196 



NATURE 



\_7an. 2, 1879 



And since eacli particle maintains constantly the same average 

 position relatively to the centre of inertia of the whole, it is 

 evident that its alternate opposite displacements and velocities 

 relatively to that centre of inertia must be numerically equal. 

 A certain particle has first a certain velocity in one direction, and 

 immediately afterwards has a numerically equal velocity in the 

 opposite direction. This change cannot take place except by its 

 transferring to the next particle the same numerical amount of 

 momentum of one direction as it receives from that particle of 

 momentum of the opposite direction. In this way constant 

 streams run in the two opposite directions, the momentum 

 flowing along one having the opposite direction to that flowing 

 along the other, and equal numerical amounts of these oppositely 

 directed momenta flowing past any given sections of the two 

 streams per unit of time. 



As a sort of parenthesis let me give the following sym- 

 bolical statement of the foregoing. Let V be the velocity of 

 flow of either of these two opposite streams, and ix. the mass 

 per unit volume of the material, and v the average numerical 

 velocity of the particles. Then since at any given instant half 

 the particles must be supposed to be moving in one direction, 

 and the other half in the opposite direction, the amount of 

 momentum of one direction passing per unit of time through any 

 ■section of unit area of the correspondingly directed stream, is 

 \ V fiv. A numerically equal amount of oppositely directed 

 momentum is flowing per unit of time through the same unit 

 section in the opposite direction. Observe that the material 

 through which these two streams are flowing is in "balance," 

 "in equilibrio." Suppose the one stream to lead out of, and 

 the other to lead into, an unbalanced mass, which mass suppose 

 not to be losing or gaining momentum except by these two 

 streims. By means of the one it loses, say ( + \ Vfi-J) per unit 

 of time. By means of the other it gains { - | Vyi. v) per unit of 

 time. The amount of positive momentum it transfers to the 

 •balanced material for unit of time is, therefore, + V y. v, and this 

 is the rate of transference of momentum from the unbalanced 

 mass to the balanced material, and through this latter. If the 

 ratio of comparison or extension, i.e., the strain of this balanced 

 material, be called c, then wXvzX. we usually call its modulus of 

 elasticity is E given by the equation eE = Vfi v. If we insert 



in this expression the proper value of F = /-^^ the velocity 

 of transmission of longitudinal vibration, we obtain a value of 

 similar to that deduced by De St. Venantforthe 



V IX 



first stage of an impact, during which a single unbalanced wave 

 of momentum is running forward through the body impinged on. 

 But the important point to notice is that the rate of transfer- 

 ence of momentum per unit area is the product of a mass per 

 unit volume ((u) and of two velocities ( V and v). In un- 

 balanced transmission these two may be in the same direction, 

 in which case the mass being accelerated is in compression, 

 or they may be in opposite directions, in which case the acce- 

 lerated mass is in tension ; or they may be at right-angles, in 

 which case the accelerated mass is in shear. In balanced trans- 

 mission if in the one stream the velocity of flow is in the 

 sa-Lie direction as that of the flowing momentum, then also in 

 the opposite balancing stream these two velocities have the same 

 directions and the material is in compi-ession, the strain being 

 double that which would occur if either of these opposing streams 

 existed by itself unbalanced in the material. Similarly for a 

 balanced state of tension and for one of shear. 



Considering these reasonings, does it not seem right to make 

 the direction sign of the force, or rate of transference of mo- 

 mentum, the same as that of the product of these two velocities. 

 The sign of the product of two vectors does not depend on the 

 absolute direction of either, or rather it does not depend on the 

 relation of either to what v.e arbitrarily choose as our standard 

 direction. It depends only on their mutual relation. Thus we 

 get a definite sign for each force not arbitrary, but real. For a 

 compression force the two vectors have different signs, and their 

 product is a multiple of - I. For tension, the two being of 

 the same sign, their product is a multiple of -h i. For shear, 

 the two being perpendicular, their product is a multiple of \' — I 

 or of - \/-i. If the direction of transference be oblique to 

 that of the momentum transferred, their product is the sum of a 

 scalar and of a vector. In this case we have compound stress, 

 that is a shear compounded with either compression or tension ; 

 and, as every one knows, it is usually convenient to consider the 



scalar and the vector parts separately. The question of the 

 mode of transmission of momentum corresponding to these 

 main kinds of stress is one of molecular mechanics, into which 

 there is no need of entering here. 



