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Library/Paris Manuscript E/19

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Paris Manuscript E — 19

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Bibliothèque de l'Institut de France, Paris

Peso sostenuto da corda, con:

a b c - n m1

a b entra ventinove volte in a c e quattro volte in a d, e quella proporzione che ha esso a b con b c, tale ha il peso che sostiene la corda n c col peso che sostiene a n del peso n; e il medesimo che sostiene c, sostiene n, perché il peso n è tutto per tutta la corda n c ed è tutto in ogni parte della sua lunghezza, cioè della ventinovesima parte l'una del peso n, che tocca a essa corda, e 'l rimanente di tal peso n, che son li 28/29, li quali si scaricano nella corda a n.

Figura analoga, con:

a e d - c - b

Il peso che si sospende ne l'angolo, darà di sé pesi alli lati d'esso angolo, li quali saranno in fra loro nella medesima proporzione, quale è quella dell'obbliquità delli lor lati in sé. Ovvero tal peso si strebuirà alli sua sostentacoli nella medesima proporzione, qual è quella delli due anguli nati dalla divisione dell'angolo, dove si sostiene tal peso; la qual divisione dell'angolo si fa dalla retta che discende nel centro del grave sospeso, cioè l'angolo a b d, tagliato dalla linia e b. Adunque essendo l'angolo e b d li 9/11 dell'angolo a b d, e l'angolo a b e li 2/11, a sente li 9/11 del peso b e d b li 2/11, ecc.

Weight sustained by a cord, with:

a b c - n m

a b goes twenty-nine times into a c and four times into a d, and the proportion that a b has with b c is the same that the weight sustained by the cord n c has with the weight that a n sustains of the weight n; and the same amount that c sustains, n sustains, because the weight n is entire along the whole cord n c and is entire in every part of its length, that is the twenty-ninth part of the weight n which touches this cord, and the remainder of that weight n, which is the 28/29, which are discharged onto the cord a n.

Analogous figure, with:

a e d - c - b

The weight that is suspended at the angle will give of itself weights to the sides of that angle, which will be among themselves in the same proportion as that of the obliquity of their sides in themselves. Or rather, such weight will distribute itself to its supports in the same proportion as that of the two angles born from the division of the angle where such weight is sustained; which division of the angle is made by the straight line that descends into the center of the suspended heavy body, that is the angle a b d, cut by the line e b. Therefore, the angle e b d being 9/11 of the angle a b d, and the angle a b e 2/11, a feels 9/11 of the weight, b e d, b the 2/11, etc.

How a Hanging Weight Divides Along a Cord and an Angle

Proportional distribution of load across the cord n c and across the divided angle a b d

Physics & Mechanics Arithmetic & Proportion Geometry

Two mechanical diagrams study how a suspended weight distributes itself among its supports. In the upper figure a weight hung from a graduated cord is shared between the segments n c and a n in the same proportion that a b bears to b c. In the lower figure a weight hung at the vertex of an angle divides between the two sides in proportion to the two angles formed when the vertical to the weight's center cuts the angle a b d.

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A weight shared proportionally along a cord

The weight hung on the cord is distributed between its parts so that the load carried by n c and the load carried by a n stand in the same ratio as a b to b c. Each part of the cord bears a twenty-ninth part of the weight n, while the remaining 28/29 is discharged onto the segment a n.

A weight at an angle divides by the two sub-angles

A weight suspended at an angle gives itself to the two sides in proportion to the two angles produced when the vertical dropped to the weight's center divides the angle a b d by the line e b. Since angle e b d is 9/11 of angle a b d and angle a b e is 2/11, the supports carry those same fractions of the load.

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