The first reason why some dog breeds have such short legs is because of a genetic mutation which is believed to have occurred early on in the evolution of domestic dogs and sometime after the ancestor of modern dog breeds diverged from wolves. This mutation resulted in a form of dwarfism called chondrodysplasia. I believe that a study has been published in the journal Science dated July 16, 2009 which researched this mutation.
The researchers found a genetic signature which is exclusive to dog breeds with such short legs. These dogs have an extra copy of the gene that codes for a growth-promoting protein called fibroblast growth factor 4. The mutation causes calcification of growth plates which curbs long bone development resulting in short legs with a curved appearance.
Sometime during the domestication of the dog there was a practical need for dogs with short legs because they were able to go down burrows to chase underground prey. The Dachshund is a classic example. The name translates to “badger-dog”. Breeders selectively bred dogs ‘suffering’ from this form of dwarfism and created more of these dogs specifically with the purpose of attacking badgers in their burrows.
So, the second reason is that breeders selectively bred dogs with very short legs for a practical reason. They were working dogs.
The third reason is that not only were these short-leg dogs useful in terms of working dogs but people found them very cute. These dogs have the mentality of a big dog with small size and a waddling gait. This provides them with a plucky appearance. This is an attractive quality. They give the owner the impression that they are toddlers i.e. small humans 😕 . Specifically, toddlers trying to work out how to walk and travel from A to B.
The fourth reason is that some dogs with short legs were selectively bred specifically to create a dog which is a bit like a baby. I’m referring to the Oriental Pekinese. They had their leg-length reduced as part of a process described by Dr. Desmond Morris as “baby-making”. They are child substitutes. And they have the appealing clumsiness, as Dr. Morris describes, it of infants struggling with their first movements.