Weights on inclined planes and quadrature of the hemisphere
Equal weights at equal obliquity stay balanced; spherical triangles and the hemisphere squared
The upper part of this leaf treats statics: a double inclined plane with a pulley and two equal weights (4, 4) illustrates that equal weights on equal balance-arms at equal obliquity neither rise nor fall, while a margin balance-schema gives three rules on parallel obliquities and equal powers. The lower part turns to geometry, stating that the sides of spherical triangles have a compound curvature facing two different centres. A hemisphere-and-plane figure shows the 'quadrature of the hemisphere' by straightening its half-periphery p m and successive circles, and a small semicircle marks the semi-diameters a b c d e n that are to be straightened.
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Equal weights at equal obliquity neither rise nor fall
Two equal weights hung on the equal arms of a balance at equal obliquities are neither heavier nor lighter than one another, so neither can make the other descend or rise. The double inclined plane with a pulley illustrates the case, with both weights marked 4.
Three rules on parallel obliquities and equal powers
First, all parallel obliquities are of equal obliquity among themselves. Second, equal weights on equal obliquities move equally, with equal velocity and time. Third, equal powers do not overcome one another. Labels: f, K d h, a c b, L e, g i.
Compound curvature of the sides of spherical triangles
The sides of a spherical triangle have a compound curvature: one concavity faces the centre of the sphere, the other faces the surface-centre of the spherical triangle.
Quadrature of the hemisphere by straightening its circles
The half-periphery p m is straightened into the line n m by motion, and the greater circle q m extended into m o at right angles to n m. The next circle a b is drawn out as line b c parallel to m o, and so on for every circle into which the hemisphere q p m is divided.