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According to Van Der Waerden's "Science Awakening", it was common for Ancient Greek mathematicians to use a board filled with sand to draw their figures, ie :

But the ancients made their geometric drawings on a board (plinthium, abacium, abacus) covered with sand. This becomes clear from the statements in Iamblichus, Apuleius and Hieronymus which Winter has brought together

The reference given is Winter's "Der Tod des Archimedes", which is however exceedingly hard to find, not being digitized or sold on any website. Would anyone happen to know what those exact statements are, by any chance?

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I do not know Winter's text, at any rate I think the references to Iamblichus and Apuleius.

For Iamblichus, the reference is definitely to Life of Pythagoras, Chapter V:

Though no one therefore attended to him, and no one was genuinely desirous of those disciplines which he endeavoured by all means to introduce among the Greeks, yet he neither despised nor neglected Samos, because it was his country, and therefore wished to give his fellow-citizens a taste of the sweetness of the mathematical disciplines, though they were unwilling to be instructed in them. With a view to this, therefore, he employed the following method and artifice. Happening to observe a certain youth, who was a great lover of gymnastic and other corporeal exercises, but otherwise poor and in difficult circumstances, playing at ball in the Gymnasium with great aptness and facility, he thought the young man might easily be persuaded to attend to him, if he was sufficiently supplied with the necessaries of life, and freed from the care of procuring them. As soon as the youth, therefore, left the bath, Pythagoras called him to him, and promised that he would furnish him with every thing requisite to the support of his bodily exercise, on condition that he would receive from him gradually and easily, but continually, so that he might not be burthened by receiving them at once, certain disciplines, which he said he had learnt from the Barbarians in his youth, but which now began to desert him through forgetfulness and the incursions of old age. But the young man immediately acceded to the conditions, through the hope of having necessary support. Pythagoras, therefore, endeavoured to instruct him in the disciplines of arithmetic and geometry, forming each of his demonstrations in an abacus, and giving the youth three oboli as a reward for every figure which he learnt. This also he continued to do for a long time, exciting him to the geometrical theory by the desire of honour; diligently, and in the best order, giving him (as we have said) three oboli for every figure which he apprehended. But when the wise man observed that the elegance, sweetness, and connexion of these disciplines, to which the youth had been led in a certain orderly path, had so captivated him that he would not neglect their pursuit though he should suffer the extremity of want, he pretended poverty, and an inability of giving him three oboli any longer. But the youth on hearing this replied, “I am able without these to learn and receive your disciplines.” Pythagoras then said, “But I have not the means of procuring sufficient nutriment for myself.” As it is requisite, therefore, to labour in order to procure daily necessaries and mortal food, it would not be proper that his attention should be distracted by the abacus, and by stupid and vain pursuits. The youth, however, vehemently abhorring the thought of discontinuing his studies, replied: “I will in future provide for you, and repay your kindness in a way resembling that of the stork: for I in my turn will give you three oboli for every figure.”

For Apuleius, the reference is to Pro se de magia liber (Apology) 16.7:

Quem tu librum, Aemiliane, si nosses ac non modo campo et glebis, uerum etiam abaco et puluisculo te dedisses, mihi istud crede, quanquam teterrimum os tuum minimum a Thyesta tragico demutet, tamen profecto discendi cupidine speculum inuiseres et aliquando relicto aratro mirarere tot in facie tua sulcos rugarum.

i.e.

And if you, Aemilianus, had known this book [the book is the lost Archimedes' Catoptrics] and had given yourself not only to the field and the glebes, but also to the abacus and the sand, believe me, even with this monstrous appearance of yours as a Tiestean mask, you would, no doubt, for the passion of study, have gone to the mirror and sometimes, having stopped the plow, you would have beheld on your face the many furrows of wrinkles.

The reference to Hieronymus is not very clear to me, and I am also not entirely sure which Hieronymus is being talked about.

Thanks to njuffa who identified the correct reference: it is the commentary to the Chapter 4 of the Book of Ezekiel of St. Jerome, that can be found in Opere D. Hieronymi Quintus Tomus Commentarios in prophetas quos maiores vocant, continet, Basel 1553, p. 387:

Symmachus manifestius interpretatus est πλινϑεϊον, quem nos laterculum et abacum appellare possumus. In cuius puluere solent geometræ, γραμμας, id est, lineas radiosque describere.

i.e.

