surfresearch.com.au  leonardo da vinci : hyrodynamics, 1515

"Three ships of uniform breadth, length and depth, when propelled by equal powers, will have different speeds of movement (Figure 83); for the ship which presents its widest part in front is swifter, and it ...

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.... resembles that shape of birds and fishes such as the mullet.
And this ship opens wIth its sIdes and in front of it a great quantIty of water which afterwards with its revolution presses against the last two-thirds of the ship.
The ship 'dc' does the opposite, and "fe" has a movement midway between the two above (G.50v)".

Not surprisingly, an erroneous theory has erroneous implications in practice.

Fig 83 "Hydrodynamic design of Boats and Fishes", based on Institut de France, Paris, 1510-1515, 50v.

"The arrow shot from the centre of the earth to the highest point of the elements will ascend and descend by the same straight line, although the elements may be in rotation around the centre.
The gravity which descends through the elements when they are in rotation always has its movement correspond to the direction of the line which extends from the commencing point of the movement to the centre of the world (G.54v). ..it comes about that a stone thrown from a tower does not strike the side of the tower before reaching the ground (G.55r)."

The fall is vertical in relation to the revolving spheres of fire, air, water and earth, but relative to exterior space it can be seen to pursue a spiral path.
A object which fell for twenty-four hours would make a complete revolution in a spiral judged from a static viewpoint (see Figure 78).
His examples and analysis are those which Oresme adduced in favour of the earth's rotation.
It is slightly disconcerting to find that Oresme, ...

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... having martialled an impressive array of evidence to indicate that the world revolved, did not consider the matter proven.
However, the equivocation of Oresme's conclusion did not prevent the efficacy of his, 'experimental' arguments from impressing Leonardo.

The subtle duality of a movement which was at once relatively vertical and absolutely spiral is typical of the complicated relativities which operated in natural science.
Leonardo saw nature as weaving an infinite variety of elusive patterns on the basic warp and woof of mathematical perfection.
Nowhere could nature's endless variations on geometrical themes be seen more matvellously than in the dynamics of water, above all in the configuration of vortices.
As a foundation for his studies he outlined a basic classification of natural spirals, the first three of which were variations on the basic schema already illustrated in Figure 78.
Altogether, there were four varieties, namely convex spiral, planar spiral, concave spiral and the fourth is the columnar spiral (Figure 84).
Each of these possessed its own dynamic properties and reacted to opposing forces in a different way.
The peculiar form and efficacy of circulatory force in a vortex came from what he called "a circumstance worthy of note"; "The spiral or rotary movement of every liquid is so much the swifter as it is neal'er the centre of its revolution", unlike a wheel in which the movement "is so much slower as it nears the centre"(C.A.296vb).
A related peculiarity was the way in which a rapid vortex tends to acquire a void at its centre: "The lateral weight of the vortex-circulation is two-fold ... and such duplication of weight firstly comes into being in the revolving movement of the water and secondly is created on the sides of this concavity, supporting itself there ... It makes the concavity in the form of a pyramid and makes it so much the more swiftly as the pyramid is more pointed"(C.A.296vb).
The "concave spiral" in water thus prettily combined a pyramidal law of the type which had so delighted him in the 1490s with the revolving motion which is found so ubiquitously in his late science.
This combination gave the vortex its uniquely concentrated force.
The vortex was a natural power-drill, gouging remorselessly into underlying surfaces, sucking fragmented particles into its whirling mass and then projecting them into the surrounding space with violent impetus: "It strikes and hollows out the bed in a sudden chasm, for, in addition to the force of the impact, there is joined the spiral quality made by the said revolution, by means of which those things disturbed by the impact are stirred up and carried away"(F .17v).
This "spiral quality" could have astonishing effects, both in remorseless power and geometrical regularity: "Solid rock of Mugnone, hollowed out into the form of vases by the force of the water, is of such precision that it appears to be handiwork" (Figure 85).

If the "concave spiral" was one of water's most characteristic configurations, it was only one of many. Impetus nowhere had more obviously geometrical consequences than in fluids, but nowhere was its action more subject to a tantalizing variety of intersecting variables.
To outline the ...

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Convex spiral.         Planar spiral.       Concave spiral.       Columnar spiral.

Image left: "Four varieties of spiral ", based on Institut de France, Paris, 1513-1514, 42r. Fig 84. page 308.

It is not hard to understand the aesthetic qualities which drew him "to investigate the many beautiful movements which result from the penetration of one element into another" (F.34v).
And a number of his analyses ...

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... are undeniably impressive pieces of writing.
But the total effect of his writings on water is to my mind rather discouraging.
An excessive accumulation of descriptive details all but obliterates a framework of dynamic law which could be adequately stated in a fraction of the space.
When he proudly informed the reader that "in these eight pages there are seven hundred and thirty conclusions on water" (Leic. 26v), we may feel that the boundary between dedication and obsession has been overstepped, just as it had been in his most repetitive pages of geometrical variations (Plate 83).
We cannot but be grateful, however, that his obsession resulted in one of his most miraculous drawings, illustrated in Plate 84.

