"A cylindrical surface is any surface generated by a moving straight line which constantly intersects a given fixed curve and which remains parallel to a fixed straight line. The fixed curve is called the directrix, and the generating line is known as the generatrix. Thus a cylinder is a ruled surface, that is, the surface is comprised of straight lines (the generators), all of which are parallel." (William L Schaaf, "Analytic Geometry; a college course guide", 1962)
"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)
"It is possible, however, to determine the general shape and salient features of a curve by a more searching method. This is accomplished by examining more closely the equation of the curve, rather than by computing a more extensive table of values. Certain characteristic features of curves can be determined by inspection of the form of the equation, and by computing only certain special values of the variables. These characteristic properties include the intercepts, extent, symmetry, and asymptotes of the curve." (William L Schaaf, "Analytic Geometry; a college course guide", 1962)
"It is to be understood that by an algebraic equation is meant an equation which involves only integral and fractional powers of the variables x and y. If the equation is of degree higher than two, the equation represents an algebraic higher plane curve. Any equation which is not algebraic is known as a transcendental equation; functions defined by such equations are said to be transcendental functions. For example, y = cos x, y = a log x, and y = k"" are transcendental functions." (William L Schaaf, "Analytic Geometry; a college course guide", 1962)
"The empirical data are usually available in the form of a table exhibiting pairs of corresponding values of the variables. Since the given values do not lie exactly along a straight hne, a parabola, an exponential curve or a power curve, it is necessar}' to find a curve that fits the given data approximately. This process is called curve fitting, and the equation of the curve which fits the data approximately is called an empirical equation. When fitting a curve to a set of given points, we must assume a general form for the equation which we are going to use to represent the curve." (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)
"Mathematical statistics provides an exceptionally clear example of the relationship between mathematics and the external world. The external world provides the experimentally measured distribution curve; mathematics provides the equation (the mathematical model) that corresponds to the empirical curve. The statistician may be guided by a thought experiment in finding the corresponding equation." (Marshall J Walker, "The Nature of Scientific Thought", 1963)
"The mathematicians and physics men Have their mythology; they work alongside the truth, Never touching it; their equations are false But the things work. Or, when gross error appears, They invent new ones; they drop the theory of waves In universal ether and imagine curved space." (Robinson Jeffers," The Beginning and the End and Other Poems, The Great Wound", 1963)
"Mathematical statistics provides an exceptionally clear example of the relationship between mathematics and the external world. The external world provides the experimentally measured distribution curve; mathematics provides the equation" (the mathematical model) that corresponds to the empirical curve. The statistician may be guided by a thought experiment in finding the corresponding equation." (Marshall J Walker, "The Nature of Scientific Thought", 1963)
"The mathematicians and physics men Have their mythology; they work alongside the truth, Never touching it; their equations are false But the things work. Or, when gross error appears, They invent new ones; they drop the theory of waves In universal ether and imagine curved space." (Robinson Jeffers," The Beginning and the End and Other Poems, The Great Wound", 1963)
"When a curve approaches an axis or any other straight line in this fashion, the line is said to be an asymptote of the curve. As a working definition, we may say: An asymptote of a curve is any straight line which a curve approaches continuously as the curve moves on to infinity." (William L Schaaf, "The Calculus, a college course guide", 1963)
"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 mathematical models for many physical systems have manifolds as the basic objects of study, upon which further structure may be defined to obtain whatever system is in question. The concept generalizes and includes the special cases of the cartesian line, plane, space, and the surfaces which are studied in advanced calculus. The theory of these spaces which generalizes to manifolds includes the ideas of differentiable functions, smooth curves, tangent vectors, and vector fields. However, the notions of distance between points and straight lines" (or shortest paths) are not part of the idea of a manifold but arise as consequences of additional structure, which may or may not be assumed and in any case is not unique." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)
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