August 27, 1886.] 



SCIENCE. 



181 



ematics, which nothing but publication in full 

 could render intelligible, and that only to the 

 mathematicians among our readers. To such, its 

 full publication in the ' Proceedings ' will prove of 

 the greatest value. 



The fourth part of the paper was devoted to 

 consideration of some of the applications of multi- 

 ple algebra. From this we quote the following : 

 " First of all, geometry, and the geometrical 

 sciences which treat of things having position in 

 space, — kinematics, mechanics, astronomy, crys- 

 tallography, — seem to demand a method of this 

 kind, for position in space is essentially a multiple 

 quantity, and can only be represented by simple 

 quantities in an arbitrary and cumbersome man- 

 ner. For this reason, and because our spatial in- 

 tuitions are more developed than those of any 

 other class of mathematical relations, these sub- 

 jects are especially adapted to introduce the stu- 

 dent to the methods of multiple algebra. Here 

 nature herself takes us by the hand, and leads us 

 along by easy steps, as a mother teaches her child 

 to walk. In the contemplation of these subjects, 

 Mobius, Hamilton, and Grassmann formed their 

 algebras, although the philosophical mind of the 

 last was not satisfied until he had produced a sys- 

 tem unfettered by any spatial relations. It is 

 probably in connection with these subjects that 

 the notions of multiple algebra are most widely 

 disseminated. Maxwell's ' Treatise on electricity 

 and magnetism ' has done so much to familiarize 

 students of physics with quaternion notations, 

 that it seems impossible that this subject should 

 ever again be entirely divorced from the methods 

 of multiple algebra. I wish that I could say as 

 much of astronomy. It is, I think, to be regretted, 

 that the oldest of the scientific applications of 

 mathematics, the most dignified, the most conser- 

 . vative, should keep so far aloof from the youngest 

 of mathematical methods ; and standing, as I do 

 to-day, by some chance, among astronomers, al- 

 though not of the guild, I cannot but endeavor to 

 improve the opportunity by expressing my con- 

 viction of the advantages which astronomers 

 might gain by employing some of the methods of 

 multiple algebra. A very few of the fundamental 

 notions of a vector analysis, the addition of vec- 

 tors and what quaternionists woiild call ' the 

 scalar part and the vector part of the product of 

 two vectors ' (which may be defined without the 

 definition of the quaternion), — these three no- 

 tions, with some four fundamental properties 

 relating to them, are sufficient to reduce enor- 

 mously the labor of mastering such subjects as 

 the elementary theory of orbits, the determina- 

 tion of an orbit from three observations, the dif- 

 ferential equations which are used in determining 



the best orbit from an indefinite number of obser- 

 vations by the method of least squares, or those 

 which give the perturbations when the elements 

 are treated as variable. In all these subjects, the 

 analytical work is greatly simplified, and it is far 

 easier to get the best form for numerical calcula- 

 tion than in the use of the ordinary analysis." 



Then followed illustrations of the various meth- 

 ods of applying multiple algebra to different 

 classes of problems, and the paper closed as fol- 

 lows : " But I do not so much desire to call your 

 attention to the diversity of the applications of 

 multiple algebra, as to the simplicity and unity of 

 its principles. The student of multiple algebra 

 suddenly finds himself freed from various restric- 

 tions to which he has been accustomed. To many, 

 doubtless, this liberty seems like an invitation to 

 license. Here is a boundless field in which caprice 

 may riot. It is not strange if some look with 

 distrust for the result of such an experiment. 

 But the further we advance, the more evident it 

 becomes that this, too, is a realm subject to law. 

 The more we study the subject, the more we find 

 all that is most useful and beautiful attaching 

 itself to a few central principles. We begin by 

 studying ' multiple algebras ;' we end, I think, by 

 studying 'multiple algebra.'" 



SEAT OF THE ELECTROMOTIVE FORCE. 



Professor Brackett's address was essentially 

 a resume of the history of the investigations to 

 find the source of the current in galvanic bat- 

 teries. No attempt was made to settle the ques- 

 tion, which has been so long a bone of contention. 



The address was so purely historical in its 

 natvire, and, withal, was so condensed and con- 

 cise, that any abstract would be necessarily little 

 more than an index of its contents. Those who 

 are interested in the subject must await its publi- 

 cation in full in the ' Proceedings ' of the associa- 

 tion. 



Galvani's two accidental discoveries were made 

 in 1789 : the one was the influence of an electrical 

 machine in causing contractions in a frog's legs, 

 and the other the production of sufficient elec- 

 tricity to cause the contraction by touching two 

 joined strips of copper and zinc to the moist 

 animal tissues. Naturally from these results there 

 arose a theory of the identity of nerve-force and 

 electricity, — the so-called animal variety of elec- 

 tricity. While this controversy, soon to subside, 

 was started among physiologists, a much more 



Abstract of an address delivered before the section of 

 physics of the American association for the advancement 

 of science at Buffalo, Aug. 19, 1886, by Prof. C. F. Brackett, 

 of Princeton, vice-president of the section. 



