One thing of interest in his design was his choice of a negative lens as an eyepiece. it has a smaller  field of view instead of a positive lens . Some  biographers have suggested that Galileo never bothered to look at a positive lens as an eyepiece. We suggest otherwise. When he did his experimenting there were a lot more short focal length positive lenses around than negative ones. They would have been used as hand magnifiers,  on a desk tops, or in ones pocket. They would be a lot cheaper than a pair of eyeglasses with matched lenses.

Although the positive lens for the same magnification had a wider field of view the image was upside down. As an entrepreneur as well  as a physicist he was really designing these telescope for none technical customers  who were going to use them  for terrestrial use only. They also had to be simple to build.  Erecting eyepieces require a minimum of  two lenses  and were  much more complex to design and build .There was  also a practical need for short telescopes which the  negative lens did for them.  Galileo had to go out of his way the use the negative lens only and we have to believe that he chose the lens type that gave an upright image and sacrificed the better field of view.  Galileo ground and polished the necessary lenses tested his results and had his high powered telescope in comparatively short period of time.

Galileo was acting as a scientist when he turned it up to the sky and stumbled onto  discoveries of a life time for at least a few weeks he was  gathering data as fast as he could and he would have been in a wild race  to gather this data  before some one else did (he new that he wasn't going to be the only the only scientist looking up there). Probably he wasn't thinking much about a new telescope design. He was also interested in getting a new job.

Jim & Rhoda Morris