=1862.= The reason why I impute any defect to geometry, is, because its original and fundamental principles are deriv’d merely from appearances; and it may perhaps be imagin’d, that this defect must always attend it, and keep it from ever reaching a greater exactness in the comparison of objects or ideas, than what our eye or imagination alone is able to attain. I own that this defect so far attends it, as to keep it from ever aspiring to a full certainty. But since these fundamental principles depend on the easiest and least deceitful appearances, they bestow on their consequences a degree of exactness, of which these consequences are singly incapable.--HUME, D. _A Treatise of Human Nature, Part 3, sect. 1._ =1863.= I have already observed, that geometry, or the art, by which we fix the proportions of figures, tho’ it much excels both in universality and exactness, the loose judgments of the senses and imagination; yet never attains a perfect precision and exactness. Its first principles are still drawn from the general appearance of the objects; and that appearance can never afford us any security, when we examine the prodigious minuteness of which nature is susceptible.... There remain, therefore, algebra and arithmetic as the only sciences, in which we can carry on a chain of reasoning to any degree of intricacy, and yet preserve a perfect exactness and certainty.--HUME, D. _A Treatise of Human Nature, Part 3, sect. 1._ =1864.= All geometrical reasoning is, in the last resort, circular: if we start by assuming points, they can only be defined by the lines or planes which relate them; and if we start by assuming lines or planes, they can only be defined by the points through which they pass.--RUSSELL, BERTRAND. _Foundations of Geometry (Cambridge, 1897), p. 120._ =1865.= The description of right lines and circles, upon which Geometry is founded, belongs to Mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn.... it requires that the learner should first be taught to describe these accurately, before he enters upon Geometry; then it shows how by these operations problems may be solved. To describe right lines and circles are problems, but not geometrical problems. The solution of these problems is required from Mechanics; by Geometry the use of them, when solved, is shown.... Therefore Geometry is founded in mechanical practice, and is nothing but that part of universal Mechanics which accurately proposes and demonstrates the art of measuring. But since the manual arts are chiefly conversant in the moving of bodies, it comes to pass that Geometry is commonly referred to their magnitudes, and Mechanics to their motion.--NEWTON. _Philosophiae Naturalis Principia Mathematica, Praefat._