SEC. VIII.] OF NATURAL PHILOSOPHY. from A to B ; E, F, G three physical points of the quiescent medium sit uate in the right line AC at equal distances from each other ; Ee, F/, G^, equal spaces of extreme shortness, through which those points go and return with a reciprocal motion in each vi bration ; e, >, y, any intermediate places of the same points ; EF, FG physical lineolae, or linear parts of the medium lying betAveen those points, and successively transferred into the places t0, 0y, and ef, fg. Let there be drawn the right line PS equal to the right line Ee. Bisect the same in O, and from the centre O, with the interval OP, describe the circle SIPi. Let the whole time of one vibration ; with its proportional parts, be expounded by the whole circumlerence of this circle and its parts, in such sort, that, when any time PH or PHS/i is completed, if there be let fall to PS the perpendicular HL or hi, and there be taken E£ equal to PL or PI, the physi cal point E may be found in e. A point, as E, moving acccording to this law with a reciprocal motion, in its going from E through e to e, and returning again through e to E, will perform its several vibrations with the same de grees of acceleration and retardation with those of an oscil lating pendulum. We are now to prove that the several physical points of the medium will be agitated with such a kind of motion. Let us suppose, then, that a medium hath such a motion excited in it from any cause whatsoever, and consider what will follow from thence. In the circumference PHSA let there be taken the equal arcs, HI, IK, or hi, ik, having the same ratio to the whole circumference as the equal right lines EF, FG have to BC, the whole interval of the pulses. Let fall the perpendicu lars IM, KN, or wi, kn ; then because the points E, F, G are successively agitated with like motions, and perform their en tire vibrations composed of their going and return, while the pulse is transferred from B to C ; if PH or PHS/t be the time elapsed since the beginning of the mo tion of the point E, then will PI or PHSi be the time elapsed since the beginning of the motion of the point F, and PK or PHSA; the time elapsed since the beginning of the motion of the point G; and therefore Ee, F0, Gy, will be respectively equal to PL, PM, PN, while the points are going, and to PI, Ptn, Pn, when the points are returning. Therefore ey or EG 4- Gy — Et will, when the points are going, be equal to EG — LN