That surface will always be divided into equal parts which receives within its bounds another surface half its size (subdouble to it), whether it be similar to the greater or varied - it makes no difference. And so it will be divided in whatever proportion one wishes: that is, if it occupies the third, the remainder will be the two thirds, and if it occupies the 2/3, the remainder will be a third, etc.

False.

d c - f e - b n a

This surface is worth half of its greatest circle, because that greatest circle is double the smallest circle. It follows, then, that the lune is half of the whole greatest circle; and if I remove from the lune the two portions a b and restore an equal quantity to that lune with the 4 portions c d e f, it follows that the surface c d e f together with n is worth the entire lune. And here below follows the remainder of this demonstration.

m - o n

The semicircle n m o is worth half of the greatest circle above, of which the lune is the half; and if from that semicircle one removes the two portions n o e, and from the lune one removes as much by taking away its two other portions a b, there will remain the surface n above equal to the triangle m below.

a - b

a is worth b.

b a - c

a b c d with its 4 portions is worth the lune above without its middle square. - And it is squared by filling the concave with the convex.

a - d c - b

a is worth b, and a c is worth c b, and c b is worth d, and the circles are each quadruple the other.

d -b -a -c; d -b S - t o a - g - m n - f
To find the angle of contingence by means of an instrument.
Let, then, the sphere be where the angle of reflection o n m is seen, and let the point a be the luminous [source], and b be the eye, and o be the angle sought, at which the image of that luminous source is seen. Now you will take a strip of thin wood, less than half a finger wide, and let it be d f, in which there is a narrow channel, and let this be fixed with a fine needle or little pin over the centre of that circle o n m, passing through that channel of the rule. Then join two other little laths equal to one another, as long as you please, and let these be joined in the manner of a compass at the same pole, which is fixed at the head of the aforesaid rule d f. And, this done, you will join the strip S g to the head of the strip d S at the pole S, after the manner of a compass that opens and closes, and you will make its channel, as you did to the strip d f, and fix a needle at the luminous point a that passes through the said channel of the little lath S g. Now you must take the end of the little lath S g at g and move it up and down about the pole a (where the said pin is fixed in place of the luminous source) until you see the circumference of the circle at the angle of contingence o, made by the division of the two laths; and by the equal angles generated within the square S b d o the perfection of the work is proved, that is, the upper angles are equal to one another, and the lateral ones are equal to one another, and the like is confirmed to be so in the angles of contingence, o t, etc.
d o is set between the eye and the luminous source, with height and nearness to the eye, and the luminous source at pleasure, because it makes no difference, provided that d S and d b are equal to one another and that the final meeting of the 2 channels be upon the circumference of the circle.
Painting.
When the running man wishes to consume the impetus that carries him, he prepares himself for a contrary impetuosity, which is generated by leaning backward. It is proved, because if the impetus carries the moving body with a power of 4 and the moving body wishes to return and fall backward with a power of 4, then the one power consumes the other contrary to it and the impetus is consumed.
Painting.
The surface of every body partakes of the colour of its object.
The colours of illuminated objects are imprinted on one another's surfaces in as many various places as there are varieties in the situations of such objects.

c b a - p o - r n - f e d n

o is the blue object, illuminated, and it sees alone, without other company, the space b c of the white sphere a b c d e f, and tinges it with a blue colour. m is the yellow object, which illuminates the space a b in company with the blue o and tinges it a green colour (by the second [proposition] of this, which proves that blue and yellow make a most beautiful green, etc.). The rest will be told in the book of painting, and in this book it will be proved by making the images of bodies and the colours of the things illuminated by the sun penetrate through a small round aperture into a dark place, onto a flat wall white in itself, etc. But everything will be upside down.