"A type of picture-graph less commonly used than formerly is the pictorial representation of an object which has been arbitrarily subdivided to show certain numerical relationships; as, for example, the pictorial representation of the food values of beefsteak. This is a very poor type of graphic representation, and should definitely be avoided. The irregular outline of the picture as a whole, and of each of the shaded areas, makes a comparison of the areas difficult, if not altogether impossible; the shading only to the confusion." (William L Schaaf, "Mathematics For Everyday Use", 1942)
"Geometric lines are fictions in the sense that, while we draw them, we think of them as having no width, simply length. Lines may be curved or straight." (William L Schaaf, "Mathematics For Everyday Use", 1942)
"Graphs showing time changes, or the increases and decreases in the amount of something over a period of time, are generally of two kinds: (1) vertical bar graphs, and (2) broken- or smooth-line graphs. Both kinds differ from the categorical charts [...] in that they have two scales instead of only one; that is why it is preferable to call them graphs rather than charts, although these terms are used rather freely and interchangeably, and there is no standard convention." (William L Schaaf, "Mathematics For Everyday Use", 1942)
"If a number or a quantity is thought of as being broken up, or subdivided into any number of equal parts, and then a certain number of those parts is considered separately in relation to the total number of such parts, we arrive at the idea of a fraction." (William L Schaaf, "Mathematics For Everyday Use", 1942)
"In [...] horizontal bar-charts, showing comparisons between different kinds of things, or between different places, only one numerical scale is required, viz., the scale representing the amounts involved. No other numerical scale is needed, since we are dealing with various categories. While not always the most effective device for exhibiting such comparisons, horizontal bar-charts are simple and convenient." (William L Schaaf, "Mathematics For Everyday Use", 1942)
"Many forms in Nature exhibit the geometric property of symmetry. Anything symmetrical, whether natural or manmade, is usually pleasing in appearance, since it is 'balanced', and appeals to the eye. Symmetry is one of the most important principles of ornament, design and architecture. It is not, however, the only one; others are repetition and rhythm." (William L Schaaf, "Mathematics For Everyday Use", 1942)
"The mathematical concept of chance, or probability, must not be confused with the psychological notion of likelihood. If a coin upon being tossed six times in succession has come up 'heads' each of the six times, we may be impelled to feel that upon the seventh toss it is 'more likely' to turn 'tails' in view of the six previous heads; but this is only an emotional reaction and not a mathematical probability. Mathematically [...] on any single throw, the chances are even for heads or tails, irrespective of what the previous trials may have been. In the long run, the greater the number of trials, the more nearly equal will become the number of heads and tails." (William L Schaaf, "Mathematics For Everyday Use", 1942)
"Practical geometry deals with the nature and properties of various geometric forms, such as rectangles, triangles, circles, etc., and emphasizes especially the measurement of such figures. Hence the chief value of practical geometry is in problems of design and construction." (William L Schaaf, "Mathematics For Everyday Use", 1942)
"The operation of adding two numbers is essentially a process of grouping, or, more accurately, regrouping. When we add two numbers we do not increase anything; we regroup the numbers in accordance with the standard pattern or number system based on groups of ten." (William L Schaaf, "Mathematics For Everyday Use", 1942)
"When statistical data are of such a nature that it is permissible to assume that 'in-between values' vary continuously and uniformly (or very nearly so) from one observed or measured value to the next, a modification of the broken-line graph may be used. Instead of connecting the plotted points with straightline segments, a 'smooth' curved line is drawn between the points [...]. Such curvedline graphs may be drawn either 'free hand' or with the aid of drafting instruments known as French curves." (William L Schaaf, "Mathematics For Everyday Use", 1942)
"You may expect to find graphs anywhere: in books, in periodicals, in newspapers, in pamphlets, on show cards in advertisements, in business reports, and so on. Their use, however, is sometimes limited. For one thing, they are of necessity less accurate than the figures on which they are based, which, of course, doesn’t matter too much in many cases. In the second place, they are sometimes misleading, which may or may not be intentional. It is also possible that the reader of a chart or graph may misinterpret it." (William L Schaaf, "Mathematics For Everyday Use", 1942)
"Mathematics is on the artistic side a creation of new rhythms, orders, designs, harmonies, and on the knowledge side, is a systematic study of various rhythms, orders." (William L Schaaf, "Mathematics: Our Great Heritage: Essays on the Nature and Cultural Significance of Mathematics", 1948)
"Probably no symbol in mathematics has evoked as much mystery, romanticism, misconception and human interest as the number π." (William L Schaaf, "Nature and History of π", 1967)
"Emerson once said that it didn’t matter much where a man was, so long as you knew the direction in which he was moving. In somewhat the same way, the significance of a graph or curve often lies not so much in what height a point on the curve has reached, as it does in how fast its height is changing, and whether it is increasing or decreasing."
"Once more we must remind the reader not to confuse the amount of change with the rate of change. A function may increase or decrease by a very small amount in a short interval, and yet be changing very rapidly - just as a bullet may travel only a small distance in one thousandth of a second and yet be moving at a very high speed."
"The creation of Analytic Geometry by Descartes in the early part of the seventeenth century was a milestone of tremendous significance. Indeed, it was the beginning of modern mathematics in the broad perspective of history. ('Modern' mathematics in the sense of contemporary mathematics did not commence until about 1900, with the advent of functional analysis, abstract spaces, set theory, and symbolic logic.) This great step of recognizing the relation between the numbers of algebra and the entities of geometry once having been taken, it is perhaps not surprising that further advances were soon to follow, culminating in the invention of the Calculus by Isaac Newton and by Gottfried Leibniz. This was a classic illustration of nearly simultaneous, but presumably independent creation." (William L Schaaf, "The Calculus, a college course guide", 1963)
No comments:
Post a Comment