The two studies at the top of the page belong to an extensive series dedicated to the turbulent effects of interruption in a fast flow.
Sometimes obstructions were placed laterally at the margins of the stream, creating between them a brilliant interlace of curvilinear percussions (e.g. F .89r).
In other drawings, as here, rudder-like obstructions cut viciously into the flow at different depths and at different angles with an incredible variety of results.
Here the water rushes onwards in a series of gurgling spirals and sweeping curves, like twisted pennants blown in a fierce wind.
And the parallels with the natural movement of hair, which we have noticed before, are particularly apparent in the horse's-mane pattern in the second demonstration.
The main drawing on the page is less hair-like and more floral in nature, resembling a bouquet of aquatic blossoms, the translucent equivalent of the Star of Bethlehem (see Plate 74).
It is the most complete of all his water drawings.
It is to his hydrodynamics what the "great lady" anatomy is to his science of the human body, that is to say, a composite study in which causes and effects from many separate analyses are fused together in an astonishing synthesis.
A set of drawings in Manuscript F and at Windsor (especially 12661-2) represent preliminary stages in this synthesis, as the components of turbulent water and submerged air unfold at first separately and then in conjunction.
He explained that there were three factors to be taken into account: the primary motion of the falling column of water; the secondary motion of the accidentally submerged air; and the reflex motion of the main mass of water in the pool.
The vortex patterns of water alone were intricate enough, but the admixture of air bubbles contributed additional complications: "Of the eddies in water, all those which begin at the surface are filled with air; those which have their origin within the water are filled with water and these are more lasting because water within water has no weight" (C.A.42ra).
In the drawing we can see the deeper eddies of "water within water" happily pursuing their revolving impetuses to uninterrupted conclusions, while those mixed with the bubbles are thrust violently upwards to the surface, where they "speedily perish in exploding rosettes."

Leonardo took special pleasure in the behaviour and form of the bubbles:
"The air which is submerged together with the water. ..returns to the ...

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... air, penetrating the water in sinuous motions, changing its substance into a great number of shapes. ..When the air enclosed within the water has arrived at the surface it immediately forms the figure of a hemisphere, and this is enclosed within an extremely thin film of water.
This occurs of necessity because water always has cohesion in itself. ..and this air having reached the opening of the surface of the water and not finding there any weight of water to press it upwards, raises its head through the surface of the water with as great a weight of water joined to it as the said tenacity can support; and it stops there in a perfect circle as the base of a hemisphere, which has the said perfection because its surface has been uniformly expanded by the uniform power of the air.
The bubble settles as the impetus of its emergence is expended, and "because the part of the water with which this air is clothed is heavier where it is more perpendicular to the centre of the circle which forms the base of the hemisphere. ..it lowers itself more"at the top of its curve, in accordance with the rule "that part of a thing supported at its extremities is so much weaker as it is more distant from its foundation".
This less than hemispherical profile is structurally unstable, and it eventually "breaks. ..in the third part of its curve; this is proved with the arches of walls, and therefore I will not treat of it in these notes, but will place it in the book where it is necessary" (Leic. 2sr).
We have seen many such analogies between the worlds of the natural and human engineer, but none is more delightful than this analysis of an air-bubble's fragile architecture.

Such considerations of hydrodynamic turbulence, "infinite" though they were, only comprised the first of fifteen projected sections in an extensive treatise on water in all its aspects:

"Book 1 of water in itself; book 2 of the sea; book 3 of underground channels [vene]; book 4 of rivers; book S of the nature of the depths: book 6 of the objects [obstructions etc.]; book 7 of gravels; book 8 of the surface of water; book 9 of the things which move in it; book 10 of the means of renovating rivers; book II of conduits; book 12 of canals; book 13 of machines turned by water; book 14 of how to make water ascend; book IS of things which are consumed by water (Leic. ISV)".

Needless to say neither this exhaustive treatise nor the alternative schemes outlined elsewhere in his late manuscripts was brought to conclusion, and the surviving notebooks do not contain any single "book" which can be regarded as complete.
However, some of the pages in the Leicester Codex, containing large blocks of continuous text and neatly disposed illustrations in the margins can be taken to indicate the kind of work he had in mind. The pattern of his projected treatise is clear from the headings; it was to progress from "pure" hydrodynamics, through the geographical study of the earth's irrigation, to questions of hydraulic engineering, military and civil.
He intended to explain, for example, "how ...

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... a river may be diverted by a few stones if one understands the line of its current'' (Leic.27v), using the corkscrew vortices to work on man's behalf, in contrast to the labour-intensive efforts of the Florentines to divert the Arno.
The power which could be placed in man's control was immense.
It was a power which promised untold benefits but also threatened untold harm in the wrong hands. One of his inventions, which enabled men to swim under water, he hesitated to ''divulge, on account of the evil nature of those men'' who might put it to destructive use in sinking boats (Leic.22v).

"Study of a life preserver,  Ms Bf. 81 v, c 1487-1490."

"Study of diving suit,  CA f. 909 v, c. 1485-1487."

This manuscript originally contained 114 leaves, but it was mutilated by Guglielmo Libri and now only contains 63. Some of the missing leaves (sold to Lord Ashburnham by Libri) were returned to the Institut de France in 1891. The manuscript contains studies by Leonardo on proportions and on movement, especially that of water. In many notes, he stressed the painter?s need for a high degree of scientific training.

fols. 59v-60r: Studies of hydrodynamics; sketches of waterfalls and whirlpools.

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