Plate 2.

Fig. 3.

a body would fall from A to X, is to the time in which it would move over AX with its last acquired velocity, as half the circumference of a circle is to its diameter.

DEF. VI. If a circle, as FCH, be rolled along the line AB, till it has turned once round; the point C in its circumference, which at first touched the line at A, will describe the curve line ACXB, which curve is called a Cycloid. The right line AB is its base: the middle point X is its vertex : a perpendicular, as XD, let fall from thence to the base, is its axis: and the circle ICH, or any other as XGD, equal thereto, is called the generating circle.

LEM III. If on XD, the axis of the cycloid, as a diameter, the generating circle XGD be described; and if from and if from a point in the cycloid, as C, the line CIK be drawn parallel to the base, the portion of it CG will be equal to the arc GX.

Because the generating circles FCH, DGX, are equal (the diameter HF being drawn) KG is equal to CI; whence, adding GI to both, KI will be equal to CG: and K1, by the construction, is equal to DF; therefore CG is equal to FD By the description of the cycloid, the arc CF is equal to the line AF: and by the construction the arc CF is equal to DG: therefore AF is equal to DG: but, by the description of the cycloid, AFD is equal to DGX, consequently, FD is equal to GX: and CG was proved to be equal to FD: therefore CG is equal to GX.

LEM. IV. A tangent to the cycloid at the point C is parallel to GX a chord of the circle DGX.

Draw ck, parallel to the base and indefinitely near to CK meeting the cycloid in c, the axis in k, and the circle in g. Let Cu and Gn, parallel to the axis, meet ck in u and n, and from T, the centre of the circle XGDM, draw the radius TG. Since cg Since cg is equal (Lem. III.) to gX, gk being added to both, ck will be equal to Xg+gk: therefore cu the excess of ck above CK is equal to Gg+gn, the excess of Xg+gk above XG +GK. And, if we suppose ck to approach towards CK, as Gg and gn vanish, the triangles Ggn and GTK become similar; for the angle gGn is then equal to the angle гGK, since both have the same angle nGT, or its alternate GTK, as their complement. Whence Gg is to gn as TG to TK, and (El V. 18.) Gg+gn to gn, as TG+ IK or DK to TK; but Gn is to gn as GK to TK; therefore Gg+gn is to Gn as DK is to GK, that is, (El. VI. 8.) as

GK

GK to XK. And consequently cu (shown to be equal to Gg + gn) is to Gn, or Cu, as GK to XK and if the chord Cc be drawn, the triangles Cuc, XKG, will be similar so that the chord Cc (as the points C and c coincide) becomes parallel to XG; therefore the tangent of the cycloid at C is parallel to XG.

Fig. 3.

LEM. V. If from a a point of the cycloid, as L, the line LMK be Plate 2. drawn parallel to the base AB, the arc XL of the cycloid, will be double of XM the chord of the circle corresponding thereto.

Draw the line Sk parallel and indefinitely near to LK crossing the circle in R, and the chord XM produced, in P: join the points X and R; on MP let fall the perpendicular RO; and draw MN, XN, tangents to the circle at M and X. Then will the lines XN and kS, being each perpendicular to the diameter DX, be parallel and the triangles MNX, MPR, having their angles at M vertical, and at P and X alternate, will be similar. But the tangents NX and NM are equal; (El. III. 36.) whence the lines PR and RM are also equal: the triangle RMP is therefore isosceles; and RO being perpendic ular to its base MP, MO (El I. 26.) is equal to OP; whence MP is equal to twice MO. The indefinitely small arc LS of the cycloid will not assignably differ from a portion of a tangent drawn through the point L. LS may therefore (Lem. IV.) be said to be parallel to MP, and consequently (from the parallelism of ML and PS) equal to it: it is therefore equal also to twice MO. But LS is the difference between the cycloidal arcs XL and XS; and MO is the difference between the chords XM and XR for since XO and XR are indefinitely near to each other, RO which is perpendicular to one of them, may be considered as perpendicular to both the indefinitely small difference therefore between any two arcs of the cycloid is twice that which is between the two corresponding chords of the circle; and the same is true when the magnitude of the difference is assignable, because such difference is compounded of indefinitely small parts. Now, any arc whatsoever may be considered as a difference between two arcs, and consequently any arc, as XL, is double of the corresponding chord XM.

