m n, o r, r o, a b

black water - white water

Here it is asked, when the two cisterns m and n pour their waters into the two vessels a b, of what color they will be, the water of cistern m being black and that of cistern n white.

And try first making the conduits o r covered, so that they pour only through the mouth. And then try them uncovered. And this same experiment can be done with wind of various odors.

On the perfected balance. If you had hung your weight on one of the arms of the balance on this side of the fulcrum, and wished to hang another weight beyond the fulcrum, on the other arm, wherever you please, and that this second weight were such as to resist the first, you will do thus: multiply the degrees of the first weight against the degrees of the arm that sustains it, and then divide that product by the degrees of the second arm. And that quotient which results will be the degrees of the second weight, which in such a position makes resistance to the first weight.

And know that the arm of a balance is always understood to be from the center of the fulcrum of that balance up to the middle of the place where the weight is attached.

Nature of motions [added:] which is properly change of place. Motions are of two natures, that is, spiritual and material. The material divides into two parts, of which one is primitive and the other derivative. The primitive is that which is made by the instruments, holding within them the thing they move. The derivative is that which the moving thing makes once it has left the instrument. And among such movements the derivative motion is found always to be in proportional quantity with the primitive motion, both in quantity of motion, and in quantity of time and in celerity.

a 7 5 n, r, 5 7

Multiply the degrees of one of the weights attached to the balance against the degrees of the arm that stands above it, and divide by the opposite degrees placed beyond the fulcrum, where you wish to attach the weight. And what results will be the weight that resists 5.

I wish to hang a weight at a, which is 5 pounds, and let it be distant from the fulcrum 7 degrees. I ask what weight I must place at 5 degrees beyond the fulcrum. You will do thus: always place as much weight beyond the fulcrum as are the degrees on this side of the fulcrum, which are 7; give 7 pounds on the far side. This just stated is the simple and ordinary case. But now, having hung 5 degrees of weight at seven degrees of the balance on this side of the fulcrum, I would now like to place a weight beyond the fulcrum, at one degree, that would make resistance to the said 5 degrees of weight. Do thus: multiply 5 degrees of weight by 7 degrees of space on this side of the fulcrum, making 35, which you divide by that one degree of space that is beyond the fulcrum, yielding 35; and these 35 are the degrees of weight that must be placed beyond the fulcrum, at the first degree of space, against the 5 degrees of weight and 7 degrees of space, and it is right. And if you wished, against the said 5 degrees of weight, to hang another weight to resist it, and it were at 2 degrees beyond the fulcrum, there results the half of 35, that is 17 and ½. Whence you see that 17 and ½, at two degrees beyond the fulcrum, resists 5 placed at 7 degrees on this side of the fulcrum. So always multiply the weight attached on one side of the balance against the degrees of the arm that is above it, and divide by the degrees of the opposite arm, where you wish to attach your weight.