First: r p – 2 9 7 3 – S q

Second: o n m L

Third: c a – h g f e – d b

Fourth: x – v t – y

Fifth. In this last figure the largest circle is worth four smaller circles; it follows that one of the two largest portions is worth four of the smallest portions. Therefore the two largest portions are worth the eight smallest portions, and for this reason the two pentagonal surfaces are worth the two squares enclosed in the smaller circles.

Sixth: a – d e – b. This largest circle is divided into four equal but dissimilar parts, since a and b are two circles equal to one another, and c and d are two curvilinear triangles. That these four parts of the largest circle are equal is proved thus: the diameter of the greater circle is double the diameter of the smaller circle; and the proportion of circle to circle is the same as the proportion of square to square formed by multiplying the diameters of those circles by themselves. Circles of double and quadruple proportion make (a figure follows).

The equality of the inclination preserves the equality of the weights. The proportion of weights placed on an obliquity of the same proportion... If the proportions of the weights and of the obliquity on which they rest are equal but inverse, those weights will remain equal in gravity and in motion.

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2 4. If the... The site of equality is when the plumb-line of the balance's pole stands perpendicular over the straight line of the balance. The site of equality is when the straight line of the balance's arms receives upon itself the plumb-line of its pole at right angles. When the plumb-line of the balance falls by a perpendicular line upon the arms of the...

4. 1 1/8 1/2. Multiply the arm of the counterweight by as many times the friction of the pivot as the number of times it contains within itself the arm of that friction. Multiply the arm of the counterweight by four times: this means the arm of the counterweight should be multiplied by as many times the arm of the friction, which is divided into four so as to equalize itself in proportion with the friction of the weight, which is equal to a quarter of the rubbed weight.

Through the whirling courses of the winds, great quantities of flocks of birds were seen coming from distant lands; and these showed themselves with almost imperceptible perception, because in their wheelings sometimes a whole flock was seen edge-on, that is at its least thickness, and sometimes at its greatest breadth, that is full face. The beginning of their appearance was in the form of an imperceptible cloud, and the second and third squadrons became the more noticeable the closer they came to the eye of the beholder. The nearest of the said flocks descended low by an oblique motion and settled upon the dead bodies carried by the waves of this deluge, and fed upon them. This they did until the lightness of the swollen dead bodies began to fail, whereupon with slow descent they went sinking to the bottom of the waters. Water does not weigh upon its bottom unless it strikes it. Every heavy body weighs of itself only along the path of its motion toward the place to which it moves. The rain that descends to earth changes the position of the earth (which of itself clothed the centre of the world) from its place. The spherical heavy body that rolls upon a flat place weighs in two respects, of which one is toward the place where it moves, the other toward the centre of the world; slow motion weighs it down toward the centre of that world, and swift motion lightens it above that centre.