Robert H. Smith 

 {To be coniimied.') 



Leibnitz's Mathematics 



In perusing some old files of Nature I came upon the fol- 

 lowing sentence in a letter from Prof. Tait (vol. v. p. 81) 

 in reference to the invention of the Differential and Integral 

 Calculus; — "Leibnitz was, I fear, simply a thief as regards 

 mathematics." Prof. Tait has more than once intimated or 

 expressed a similar opinion. 



In reply to this imputation Dr. Ingleby says (Nature, 

 vol. V. p. 122) ; — "I do not object to the Professor calling a 

 spade a spade ; but I assure him that this charge is made just 

 twenty years too late. It is exactly that time since the last 

 vestige of presumption against the fair fame of the great German 

 was obliterated. If Prof. Tait does not understand me, or, 

 understanding me, disputes the unqualified tr^th oi my s,\.2iiQxatTaX, 

 I promise to be more explicit in a future letter. But I incline 

 to think the question is not susceptible of /w<?/" until the Council 

 of the Royal Society, who so grossly disgraced themselves in 

 1 712, shall do the simple act of justice and reparation required 

 of them, viz., publish the letters and papers relating to this con- 

 troversy, which since that date have slumbered in the secret 

 archives." 



Prof. Tait, as far as I know, never responded to the challenge, 

 and I presume there is but one inference to be drawn from his 

 silence. 



In a late reading of an account of this controversy from the 

 German standpoint, my interest in the subject has been re- 

 awakened, and I feel a strong desire to see the whole question 

 thoroughly ventilated. Such a consummation must surely be 

 wished by every fair-minded man, and in the name of justice I 

 would ask Dr. Ingleby to be more explicit and do what lies in 

 his power to remove the imputation which has been attempted 

 to be fastened upon Leibnitz. 



This question will not down at the bidding of any one, and 

 the documentary evidence alluded to by Dr. Ingleby must sooner 

 or later see the light. Let us have the matter at once and for 

 ever definitely and honourably settled. A. B. Nelsox 



Danville, Ky., U.S.A., November 27, 1878 



[It is not to be absolutely presumed that, when a busy scien- 

 tific man lets pass such challenges, he has given up his point. 

 The question has now lost all but a species of antiquarian 

 interest : — still it is worth clearing up. We might begin by 

 asking Mr. Nelson and other defenders of Leibnitz to explain 

 the very singular appropriation which Leibnitz made of 

 "Gregory's Series" after having acknowledged whence he got 

 it.— Ed.] 



Commercial Crises and Sun-Spots 



A suggestion is made by Mr. John Kemp, in Nature, 

 vol. xix. p. 97, to test the relation of sun-spots to the variation 

 in weight of the cereal grains. Probably the difficulties of 

 giving such a test scientific precision are insurmountable. No 

 doubt these grains do vary in weight from year to year. Of 

 some samples of oats, of crop 1877, contributed by me to the 

 South' Kensington Museum, the pound contained 13,642 grains, 

 while the pound of crop 1878 contained 16,870. But there are 

 many varieties of oats, barley, and wheat in general cultivation, 

 each producing grains differing in weight from the others. In 

 an inquii-y which I made regarding the weight of the sterling, 

 average grains of wheat of crop 1876 from the south of England 

 were found, in an air-dry condition, to weigh as follows : 

 Talavera, I'oi gi". troy; Chidham white, '76; Sherrift's bearded, 

 •86 ; Kessingland red, "92 ; Nursery red, 76 ; Trump white, 

 •81 ; Red rivet, i"00; Lammas red, "89; Hunter's white, '75. 

 And different ears of a given variety of wheat have grains of 

 different weight. If six or eight culms come up on one stool, 

 the largest ears have the heaviest grains. In general, the larger 

 flower-cups in an ear, contain the heavier grains, Then, there 

 is scarcely such a thing to be found as a crop of one pure 

 variety. Any variety rapidly gets mixed with others. And, 

 ^ supposing that a plot were set aside for a pure variety, year after 



