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In his work On the Sphere and Cylinder, Archimedes proved that the ratio of the volume of a sphere to the volume of the cylinder that contains it is 2:3. In that same work he also proved that the ratio of the surface area of a sphere to the surface area of the cylinder that contains it, together with its circular ends, is also 2:3.
And although he made many excellent discoveries, he is said to have asked his kinsmen and friends to place over the grave where he should be buried a cylinder enclosing a sphere, with an inscription giving the proportion by which the containing solid exceeds the contained. Πολλῶν δὲ καὶ καλῶν εὑρετὴς γεγονὼς λέγεται τῶν φίλων δεηθῆναι καὶ τῶν συγγενῶν ὅπως αὐτοῦ μετὰ τὴν τελευτὴν ἐπιστήσωσι τῷ τάφῳ τὸν περιλαμβάνοντα τὴν σφαῖραν ἐντὸς κύλινδρον, ἐπιγράψαντες τὸν λόγον τῆς ὑπεριχῆς τοῦ περιέχοντος στερεοῦ πρὸς τὸ περιεχόμενον.
Because expressions for the volume and surface area of a cylinder were known before his time, Archimedes’’ results established the first exact expressions for the volume and surface area of a sphere.
English translation by Earnest Cary in Dio’s Roman History, Volume II: Fragments of Books XII-XXXV, The Loeb Classical Library, Harvard University Press, Cambridge, 1914.
Marcellus straightway mourned on learning this [Archimedes’ death], and buried him with splendour in his ancestral tomb, assisted by the noblest citizens and all the Romans; Ἐθρήνησε δὲ Μάρκελλος τοῦτο μαθὼν εὐθέως,
Λαμπρῶς τε τοῦτον ἔκρυψεν ἔν τάφοις τοῖς πατρώοις
Σὺν τοῖς ἀρίστοις πολιτῶν καὶ τοῖς ‘Ρωμαίοις πᾶσι˙
Translation by Michael Grant in Cicero-On the Good Life, Penguin Books, New York, 1971, Pages 86-87.
But from Dionysius’s own city of Syracuse I will summon up from the dustwhere his measuring rod once traced its linesan obscure little man who lived many years later, Archimedes. When I was questor in Sicily [in 75 BC, 137 years after the death of Archimedes] I managed to track down his grave. The Syracusians knew nothing about it, and indeed denied that any such thing existed. But there it was, completely surrounded and hidden by bushes of brambles and thorns. I remembered having heard of some simple lines of verse which had been inscribed on his tomb, referring to a sphere and cylinder modelled in stone on top of the grave. And so I took a good look round all the numerous tombs that stand beside the Agrigentine Gate. Finally I noted a little column just visible above the scrub: it was surmounted by a sphere and a cylinder. I immediately said to the Syracusans, some of whose leading citizens were with me at the time, that I believed this was the very object I had been looking for. Men were sent in with sickles to clear the site, and when a path to the monument had been opened we walked right up to it. And the verses were still visible, though approximately the second half of each line had been worn away. Non ego iam cum huius vita, qua taetrius miserius detestabilius escogitare nihil possum, Platonis aut Archytae vitam comparabo, doctorum hominum et plane sapientium: ex eadem urbe humilem homunculum a pulvere et radio excitabo, qui multis annis post fuit, Archimedem. Cuius ego quaestor ignoratum ab Syracusanis, cum esse omnino negarent, saeptum undique et vestitum vepribus et dumetis indagavi sepulcrum. Tenebam enim quosdam senariolos, quos in eius monumento esse inscriptos acceperam, qui declarabant in summo sepulcro sphaeram esse positam cum cylindro. Ego autem cum omnia conlustrarem oculisest enim ad portas Agragantinas magna frequentia sepulcrorum -, animum adverti columellam non multum e dumis eminentem, in qua inerat sphaerae figura et cylindri. Atque ego statim Syracusaniserant autem principes mecumdixi me illud ipsum arbitrari esse, quod quaererem. lnmissi cum falcibus multi purgarunt et aperuerunt locum. Quo cum patefactus esset aditus, ad adversam basim accessimus. Apparebat epigramma exesis posterioribus partibus versiculorum dimidiatum fere.
So one of the most famous cities in the Greek world, and in former days a great centre of learning as well, would have remained in total ignorance of the tomb of the most brilliant citizen it had ever produced, had a man from Arpinum not come and pointed it out! Ita nobilissima Graeciae civitas, quondam vero etiam doctissima, sui civis unius acutissimi monumentum ignorasset, nisi ab homine Arpinate didicisset.
The Romans were so uninterested in mathematics that Cicero’s act of respect in cleaning up Archimedes’ grave was perhaps the most memorable contribution of any Roman to the history of mathematics.Cicero himself lends credence to Simmons’ assessment when he writes (Tusculan Disputations, Book I, Section 5):
Among them [the Greeks] geometry was held in highest honor; nothing was more glorious than mathematics. But we [the Romans] have limited the usefulness of this art to measuring and calculating. In summo apud illos honore geometria fuit, itaque nihil mathematicis inlustrius; at nos metiendi ratiocinandique utilitate huius artis terminavimus modum.
Call Archimedes from his buried tomb
Upon the plain of vanished Syracuse,
And feelingly the Sage shall make report
How insecure, how baseless in itself,
Is the Philosophy, whose sway depends
On mere material instruments;how weak
Those arts, and high inventions, if unpropped
By virtue.He, sighing with pensive grief,
Amid his calm abstractions, would admit
That not the slender privilege is theirs
To save themselves from blank forgetfulness!