Dividing a Pyramid in Septuple Proportion
Cuts of a pyramid giving 1/8, 1/64, 1/512, with circular water-tubes and river sediment
The left column studies how a pyramid cut parallel to its base at the middle of its length is divided in 'septuple' (sevenfold) proportion, and a pyramid taken with three successive cuts (a e d c b) yields fractions of one eighth, one sixty-fourth and one 512th, worked through in the arithmetic and illustrated by pyramids and polyhedra at the head of the sheet. Other passages treat water held in circular tubes variously cut and bent, and where its centre of gravity then lies. The right column compares snow settling on rounded surfaces to river-borne earth clothing the shapes of a riverbed, and a final note observes that stones on an arc press not on the centre of its chord but on the centre of the semicircle.
On this page
A pyramid cut parallel to its base divides in septuple proportion
A pyramid cut parallel to its base at the middle of its length is left divided in septuple (sevenfold) proportion, keyed here with the figures 8 and 64.
Three cuts of a pyramid (a e d c b) and their fractions
A pyramid lettered a - e - d - c - b is cut three times: a b halved at c gives one eighth of the whole with a c; a c divided at d leaves a d one eighth of a c and one sixty-fourth of a b; a d divided at e leaves a e one 512th.
Proportional halving: 1/8, 1/64, 1/512
The arithmetic beside the pyramid works the proportions 64, 8 and 512, confirming that each successive cut of the solid gives a term that is one eighth of the last - one eighth, one sixty-fourth, one 512th of the whole.
Water in circular tubes variously cut and bent
Figures of circular tubes examine water within a uniform pipe bent into a circle: with the diameter set toward the centre of the world the water balances with equal opposite weights about the diametral line, but where the tube is cut and set obliquely the water's centre of gravity moves off-centre toward the lower opening.
Snow on rounded surfaces likened to river sediment
The right column compares the snow that falls on various rounded surfaces, clothing them with thicknesses that vary with their obliquity, to the earth carried by river floods, which after the water settles descends upon the rounded shapes of the bed and clothes them in the same manner.
Polyhedra at the head of the sheet
The top of the sheet is filled with sketched solids - several tetrahedra, a skeletal open-frame polyhedron, and cubes - accompanying the study of how a pyramid is divided by parallel cuts.