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SCIENTIFIC THOUGHT. 



36. 

 Invariants. 



to a unification on a higher level. But the distinc- 

 tion mentioned above led to another most remarkable 

 line of thought and research which tends more and 

 more to govern mathematical doctrine. The methods 

 of projection are based upon the motion or upon the 

 transformation of figures. Under such a process some 

 relations remain unaltered or invariant, others change. 

 As analytical methods in the hands of Pllicker and 

 others began to accommodate themselves more closely to 

 geometrical forms, as an intimate correspondence was 

 introduced between the figure and the formula, it became 

 natural to study the unalterable properties of the figure 

 in the invariant elements of the formula. This is the 

 origin and meaning of the doctrine of Invariants.^ It 

 is the great merit of the English school of mathe- 

 maticians, headed by Boole, Cayley, and Sylvester, both 

 to have first conceived the idea of a doctrine of invariant 



1 '• In anj subject of inquiry 

 there are certain entities, the 

 mutual relations of which, under 

 various conditions, it is desirable to 

 ascertain. A certain combination 

 of these entities may be found 

 to have an unalterable value when 

 the entities are submitted to cer- 

 tain processes or are made the 

 subjects of certain operations. 

 The theory of invariants in its 

 widest scientific meaning deter- 

 mines these combinations, eluci- 

 dates their properties, and expresses 

 results when possible in terms of 

 them. Many of the general prin- 

 ciples of political science and 

 economics can be expressed by 

 means of invariantive relations 

 connecting the factors which 

 enter as entities into the special 

 problems. The great principle of 

 chemical science which asserts that 



when elementary or compound 

 bodies combine with one another 

 the total weight of the materials 

 is unchanged, is another case in 

 point. Again, in physics, a given 

 mass of gas under the operation 

 of varying pressure and tempera- 

 ture has the well-known invariant, 

 pressure multiplied by volume and 

 divided by absolute temperature. 

 Examples might be multiplied. In 

 mathematics the entities under ex- 

 amination may be arithmetical, 

 algebraical, or geometrical ; the 

 processes to which they are sub- 

 jected may be any of those which 

 are met with in mathematical 

 work. It is the principle which 

 is valuable. It is the idea of in- 

 variance that pervades to-day all 

 branches of mathematics." (Major 

 P. A. MacMahon, Address, Brit, 

 Assoc, 1901, p. 526.) 



