Sidereus Nuncius
The first telescopes appeared in the Netherlands in 1608 when Middelburg spectacle-maker Hans Lippershey tried to obtain a patent on one. By 1609 Galileo had heard about it and built his own improved version. He probably was not the first person to aim the new invention at the night sky but his was the first systematic (and published) study of celestial bodies using one. One of Galileo's first telescopes had 8x to 10x linear magnification and was made out of lenses that he had ground himself. This was increased to 20x linear magnification in the improved telescope he used to make the observations in "Sidereus Nuncius".
Leonard Digges (scientist)
The writings left by Digges' colleague William Bourne contain further detail on the experiments with lenses and mirrors conducted by Leonard Digges and his son. It is these details which led some researchers, most notably Colin Ronan, to claim that Leonard Digges invented a functioning telescope sometime between 1540 and 1559. The description seemed to suggest that Digges created a rudimentary instrument incorporating lenses and a concave mirror, in a manner rather different from a modern reflecting telescope. However, the construction of lenses to the required optical precision would have been very difficult in the 16th century, and the construction of an adequate mirror would have been much harder still. It is doubtful that Digges built a successful instrument, and the optical performance required to see the details of coins lying about in fields, or private activities seven miles away was far beyond the technology of the time.