Against mutual attractive and repulsive forces, it has been usual to object, that they are no better than the occult qualities of the Peripatetics, and that they induce action at a distance. The same objection has been made to the Newtonian theory of gravity ; but the answer is easy, we observe the effects, which are sufficiently manifest. We must admit for them an adequate cause. Whether that be the immediate act of the Creator, or some mediate instrument which he employs, we are unable to determine. With respect to action at a distance, there is, at least, nothing more • occult in•that, than in the production of motion by immediate impulse. Newton has given a satisfacto ry explanation of the phenomena of light ; and has reduced mechanical astronomy to rigid calculation, without employing impulse ; and it is highly proba ble, that we maybe equally successful in other de partments of nature.
It has been objected, that the theory itself admits of a hreach of continuity, in passing suddenly from repulsion to attraction ; but this we have already shown to take place, by passing through all the in termediate degrees, in the same manner as the change is made from positive to negative quantities, by a con tinual subtraction.
It may be objected, that the complication of the curve, made up of many arches, repulsive and attrac tive, is no better than the old doctrine of the arbitra ry qualities and substantial forms. Boscovich an swers, that repulsion is but a negative attraction, as may be illustrated by algebraic equations, and geo : and again, that, supposing us entirely ignorant of the law of mutual forces, it is at first much more likely, that the curve, which expresses it, is of a high than of a low order ; that is, it is much more likely that it frequently intersects the axis, or has frequent flexures, than otherwise; seeing that the higher orders of lines are so much more numerous than the lower: But, independent of this conjecture, ' the form of the curve has been derived by positive argument from the phenomena ; and it is well known, that there are many curves which, from their nature, must form frequent flexures and intersections with the axis. To our minds, the mutual congruity of straight lines, upon which, by the way, the whole of our geometry depends, makes them appear the sim plest of any, and others to be the more complicated, only as they remove the more from the right line. But all continued lines of uniform nature are equally simple • and a mind may be conceived, to which the parabola, for instance, might appear as essentially simple, as to us appears the straight line. But besides
this general reply to the objection before stated, Bos covich has chewed, in his Dissertation de Lege Viri um, that this curve is uniform and regular, and may be expressed by one general algebraic equation.
For this purpose, six conditions are proposed, as requisite to the complete expression of the law of forces. -1st, The curve must be regular and simple, and not composed of an aggregate of different curves. 2d, It must cut 'the axis CAC' only in certain given points, at equal 'distances on each side, as AE' AE, AG' AG, AI' AI, and so forth. 3d, To every ab sciss there must be a corresponding ordinate. 4th, To equal abscisses on either side, equal ordinates must correspond. 5th, The straight line AB must be an asymptote to the curve on either side, and the asymptotic area BALD must be infinite. Gth, The arch;intercepted between any two intersections, may be varied at pleasure, may recede to any dis tance from the axis, and may approach at pleasure to an arch of any other curve, cutting, touching, or osculating it, in any place, or in any way that may be proposed.
I. That these conditions may be fulfilled, he finds an algebraic formula which contains his law, calling the ordinate as usual y, and the abscissa =x, he takes xx=z. Let all the values of AE, AG, AI, &c. be taken with the negative sign, and let the sum of the squares of these values be called a ; of the products of every two squares be called b ; of every three c, and so on ; and let the product of all of them be called f, and let the number of values be called m. Now put 9 &c. +f= P.
If we suppose P=---0, it is clear that all the roots of the equation will be real and positive, namely the squares of the quantities AE, AG, AI, which are the values of z, and since it is plain that the values of x are as well AE, AG, AI, positive, as AE', AG', Al' negative.
II. Next, let any given quantity be taken for z, only that it may not have a common divisor with P ; and z vanishing, it will also vanish ; and x being made an infinitesm of the first order, it will also become an infinitesm of the same, or a lower order, as any formu la &c. b, which being put —0, may have any number of imaginary, and any num ber of, and whatever real roots ; (but none of them —AE, AG, &c. either positive or negative ;) if then the 'whole be multiplied by z, let that be called Q.