:

COR. Since when the arc XL becomes XB, the corresponding chord XM becomes XD the diameter of the circle DMX; it is obvious, that the semicycloid BX, or AX, is equal to twice DX the diameter of the generating circle DMX.

LEM. VI. If a body descends in a cycloid, the force of gravity, so far as it acts upon the body in causing it to descend along the cycloid, will be proportional to the distance of the body from the lowest point of the cycloid.

Let the cycloid be AXB, whose base is AB, and its axis DX; on which last, as a Plate 2. diameter, describe the generating circle DOX: draw the chords OX and QX; through Fig. 4

Plate 2.
Fig. 4.

the points O and Q, and parallel to the base AB, draw the lines LS and MR; draw also
the tangents LV and MY. Then because (by Lem. IV.) the tangent LV is parallel to
OX, and the tangent MY parallel to QX, it is obvious that gravity exerts the same
power upon a body descending in the cycloid at L (because it then dsscends in the tangent
LV) as it would do upon the same body descending along the chord OX: and, for the like
reason, it exerts the same force upon it when it comes to M, that it would do if it were
descending along QX: but (from Prop. XXXV) the power or force of gravity upon
bodies descending along the chords OX and QX, are as the lengths of those chords; that is,
by Lem. V. (halves being proportional to their wholes) as the length of the cycloidal arcs
LX and MX. The force therefore of gravity upon a body descending in the cycloidal at
the point L is to its force upon the same when at M (as may be said of
any other cor-
responding points) as the space or distance it has to move over in the former case, before it
reaches the lowest point X, to that which it has to pass over in the latter, before it arrives
at the same point.

PROP. XLV. If a pendulum be made to vibrate in a cycloid, all its vibrations, however unequal in length, will be performed in equal times.

The force of gravity, (by Lem VI.) so far as it causes a body to descend in a cycloid, is proportional to the distance of that body from the lowest point: imagine then that body to be a pendulum vibrating in the cycloid, and from whatever point it sets out, it will (by Lem. II.) come to the lowest point in the same time and consequently, since the same may be easily inferred in its ascending from that point, all its vibrations, be they large or small, will be performed in the same time.

:

SCHOL. This proposition is demonstrated only on the supposition that the whole mass of the pendulum is concentrated in a point, for it cannot otherwise take place, because as the string varies in its length, the centre of oscillation of a body will vary On this account, therefore, pendulums vibrating in circular arcs are now always used, for the same arcs will be always described in the same time.

PROP. XLVI. To make a pendulum vibrate in a given cycloid.

Let AXB be the given cycloid; its base AB, its axis DX, and its generating circle. DQX, as before: produce XD to C, till DC is equal to DX: through C draw the line EF parallel to AB, and take CE and CF, each equal to AD or DB; and on the line CE as a base, and with the generating circle AGE equal to DQX, describe the semicycloid CIA, whose vertex will therefore touch the base of the given cycloid in A. And on the line CF also as a base, descr be an equal semicycloid CB. Let the semicycloids CA, CB, represent thin plates of metal bent to their figure, and on the point C, hang the pendulum

СІР

CIP by a flexible line equal in length to the line CX. The upper part of its string (as
Cr, in its present situation in the figure) as it vibrates, will then apply itself to the
cycloidal cheeks CA and CB, and a ball at P will oscillate in the given cycloid AXB.
Draw TG and PH each parallel to the base AB, and draw AG and DH. Then (L.em. V.
Cor) AC is equal to twice AE; and by construction, twice DC, that is, twice AE, is
equal to CX; therefore AC is equal to CX. Also, by construction, CTP is equal to CX,
that is, to AFC: whence, taking away CT, AT is equal to TP. By Lem. IV. GA IS
parallel to TP; and, by construction, AK is parallel to GI; therefore GA is equal to
TK, and GT to AK; but (Lem. V.) GA is half TA; therefore TK is equal to half
TA: since therefore it has been proved that TA is equal to TP, TK is equal to half
TP, that is, to KP. Hence it is manifest, that the parallel lines GT, PH are equally
distant from AD, the arc GA equal to the arc DH, the chords GB and DH parallel, and
GE equal to HX And, because GA has been shewn to be parallel to TK, and also to
DH, KP and DH are parallel; whence KD is equal to PH. But (Lem. III.) GT,
that is, AK is equal to the arc AG and by the description of the semicycloid CTA,
AKD is equal to AGE; therefore KD is equal to EG, that is PH is equal to HX.
And (by Lem. III) if PH be equal to HX, P is a point in the cycloid AXB. The ball
of the pendulum therefore being at that point, is in the given cycloid.

