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Paris Manuscript F — 82

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Bibliothèque de l'Institut de France, Paris

Diagramma della terra, con:

a b c - o - r - n

Se la superfizie dell'acqua de' fiumi correnti è sperica o no, o s'elli è necessario in equal moto d'acqua che la sua superfizie sia diritta o curva.

Po essere retta e curva pur che nel suo moto acquisti vicinità al centro del mondo, perché se così non facessi, essa acqua resterebbe immobile, se già non fussi corso refresso, e di questo non tengo conto perché moto violente.

Provasi. o c r sia un monte da la cima del quale discende un fiume e corre insino alla bassezza della sua spera, e essa acqua po discendere per la retta c o a e per la curva c b a, perché l'altezza sua dal centro del mondo è più lunga per la linia c n che per o n o a n. El simile fa la curva ch'è più alta in c n che in b n o a n, sì che si conclude non sia necessario al moto dell'acqua l'essere più curvo che retto.

Schema di arco e corda, con:

a d - b - c

L'acqua dell'arco è più declinante in superfizie che in fondo per non essere d'equal grossezza.

L'acqua1 che si move per fondo convesso sarà più tarda che movendosi per fondo piano in sé, e ancor ch'ell'abbia un principio e fine comune, sarà tanto più tarda la curva, quanto ell'è più lungo il suo arco che la sua corda.

L'acqua che da u[n] suo principio si move al fine, sarà tanto più tarda per arco che per corda, quanto esso è più lungo l'arco che la corda.

Dice qui l'avversario ch'e moti proposti saran fatti con equal tempo, imperò che, se ben l'acqua corre più tarda per a d che per a b, ch'ella ristora il moto dal d c, ch'è più repente che b c. Qui si risponde che l'acqua d c è sottile e veloce, e che l'acqua a d è tarda e grossa, e sebben giugne al fine dell'acqua in c dal d, come in c dal b, quella del d c è tanto più sottile e meno acqua2, che l'acqua che viene dal b, quanto la linia d c è meno obbliqua che la linia c b, sì che in pari tempo l'acqua d c è men che del b c.

Diagram of the earth, with:

a b c - o - r - n

Whether the surface of the water of running rivers is spherical or not, and whether, in equal motion of water, its surface must be straight or curved.

It can be straight and curved, provided that in its motion it gains nearness to the centre of the world; for if it did not do so, that water would remain motionless, unless there were a rebounding current — and of this I take no account, because it is violent motion.

It is proved. Let o c r be a mountain from whose summit a river descends and runs down to the lowest part of its sphere; and that water can descend by the straight line c o a and by the curved line c b a, because its height from the centre of the world is greater along the line c n than along o n or a n. The curve does likewise, being higher at c n than at b n or a n; so it is concluded that it is not necessary to the motion of water to be more curved than straight.

Scheme of an arc and a chord, with:

a d - b - c

The water of the arc is more declining at the surface than at the bottom, through not being of equal thickness.

Water that moves over a convex bottom will be slower than moving over a bottom flat in itself; and though it have a common beginning and end, the curve will be the slower the longer its arc is than its chord.

Water that moves from its start to its end will be slower by arc than by chord, by as much as the arc is longer than the chord.

Here the adversary says that the proposed motions will be made in equal time, since, though the water runs slower along a d than along a b, it makes up the motion along d c, which is more sudden than b c. Here it is answered that the water d c is thin and swift, and the water a d is slow and thick; and although it reaches the end of the water at c from d as from b, that of d c is so much thinner and less water than that which comes from b, as the line d c is less oblique than the line c b; so that in equal time the water d c is less than that of b c.

Is a River's Surface Spherical? Water Along Arc and Chord

Whether flowing water runs straight or curved toward the world's centre, and which path is slower

Hydraulics Physics & Mechanics Geometry

This hydraulic page asks whether the surface of a flowing river is spherical, straight or curved, arguing that water may run either way so long as its motion brings it nearer the centre of the world. In a mountain diagram (o c r), a river may descend by the straight path c o a or the curved c b a. A second figure compares an arc and its chord: water is slower over a convex bed than a flat one, and slower along the longer arc than the shorter chord, refuting an 'adversary' who claims the two paths take equal time.

On this page

Whether a flowing river's surface is spherical

Leonardo asks whether running water's surface must be straight or curved, concluding it may be either provided its motion gains nearness to the centre of the world. On a mountain o c r a river may descend by the straight line c o a or the curved c b a.

Water is slower by arc than by chord

Water over a convex bed runs slower than over a flat one, and slower along the longer arc than the shorter chord. Against an 'adversary' who claims equal time, Leonardo argues that the water along d c is thin and swift while that along a d is slow and thick, so in equal time d c carries less water.

Reader's Edition — described with AI from the e-Leo transcription.