The discovery in 1998 that the universe is paradoxically accelerating its expansion has led some cosmologists to question the correctness of the non-Euclidean geometric theory of gravity, General Relativity. Physically assigning the term Dark Energy to the Cosmological Constant, sometimes viewed as a constant of integration, as the source of this acceleration has only produced even more questions. In the 17th century, there was also a great paradox between two views for the geometric constituents of a line, heterogeneous (made of points) versus homogeneous (made of infinitesimal segments). Evangelista Torricelli elucidated his logical reasoning on why lines must be made of infinitesimal segments instead of points and created one particular fundamental example among many. In this paper, using primitive notions called homogeneous infinitesimals and a new choice axiom, I produce unknown corollaries to Torricelli's argument. With these primitive notions I can correct Leibniz's notation in order to falsify the relationship between infinitesimals and the Archimedean axiom, scale factors/metrics, redefine the Fundamental Theorem of Calculus, differential forms, n-spheres, Gaussian curvature as well as redefine the relationship between real numbers and infinitesimals. I hypothesize that the voluminal elements of the Ricci tensor are a logically flawed view of homogeneous infinitesimals and metrics are an imperfect measuring paradigm. This allows the conjecture that the intractability of Dark Energy is due to the points of coordinate systems within General Relativity actually being a logically flawed heterogeneous interpretation of homogeneous geometry. I propose that Euclidean and non-Euclidean geometry, and the physics equations based upon them (such as relativity and quantum mechanics), should be rewritten from the perspective of homogeneous infinitesimals. I introduce the logical resolutions to geometrical paradoxes in this paper in order to pave the way for the physical logic.