October 2, 1891.] 



SCIENCE. 



187 



while Euler, Peacock, De Morgan, and others have devel- 

 oped it more as a double algebra. 



Up to this point i had been regarded as a scalar operator 

 merely, and the corresponding geometry only plane, 

 though attempts had been made without much success 

 to extend the treatment into three-dimensional space. It 

 remained for Hamilton to accomplish this by the sim- 

 ple device of making i a directed operator, or handle, 

 perpendicular to the plane of rotation, which opened 

 the way for any number of similar operators differing in 

 direction, but, as to their other properties, simply square 

 roots of minus one. In order to produce a convenient alge- 

 bra on this basis, Hamilton was obliged to take the further 

 step of giving to all vectors the properties of / — 1, and thus 

 the calculus of quaternions was produced, a non-commuta- 

 five quadruple algebra. These ideas have been generalized 

 still farther by Unverzagt in his " Theorie der goniometris- 

 chen und der longimetrischen Quaternionen." In this book 

 the author first develops a trigonometry based on a general 

 instead of a right-angled triangle, and then shows that the 



X 



operator J = ( — !)"■ (in which A is the fundamental angle, 

 taking the place of y) takes in this trigonometry the place of 

 i in De Moivre's theorem generalized. He then takes three 

 units y,, jg, ^3, corresponding to Hamilton's i, j, k, and 

 forms a generalized quaternion, based on some angle A, 



which reduces to the ordinary system when X = -^. The 

 case particularly discussed is that in which A = 0. 



The theory and laws of linear, associative algebras, which 

 includes quaternions as a particular case, have been thor- 

 oughly treated by Peirce in his work bearing that title. 



We turn now to the other line along which multiple alge- 

 bras have been developed. In 1827 MObius published his 

 " Barycentrische CalcUl," in which points are the ultimate 

 units, to which any desired weights may be assigned. He 

 gave the laws of combination of these units so far as addi- 

 tion and subtraction are concerned, but did not proceed to 

 multiplication: in fact, he distinctly states that they can be 

 multiplied only by numbers. He then proceeds to treat 

 analytical geometry on this basis. His treatment of points, 

 so far as it goes, is on the same plan afterwards indepen- 

 dently developed by Grassmann. 



In 1844, one year after Hamilton's first announcement of 

 his discovery, Grassmann published his "Ausdehnuugslehre," 

 which contains a complete and logical exposition of his new 

 algebra for any number of independent units, and hence, 

 geometrically interpu^ted, for space of any dimensions. 

 This book was so abstract and general in form, and so un- 

 like the ordinary language of mathematics, that it attracted 

 hardly any notice, and the author was obliged to recast and 

 republish it in 1862. Grassraan's algebra is non-linear, and 

 only partially associative, so that it differs fundamentally 

 from all those discussed by Peirce. The 4/ — 1 plays no part 

 whatever in the theory, and Grassman's vector is a vector 

 pure and simple, i.e., a quantity having direction and mag- 

 nitude, and not, as in quaternions, a j;e)'so?'-vector, combin- 

 ing the properties of a vector and of the 4/ — 1. The funda- 

 mental notion of Grassmann's multiplication is extension or 

 generation; the product pi p^ is the line generated by a 

 point moving straight from pi io p.y, etc. 



In this great invention of Grassnian we have a multiple 

 algebra which is the natural language of geometry and 

 mechanics, dealing in a manner astonishingly simple, con- 

 •cise, and expressive with these subjects, and certain, it ap- 

 pears to me, to gain constantly in the appreciation of math- 



ematicians as it is more generally understood and used. 

 The fact of its perfect adaptability to n-dimensional space is 

 an additional argument in its favor for those who are inter- 

 ested in that line of investigation. 



