Graph

“ Such tables are usually of single entry, i.e. to a certain value of one variable there corresponds one, and only one, value of the other variable. To construct the graph, as it is called, of such a table, Cartesian co-ordinates are usually employed. Two lines or axes at right angles to each other are chosen, intersecting at a point called the origin; the horizontal axis is the axis of abscissae, the vertical one the axis of ordinates. Along one, say the axis of abscissae, distances are taken from the origin corresponding to the values of one of the variables ”

“ Representation of Volume by Area.—An important plane graph is that which represents the volume of a solid figure.
Suppose that we take a pair of parallel planes, such that the solid extends from one to the other of these planes. The section by any intermediate parallel plane will be called a “cross-section.” The solid may then be regarded as generated by the cross-section moving parallel to itself and changing its shape, or its position with regard to a fixed axis to which it is always perpendicular, as it moves. ”

“ Whether your data can be suitably represented by a stem and leaf display depends on whether they can be rounded without loss of important information. Also, their extreme values must fit into two successive digits, as the data in Figure 5 fit into the 10,000 and 100,000 places (for leaves and stems, respectively) . Deciding what kind of graph is best suited to displaying your data thus requires good judgment. Statistics is not just recipes! ”

“ The large numbers refer to explanations following the graph.
SAMPLE ONLY
The figure title describes the graph's information. Listed above the graph, the “N” shows the number of visitors responding and a description of the chart's information. Interpret data with an “N” of less than 30 with CAUTION! as the results may be unreliable. Vertical information describes categories. Horizontal information shows the number or proportions in each category. In most graphs, percentages provide additional information. ”

“ If we represent the corresponding value of u by an ordinate drawn from the line, the extremities of all such ordinates will lie on a curve which will be the graph of u with regard to x. Interpolation therefore consists in determining the length of the ordinate of a curve occupying a particular position, when the lengths of ordinates occupying certain specified positions are known. If u is a function of two variables, x and y, we may similarly represent it by the ordinate of a surface, the position of the ordinate being determined by the values of x and of y jointly. ”

“ Figure 7 inappropriately shows a line graph of the card game data from Yahoo. The drawback to Figure 7 is that it gives the false impression that the games are naturally ordered in a numerical way when, in fact, they are ordered alphabetically. ”

“ For instance, if we say that shall be the arc of smallest numerical value whose tangent is , that is, shall take on only values between and , then we are limited to the branch passing through the origin, and the condition for continuity is satisfied. (9) Similarly,
is found to be a many-valued function. Confining ourselves to one branch of the graph of
we see that as approaches zero from the left, approaches the limit and as approaches zero from the right, approaches the limit Hence the function is discontinuous when Its value for can be assigned at pleasure. ”

“ But ordinary co-ordinate paper, on a small scale, is the best form in which to first arrange the chart. It can be purchased in sheets at a small price, and is invaluable for both student and instructor. In fact, no lesson is more stimulating to a class than to give them the data of a subject and ask them to put it into graphic form. For the first time they begin to realize that statistics are not dry ”

“ The quicker and surer method, however, in driving home the essential points of statistics is their graphic presentation. For the majority of business men the graphic method will accomplish what statistical tabulation fails to do. ”

“ In this day of computers, researchers all too often see only the results of complex computer analyses without ever taking a close look at the data themselves. This is all the more unfortunate because computers can create many types of graphs quickly and easily. ”

“ These graphs represent the normal form of distribution usually expected when any fundamental ability is measured in a random group. If the number of cases at each unit of measurement are plotted by a point placed relatively as far above the scale, used as a base line, as the number of cases found at that unit of the scale, it will be discovered that these points arrange themselves in the form of a symmetrical curve high at the middle and flaring out 257along the base-line scale. ”

“ The geometrical solution of arithmetical and algebraical problems is usually termed graphical analysis; the application to problems in mechanics is treated in Mechanics, § 5, Graphic Statics, and Diagram. ”

“ Each individual manager selects the particular set of facts which he thinks will best show the trend of the business. The complexity of the business usually determines the number of reports.
Use of Graphs in the Presentation of Statistics
Chart Showing Comparison of Sales
with Cost of Advertising
Long experience or special aptitude may qualify a man to grasp readily the full significance of statistical compilations in a variety of forms without the use of charts or graphs. ”

“ Anyone who understands Graphology can tell by simply examining your handwriting just what sort of a person you are. ¶ There is no method of character reading that is more interesting, more trustworthy, and more valuable than that of Graphology, and it is the aim of this volume to enable anyone to become a master of this most fascinating art. ”

“ One may afterwards follow out each recognized lineation into further complexities or joinings, as you might trace out a railway line with its various junctions in a map.
Diagrammatic Analysis of Lineations in a Restricted Section.
A photographic enlargement, meant for forensic use, ought not to be marked or soiled in any way, but dots of coloured chalk or ink might be placed along the margins to denote where imaginary ruled lines might begin or end. One might also use glazed tissue paper, ruled in squares, or with eccentric circles like the mileage lines in a map of London. ”

“ A little instruction in the elements of chartography—a little practice in the use of the compass and the spirit level, a topographical map of the town common, an excursion with a road map—would have given me a fat round earth in place of my paper ghost; ”

“ The arithmetical symbols are written diagrams and the geometrical figures are graphic formulas; ”

“ See "The Black Star Passes", Ace Books, F-346.
I
Three men sat around a table which was littered with graphs, sketches of mathematical functions, and books of tensor formulae. Beside the table stood a Munson-Bradley integraph calculator which one of the men was using to check some of the equations he had already derived. The results they were getting seemed to indicate something well above and beyond what they had expected. ”

