Centres of gravity of a cone and its inscribed pyramid
Natural and accidental centres of gravity of bodies resolved into parts
This double page continues Leonardo's study of centres of gravity, distinguishing the 'natural' centre of a body from its 'accidental' centre and showing how each can be recovered from the parts into which a body may be resolved. On folio 111v he labels the natural centres of a wedge, a truncated wedge and a pyramid (points c, b, d), while on folio 108r he treats a conical body a b c together with the largest pyramid inscribed in it, marking its centres of magnitude and of natural and accidental gravity. Interleaved geometry relates a square whose sides touch a circle to the square touched at its corners, and compares right triangles built on a quarter-circumference. Small diagrams of boxes, cones and geometric constructions fill both leaves.
On this page
Finding a body's centre of gravity from its resolved parts
Leonardo states a set of reciprocal rules: from the accidental centre of gravity of the two parts into which a body can be resolved, one finds the natural centre of the whole; and from the natural centres of the parts, the accidental centre of the whole. He then combines natural and accidental centres of the two parts to recover the accidental centre of the whole body.
Natural gravity centres of a wedge, truncated wedge and pyramid
A caption to the left of the figure names the points: c is the natural centre of gravity of the wedge a c r m; b is the natural centre of the truncated wedge a n c c; and d is the natural centre of the pyramid c n r. The letters key the drawn solids to their computed centres.
Centres of the conical body a b c and its inscribed pyramid
On folio 108r the cone a b c is analysed: s t marks the centre of its magnitude, o p its accidental centre of gravity, and d e its natural centre. K h is the natural centre of the largest pyramid that fits within the cone and g i the pyramid's accidental centre; from the cone's natural centre and that of its smaller pyramid he derives the natural centre of the larger inscribed pyramid.
Square on a circle and quarter-circumference right triangles
The square whose sides touch the circle is stated to be double the square whose corners are touched by that circle, and the triangle a b c is halved by the line d e. With angle a a right angle, the curve n m o is a quarter of a circle's circumference, side r t is double a r, n o equals a r, and so the right triangle a r t is double the right triangle a n o.