luminous - d blue - e yellow a - b blue - c yellow. The shadow made by the luminous source e, which is yellow, comes to be blue, because it is the shadow of the body a cast upon the floor at b, where it is seen by the blue luminous source. And thus the shadow made by the luminous source d, which is blue, will be yellow at the site c, being seen by the yellow luminous source. The field surrounding those shadows b c will be (besides its natural colour) tinted with a colour mixed of yellow and blue, because it is seen and illuminated by the yellow luminous source and the blue luminous source at one and the same time.

That which makes the shadow does not see it, because shadows are made by the luminous source that girds, or rather surrounds, the shadows.

red green blue yellow. Shadows of various colours according to the lights that see them. That light which makes the shadow does not see it. — Of the world. No part of the earth is uncovered by the wearing-away of the course of the water that was not already a surface of earth seen by the sun. — Of weights.

If the cord, which descends from the pulley to the weight and from that weight returns to your pulley — set at the same height as the first, but somewhat distant — then the heavy body, pulled by the two ends of that cord, at every degree of height gains degrees of accidental weight.

The accidental weight generated in the angle of the cord that holds the natural weight is greater in proportion as the angle of that cord is more obtuse.

The accidental weight is infinite. Therefore no cord can lift it, none. Therefore such a proportion holds among the accidental weights generated in the angle of that cord as holds among the said angles themselves. — There are 10 principal motions of the hand, that is: inward, outward, right and left, rotational, up or down, closing and opening, and the dilation and contraction of its fingers.

Whether waters of various weights can have their surface in one same sphere, being contiguous at their lateral bounds. The heavy water will join with its lateral bounds beneath the lighter water and will make a spherical surface beneath the surface of the lighter water, and the lighter water will make itself a spherical mantle of equal thickness above the heavier water.

Having finished, opposite here, various ways of squaring circles — that is, to give squares of capacity equal to the capacity of the circle, and given the rules for proceeding to infinity — I now begin the book called On the Game of Geometry (De ludo geometrico), and I give also a method of infinite process. — Lunes, sickles and sectors for the value of a circle.

a b - c. a b is worth c, subduple to the larger.

Four quarters make one whole.

The 4 semicircles are two wholes of the value of the half-maximum; the remainder of the maximum is worth another circle equal to the first.

a b c. a b is worth c, and the circles are of double proportion, and the removed parts are equal, and so on.

It lacks the 4 maxima, the remainder is worth the maximum square of the maximum circle.

n n - o o. This is worth the circular parallel and worth the hatched part. Remove the 4 maxima and the remainder is a square.

If you want the half of a sickle (falcata), draw out a subduple sickle.

a b — c d. b a is worth c d.

a b - c. a is worth half a portion, that is a is worth b and also worth c.

a. a is worth the remainder of a circle reduced into a parallel strip, from which the 4 maxima are removed. Let whatever part of a circle is asked be given between two parallel lines of equal length.

a b. The method is given in a b with the help of one proposition of the Geometric Elements, which changes the angle of the equilateral triangle onto one of the sides of the base without losing the first value.

a b. Transformation of a circle equal to the circle subduple to the maximum and double to the minimum. That is, a b is worth a circle double the minimum and subduple to the maximum.

The removed part is worth 2 circles subquadruple to the maximum, and, joined together, they are worth half of the maximum. Therefore the remainder of that maximum is worth those 2 circles, or, I mean, they are worth a circle subduple to that maximum. It follows that, removing the square, the sickles remain in place of the 4 maxima of the maximum circle. Therefore this figure + is squared, because it is a circle that has less the 4 maxima.

a b. a b is worth the square above, and the supplements below are worth the supplements above.

a b c. The 4 semicircles are worth the remainder of the maximum circle, and are worth a circle subduple to that maximum circle. — Therefore the remainder is worth a circular parallel, and it follows that a b c is worth the maximum square of the maximum circle.

Here in the 4 maxima the half is lacking, removing from them the 4 maxima of a subduple circle.

These are worth a circle that is worth half of the circle a b.

The 4 sickles are worth half of the 4 maxima.

A maximum divided into 3 equal parts.

a a - b b. A maximum divided into 2 equal parts.

Variety of sickles and divisions of a maximum into 4 equal parts.

Equal sickles.

A maximum divided into 4 equal parts.

A maximum divided in half.

a b. On lending a known part to an unknown part and then taking it back — that is, lending b to a. a is the fourth part of a maximum. Adding b to it, you have made a sector of a circle, to which one can then give an equal circle, and make use of that circle in its transformations ... and then take back what was lent ... and the remainder stays known.

with: n - lent and removed. Here, in the borrowing and returning, I have varied the periphery of the circle n, as we shall show in its place, and so on. The circle n stands in place of the fourth part of a maximum portion. Therefore we shall employ this empty circle in place of a maximum portion subquadruple to the origin of this n.