my own invention? You have made nearly as much progress in the Principia, as in Euclid's Elements. A. Pray then proceed; or rather expect me as a willing hearer to-morrow. DIALOGUE VI. B. I THINK you will perceive, that if a body move with uniform velocity in the circumference of a circle, all the attractive force must be directed to the centre of that circle. A. I understand, from the uniform velocity of the motion, that equal parts of the circumference must always be passed over in equal times; and I know that equal triangular areas are also described in the same times. But equal triangles, which have equal bases, must also have equal altitudes; and in the present case these triangles of equal altitude all have their vertices in one point; and as their bases form the circumference of a circle, that point is its centre. XXIII. B. You have proved it sufficiently. Now sup-Prop. pose two circles to be drawn, one within the other, having the same centre; and two bodies to revolve in their circumferences with different, but separately uniform velocities; the degrees or quantities of centripetal force, by which these bodies are retained in their orbits, will be proportional to the squares of any two arcs described by them in the same time, divided by the radii of the respective circles. A. I do not quite understand this. Prop. XXIII. A. B. It will become plainer as we go on. Suppose the body C to describe the arc CD in the same time that the body A takes to describe the arc AB; what I affirm is, that the centripetal force operating upon C will be to the centripetal force operating upon A, as the square of CD, divided by SC, to the square of AB, divided by SA. A. But what is meant by the squares of CD and AB, which are not straight lines? B. You may consider them as "stretched into "longitude:" but indeed the whole expression is arithmetical. May not curves, as well as straight lines, be conceived to be divided into small parts, of which so many shall be contained in the curve, and so many in the straight line? A. I will allow that to be conceivable. B. And then you know the square of the curve (of a circular arc for instance, as here) is only the number of those parts which it contains, multiplied by itself; and the product of this multiplication may again be divided by the number which the radius contains of the same parts. A. I think I see now what is to be proved. B. Here then is the demonstration, which I will divide into three propositions. First, the comparative intensities of any two continually acting forces are proportional to the lines through which two bodies, severally subjected to the operation of those forces only, would describe in one and the same indefinitely small portion of time. A. I do not understand how any thing can be done in an indefinitely small portion of time. Will not these lines be themselves indefinitely small? and then how can one be greater than the other? B. It is certainly true that they will be indefinitely small but I can easily show you that of two such quantities one may be greater, in any ratio, than the other. Thus, let the line AB be divided into any number of equal parts, as AC, CD, DB: from its extremity B, let BE be drawn at random, and let CF and DG be drawn parallel to it, and also a straight line joining the points A and E. Now, by Prop. XIV., DG is double, and BE triple of CF; and if the line AE, by which those lines are bounded, be made to turn upon the point A, even till it coincides in position with AB, the line BE, though indefinitely diminished, will still be the triple, and DG the double, of CF; and in that same ratio all the three lines will ultimately vanish together. Does it not then appear that the ratios of lines to each other may exist and be estimated, independently of the lines themselves having any actual definite magnitude? A. It seems so certainly. But why is it necessary to make the time, and the lines, indefinitely small? Might not the ratio of the forces to each other be represented by that of the lines which they caused the bodies to describe in any one definite portion of time, as a minute, or a second? B. It is true that they might. But I think the Prop. truth of the proposition is more obvious when the time is considered as indefinitely small, because then the continual force may be regarded as a single impulse. However, it is necessary at all events that the time should be taken indefinitely small; first, because, as the body approaches the point towards which it is attracted, the force may be supposed to vary in intensity; but by making the time, and therefore the alteration of the body's distance from that point, indefinitely small, we exclude all errors that might otherwise arise from any such variation of the force. And secondly, the same thing is necessary in order to the application of the proposition, as you will soon see. My second proposition is, that when a body moves in the circumference of a circle, if at any instant the centripetal force were to cease, the straight line in which the body would thenceforward move is at right angles to the radius, or line drawn from the centre to the place of the body at the instant of the change. Which I prove thus: if this line made any other angle with the radius than a right angle, a perpendicular might be drawn to it from the centre, not coinciding with the radius; and this perpendicular, together with the radius, and the part of the first mentioned straight line intercepted between them, would form a right-angled triangle, in which the radius would be the hypothenuse, and therefore the greatest side; consequently a definite part of |