Symmachus translates more clearly as πλινϑεϊον what we may call laterculum or abacum, which is the instrument in whose sand the geometers make their "γραμμας", i.e., drawings formed by lines and rays.

The passage of the Bible traslated by Symmachus is

You also, son of man, take a clay tablet and lay it before you, and portray on it a city, Jerusalem.

Note that Jerome is not a mathematician, so it is doubtful that he had direct knowledge of the instrument he is talking about. This is my personal impression, corroborated by the passage "drawings formed by lines and rays." Here, lines and rays does not make much sense, one would rather expect something like "lines and circles"; my personal impression is that Jerome has incorrectly interpreted some passage from Virgil or Cicero (see below) and confuses the meaning of the term "radius," which can also denote the radius of a circle, but mainly designates the rod for drawing diagrams in the sand.


There is also another Hieronymus, Hieronymus of Rhodes, belonging to the school of Aristotle and therefore also called Hieronymus Peripateticus. He is cited by Diogenes Laertius as the source of the account concerning Thales' journey to Egypt and the measurement of the pyramids, and he is also often quoted by Cicero (although it may be doubted that he is the same Hieronymus, and not an even more obscure mathematician) as a philosopher. And speaking of Cicero, although I was unable to find the reference to the correct Hieronymus, may I add an interesting quote from De natura deorum 2, 18

Interea Vellei noli quaeso prae te ferre vos plane expertes esse doctrinae. conum tibi ais et cylindrum et pyramidem pulchriorem quam sphaeram videri, novum etiam oculorum iudicium habetis. Sed sint ista pulchriora dumtaxat aspectu -- quod mihi tamen ipsum non videtur; quid enim pulchrius ea figura, quae sola omnis alias figuras complexa continet, quaeque nihil asperitatis habere, nihil offensionis potest, nihil incisum angulis nihil anfractibus, nihil eminens nihil lacunosum; cumque duae formae praestantissimae sint, ex solidis globus (sic enim σφαῖρα interpretari placet), ex planis autem circulus aut orbis, qui κύκλος Graece dicitur, his duabus formis contingit solis ut omnes earum partes sint inter se simillumae a medioque tantum absit extremum, quo nihil fieri potest aptius -- sed si haec non videtis, quia numquam eruditum illum pulverem attigistis, ne hoc quidem physici intellegere potuistis, hanc aequabilitatem motus constantiamque ordinum in alia figura non potuisse servari?

i.e.

In the mean while, Velleius, let me entreat you not to be always saying that we are utterly destitute of every sort of learning. The cone, you say, the cylinder, and the pyramid, are more beautiful to you than the sphere. This is to have different eyes from other men. But suppose they are more beautiful to the sight only, which does not appear to me, for I can see nothing more beautiful than that figure which contains all others, and which has nothing rough in it, nothing offensive, nothing cut into angles, nothing broken, nothing swelling, and nothing hollow; yet as there are two forms most esteemed, the globe in solids (for so the Greek word σφαῖρα, I think, should be construed), and the circle, or orb, in planes (in Greek, κύκλος); and as they only have an exact similitude of parts in which every extreme is equally distant from the centre, what can we imagine in nature to be more just and proper? But if you have never raked into this learned dust to find out these things, surely, at all events, you natural philosophers must know that equality of motion and invariable order could not be preserved in any other figure.

Here the "learned dust" (or sand), as in Apuleius, refers to the instruments of geometers, and then it is a classical locution to refer to mathematical knowledge.

