16 



SCIENCE. 



[Vol. VII., No. 152 



is very curious to find a compendium of logic with 

 the syllogism left out. Hamlet is even less neces- 

 sary to his play than the syllogism to logic. It is 

 true, however, that the syllogism is an easy matter 

 compared with inversion and contra-position. 

 There is hardly a boy who is not greatly surprised 

 to find that when he has proved that an isosceles 

 triangle has two equal angles, it still remains to be 

 proved that a triangle having two equal angles is 

 isosceles. As De Morgan lias pointed out, Euclid 

 himself was apparently not aware that it follows 

 every time from .4 implies B that non-B implies 

 non-A. 



In regard to 1 his rule of inversion.' when 

 three or more propositions are involved, Mr. 

 Halsted has fallen into a slight inaccuracy. In 

 the first place, if the term ' contradictory ' is to be 

 applied to three terms at all, it should be used in 

 the same sense as when applied to two terms; the 

 three terms should together cover the whole field, 

 and they should not overlap. The word is a bad 

 one for this purpose, however, and it is just as well 

 to keep the two properties — that of being exhaus- 

 tive and that of being incompatible — distinct. 



In the second place, there is a redundancy in 

 the rule as given by Mr. Halsted. From the three 

 propositions, 1 



X implies x, 

 Y implies y, 

 Z implies z, 

 it may be inferred that 



x implies X, 

 y implies Y, 

 z implies Z, 



provided that the subjects cover the whole field, and 

 the predicates are incompatible. It is not neces- 

 sary that the subjects should be known to be in- 

 compatible, though it follows from the premises 

 given that they are so, but also that the predicates 

 arc exhaustive. From the first two we have 



X Y implies x y ; 

 and. since there is no x y, there cannot be any X Y 

 cither. 



ft is very well worth while to have formulated 

 the reasoning involved, instead of going through 

 all the separate steps every time there is occasion 

 for it. as the usual hooks on geometry do. 



The conclusion docs not follow if it is given 

 that the subjects are incompatible, and that the 

 predicates together till the universe. The nature 

 of the argument is most clearly seen in space. 

 Lange believes that the logical laws of thought are 

 derived from space-conceptions. Suppose there is 

 a table painted in various colors, but so that 

 the red is all in the violet, 

 the yellow is all in the blue, 

 and the orange is all in the green ; 



1 Thy h'ttcrn .stand Tor yithor tyrms or propositions. 



and suppose, also, that the red, the yellow, and the 

 orange together cover the whole table, and that 

 the violet, the blue, and the green do not overlap : 

 it follows that 



red = violet, 

 yellow=blue, 

 orange = green. 

 To show how a somewhat complicated argument 

 can be simplified by having this type of reason- 

 ing at command, we add a real illustration from 

 algebra. In Descartes' method of solution of the 

 biquadratic equation, the following relations are 

 seen to hold between its roots and those of the 

 auxiliary cubic : — 



Roots of the Roots of the 



biquadratic. cubic. 



* -i i -i 7 . \ All real and 



All real impli~~ 



Two real (unequal) implies 



positive. 

 One positive, 

 two imaginary. 



Two real (equal) implies j «ga«'^° 



imvlies ^ 0ne P° sitiye > two 

 impaes i unequal negative. 



All imaginary 



But the division on the left is exhaustive, and the 

 classes on the right are mutually exclusive : hence, 

 by a purely logical tour de force, these propositions 

 can all be inverted, and the desired inferences 

 from the roots of the cubic to the roots of the 

 biquadratic can be obtained at once. 



Mr. Halsted's reviewers have pointed out before 

 that he is deficient in a certain natural and 

 becoming modesty. 'Two formative years' of his 

 life is too high-sounding a phrase to be applied to 

 any but a very great mathematician, like Professor 

 Cayley, for instance. 



CEREBRAL EXCITABILITY AFTER DEATH. 



The problems of brain physiology are so com- 

 plex, and our means of studying them, especially 

 in the human subject, so insufficient, that it is 

 not to be wondered at if rather out-of-the-way 

 and venturesome experiments are sometimes 

 undertaken by the anxious physiologist ; as, wit- 

 ness the actual stimulation of the exposed brain 

 in a patient whose death seemed certain. Such 

 an experiment is not apt to be repeated ; and a few 

 French physicians have now wisely set to work 

 to study the results of stimulating the cerebrum, 

 exciting the sense-organs, and subjecting the whole 

 body to a vigorous examination in the case of 

 criminals who have suffered death by decapitation. 1 

 Such investigations are not new ; but the results 

 have been, as a rule, cither entirely negative, or 

 brought out only a few rather obvious facts. In 

 t he experiments about to he described, the methods 

 1 Revue scientifique, Nov. 28. By J. V. Laborde. 



