132 SCIENCE PROGRESS 



1 fi 3 



the three operative cube roots X , K , and K , all of which are different. 

 But these operative roots are real and do not involve the imaginary unit. 

 In the sense of i'o5, however, the algebraic roots of K^i are given by the 

 operative roots of K^. 



7*3. This paper has been the briefest possible introduction to a geometry, 

 suggested by the if-operations, which is almost as simple and elegant as 

 Hamiltonian geometry, but which has the advantage of not being based upon 

 any preliminary assumption — indeed, I repeat, the whole matter here set 

 forth is nothing but ordinary trigonometry put in the form of operator and 

 operand. I hope to show in a subsequent paper its relation both with 

 Hamilton's and with Grassmann's systems, as well as with "complex numbers." 

 Years ago I attempted to prove (i) that these systems can be combined and 

 consolidated by the introduction of a single scalar unit which is common to 

 both but the existence of which was overlooked by the authors and, apparently, 

 by subsequent writers. The present results obviously suggest further develop- 

 ments in the same direction ; but the exposition requires another paper. 



7" 4. The theorem i'o2 may be proved in several ways. It is the pro- 

 perty of four numbers which possess, two and two, the same modulus. 

 Geometrically, it can be derived at once from Ptolemy's Theorem — that the 

 product of the diagonals of a quadrilateral in a circle equals the products of 

 opposite sides, two and two (M. Raima's Almagest, 1813, vol. i, page 29, or 

 H. M. Taylor's Euclid, III, 37B). See also my paper 4 — examples. 



7*5. As K^ occurs frequently in geometry, many geometrical inter- 

 pretations of it may be given. Perhaps that of 4-0 is the most fertile ; but 

 it may also be presented as the operative ratio of the co-ordinates of rotation, 

 as indicated in 7'i4. This interpretation is connected with that of 4*0 by 

 the method of 6"4. 



7*6. Operative analysis seems to reduce everything to the two funda- 

 mental processes, iteration (including inversion) and change of base, and I 

 venture to think that it will eventually replace our present methods of work- 

 ing. Functions are not mere jumbles of quantities put together anyhow, 

 but possess, so to speak, molecular as well as atomic structure. Operative 

 procedure enables us to apply group-theory immediately to every problem. 



7*7. The fact that ^° does not equal numerical unity applies to all 

 cancelled operations (except 0°), such as A°, 2°, -0°. I hope to show later 

 how recognition of this fact consolidates Boole's differential symbolism, and 

 how it can be applied in many directions, as for determinants, invariants, 

 series, and the Calculus. 



7'8. The names Real Quaternion or Operative Quaternion may be suggested 

 for K^ ; but it is better to reserve decision on this for the present. 



May 13, 1921. 



THE WRITER'S PAPERS 



1. The Algebra of Space (G. Philip & Sons, London, 1901). 



2. " Verb-Functions" {Proc. R. Irish Acad., XXV, A, 1905). 



3. " The Solution of Equations by Operative Division " {Science Progress, 



Vol. X, Nos. 38, 39, 40, 1915-16). 



4. " Operative Algebra : Operative Involution " {Science Progress, 



Vol. XIII, No. 50, 1918). 



5. "Isosceles Trigonometry" {Science Progress, Vol. XIII, No. 51 



1919). 



