Archimedes' Cattle Problem

Classic Problem
The sun god had a herd of cattle consisting of bulls and cows, divided into $4$ herds according to their colour:
 * white
 * black
 * dappled
 * brown.

(Accounts vary as to their actual colours, which are arbitrary.)

Among the bulls:


 * the number of white ones was one half plus one third the number of the black greater than the brown
 * the number of the black ones was, one quarter plus one fifth the number of the spotted greater than the brown
 * the number of the spotted ones was one sixth and one seventh the number of the white greater than the brown.

Among the cows:


 * the number of white ones was one third plus one quarter of the total black cattle
 * the number of the black ones was one quarter plus one fifth the total of the spotted cattle
 * the number of spotted cattle was one fifth plus one sixth the total of the brown cattle
 * the number of the brown cattle was one sixth plus one seventh the total of the white cattle.

What was the composition of the herd?

Also known as
This problem is also known as:


 * the bovinum problema (that is, cattle problem in Latin)


 * Archimedes' Reverse