Mechanical elements: loaded beams and wheels

Descent of an inclined beam: arc versus chord

Leonardo opens the mechanical elements by treating a beam raised obliquely by one end. Its descent, he argues, is necessarily made more along an arc than along a straight chord. Only if forced to fall along the chord does the beam behave as the hypotenuse of a right angle.

The beam as hypotenuse of the right angle

Treating the beam as the hypotenuse of a right triangle, Leonardo says it is more powerful at one end than the other in proportion as the two sides of its rectangle differ in length. Where the two sides are equal, equal weights at the ends resist one another equally; if unequal, the greater weight moves the lesser.

Equal and unequal weights on the hypotenuse

When the sides of the right angle are unequal, equal weights fixed at the opposite ends of the hypotenuse set it in motion. In that case the lesser weight placed on the shorter side can move the greater weight placed on the longer side.

A load on the contact of two touching wheels

A weight resting on the point where the rims of two wheels touch discharges itself onto their poles along lines running from those poles to the point of contact. Of two wheels touching rim to rim, the one of smaller circle bears more of the load. The auxiliary lines n o and m p are drawn to answer an imagined opponent.

The contact of two circles and the radius

The contact of two circles lies on the straight line joining their centres. Every straight line drawn from a centre to the circumference cuts that circumference at equal angles. These geometric propositions underpin the analysis of loaded wheels.

Every weight's power aims at the centre of the world

On folio 194r Leonardo repeats that the power of a weight resting on a wheel's rim, suspended from a single support, is always directed straight to the centre of the world. A weight descends faster along a straighter line and slower along a more oblique one; making its motion oblique effectively loads it with extra weight.

0195 v

Figura con lettere.

A lato della figura:

Figura con lettere e numeri.

A lato della figura:

Figura con numeri.

A lato della figura:

Figura con scritte.

A lato della figura:

Figura con numeri.

Figura con lettere:

Nel margine destro del foglio in alto figura.

Sopra la figura:

A lato della figura:

0194 r

Nel margine destro figura con lettere:

Figure with letters.

Beside the figure:

Figure with letters and numbers.

Beside the figure:

Figure with numbers.

Beside the figure:

Figure with writing.

Beside the figure:

Figure with numbers.

Figure with letters:

A figure in the right margin at the top of the sheet.

Above the figure:

Beside the figure:

In the right margin, a figure with letters:

Mechanical elements: loaded beams and wheels

On this page

Descent of an inclined beam: arc versus chord

The beam as hypotenuse of the right angle

Equal and unequal weights on the hypotenuse

A load on the contact of two touching wheels

The contact of two circles and the radius

Every weight's power aims at the centre of the world