### The kiss precise

#### Frederick Soddy

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.