We have now traced the development of our subject 

 from its elementary beginnings through a long period in 

 which it was in the rhetorical stage, approaching at inter- 

 vals here and there to the syncopated; then, on the revival 

 of Jearning in Europe after the dark ages, we have seen its 

 comparatively rapid progress through the syncopated stage 

 to the purely symbolical, when it was at last in a shape 

 suitable for the astonishing progress of the last two hundred 

 years. Finally, in the present century, we have noted the 

 appearance, as in the fulness of time, of multiple algebras 

 from different and independent sources, whose realm is that 

 of the future. 



NOTES AND NEWS. 

 The astronomers sent to the Sandwich Islands recently on the 

 part of the International Geodetic Association of Europe and the 

 United States Coast and Geodetic Survey, in order to make a more 

 exhaustive study of the changes of latitude, have located their 

 observatories at Waikiki, near Honolulu. It is proposed to observe 

 during the year about sixty five pairs of stars, chosen on account 

 of their well-determined proper motions, and to make in all not 

 far from twenty-five hundred observations of the latitude. The 

 results, compared with those made simultaneously in Europe and 

 America, will settle definitely the question whether there is a real 

 motion of the pole. At the suggestion of the American representa- 

 tive, the force of gravity will be measured every night that latitude 

 observations are made. This may throw light on one of the theo- 

 ries proposed to explain the changes of latitude, viz., that of large 

 transfers of matter beneath the earth's surface. The new pendu- 

 lums made at the Coast and Geodetic Survey Office in Washington, 

 and which are similar to those taken to Alaska by Professor Men- 

 denhall last spring, will be employed at Waikiki. They are of 

 fine workmanship, and are capable of detecting changes that do 

 not exceed one hundred-thousandth part of the quantity measured. 

 Besides the observations at th« regular station, a number of 

 magnetic determinations will be made at other points in the 

 Islands, — notably at Kealakeakua Bay, where Captain Cook ob- 

 served the declination more than a hundred years ago, and at 

 Lahaina, where De Freycinet had an observatory for pendulum 

 and magnetic work in 1819. The re-occupation of these points 

 will show the change of the needle during the past century, and 

 will be of great value in determining tlie secular variation. It is 

 intended also to seize the opportunity now presented to measure 

 the force of gravity on the summit of Mauna Kea (14 000 feet 

 elevation). Observations made at the top of Haleakala (10,000 

 feet) in 1887 showed conclusively that the mountain was solid. 

 This fact received additional support from the zenith observations 

 at the sea-level north and south of the mountain. The large 

 deviation of the plumb line (29") brought to light in that work has 

 now been exceeded on Hawaii, where 1' 36" has been discovered 

 at the south point of the island (Ka Lae). This fact, recently 

 communicated by Surveyor General Alexander, makes the ques- 

 tion of the force of gravity at the summit of Mauna Kea one of 

 double interest, and it is desirable, both from a geological and 

 geodetic standpoint, that pendulum observations be made on top 

 of one of the mountains. Doctor Marcuse, who is from the Royal 

 Observatory at Berlin, observes for latitude on the prirt of the 

 European association, and Mr. Preston, who made the observations 

 at the summit of Haleakala four years ago, is from the United 

 Stales Coast and Geodetic Survey, and makes gravity and mag- 

 netic determinations. He also, as the representative of the United 

 States, observes for latitude in connection with Dr. Marcuse, in 

 the international geodetic work. The ob-ervers had the good 

 fortune to arrive at Honolulu on the day preceding the transit of 

 Mercury (9th of May), and male successful observations of the 

 phenomenon. The second contact was also observed by Mr. Lyons 

 of the government survey. The two interior contacts were noted 

 by local mean time (Waikiki 8" east of Honolulu) as follows: — 

 H. M. S. H. M. S. 



Mr. Lyons 1 36 83 — — — 



Mr. Preston 36 5:3 6 10 50 



Dr. Marcuse 37 3 11 23 



The station was in latitude 31° 16' 81" north, and in longitude 137° 

 49' 30" west. The mean observed times of contact are in botlv 

 cases about a minute less than the computed ones. 



