In 1900, no one knew the Milky Way was a spiral, and there was good evidence that the spiral nebulae lay inside the Milky Way.
Lord Rosse found the first celestial spiral in 1845; he then found many other spirals. "Rosse's discoveries renewed speculation that some of the nebulae were external galaxies. They also gave rise to the first suggestion, in 1852 by American astronomer Stephen Alexander, that the Milky Way itself was a spiral - a speculation that took ninety-nine years to confirm.
"During the latter half of the 1800s, however, in one of those curious backward steps that astronomy occasionally takes, several events conspired to undermine belief in external galaxies." - The Universe at Midnight by Ken Croswell, page 19.
Croswell goes on to list these as:
1. The 1864 discovery that the Orion Nebula is made of gas, which suggested that ALL nebulae, including the spirals, were gaseous objects similar to the Orion Nebula.
2. The 1869 discovery that the spiral nebulae lay away from the Milky Way's plane, suggesting they were part of our Galaxy.
3. The 1885 "nova" in the Andromeda Nebula. If the Andromeda Nebula lay outside our Galaxy, then that "nova" would have been impossibly powerful.
"These observations convinced astronomers that the spiral nebulae lay within the Milky Way. `The question whether nebulae are external galaxies hardly any longer needs discussion,' wrote British author Agnes Clerke in 1890. `It has been answered by the progress of discovery. No competent thinker, with the whole of the available evidence before him, can now, it is safe to say, maintain any single nebula to be a star system of coordinate rank with the Milky Way.' In short, there was only one galaxy in the universe: our own." - The Universe at Midnight by Ken Croswell, page 21.
In the 1910s, Vesto Slipher's discovery of the high velocities of the spiral nebulae suggested they lay far beyond the Milky Way. The following decade, Edwin Hubble's discovery of Cepheids in nearby spirals demonstrated this beyond all doubt.
Finally, two corrections: Henrietta Leavitt discovered the period-luminosity relation for Cepheids in 1907, not 1908. And the discovery that the Milky Way is a spiral came in 1951 from optical observations, not radio observations. References to both assertions are available upon request.