Why a Weight Falls Faster Along an Arc Than a Chord
Folio 1v: counting degrees of velocity along arc versus chord
Leonardo compares the descent of a weight p along a circular arc with its descent along the chord of that arc, using lettered points p, r, n and v. He reasons that along the arc the fall is partly perpendicular (p to r) and partly a 'reflected' motion (r to n) worth 7/8 of the incident speed, while the chord, cutting the right angle r p v in half, is descended twice as slowly. Reckoning in degrees of velocity, he finds the arc totals 15 and the chord 8, so the arc is faster by 7/8. A small wheel-and-inclined-plane apparatus is drawn at the foot of the page.
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Descent along the arc p r n versus the chord p n
The weight p descends more quickly by the arc than by the chord: for half the arc it falls from p to r along the perpendicular, and the rest of the path (r to n) is 'reflected' motion worth 7/8 of the incident speed. Along the chord p n the motion is half as fast, so the arc wins.
Degrees of velocity: arc 15, chord 8, faster by 7/8
Reckoning the descent in degrees of velocity, the weight moves along the arc in 8 and 7 degrees (15 total) and along the chord in 4 and 4 degrees (8 total). The excess of 15 over 8 is 7, so the arc is faster by 7/8.
The chord bisects the right angle r p v
Because the chord p n cuts the right angle r p v exactly in half, the descent along it is taken to be half as fast as along the perpendicular line p r. This geometric halving underpins the whole comparison of arc and chord.