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PROP. II. THEOR.
Every body, which is moved in any curve line described in a plaiu,
and by a radius drawn to an immoveable point, describes areas about that point, proportional to the times, is urged by a centripetal force tending to the same point.
For every body, which is moved in a curve line, is turned from is rectilineal course, by some force acting on it (by law 1), and that force, by which a body is turned from a rectilineal course, and is made to describe the equal least possible triangles SAB, SBC, SCD, &c. see fig. to prec prop., about an immoveable point S, in equal times, acts, in the place B, according to a line parallel to co (40. 1 Eu. and Law 2), or, according to the line BS; and, in the place C, according to a line parallel to dD, or, according to the line CS, &c. Therefore it always acts according to lines tending to that immoveable point S.
Cor. 1. In non-resisting spaces or mediums, if the areas be not proportional to the times, the forces do not tend to the concourse of the radius, but deviate therefrom, in consequentia, or towards the part to which the motion is directed, if the description of the areas be accelerated; but in antecedentia, if retarded.
Cor. 2. Even in resisting mediums, if the description of areas be accelerated, the directions of the forces deviate from the concourse of the radiuses, towards the part, to which the motion is made.
Scholium.—A body may be urged by a centripetal force compounded of several forces. In this case, the sense of the proposition is, that the force, which is compounded of all, tends to the point S. Moreover, if any force act according to a line perpendicular to the described surface, this will cause, that the body deviate from the plain of its motion, but will neither increase nor diminish the quantity of the described surface, and is therefore to be neglected in the composition of forces.
And since the equable description of areas is an index of the centre, which that force respects, by which a body is most affected, and by which it is drawn from a rectilineal motion, a retained in its orbit; the equable description of areas, is used in this tract, as the index of the centre, about which, all curvilineal inotion is performed in free spaces.
PROP. III. THEOR.-See Note."
The centripetal forces, of bodies, which describe different circles
with an equable motion, tend to the centres of the circles, and are to each other, as the squares of arches described together, applied to the radiuses of the circles.
These forces tend to the centres of the circles, by prop. 2 and cor. 2 prop. 1 Nat. Ph.; and are to each other, as the versed sines of the least possible arches, described in equal times (Cor. 4. 1 Nat. Ph.), or, which is equal (Lem. 7 Nat. Ph. 31. 3, Cor. 1 to 8. 6 & 17. 6 Eu.), as the squares of the same arches applied to the diameters of the circles; and therefore, since these arches, are ås arches described in any equal times, and the diameters of circles, are as their radiuses, these forces are to each other, as the squares of any arches described together, applied to the radiuses of the circles.
Cor. 1. Therefore, since these arches, are as the velocities of the bodies, the centripetal forces are as the squares of the velocities, applied to the radiuses of the circles; or, in the language of geometers, in a ratio compounded of the duplicate ratio of the velocities and the inverse simple ratio of the radiuses.
Cor. 2. And, since the periodick times, are in a ratio compounded of the direct ratio of the radiuses and the inverse one of the velocities; the centripetal forces are inversely as the squares of the periodick times applied to the radiuses of the circles; that is, in a ratio, compounded of the direct ratio of the radiuses and the inverse duplicate one of the periodick times.
Cor. 3. Whence, if the periodick times be equal, and therefore the velocities be as the radiuses; the centripetal forces are as the radiuses : and the contrary.
Cor. 4. If the periodick times, and therefore the velocities, be in a subduplicate ratio of the radiuses; the centripetal forces are equal : and the contrary.
Cor. 5. If the periodick times be as the radiuses, and therefore the velocities equal; the centripetal forces are inversely as the radiuses : and the contrary.
Cor. 6. If the periodick times be in a sesquiplicate ratio of the radiuses, and therefore the velocities in an inverse subduplicate ratio of the radiuses; the centripetal forces are inversely as the
squares of the radiuses : and the contrary. Cor. 7. And universally, if the periodick time be as any power R" of the radius R, and therefore the velocity inversely as R*1; the centripetal force is inversely as R*-: and the contrary.
Cor. 8. All the same things, concerning the times, velocities, and forces, with which bodies describe similar parts of any similar figures, having their centres similarly posited in those figures, follow from the demonstration of this proposition and its corollaries, applied to these cases. And it is applied, by substituting the equable description of areas, for equable motion, and the distances of the bodies from the centres, for the radiuses.
Cor. 9. From the same demonstration, it follows also ; that the arch, which a body, by revolving uniformly in a circle with a given centripetal force, describes in any time, is a mean proportional between the diameter of the circle, and the descent of the body performed in the same time by falling with the same given force.
