On Coordinates (1950-1974)

"In order to describe the magnitude and direction of the force at any point in the field, a coordinate system is necessary not only to identify the position of the point in question but also to provide suitable components of the force at the point; these components considered together give both the magnitude and the direction of the force at the point selected. The coordinate system can be chosen to suit any particular problem and the form of the result." (William J Gibbs, "Conformal Transformations in Electrical Engineering", 1958)

"In algebra, when we plot the graph of an equation, the curve so obtained represents the locus of the equation. In analytic geometry, the word 'curve' refers to any locus; it may be a straight line, a curved line, or a group of lines. We therefore see that: (1) The locus of an equation is a curve containing all those points, and only those points, whose coordinates satisfy the given equation. (2) The equation of a locus is an equation such that the coordinates of every point on the locus satisfy the equation, and every ordered pair of numbers which satisfy the equation are the coordinates of a point on the locus." (William L Schaaf, "Analytic Geometry; a college course guide", 1962)

"Very little can be learned about the nature of a surface merely by locating points on the surface. Instead, we study (1) its intercepts and traces in the coordinate planes; (2) its extent; (3) its symmetry, and (4) plane sections." (William L Schaaf, "Analytic Geometry; a college course guide", 1962) 

"When either the translation or the rotation transformations are applied to an equation, they change the coordinates of all points in the plane (except the origin in the case of a rotation). These transformations actually move the axes with relation to the curve, but they do not alter the shape of the curve represented by the original equation and by the transformed equation. Beside this constant or unvarying feature, i.e., the preservation of the geometric form of the curve, there are also certain algebraic expressions whose values remain unchanged in both equations. Mathematical forms which are preserved or do not vary when other changes take place are called invariants." (William L Schaaf, "Analytic Geometry; a college course guide", 1962)

"When plotting or drawing the locus of a given equation, the position selected for the coordinate axes is, of course, immaterial. Hence, after the locus has been drawn, we may change the position of the axes arbitrarily, as we wish. Naturally, if the locus remains where it was drawn, but the position of the axes is changed, then the equation will have to be changed to correspond to the new position of the axes if the equation is to describe the same locus. In other words, a given locus may be described by more than one equation; each of the equations describes the locus in question with reference to a different set of axes."(William L Schaaf, "Analytic Geometry; a college course guide", 1962)

"From a pessimistic viewpoint, it can be stated that there is no good general way of structuring a system. However, from an optimistic point of view one can say that a number of good ways of structuring systems exist and that some are better than others for any particular system. In this and the following sections, there will be a presentation of a number of structuring approaches that have merit and have been employed successfully, including functional structuring, equipment structuring, and use of various coordinate systems." (Harold Chestnut, "Systems Engineering Tools", 1965)

"A manifold can be given by specifying the coordinate ranges of an atlas, the images in those coordinate ranges of the overlapping parts of the coordinate domains, and the coordinate transformations for each of those overlapping domains. When a manifold is specified in this way, a rather tricky condition on the specifications is needed to give the Hausdorff property, but otherwise the topology can be defined completely by simply requiring the coordinate maps to be homeomorphisms." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"A manifold, roughly, is a topological space in which some neighborhood of each point admits a coordinate system, consisting of real coordinate functions on the points of the neighborhood, which determine the position of points and the topology of that neighborhood; that is, the space is locally cartesian. Moreover, the passage from one coordinate system to another is smooth in the overlapping region, so that the meaning of 'differentiable' curve, function, or map is consistent when referred to either system." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"The main object of study in differential geometry is, at least for the moment, the differential manifolds, structures on the manifolds (Riemannian, complex, or other), and their admissible mappings. On a manifold the coordinates are valid only locally and do not have a geometric meaning themselves." (Shiing-Shen Chern, "Differential geometry, its past and its future", 1970)