September 9, 1892.] 



SCIENCE. 



149 



the forest-trees to-day? The records are silent upon these points. 

 A discovery that I made in the vicinity of Tokio last year leads 

 me to believe that possibly the traces of a race of men previous to 

 the Aino occupation have been found." Again I say : "The next 

 question arises as to whether the deposits are Aino or pre-Aino. 

 The race who left these remains were pot-makers par excellence. 

 It is generally admitted by ethnologists that the art of pottery 

 once gained is never lost. It is a fact, however, that neither the 

 Esquimaux, Aleutians, Kamtchadales. nor the Ainos are essentialy 

 earthen pot makers." And, again, having shown incootestible 

 proofs of the evidences of cannibalism in these deposits, I ask, 

 "Were the Ainos cannibals? Repeated inquiries among eminent 

 Japanese scholars and archseologists, like Mr. Kanda, Mr. Nina- 

 gawa, and others, as to this question, are always answered in the 

 same way. Not only were they not cannibals, but they are reported 

 as heing so mild and gentle tliat murder was never known to have 

 occurred. So monstrous a habit would certainly have been known 

 and recorded, particularly in the painstaking annals of early his- 

 torians. " 



In the Proceedings of the American Association for the Advance- 

 ment of Science for 1878 occurs in tire list of papers read by title 

 the following one of mine, entitled '• Evidences of Cannibalism in 

 a Nation before the Ainos in Japan." A foot note states that this 

 paper was published in the Tukio Times 



In the year 1879 the University of Tokio published my memoir 

 on the "Shell Mounds of Omori," illustrating the various forms 

 of pottery, bone implements, etc., by seventeen folded plates. 

 While this memoir is devoted exclusively to a minute description 

 of the Omori deposits as a basis of comparison with material that 

 I had on hand for the description of other shell-heaps, yet I urged 

 the evidence of the deposits not having been made by Ainos, but 

 hy a race anterior to the Ainos, and cited especially the evidences 

 of cannibalism as bearing on this point. 



Twelve years ago I had occasion to criticise and controvert 

 (American Naturalist, September, 1880), in the most emphatic 

 manner Professor Milne's views as published in the Transactions 

 of the Asiatic Society of Japan. At the same time I also showed, 

 as I believed, the fallacy of the views of Henry von Siebold on this 

 question. Thus in various publications in 1877, 1878, 1879, and 

 1880 I have urged the existence of a pre-Aino race in Japan. 



Had Mr. Hitchcock taken the trouble to give proper credit to 

 others who had worked in this field, he would have found addi- 

 tional support to the position he takes ; as it is, his paper is marred 

 by misapprehension and by the injustice of these omissions. 



Edwaed S. Morse. 



Saleui, Mass., Aug. 30. 



On the Fundamental Hypotheses of Abstract Dynamics; 

 From Another Point of Vievy. 



There is at present very little agreement among physicisfe or 

 philosophers as to the nature of the hypotheses or laws upon 

 which dynamics is based. On Aug. 5 Professor MacGregor ex- 

 pounded one view of the matter in these columns; but as I cannot 

 but think his view contains some logical imperfections, I wish 

 to lay before your readers a different view with which to compare 

 it. For this is not a question to be settled by authority; the 

 arguments on either side are after all simple enough, and, having 

 studied them, any man of average attainments is capable of 

 weighing them and forming his own opinion. 



The principles of abstract (subjective) geometry may be deduced 

 from definitions of the terms "Position" and "Direction,"' to- 

 gether with certain axioms asserting the conceivability of geo- 

 metrical figures and constructions. Even without these axioms 

 a symbolic geometry might be deduced, whose conclusions, bow- 

 ever, would be mere truisms, or verbal assertions, till they were 

 given a meaning by the axioms. To proceed to the objective 

 geometry of material space, we require in addition certain in- 

 ductions ; which, however, are so complete that no practical 

 doubt remains as to their validity. 



' See my " Foundations of Geometry," Deighton, Bell, & Co., Cambridge, 

 Eng., 1891. 



In the same way we may treat kinematics from three different 

 points of view. Symbolically, it is sufiiciont to define Time im- 

 plicitly by the assertion, " The positions of points are all con- 

 tinuous single-valued functions of the Time." This definition 

 may be given a subjective meaning by the axiom, " Particles are 

 conceivable in Time," and an objective meaning by an induction 

 proving that "material particles exist only in Time," i.e., their 

 positioxis are continuous single-valued functions of a certain 

 variable, which we may call Time. 



To proceed to kinetics symbolically, wc require definitions of 

 Mass and Force. The only connotation symbolically required for 

 the former term is "Mass is not a function of Space or Time." 

 The latter term may be defined imjilicitly by assertions equivalent 

 to Newton's laws of motion, which may be stated thus : — 



1. The resultant force on any particle in any direction, referred 

 to a given set of axes, is the product of the measures of its mass 

 and its acceleration in that direction. 



2. All forces go in pairs between pairs of particles, equal forces 

 in opposite directions acting on the particles respectively in the 

 line joining them. (Such a pair of forces may be spoken of as a 

 stress.) 



It is evident from 1, since mass is not a function of space or 

 time, that forces, like accelerations, are vectors, and may be com- 

 pounded by the parallelogramic law. Paragraph 1, however, 

 only speaks of resultant forces, and the actual, or acting, forces 

 on any particle would remain entirely arbitrary but for paragraph 

 2, which must be read in conjunction with 1. Professor Mac- 

 Gregor asserts that paragraph 2 is not consistent (i.e., m,ight be 

 inconsistent) with 1. So far from this being the case, I propose 

 to show that it still leaves the term Force to some extent arbi- 

 trary. The stresses between particles are not completely deter- 

 mined, even with reference to a given set of axes; and, moreover, 

 both Force and Stress are relative to the axes chosen . 



In geometry and kinematics both position and dii'ection are 

 relative terms. To determine a position we require to know its 

 distance and direction from a given position. To know its direc- 

 tion we require to know the inclination of that direction to two 

 given (independent) directions, and, in addition, which side it is 

 of the plane determined by them. 



Suppose, then, we have a set of particles numbered from 1 to n. 

 Choose the first particle as origin of a system of rectangular co- 

 ordinates; the direction 12 as that of the axis of x; the direction 

 at right-angles to this in the plane 123, and on that side of the 

 line 12 on which the particle 3 lies, as that of the axis y; and the 

 direction perpendicular to the plane 123, on that side of it on which 

 the particle 4 lies, as that of the axis z. Thus we have deter- 

 mined a set of axes completely, and in doing so we have made 

 the six arbitrary assumptions: — 



Xi — 2/i = Zi = , 



!/„ = z, = , 



23=0. 



Now let F^s be the stress between the particles )• and s, being 

 positive if they attract, negative if thej' repel one another. Then 

 considering forces acting on particle 1 we have the equations — 



F12 -^ 



+ •-^10 



-I- 



m x^, 



and two similar equations with y and z (r, „ being the distance 

 between the particles). Thus in all we have 3 n equations be- 



n -n — 1 



tween — — quantities i^j a, i?"!,, etc. But these equations 



may not all be independent. As, however, they contain (3 ?! — 6) 

 independent variables, .r,, x^, y^, etc. (the other six having been 

 arbitrarily equated to zero), there will in general be (3 n — 6) of 

 them independent. If they only just sufficed to determine the 

 quantities F^^, F^, etc., we should have 



= 3 71 — 6. 



Whence ?i =; 3 or 4. Therefore, if n is gi-eater than 4 (which, of 

 course, it is), the equations must be insufficient to determine the 

 quantities; that is, the stresses lemain to some extent arbitrary; 



