x z y - Q p o a b c d e - f g h i K
l m n - 1 2 3 4 5 6 7 8
Q p o a b c d e are 8 empty buckets, which have as many facing them that are, like them, equally distant from the pole of the wheel: that is f g h i K l t m n. The 8 above are all empty, the other 8 below are all full; but of the 8 above, which shall be named after the characters inside them, know that 4 against 5 has a triple proportion of weight for the distance it has from the perpendicular of the pole beyond 5; but for the distance the water has that is in bucket K, which weighs by itself as much as the counterweight pressing it, and which bucket is 7 times more distant from the perpendicular of the pole than the ball 4, that bucket stands equal in the balance with 7 similar balls of 4, of which there are only 2, one above the pole and the other below it; and so the 8 above stand as the 8 below. Besides this, 3 exceeds 6 by 1/3, and 2 is more distant than 7 by 1/6, and 1 is more distant than 8 by 1/10.
1/2 - 1/2.
4 exceeds 5 by 3, and f exceeds it by 5.
3 exceeds 6 by 2/3, and g exceeds it by 1 3/4.
2 exceeds 7 by 2/5, and h exceeds it by 5/6.
1 exceeds 8 by 2/11, and i exceeds it by 1/2.
Hence we shall say that 1 2 3 4 exceed in weight 5 6 7 8 by 4 1/6 pounds, each weight being one pound, and likewise the water of each bucket another pound. And if 1 2 3 4 exceeds 5 6 7 8 by 4 1/6 pounds, f g h i exceeds 1 2 3 4 by 3 5/6 pounds.
Now there remain to us K l m n full of 4 pounds of water and b c d e which are empty, so that when a f g h i exceeded nothing, these 4 would suffice. Although I have set weights under 1 2 3 4 equal to them, I can subtract from them the water in the buckets against its counterweight; but by as much less than the counterweight as the centre of lightness of the water is more remote than that of the lead, being of greater volume.

1/6 2/6 4/6
2/3 3/4 1/2

m - n

The appendages always stand perpendicular and with weights equally distant from the centre of the wheel. Now arrange that such appendages, as they turn, should pull with cords and weights out of equality, or that they close or open holes for the water.