The kiss precise
For pairs of lips to kiss maybe
Involves no trigonometry.
'Tis not so when for circles kiss
Each one the other three.
To bring this off the four must be:
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.
Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance form the center.
Though their intrigue left Euclid dumb
There's now no need for rule of thumb.
Since zero's bend's a dead straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.
To spy out spherical affairs
An oscular surveyor
Might find the task laborious,
And now besides the pair of pairs
A fifth spere in the kissing shares.
Yet, signs and zero as before,
For each to kiss the other four
The quare of the sum of all five bends
Is thrice the sum of their squares.
A final verse by Thorold Gosset
And let us not confine our cares
To simple circles, planes and spheres,
But rise to hyper flats and bends
Where kissing multiple appears.
In n-ic space the kissing pairs
Are hyperspheres, and Truth declares-
As n+2 such osculate
Each with an (n+1)-fold mate.
The square of the sum of all the bends
Is n times the sum of their squares.