THE FOURTH DIMENSION. 



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our unit of time, the period of the rotation of the earth about its 

 axis, namely, the day, or a definite portion of a day. The zero-point 

 of time is regarded in Christian countries as the year of the birth of 

 Christ, and the positive direction of time is the time subsequent to 

 the birth of Christ. These data fixed, all that is necessary to estab- 

 lish and distinguish any definite point of time amid the infinite to- 

 tality of all the points of time, is a single number. Of course this 

 number need not be a whole number, but may be made up of the 

 sum of a whole number and a fraction in whose numerator and de- 

 nominator we may have numbers as great as we please. We may, 

 therefore, also say that the totality of all conceivable numerical 

 magnitudes, or of only such as are greater than one definite number 

 and smaller than some other definite number, is one-dimensional. 



We shall add here a few additional examples of one-dimen- 

 sional magnitudes presented by geometry. First, the circumference 

 of a circle is a one-dimensional magnitude, as is every curved line, 

 whether it returns into itself or not. Further, the totality of all 

 equilateral triangles which stand on the same base is one-dimen- 

 sional, or the totality of all circles that can be described through 

 two fixed points. Also, the totality of all conceivable cubes will be 

 seen to be one-dimensional, provided they are distinguished, not 

 with respect to position, but with respect to magnitude. 



In conformity with the fundamental ideas by which we define 

 the notion of a one-dimensional manifoldness, it will be seen that 

 the attribute //<?-dimensional must be applied to all totalities of 

 things in which two numbers are necessary (and sufficient) to dis- 

 tinguish any determinate individual thing amid the totality. The 

 simplest two-dimensioned complex which we know of is the plane. 

 To determine accurately the position of a point in a plane, the sim- 

 plest way is to take two axes at right angles to each other, that is, 

 fixed straight lines, and then to specify the distances by which the 

 point in question is removed from each of these axes. 



This method of determining the position of a point in a plane 

 suggested to the celebrated philosopher and mathematician Des- 

 cartes the fundamental idea of analytical geometry, a branch of 

 mathematics in which by the simple artifice of ascribing to every 



