CHAPTER II RECTANGULAR COÖRDINATES AND THE POWER 10. Rectangular Coördinates. Two intersecting algebraic scales, with their zero points in common, may be used as a system of latitude and longitude to locate any point in their plane. The student should be familiar with the rudiments of this method from the graphical work of elementary algebra. The scheme is illus trated in its simplest form in Fig. 18, where one of the horizontal lines of a sheet of squared paper has been selected as one of the algebraic scales and one of the vertical lines of the squared paper has been selected for the second algebraic scale. To locate a given point in the plane it is merely necessary to give, in a suitable unit of measure (as centimeter, inch, etc.), the distance of the point to the right or left of the vertical scale and its distance above or below the horizontal scale. Thus the point P, in Fig. 18, is 2 units to the right and 31⁄2 units above the standard scales. P2 is 3 unitș to the left and 2 units above the standard scales, etc. Of course these directions are to be given in mathematics by the use of the signs "+" and “ " of the algebraic scales, and not by the use of the words "right" or "left," "up" or "down." The above scheme corresponds to the location of a place on the earth's surface by giving its angular distance in degrees of longitude east or west of the standard meridian, and also by giving its angular distance in degrees of latitude north or south of the equator. The sort of latitude and longitude that is set up in the manner described above is known in mathematics as a system of rectangular coördinates. It has become customary to letter one of the scales XX', called the X-axis, and to letter the other YY', called the Y-axis. In the standard case these are drawn to the right and left, and up and down, respectively, as shown in Fig. 18. The distance of any point from the Y-axis, measured parallel to the X-axis, is called the abscissa of the point. The distance of any point from the X-axis, measured parallel to the Y-axis, is called the ordinate of the point. Collectively, the abscissa and ordinate are spoken of as the coördinates of the point. Abscissa corresponds to the longitude and ordinate corresponds to the latitude of the point, referred to the X-axis as equator, and to the Y-axis as standard meridian. In the standard case, abscissas measured to the right of YY' are reckoned positive, those to the left, negative. Ordinates measured up are reckoned positive, those measured down, negative. Rectangular coördinates are frequently called Cartesian coördinates, because they were first introduced into mathematics by René Descartes (1596-1650). The point of intersection of the axes is lettered O and is called the origin. The four quadrants, XOY, YOX', X'OY', Y'OX, are called the first, second, third, and fourth quadrants, respectively. A point is designated by writing its abscissa and ordinate in a parenthesis and in this order: Thus, (3, 4) means the point whose abscissa is 3 and whose ordinate is 4. Likewise (−3, 4) means the point whose abscissa is (-3) and whose ordinate is (+ 4). Unless the contrary is explicitly stated, the scales of the co ördinate axes are assumed to be straight and uniform and to intersect at right angles. Exceptions to this are not uncommon, however, of which examples are given in Figs. 19 and 22. The use of two intersecting algebraic scales to locate individual points in the plane, as explained above, is capable of immediate enlargement. It will be explained below that a suitable array, or set, or locus of such points may be used to exhibit the relation between two variables laid off on the two scales, or between a variable laid off on one of the scales and a function of the variable laid off on the other scale. This fact has already been explained from another point of view at the close of the preceding chapter. 11. Statistical Graphs. From work in elementary algebra the student is supposed to be familiar with the construction of statis FIG. 19.-Barograph Taken During a Balloon Journey. The vertical scale is atmospheric pressure in millimeters of mercury. tical graphs similar to those presented in Figs. 19 to 32. The student will study each of these graphs and the following brief descriptions before making any of the drawings required in the exercises that follow. Fig. 19 is a barograph, or autographic record of the atmospheric pressure recorded November 24, 1907, during a balloon journey from Frankfort to Marienburg in West Prussia. One set of scales consists of equal circles, the other of parallel straight lines. The zero of the scale of pressure does not appear in the diagram. Note also that the scale of pressure is an inverted scale, increasing downward. The scale of time is an algebraic scale, the zero of which may be arbitrarily selected at any convenient point. The scale of pressure is an arithmetical scale. The zero of the barometric scale corresponds to a perfect vacuum-no less pressure exists. 4 P.M. FIG. 20.-Graphical Time-table of Certain Railway Trains between Chicago and Minneapolis. FIG. 21.-Graphical Time-table of Passenger Trains between Chicago and Los Angeles. Fig. 20 is a graphical time-table of certain passenger trains between Chicago and Minneapolis. The curves are not continuous, as in the case of the barograph, but contain certain sudden jumps. What is the meaning of these? What indicates the speed of the FIG. 22.-Upper Curve, Elevation of Water in a Well on Long Island Lower curve, elevation of water in the nearby ocean. trains? Where is the fastest track on this railroad? What shows the meeting point of trains? If the diagram, Fig. 20, be wrapped around a vertical cylinder of such size that the two midnight lines just coincide, then each train line may be traced through continuously from terminus to terminus. |