Clifford's well-known chain of theorems concerning circles in a plane may be stated as follows (the first two links being trivial):
(C2) Given any point A, and through it two circles C1 and C2, these meet again in a point A12, say.
(C3) Given three circles C1, C2, C3 through A, the three points A23, A13, An lie on a circle C123, say.
(C4) Given four circles C1,…,C4 through A, the four circles C234,…,C123 meet in a point A1234 (Wallace(1)).
(C5) Given five circles C1,…,C5 through A, the five points A2345,…,A1234 lie on a circle C12345 (Miquel (2)).
And so on (Clifford(3)).
Grace (4) showed that in three dimensions [3], there is no direct analogue to Clifford's chain, but in [4] he noted the following:
(G4) Given any point A in [4], and through it four hyperspheres S1,…, S4, these meet again in a point A1234, say.
(G5) Given five hyperspheres Sl,…,S5 through a point A, the five points A2345, A1234 lie on a hypersphere S12345, say.
(G6) Given six hyperspheres Sl,…, S7 through A, the six hyperspheres S23456,…, A12345 have in common a point A123456.
This leads naturally to the conjecture
(G7) Given seven hyperspheres Sl,…, S7 through A, the seven points A234567,…, A123456 lie on a hypersphere S1234567.