Wheel for Perpetual Motion: Proof of Impossibility
Five balls against three, and the perpendicular that falls between equal angles
Another disc of spiralling channels carrying balls, keyed to the letters b p q and a n m, is offered as a further proof. Leonardo invokes the fifth figure of the third book of his demonstrations to show the supposed motion is impossible, and answers the objection that a ball running from a to b would leave five balls near the pole against three farther out. He notes geometrically that the circle opposite a n m cuts the other circles at their lowest points, and that a perpendicular dropped through such an intersection falls between equal angles.
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The impossibility of the believed motion
Using the fifth figure of the third book of demonstrations, Leonardo shows the motion supposed above is impossible. To the objection that if the little ball a ran to b there would be five balls near the pole against three farther out, whose distance would give them power to pull down their part of the wheel, he answers that the following figure with the same check proves it impossible too.
Circle a n m and the perpendicular between equal angles
The circle opposite a n m cuts the other circles at the point of their lowest part. If a perpendicular line is let fall at the point of that intersection, it will be found to fall between equal angles.
Disc of spiralling ball-channels
The device is drawn as a disc filled with curved spiral channels radiating from the hub, with small balls set along them, a variant of the overbalancing wheels on the neighbouring pages.