68 THE FOURTH DIMENSION. 



point in a plane two numerical values, determined by its distances 

 from the two axes above referred to, planimetrical considerations 

 are transformed into algebraical. So, too, all kinds of curves that 

 graphically represent the dependence of things on time, make use 

 of the fact that the totality of the points in a plane is two-dimen- 

 sional. For example, to represent in a graphical form the increase 

 in the population of a city, we take a horizontal axis to represent 

 the time, and a perpendicular one to represent the numbers which 

 are the measures of the population. Any two lines, then, whose 

 lengths practical considerations determine, are taken as the unit of 

 time, which we may say is a year, and as the unit of population, 

 which we will say is one thousand. Some definite year, say 1850, 

 is fixed upon as the zero point. Then, from all the equally distant 

 points on the horizontal axis, which points stand for the years, we 

 proceed in directions parallel to the other axis, that is, in the per- 

 pendicular direction, just so much upwards as the numbers which 

 stand for the population of that year require. The terminal points 

 so reached, or the curve which runs through these terminal points, 

 will then present a graphic picture of the rates of increase of the 

 population of the town in the different years. The rectangular axes 

 of Descartes are employed in a similar way for the construction of 

 barometer curves, which specify for the different localities of a 

 country the amount of variation of the atmospheric pressure during 

 any period of time. Immediately next to the plane the surface of 

 the earth will be recognised as a two-dimensional aggregate of 

 points. In this case geographical latitude and longitude supply the 

 two numbers that are requisite accurately to determine the position 

 of a point. Also, the totality of all the possible straight lines that 

 can be drawn through any point in space is two-dimensional, as we 

 shall best understand if we picture to ourselves a plane which is 

 cut in a point by each of these straight lines and then remember 

 that by such a construction every point on the plane will belong to 

 some one line and, vice versa, a line to every point, whence it fol- 

 lows that the totality of all the straight lines, or, as we may call 

 them, rays, which pass through the point assigned are of the same 

 dimensions as the totality of the points of the imagined plane. 