Scholium. The case of the sixth corollary of this proposition, namely, that of the periodick times being in a sesquiplicate ratio of the distances, or, which is the same, of the squares of the periodick times being as the cubes of the distances, obtains in the planetary bodics, as has been observed by Kepler, see the third law discovered by him, mentioned in these elements of Natural Philosophy, in the preparatory observations; and therefore those things, which relate to a centripetal force, decreasing in a duplicate ratio of the distances from the centres, are more particularly explained in these elements.
PROP. IV. THEOR.
If a body in a non-resisting space, be revolved in any orbit, about
an immoveable centre, and describe any arch just nascent in the least possible time, and the sagitta be understood to be drawn, which may bisect the chord, and, being produced, may pass through the centre of force; the centripetal force in the middle of the arch, is as the sagitta directly, and the square of the time inversely.
For the sagitta in a given time is as the force (Cor. 4. 1 Nat. Ph.), and by increasing the time in any ratio, because the arch is increased in the same ratio, the sagitta is increased in a ratio which is duplicate of that ratio (Cor. 2 and 3 Lem. 11 Nat. Ph.), and therefore is as the force and square of the time jointly, Taking from each the duplicate ratio of the time, the force is, as the sagitta directly, and the square of the time inversely.
The same may also be demonstrated from Lem. 10. Nat. Ph. thus :
The spaces, which a body describes, by any finite force urging it, whether that force be immutable or continually increased or diminished, are, in the very beginning of the motion, in a duplicate ratio of the times (Lem. 10. Nat. Ph.), and therefore, the forces being varied, as the forces and squares of the times jointly. Taking from each the duplicate ratio of the times, the forces are, as the spaces described directly, and the squares of the times inversely, and these spaces are as the sagittas mentioned in this proposition, as is manifest from Cor. 4. 1 Nat. Ph. Cor. 1. If a body P, in re
R volving round a centre S, describe a curve line APQ and a
N right line RPN touch that curve in any point P, and from any other point of the curve Q, a right line QR be drawn parallel to the distance SP, and a perpendicular QT be drawn to that listance SP: the centripetal force
PʻxQT is inversely as the solid
if that quantity of this
QR solid be always taken, which it has ultimately when the points P and Q coincide.
For QR is equal to the sagitta of double the arch QP, in the middle of which is P; and double the triangle SQ , or SP QT, is proportional to the time in which that double arch is described (1 Nat. Ph.), and therefore may be used, as an exponent of the time.
Cor. 2. By a similar reasoning, a perpendicular SY being let fall, from the centre of the force S, on a tangent of the orbit
SYoxQP PR, the centripetal force is inversely as the solid
QR for the rectangles SYxQP and SPxQT are equal, being each equal to double the triangle SQP.
Cor. 3. If the orbit be a circle, or contains the least possible angle of contact with a circle, having the same curvature, and the same railius of curvature at the point of contact P, and if PX be the chord of this circle, drawn from the body through the centre of force; the centripetal force is inversely as the solid
SYRXPX. For QP is equal to the rectangle PXxQR (Schol.
QP Theor. 5. 4 Eu.), and therefore PX is equal to
QR be substituted for it, in expressing the quantity of the solid, mentioned in the preceding corollary.
Cor. 4. The same things being supposed, the centripetal force is, as the square of the velocity directly, and that chord inversely. For the velocity is inversely as the perpendicular SY (Cor. 1. 1 Nat. Ph).
Cor. 5. Hence, if any curvilineal figure APQ be given, and in it a point S be also given, to which the centripetal force is continually directed; the law of the centripetal force may be found, by which, any body P, being continually drawn off from a rectilineal course, will be detained in the perimeter of that figure, and, in revolving, describe it. Namely, either the solid SPʻxQT
or the solid SY®XPX, should be computed, as QR inversely proportional to this force.
Scholium. Although the method of investigating centripetal forces, given in the preceding corollary, being the fifth of this proposition, is general, extending itself to any given curvilineal figure, and any point therein ; yet as the principal object of these elements, is the investigation of those laws, which actually prevail in nature, and as Kepler has, from actual observation, ascertained, that the primary planets in their revolutions about the sun, describe ellipses, the sun being in one of the focuses ; see the second law discovered by him, mentioned in these elements of natural philosophy, in the preparatory observations ; and as the same law has been found, as far as observations have been made, to prevail in the motions of the secondary planets round their primaries. The investigation of the law producing a motion in an ellipse, round a focus, as the centre of force, is given in the next proposition; the like investigation, as respects a motion in a hyperbola or parabola, round a focus, as the centre of force, being given in the two following propositions.