Have difficulty in understanding the small text

1) What does "and these" mean? It looks like a grammatical error. But since this report tops the Nature magazine's headlines today, it is less likely a fault. But what does it mean?
2) Does "the remaining portions shall fit together" mean "the two cut surfaces of the remaining portions can be put exactly together (so that no cut surfaces will be exposed openly)"?
3) Does "chord" mean "a straight line connecting two points on a curve" here?
4) What does "figure" mean in "the first figure"? A diagram in the paper? But I've found no diagram there.
5) Does "successive operation" mean "following operation"?

Context:

Letters to Editor

Nature 75, 173-173 (20 December 1906) | doi:10.1038/075173c0

Cutting a Round Cake on Scientific Principles

F. G.
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Abstract

CHRISTMAS suggests cakes, and these the wish on my part to describe a method of cutting them that I have recently devised to my own amusement and satisfaction. The problem to be solved was, “given a round tea-cake of some 5 inches across, and two persons of moderate appetite to eat it, in what way should it be cut so as to leave a minimum of exposed surface to become dry?” The ordinary method of cutting out a wedge is very faulty in this respect. The results to be aimed at are so to cut the cake that the remaining portions shall fit together. Consequently the chords (or the arcs) of the circumferences of these portions must be equal. The direction of the first two vertical planes of section is unimportant; they may be parallel, as in the first figure, or they may enclose a wedge. The cuts shown on the figures represent those made with the intention of letting the cake last for three days, each successive operation having removed about one-third of the area of the original disc. A common india-rubber band embraces the whole and keeps its segments together.

MOre:
http://www.nature.com/nature/journal/v75/n1938/abs/075173c0.html#top