SCHOL. I. It is easy to conceive, that in a pendulum there must be some one point, on each side of which the monenta of the several parts of the pendulum will be equal, or in which the whole gravity of the pendulum might be collected without altering the time of its vibrations. This point, which is called the centre of oscillation, is different from the centre of gravity for if a plane, perpendicular to the string of the pendulum AB, be con- Plate 1. ceived to pass through the centre of the ball B, bisecting it; the velocity of the lower half, Fig. 18. and consequently its momentum, will, in vibration, be greater than that of the upper half : consequently the centre of oscillation must be farther from A than the centre of gravity is; and a plane passing through the centre of oscillation will divide the ball into two unequal parts, so that the greater quantity of matter above it, shall compensate for the greater velocity below it, and the momenta on each side be equal. If the pendulum be an inflexible rod, every where of equal size, it is found, that the distance of the centre of oscillation from the point of suspension is two-thirds of the length of the rod.

If, whilst a pendulum is in motion, it meets with an obstacle at its centre of oscillation sufficient to stop it, the whole motion of the pendulum will cease at once, without any jarring for the obstacle resists equal momenta above and below this point; which is therefore also called the centre of percussion.

SCHOL. 2. The vibrations of pendulums are subject to many irregularities, for which no effectual remedy has yet been devised. These are owing partly to the variable density and temperature of the air, partly to the rigidity and friction of the rod by which they are suspended, and principally to the dilatation and contraction of the materials, of which they

are

are formed. The metalline rods of pendulums are expanded by heat, and contracted by cold; therefore clocks will go slower in summer, and faster in winter. The common remedy for this inconvenience is the raising or lowering the bob of the pendulum (by means of a screw) as the occasion may require. By the last scholium it appears, that a pendulum consisting of a tube of glass or metal, every where uniform, filled with quicksilver, and 58 8 inches long, will vibrate seconds; for of 58 8 is equal to 39 2. Such a pendulum will be expanded and contracted at the same time; for when the tube is extended by heat, the mercury will also be expanded, and by rising in the tube, will raise the centre of oscillation, so that its distance from the point of suspension will be diminished, and the vibrations of the pendulum, which would have been rendered slower by the expansion of the tube, will become quicker by the expansion of the mercury and, by adjusting the tube and mercury in such a manner, that these contrary effects may be the same, a clock with such a pendulum would admit of little or no variation for a long time. Phil. Trans. No. 392. p. 40.

Plate 2.
Fig. 5.

Fig. 6.

SECT. IV.

Of the Centre of Gravity.

PROP. XLVII. In every body there is a centre of gravity, or a point about which all its parts balance each other.

Let AB be an inflexible rod, throughout uniform and of the same density: let it be supported at the point C, equally distant from its extreme points A and B, by the prop C. Let A and B be indefinitely small and equal portions of the rod AB. These portions, A and B, tend towards the centre of the earth with equal forces of gravitation. They would likewise, without obstruction, move with equal velocities for if the rod AB be moved on its prop till it come into the position DE, the velocities of the parts A, B, or F, will be as the spaces over which they pass in the same time; that is, as the arcs AD, EB or FG; which arcs are as their respective circumferences, or as their diameters or radii : whence the velocity of the part B is to the velocity of the part A, or F, as BC is to AC, or FC. And the quantities of matter in A and B are by supposition equal. Therefore, if the parts A and B were in motion, they would have equal momenta; that is, the efforts which A and B make to descend towards the earth, are equal. But these efforts counteract each other: for, whilst the portion A endeavours with a certain force to draw down one arm of the rod, the other portion B endeavours with the same force to draw down the other arm, that is, since the rod is inflexible, to raise the portion A. Therefore the portion A is acted upon by two equal forces in contrary directions, and consequently must be at rest. For the same reason, the portion B will be at rest. And the same may be shewn concerning any other equal portions, at equal distances from C, in the rod AB. Therefore the rod