“ mathematics was hard, dull work, I thought; geography pleased me more. For my other studies, as well as for dancing, I was quite enthusiastic; ”

“ But to form a table of prime numbers the process is theoretically simple and rapid, for we have only to range all the numbers in a line and strike out every second number beginning from 2, every third beginning from 3, and so on, those that remain being primes. Even when the tabular results are constructed separately, the method of differences or other methods connecting together different tabular results may afford valuable verifications. By having recourse to tables not only does the computer save time and labour, but he also obtains the certainty of accuracy. ”

“ ‘I’m sure I’m not Ada,’ she said, ‘for her hair goes in such long ringlets, and mine doesn’t go in ringlets at all; and I’m sure I can’t be Mabel, for I know all sorts of things, and she, oh! she knows such a very little! Besides, she’s she, and I’m I, and—oh dear, how puzzling it all is! I’ll try if I know all the things I used to know. Let me see: four times five is twelve, and four times six is thirteen, and four times seven is—oh dear! I shall never get to twenty at that rate! However, the Multiplication Table doesn’t signify: let’s try Geography. ”

“ If, in a certain diagram, whose geometrical history we know, we want to test whether two Lines, passing through a common point, have, or have not, 'the same direction,' we have simply to take any other point on one of the Lines, and observe whether the other Line does, or does not, pass through it. ”

“ It is perfectly possible to set, up a one-to-one correspondence between the points of a line and the points of a plane, or, indeed, between the points of a line and the points of a space of any finite number of dimensions, if the correspondence is not restricted to be continuous. ”

“ For example, a table of logarithms is a table of single entry, the numbers being the arguments and the logarithms the tabular results; an ordinary multiplication table is a table of double entry, giving xy as tabular result for x and y as arguments. The intrinsic value of a table may be estimated by the actual amount of time saved by consulting it; for example, a table of square roots to ten decimals is more Valuable than a table of squares, as the extraction of the root would occupy more time than the multiplication of the number by itself. ”

“ My own opinion, as I have intimated, is, that the second account of the origin of the trigrams and hexagrams is the true one. However the idea of the whole and divided lines arose in the mind of the first framer, we must start from them; and then, manipulating them in the manner described, we arrive, very easily, at all the lineal figures, and might proceed to multiply them to billions. ”

“ Can you construct a square of sixteen different numbers so that it shall be magic (that is, adding up alike in the four rows, four columns, and two diagonals) , whether you turn the diagram upside down or not? You must not use a 3, 4, or 5, as these figures will not reverse; but a 6 may become a 9 when reversed, a 9 a 6, a 7 a 2, and a 2 a 7. The 1, 8, and 0 will read the same both ways. Remember that the constant must not be changed by the reversal. ”

“ Here you see the pattern is a large square, and the letters are drawn very carefully in them. Did you know before there was as much figuring and measuring done in the making of the Alphabet as there is in building a house? Look at the letter E, for example, and all the circles and squares that have been measured and drawn to make it. You will find that every letter is made just as carefully. ”

“ By making things mathematical they make them thin. Take the living lines out of that landscape, simplify it to a right angle, and you flatten it out to a mere diagram on paper. Diagrams have their own beauty ”

“ In mathematics, a part often contains the whole. If we represent points by vectors drawn from a common origin, and then develop those relations between such vectors representing points, which are independent of the position of the origin,—by this simple process we may obtain a large part, possibly all, of an algebra of points. In this way the vector analysis may be made to serve very conveniently for many of those subjects which I have mentioned as suitable for point analysis. ”

“ Any fact which we know or have been told about our problem is called a datum. The number of pounds of sugar we are to buy is one datum; the price per pound is another.
The plural of datum is data. It is a good plan to write all one's data on one column or page of the paper and work one's sum on the other. This leaves the first column clear for adding to one's data if one finds out any fresh one. ”

“ A square is the whole amount of the root multiplied by itself.
A simple number is any number which may be pronounced without reference to root or square.
A number belonging to one of these three classes may be equal to a number of another class; you may say, for instance, “squares are equal to roots,” or “squares are equal to numbers,” or “roots are equal to numbers.” [2]
(4) Of the case in which squares are equal to roots, this is an example, “A square is equal to five roots of the same;” ”

“ A plane may be represented in multiple algebra as in synthetic geometry by a single letter; in the ordinary algebra it is sometimes represented by three coordinates, for which it is most convenient to take the reciprocals of the segments cut off by the plane on three axes. But the modern analyst has a more ingenious method of representing the plane. ”

“ The attempt to fashion the Great Extreme as a circle must be pronounced a failure.
But when we start from the two lines as bases, the formation of all the diagrams by a repetition of the process indicated above is easy. The addition to each of the trigrams of each of the two fundamental lines produces 16 figures of four lines; dealt with in the same way, these produce 32 figures of five lines; and a similar operation with these produces the 64 hexagrams, each of which forms the subject of an essay in the text of the Yî. ”

“ Our puzzle, therefore, resolves itself into forming a magic square with nine different prime numbers.
In Diagram A we have a magic square in prime numbers, and it is the one giving the smallest constant sum that is possible. As to the little trap I mentioned, it is clear that Diagram A is barred out by the words "every basket contained plums," for one plum is not plums. And as we were referred to the baskets, "as shown in the illustration," it is perfectly evident, without actually attempting to count the plums, that there are at any rate more than 7 plums in every basket. ”