58 



combined, zero result; e. g., assets and liabilities. In this case — 1 reverses 

 quality without altering magnitude, so that 1 -p ( — 1 ) = 0. But what is a 

 farm of — 80 acres? Imagine a farm that put with an SO acre farm gives no 

 land at all. 



Fourth, the incommensurable numbers, e. g.. the ratio of a diagonal to a 

 side of a square. These require continuous quantitj'^, and their use with 

 quantity whose partitions are limited is impossible. What is a space of 

 |, '7 dimensions, a country with ^ '7 presidents, a man with i 7 dollars in 

 his pockets? 



We recognize a number by what it can do with appropriate quantity to 

 operate upon, not by what it can not do with inappropriate quantity. The 

 interpretation of imaginary number requires quantity that has magnitude 

 and different qualities. These quantities, whether geometrical or physical, 

 may be represented by certain geometrical quantities called by Clifford 

 steps. 



The step from a position A in space to another position B has length and 

 direction. Two steps are equal that have the same length, and the same 

 direction ; i. e., the opposite sides of a parallelogram taken in the same di- 

 rection are equal steps. The sum of any number of successive steps in 

 various directions is the step from the first point of departure to the last 

 point reached ; e. g., A B + B C + C D = A D. In particular the sum of 

 two successive steps along the sides of a parallelogram is equal to the step 

 along the diagonal. As the remaining sides in the parallelogram form 

 equal steps added in reverse order, we learn that the order of successive 

 steps in a sum may be changed without altering the sum. 



Positive numbers operating on steps change lengths but not directions ; 

 — 1 reverses direction without altering length ; e. g., — 1 A B = B A. If x 

 be any real number we see by similar triangles that x ( A B + B C) = x A B 

 J-xBC. 



A valuable analysis may be developed by the use of steps and real num- 

 bers only. From its simplicity, and its value in physical applications, it 

 ought to displace ordinary analytical geometry, in technical schools at 

 least. The main difticulty is the lack of a suitable text book. 



Let us confine ourselves, now, to steps in the plane of the paper, and 

 consider the nature of the number that multiplying A produces O B. It 

 must alter the length of O A into the length of O B ; this is the tensor fac- 

 tor, an ordinary positive number. It must turn O A thus lengthened into 

 OB; this is the versor factor; the angle of this turn, reckoned as positive 



