The calculations of mass and gravity were calculated By Edwin Hubble around 1932. He showed that The Milky Way had only 1/5th the mass needed to hold the galaxy together. Thus was Dark Matter, or it's theory, born. This doesn't even address the biggest problem with the orbital dynamics of galaxy sized systems.

Consider a solar system. Mercury travels faster in it's orbit than Venus. Venus moves faster than earth, and so on. Predictably, all the planets in our solar system move at a greater velocity than Neptune. Everybody knows this and understands why.

Now consider the galaxy and it's spiral shape. In order to maintain this shape the stars at the farthest outskirts have to be moving much faster that those closer in to the center. This configuration should force the galaxy to fly apart, but it doesn't. the question is, why don't the stars fly out into intergalatic space. Spiral galaxies shouldn't exist and nobody knows why.