Vanitas vanitatum et omnia vanitas: bubbles are emptiness, non-liquid, a tiny cloud shielding a mathematical singularity. Born from chance, a violent and brief life ending in the union with the (nearly) infinite. But a wealth of phenomena spring forth from this nothingness: underwater noise, sonoluminescence, boiling, and many others. Some recent results on a “blinking bubble” micropump and vapor bubbles in sound fields are outlined. The last section describes Leonardo da Vinci’s observation of the non-rectlinear ascent of buoyant bubbles and justifies the name Leonardo’s paradox recently attributed to this phenomenon.

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The popularity of the pastime was no doubt also related to the recent major advances in the art of soap making, which had replaced the basically abrasive concoctions of antiquity and the Middle Ages with something more similar to what we know today.
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At the time this picture was painted, the tulip had recently (1593) been introduced to Holland from Turkey via Vienna and the country was about to precipitate in a veritable tulip obsession. Investing and trading in tulips reached absurd proportions and the price of rare specimens soared to incredible heights. Many fortunes were made and lost until everything came to a screeching halt with the bursting of the “tulip bubble” in 1636. An amusing account of the events is given in the classic Extraordinary Popular Delusions and the Madness of Crowds by C. Mackay, written in 1841 and reprinted many times, most recently (1995) by Wordsworth Editions (Ware, UK). Another savory tidbit in the same book—which fits in the general theme of this paper—is the chapter on “The South Seas Bubble” which roiled the British financial markets in the 1720s.
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“… for they consider Aristotle not subject to human limitations, but rather as having some divine power, and they hold it a sin if one differs even in a nail’s breadth from him,” was the complaint of G. B. Benedetti (1530–1590; Ref. 47 p. 154).
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50.
It is necessary “to assume that some incorporeal motive force is imparted by the projector to the projectile, and that the air set in motion contributes either nothing at all, or else very little to this motion of the projectile” [Ref. 51, p. 223].
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The most illustrious example is Thomas Aquinas (1225–1274) who writes: “However, it ought not to be thought that the force of the violent motor impresses in the stone which is moved by violence [as in throwing, as opposed to the “natural” movement of falling] some force (virtus) by means of which it is moved…. For, if so, violent motion would arise from an intrinsic source, which is contrary to the nature of violent motion. It would also follow that a stone would be altered by being violently moved in local motion, which is contrary to sense” (Ref. 53, p. 517; Ref. 49, Vol. 2, p. 192).
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54.
Buridan recognized that the impetus was proportional to the density and volume of the body, and was an increasing function of velocity (Ref. 49, Vol. 3, p. 49). Perhaps worried about the possible theological implications of his dismissal of the necessity of a primum mobile, he says: “In the creation of the world, God set into motion each [celestial] sphere with the velocity which His will assigned to it, and then stopped moving it; in the following time, these movements have always persisted by virtue of the impetus imparted to those spheres. This is why it is said that God rested the seventh day” (reported in Ref. 49, Vol. 3, p. 53; see also p. 42).
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57.
“L’impeto è una virtú trasmutata dal motore al mobile” (Ref. 58, p. 350).
58.
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59.
E. MacCurdy, The Notebooks of Leonardo da Vinci (Reynal & Hitchcock, New York, 1938), Vols. 1 and 2.
60.
The use of the adverb “frequently” betrays some hesitation in Leonardo’s adherence to this concept, which is confirmed by several other passages in which, echoing the old idea of the “antiperistasis” already refuted by Aristotle, he attributes an active effect of the medium through which the body moves. For example: “Impetus at every stage of time becomes less by degrees, and the prolongation of its essence is caused by the air or the water, which closes up behind the movable thing, filling up the vacuum which the movable thing that penetrates it leaves of itself. And this air is more powerful to strike and compress the movable thing by direct percussion, than is the air which is so placed as to resist the penetration of this movable thing by becoming compressed; and it is this compression of the air which diminishes the fury of the aforesaid impetus in the movable thing” (Cod. Atl. 168 v. b, Ref. 59, Vol. 1, p. 526). The same idea is expressed in Cod. Atl. 168 v. b (Ref. 59, Vol. 1, p. 526) and 219 v. a (Ref. 59, Vol. 1, p. 529). It has been argued that Leonardo’s frequent vacillation between the traditional and the new makes him the very embodiment of “the irreconcilable conflict from which the scientific revolution came to be born” (Ref. 61). Note also in the passage quoted the erroneous idea that the air ahead of the body is compressed. In this respect Leonardo states: “The impetus of the movable thing within the water is different from the impetus of the movable thing within the air, and these differences result from the varieties of the aforesaid liquids, because air is condensable to infinity and water is not” (Cod. Atl. 168 v. b, Ref. 59, Vol. 1, p. 527).
61.
M. Kemp, “The crisis of received wisdom in Leonardo’s late thought,” in Leonardo e l’Eta’ della Ragione, edited by E. Bellone and P. Rossi (Scientia, Milano, 1982), pp. 27–39.
62.
“Ogni moto naturale e continuo desidera conservare suo corso per la linia del suo principio, cioè in qualunque loco esso si varia domanda principio.”
63.
J. Roberts, The Codex Hammer of Leonardo da Vinci (Giunti Barbera, Florence, 1981).
64.
“L’aria che si sommerse insieme coll’acqua che sopra l’altr’acqua percosse, ritorna alla aria penetrando l’acqua in moto fressuoso, variando il suo corpo in moltissime forme. E questo accade, perche’ il lieve non po’ stare sotto il grieve, anzi al continuo e’ premuto dalla parte del liquido che sopra se li posa… .”
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