Proofs using geometrical intuition and Euclid axioms were almost the norm in ancient Greek mathematics. However side by side the idea of numbers was also developing for obvious practical needs (i.e. counting and measuring). With advent of algebra it became obvious that methods based on algebraic manipulation of numbers were far more powerful than the non-obvious geometrical arguments based on Euclidean axioms and this was the main reason for shift of focus from geometry to arithmetic/algebra.
Use of algebra became so much so prevalent that it became fashionable to study geometry using numbers (co-ordinate geometry). However as many mathematicians during 1800-1900 realized the idea of replacing geometrical concepts with numbers was not easy. The concept of a straight line as a smooth continuum of points was very difficult to map with the existing theory of numbers (which basically was a theory of numbers accessible via algebraic techniques i.e. the numbers considered were algebraic). Only with the development of a proper theory of real numbers by Cantor, Dedekind, Weierstrass it was possible to map the points of a line with real numbers and the set of real numbers could then be viewed as a sort of arithmetical continuum.
All this development by the way had a side effect. The charm of Euclid's Elements is no more available in standard high school curriculum. Only the bare minimum material based on Euclid axioms is covered in school syllabus and students get to learn analytic geometry and tools of calculus to deal with general curves. The beautiful theory of conics by Apollonius of Perga is a remarkable work based on Euclid axioms and sadly very few students are aware of it. I did study some of it from the book Apollonius, Treatise on Conic Sections by T. L. Heath and wrote some posts on conics.