This article argues that Spinoza’s conception of truth as an intrinsically structured deductive system justifies the geometrical method of the Ethics. Three aspects of the method he develops for forming clear and distinct ideas — its deductive character, its reliance on self-evident starting points functionally analogous to axioms, and its use of definitions as guides to essences — converge with the constitutive features of the mos geometricus. Against externalist readings and beyond existing internalist accounts, these three aspects form a unified system explaining why the Ethics requires a deductive order, undemonstrated starting points, and definitions alongside axioms.

Screw theory has demonstrated its wide applications in robot kinematics and statics. We aim to propose an intuitive geometrical approach to obtain the reciprocal screws for a given screw system. Compared with the traditional Plücker coordinate method, the new approach is free from algebraic manipulation and can be used to obtain the reciprocal screws just by inspecting the structure of manipulator. The approach is based on three observations that describe the geometrical relation for zero pitch screw and infinite pitch screw. Based on the observations, the reciprocal screw systems of several common kinematic elements are analyzed, including usual kinematic pairs and chains. We also demonstrate usefulness of the geometrical approach by a variety of applications in mobility analysis, Jacobian formulation, and singularity analysis for parallel manipulator. This new approach can facilitate the parallel manipulator design process and provide sufficient insights for existing manipulators.