Practice of Geometry, the Balance, and the Course of Rivers
Squaring polygons and solids, rules of counterweight and friction, and river flow
A densely worked double page. The right-hand face is headed 'Practice of geometry' and turns plane and solid figures into equal squares - a triangle, then a hexagon and any polygon, resolved into triangles - proving that every pyramid is a third of its prism by cutting a cube into six pyramids. The rest of the sheet develops the arithmetic of the balance in long rules and worked examples relating the arms, the given weight and its counterweight, extended to the friction of the axle, together with observations on rivers (a slower stream winds through a longer bed; floods abandon the old channel and level the bottom) and columns of arithmetic. A marginal reminder to 'look for a male squirrel' and a note of sums in florins and lire round out the page.
On this page
Squaring a triangle by lending it equal surface
To square the triangle a b c, add equal areas around it in the pieces b d a and a e c, forming the rectangle d e b c; throw away the half a n e c and the remaining half d a b n is equal to the original triangle.
Every pyramid is a third of its prism; the cube in six pyramids
Cut a cube into 6 pyramids whose bases are its faces and whose apexes meet at its centre. A cut through the centre parallel to a face halves the cube, each half equal to 3 pyramids, whence the pyramid built on the base and height of the prism is one third of it.
The rule of arm, weight and counterweight
With a b the counterweight arm, b c the weight arm and e the given weight, multiply a b by 4 as often as it contains b c, divide the weight by the total, then multiply by 4 again: the result is the counterweight d that resists the descent of the load. In the worked case one whole resists one whole.
Reckoning the friction of the axle
For a weight, lever and counter-lever all equal to 4, the friction of the pole is a quarter (4/3). Leonardo checks that 1 and 1/3 make 4/3, equal to the counterweight of the friction, and confirms that the 4 on the pole plus 4/3 of the counterweight make 16/3, whose quarter is again 4/3.
A slower river needs a longer bed; more water runs straighter
One and the same river occupies a longer, more winding bed the less water it carries and the slower it moves, and a shorter one the swifter its flow. It follows that if a river's waters are continually increased, its course becomes straighter.
How floods reshape a winding river and level its bed
The river a d b g c f in full flood would run far shorter than the trickle d b f through the same width. Floods drive rivers out of their beds into new hollows they then follow, abandoning the old channel; the greater the flood, the more level and even the bottom becomes.