ 
quently that the ratio of the circumference to the diameter 
hes between thefe two ratios. Now the fird ratio, of if 
to 1, reduced to whole numbers, gives that of 22 to 7, for 
: x :: 22 ; 7 ; which therefore is nearly the ratio of the 
M E D E S. 
circumference to the diameter. From this ratio between 
the circumference and the diameter, Archimedes compu¬ 
ted the approximate area of the circle, and he found that 
it is to the fquare of the diameter, as u is to 14. He de¬ 
termined alfo the relation between the circle and ellipfe, 
with that of their fimilar parts: and it is probable that he 
likewife attempted the hyperbola; but it is not to be ex¬ 
pected that he met with any fuccefs, lince approximations 
to its area are all that can be given by the various methods 
that have (ince been invented. Betides thefe figures, he 
determined the meafures of the fpiral, defcribed by a point 
moving uniformly along a right line, the line at the fame 
time revolving with an uniform angular motion; determi¬ 
ning tire proportion of its area to that of the circumfcribed 
circle, as alfo the proportion of their feCtors. 
Throughout the whole works of this great man, we every 
where perceive the deeped defign and the fined invention. 
He teems to have been, with Euclid, exceedingly careful 
of admitting into his demondrations nothing but principles 
perfectly geometrical and unexceptionable; and, although 
his mod general method of demondrating the relations of 
curved figures to draight ones be by infcribing polygons 
in them, yet, to determine thofe relations, he does not 
increase the number, and diminifh the magnitude, of the 
tides ot the polygon ad infinitum ; but from this plain fun¬ 
damental principle, allowed in Euclid’s Elements, (viz- 
that any quantity may be fo often multiplied, or added-to 
itfelf, as that the refult diall exceed any propofed finite 
quantity of the fame kind,) he proves that to deny his fi¬ 
gures to have the propofed relations, would involve an 
abfurdity ; and, when he deiiiondrated many geometrical 
properties, particularly in the parabola, by means of cer¬ 
tain progreffions of numbers, whofe terms are fimilar to 
the infcribed figures; this was dill done without coniider- 
ing fucli f'eries as continued ad infinitum, and then collect¬ 
ing or dimming up the terms of lucli infinite fe-ries. 
There have been various editions of the exiding writings 
of Archimedes. The whole of thefe works, together with 
the commentary of Eutocius, were found in their original 
Greek language, on the taking of Conftantinople, from 
whence they were brought into Italy ; and here they were 
found by that excellent mathematician John Muller, other- 
wife called Regiomontanus, who brought them into Ger¬ 
many ; where they were, with that Commentary, pubjifhed 
long afterwards, viz. in 1544, at Balil, being mod beau¬ 
tifully printed in folio, both in Greek and Latin, by Her- 
vagius. A Latin tra.nflation was publidied at Paris, 1557, 
by Pafcalius Hamellius. Another edition of the whole, 
in Greek and Latin, was publidied at Paris, 1615, in folio, 
by David Rivaltus, illudrated with new demondrations 
and commentaries: a life of the author is prefixed ; and 
at the end of the volume is added fome account, by way 
of redoration, of our author’s other works, which have 
been lod ; viz. The Crown of Hiero ; the Cochleon or 
Water Screw ; the Helicon, a kind of endlets (crew ; the 
Trifpadon, confiding of a combination of wheels and axles; 
the Machines employed in the defence of Syracufe ; the 
Burning Speculum ; the Machines moved by Air and Wa¬ 
ter ; and the Material Sphere. In 1675, Dr. Iiaac Barrow 
publidied a neat edition of the works, in Latin, at London, 
in quarto ; illudrated, and fuccinCtly demonftrated in a new 
method. But the mod complete of any, is the magnifi¬ 
cent edition, in folio, printed at the Clarendon prefs, Ox¬ 
ford, 1792. This edition was prepared for the prefs by 
the learned Jofeph Torelli, of Verona, and in that date 
prefented to the univerfitv of Oxford. The Latin tranfia- 
tion is a new one. Torelli alfo wro(e a preface, a com¬ 
mentary on fome of the pieces, and notes on the whole. 
An account of the life and writings of Torelli is prefixed, 
by Clemens Sibiliati. And at the end a large appendix is 
added, in two parts ; the fird being a Commentary on Ar¬ 
chimedes’s Paper upon Bodies that float on Fluids, by the 
Rev. Abram Robertfon, of Chrid-church college; and 
the latter is a large collection of various readings in the 
manufeript works of Archimedes, found in the library of 
the 
