The Kiss Precise     by Frederick Soddy

 For pairs of lips to kiss maybe Involves no trigonometry. This not so when four 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 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, The sphere is much the gayer, And now besides the pair of pairs A fifth sphere in the kissing shares. Yet, signs and zero as before, For each to kiss the other four The square of the sum of all five bends Is thrice the sum of their squares.       in Nature, June 20, 1936 Later another verse was written by Thorold Gosset to describe the even more general case in N dimensions for N+2 hyperspheres of the Nth dimension. The Kiss Precise     (generalized) 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.       in Nature, January 9, 1937.