Is there a R function or better way to sum rows and columns cumulatively?

Let´s suppose a data frame

df1 <- data.frame(A = c(1,2,3), B = c(4,5,6), C = c(7,8,9), D = c(10,11,12))

I want to sum both rows and columns cumulatively, like

mysum <- c(sum(df1[1,1:4]), 
       sum(df1[1:2,1:4]),
       sum(df1[1:3,1:4])
          )

Any thoughts to improve the code?. dplyr solution? R base? I have thousands of rows and columns. Thanks in advance