There are also many other examples in Greek and Latin literature of the use of figures drawn in sand. A very famous one is found in the episode of the dialogue with the slave in Plato's Menon, in which Plato draws figures on the ground. Another example is found in the first of three versions of Archimedes' death reported in Plutarch's Life of Marcellus (here Archimedes is drawing figures in the sand when a Roman soldier discovers and kills him). In these two episodes there is no direct reference to the use of the abacus or aβαχ (term derived from the ancient Phoenician word abak, meaning dust), but the reader is left to imagine the scene. By contrast, a lesser-known but perhaps more relevant example is found in Persius' first satire in which the author explicitly mentions the "abaco numeros et secto in puluere metas"

inde uaporata lector mihi ferueat aure,
non hic qui in crepidas Graiorum ludere gestit
sordidus et lusco qui possit dicere 'lusce,'
sese aliquem credens Italo quod honore supinus
fregerit heminas Arreti aedilis iniquas,      
nec qui abaco numeros et secto in puluere metas
scit risisse uafer, multum gaudere paratus
si cynico barbam petulans nonaria uellat.
his mane edictum, post prandia Callirhoen do.

i.e.

I want readers with cleansed ears, fired by such stuff, not
Some wretch who delights in poking fun at Greek sandals;
A one-eyed man who loves to call another man ‘One-eye’,
Who thinks he’s something, full of provincial importance,
Because at Arretium, as aedile, he ruled on half-measures,
Not some crafty fellow who’s used to jeering at maths
On the abacus, or diagrams drawn in the furrowed dust,
One ready to howl with delight when some insolent whore
Tugs at a Cynic’s beard. To them I’d recommend reading
Posters in the morning, a novel, say Callirhoe, after lunch.

"Secto in pulvere metas" means literally "conical section [drawn on the] dust". So, in this passage, Persius is referring to the "dust abacus" (which differs from the "lapilli abacus" which originates the counting frames for arithmetic calculations), a rectangular tablet, made of wood or bronze, on which green-colored powder ("pulvis hyalinus") was glued, allowing numerical symbols and geometric figures to be traced with a stick ("radius"), using it, in this way, as we use a blackboard today.

This instrument is mentioned again by Cicero in his Tuscolanae Disputationes (Quaestio V, 23):

ex eadem urbe humilem homunculum a pulvere et radio excitabo, qui multis annis post fuit Archimedes

i.e.

From the same city I shall awaken, from the dust and the stick with which one writes in the sand, a humble little man who lived many years later, Archimedes.

from which it appears that Archimedes used this type of abacus, to draw geometric figures; but also Virgil mentions it in his Eglogues (III, 40)

si quis fuit alter
Descripsit radio totum qui gentibus orbem

i.e.

who was the other, who to men drew with his stick all the sky.

A description of this type of abacus, but much later than Virgil and Cicero, is found in De Nuptiis Philologiae et Mercuri, Liber VI, De Geometria, by Martianus Capella (5th century)

Parent denique jam ingressurae artes obsequio electissimae feminarum, quae decentem quandam atque hyalini pulveris respersione coloratam velut mensulam gestitantes ad medium superi senatus locum fiducia promtiore procedunt [...] Illud quippe quod gerulae detulerunt, abacus nuncupatur, res depingendis designandisque opportuna formis; quippe ubi vel lineares ductus, vel circulares flexus, vel triangulares arraduntur anfractus.

i.e.

Finally, the arts show themselves in deference to the most chosen women, who, carrying a table of a decent sort and colored with a sprinkling of hyaline powder, confidently advance to the middle of the upper senate.[...] In fact, what the handmaids brought is called an abacus and is used to draw and delineate things into proper shapes by scraping straight, curved or triangular figures on its surface.

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    $\begingroup$ The Hieronymus referenced in Winter is the commentary of St. Jerome on chapter 4 of Ezekiel. Opere D. Hieronymi Quintus Tomus Commentarios in prophetas quos maiores vocant, continet. Basel 1553, p. 387: "Symmachus manifestius interpretatus est πλινϑεϊον, quem nos laterculum et abacum appellare possumus. In cuius puluere solent geometræ, γραμμας, id est, lineas radiosque describere." $\endgroup$
    – njuffa
    Jan 21 at 1:53
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    $\begingroup$ I don't see how "pulvere" translates to "instrument"? My Latin dictionary says that pulvis (-eris) is the sand into which mathematicians drew their figures with a stylus. $\endgroup$
    – njuffa
    Jan 21 at 12:16
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    $\begingroup$ @njuffa Sure, "pulvis" means "dust" (actually, in this context, "glass powder"), and by metonymy it also indicates mathematics; I had forgotten a piece of the translation. $\endgroup$
    – user6530
    Jan 21 at 17:45

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