** 



BOSCOVICH'S THEORY. 



757 



Boscovicl.' 

 Theory of 

 Natural 

 Pbiloxi- 

 phy. 



Applica- 

 tion of the 

 theory to 

 mechanics. 



Arches. 



Plate 

 LXV. 

 Fig. 1. 



Areas. 



Plate 

 J. XV. 



rig. i. 



1 but a thing differing from both ; from the one by its 

 power of cogit?tion, as from the other by its inertia 

 and impenetrability. 



Application of the Theory to Mechanics. 



The second department of our subject is, the ap- 

 plication which may be made of this theory to the 

 explanation of the principal laws of equilibrium, and 

 other parts of elementary mechanics. But, in the 

 first place, a few observations are to be premised re- 

 specting the curve of forces, upon which all the phe- 

 nomena depend. These observations relate to the 

 arches of the curve, to the areas intercepted between 

 it and the axis, and to the points in which the curve 

 cuts the axis. 



The arches are either repulsive or attractive, ac- 

 cording as they lie on the side of the asymptotic 

 arc EG, or on the opposite side. The arches may 

 touch the axis, or they may bend from it with a con- 

 trary flexure, as P eJ~gK, Fig. 1. 



The area corresponding to any small portion of 

 the axis may be ever so great, and that which cor- 

 responds to a great segment may be ever so small, 

 according as the curve recedes very far from the 

 axis, or approaches very near to it. It were easy to 

 demonstrate this, but we shall not occupy the reader's 

 time with it. The area included between an asymp- 

 tote and any ordinate, may be either finite or infinite. 

 The former, when the ordinate increases in a less 

 ratio than the reciprocal simple ratio of the abscisses; 

 the latter, when it increases in that or in a greater 

 ratio, as may be thus proved. Take x the ordinate 



as A a, Fig. 1, and y the absciss ag, and let y= 



or y=x ; then the fluxion of the area yx will be = 



_ n "~~ m . _' 



x n x and its fluent x -4- A, or since x 



n m ' 



Are scales 

 of velocity 



Limits. 



= y> 



: have 



x y -(- A; A being a constant 



quantity. Since the area begins in A, the beginning 

 of the abscisses, if n in be a positive number, and 

 therefore w^m, the area will be finite, and A:r : 

 But the area will be to the rectangle Aag as n to 

 n ni; which rectangle, since ag may be great or 

 small without limit, is also without limit. Its value 

 is infinite, if mz=n, for then the divisor =0; much 

 more then if m-^*n, that is, when the ordinate in- 

 creases in a greater than the reciprocal simple ratio 

 of the abscisses. This observation wasnecessary, that 

 we might have some scale of velocities in the access 

 or recess of one point from another. For, as already 

 observed, when the spaces are expressed by the ab- 

 scisses, and the forces by the ordinates, the area de- 

 scribed by the ordinate expresses the increment or 

 decrement of the square of the velocity. 



With respect to the points in which the curve 

 meets the axis, they are either points of section, as 

 E, G, I, or of contact. In the former, there is a 

 transition from attraction to repulsion, or the con- 

 trary, and these by our author are called limits. 

 These limits are of two kinds ; first, where the transi- 

 tion, by an increase of distance, is from repulsion to at- 



traction, as is the case at E, I, N, R, which are called 

 limits of cohesion, for in such a situation the points 

 resist all change of position, viz. separation by means 

 of the attractive force which immediately begins to 

 operate, and mutual approach in like manner by the 

 incipient repulsion. But in the limits of the second 

 kind, as G, L, P where the transition, by an increase 

 of distance, is from attraction to repulsion, although 

 the points in such situations do not exert any force on 

 each other, yet the smallest change of distance pro- 

 duces a very important alteration : for if they be in 

 the least separated, the repulsive fore? then acting 

 will remove them still farther asunder ; and, on the 

 other hand, if their distance be diminished in the least, 

 they will tend together more and more. Such limits, 

 therefore, are by Boscovich called limits of non-cohe- 

 sion. 



The limits of cohesion may be powerful or weak, 

 according to the angle at which the curve intersects 

 the axis, or the distance to which it removes from it. 

 t N y exhibits a limit of the former ; c N x of the latter 

 kind. The most powerful kind of limit at first, at 

 least, is, where the curve at cutting the axis has the 

 ordinate for its tangent, as X, Fig. 6. ; and, in like 

 manner, the weakest is, when the axis is the tan- 

 gent, as Y Fig. 6., both being points of contrary 

 flexure. 



This being premised, we now proceed to the con- 

 sideration of some of the combinations of the points 

 of matter, and of their mutual actions on each other. 



If two points be placed at such a distance from 

 each other, as is equal to that of some limit from the 

 beginning of the line of abscisses, as AG, AE, &c. 

 and without any kind of motion, they must evidently 

 remain there at rest, since they have no mutual ac- 

 tion. But if the points be placed out of limits of 

 that kind, they will immediately begin to approach 

 or recede by equal intervals. The force continuing 

 in one direction, will carry them to the distance of 

 the nearest limit, which will, of course, be a limit of 

 cohesion. They will arrive at that with an accele- 

 rated motion, and the squares of their velocities will 

 be proportional to the area described by the accom- 

 panying ordinate. But they will not stop at this li. 

 mit. Having arrived there with a motion continually 

 accelerated, they will go on beyond it, and, of course, 

 they will be immediately acted on by a force dircctly 

 opposite. Their motion will therefore be retarded 

 until the velocity be totally extinguished, by the area 

 under this second branch of the curve becoming 

 equ?l to that intercepted between the ordinate at the 

 original place of the point and the limit aforesaid. 

 Should the area of this second segment be too small, 

 the original motion of the points will go on; they 

 will pass the second limit of non-cohesion, if they ar- 

 rive at it with the smallest velocity. Beyond that the 

 original motion will be again accelerated by an action 

 of the same kind as at first, and the. points will pass 

 another limit of cohesion. A second retarding force 

 will now act, and may at length be equal to the ex- 

 tinction of the velocity. If that does not take place 

 exactly at a limit of non cohesion, which is scarcely 

 possible, the bodies will be returned again with a se- 

 ries of motions just the contrary of the former, and 

 they will arrive at the same position from which they 



Boscovich'a 

 Theory of 

 Natural . 

 Philoso- 

 phy. 



s> 



Limits of 

 cohesion. 



Limits of 

 non- cohe- 

 sion. 



Plats 

 LXV. 

 Fig. 6. 



Of the sys. 

 tern of two 

 points. 



Plate 

 LXV. 

 Fig. I. 





