The force with which it has thus to be constantly pulled in towards the centre, or, which is the same thing, the force with which it is tugging at whatever constraint it is that holds it in, is _mv^2/r_; where _m_ is the mass of the particle, _v_ its velocity, and _r_ the radius of its circle of movement. This is the formula first given by Huyghens for centrifugal force. We shall find it convenient to express it in terms of the time of one revolution, say _T_. It is easily done, since plainly T = circumference/speed = _2[pi]r/v_; so the above expression for centrifugal force becomes _4[pi]^2mr/T^2_. As to the fall of the body towards the centre every microscopic unit of time, it is easily reckoned. For by Euclid III. 36, and Fig. 58, _AP.AA' = AO^2_. Take _A_ very near _O_, then _OA = vt_, and _AA' = 2r_; so _AP = v^2t^2/2r = 2[pi]^2r t^2/T^2_; or the fall per second is _2[pi]^2r/T^2_, _r_ being its distance from the centre, and _T_ its time of going once round. In the case of the moon for instance, _r_ is 60 earth radii; more exactly 60·2; and _T_ is a lunar month, or more precisely 27 days, 7 hours, 43 minutes, and 11-1/2 seconds. Hence the moon's deflection from the tangential or rectilinear path every minute comes out as very closely 16 feet (the true size of the earth being used). Returning now to the case of a small body revolving round a big one, and assuming a force directly proportional to the mass of both bodies, and inversely proportional to the square of the distance between them: _i.e._ assuming the known force of gravity, it is _V Mm/r^2_ where _V_ is a constant, called the gravitation constant, to be determined by experiment. If this is the centripetal force pulling a planet or satellite in, it must be equal to the centrifugal force of this latter, viz. (see above). _4[pi]^2mr/T^2 Equate the two together, and at once we get _r^3/T^2 = V/4[pi]^2M;_ or, in words, the cube of the distance divided by the square of the periodic time for every planet or satellite of the system under consideration, will be constant and proportional to the mass of the central body. This is Kepler's third law, with a notable addition. It is stated above for circular motion only, so as to avoid geometrical difficulties, but even so it is very instructive. The reason of the proportion between _r^3_ and _T^2_ is at once manifest; and as soon as the constant _V_ became known, _the mass of the central body_, the sun in the case of a planet, the earth in the case of the moon, Jupiter in the case of his satellites, was at once determined. Newton's reasoning at this time might, however, be better displayed perhaps by altering the order of the steps a little, as thus:--