# Why Euclidean volume is used in Curved Newtonian Cosmology?

In the derivation of Newtonian Cosmology people simply replace for the mass inside a sphere as [see Liddle or Roos textbooks or APJ 1965 and 1996 papers]

$$M(r)=\frac{4π}{3} ρ\, r^3​​$$

​But this formula is based on volume of sphere in Euclidean geometry. If Universe would be curved the volume is different in say Spherical or Hyperbolic geometry. As a two dimensional example, the area of a circle of radius $r$ on a 2-sphere of radius $R$ is given by

$$A(r)=2\pi R^2\left(1-\cos\frac{r}{R}\right)$$

One may say that the $ρ$​​ is the effective one in Curved geometry, but then it should depend on $r$​​, violating homogeneity (as is evident by the 2d example in above).