In this sphere of magnitudes, and initially of spatial magnitudes first of all in classes of privileged cases (straight lines, limited plane figures, and the corresponding cases of spatial magnitudes), first of all in the empirical intuition that magnitudes divide into equal parts and are composed again of equal parts or of aggregates Mengen of like elements which decompose into partial aggregates and can be expanded into new aggregates through the addition of elements or of aggregates of such elements in this sphere, there arose the exact comparisons of magnitudes which led back to the comparison of numbers.