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The predominant theory regarding the origin of the Miniature, Dwarf and Rabbit Dachshunds is that Standard Smooth Dachshunds were bred with toy terriers or pinschers to reduce their size. These now smaller Smooth Dachshunds were then bred with the Papillon to create the longhaired variety, and with the Miniature Schnauzer to create the the Miniature Wirehaired Dachshund. In Germany these new smaller Dachshunds were then selectively bred to achieve the desired size and classified in two seperate classes based upon the size of the hole they could enter. The first being the dwarf Dachshund (zwergteckel) that measures approximately 11.8 inches around its chest and the second being the rabbit Dachshund (kaninchenteckel) that measures approximately 13.8 inches around its chest.
Whether or not the above theory is the true origin seems to have been lost in time. What we do know is that as with all breeds, intentional modifications of coat type, size, temperament etc. are generally the result of necessity. These breed variations are designed to expand the breeds usability for a specific task, or in alternate climates, and terrains.
Miniature Dachshunds mimic their bigger counterparts in all aspects except size; with a diminuitive height of only 6 inches at the withers and a requirement that weight not exceed 11lbs they really are just smaller versions of the Standard Dachshunds.
In general, Miniature Dachshunds will tend to be more active than the larger Standard Dachshunds.