Principia Mathematica (Motte Translation, 1848) — part 13 of 45

pC, that will be to the distance whicli tlie other body P acquires from the line PC as the transverse motion of the body jo to the transverse motion of the other body P. Therefore since kr is equal to the distance which the body P acquires from the line PC, and mr is to kr as the angle YC/? to. the angle YCP, that is, as the transverse motion of the body p to the transverse motion of the body P, it is manifest that the body p, at the ex- piration of that time, will be found in the place m. These things will be so, if the bodies p and P are equally moved in the directions of the lines pC and PC, and are therefore urged with equal forces in those directions. But if we take an angle pCii that is to the angle pCk as the angle YC^ to the angle YCP, and iiC be equal to kC, in that case the body p at the expiration of the time will really be in n ; and is therefore urged with a greater force than the body P, if the angle nCp is greater than the angle kCp, that is, if the orbit vpk^ move either in consequeiitia, or in antece- dentia, with a celerity greater than the double of that with which the line CP moves in consequentia ; and with a less force if the orbit moves slower in antecedentia. And the diiference of the forces will be as the interval mn of the places through which the body would be carried by the action of that difference in that given space of time. About the centre C with the interval Cn or Ck suppose a circle described cutting the lines mr, mn pro- duced in s and t, and the rectangle tmi X int will be equal to the rectan-

mk X ms gle Wjk X ms, and therefore Qnn will be equal to — . But since

the triangles pCk, pCn, in a given time, are of a given magnitude, kr and m.r, and their difference mk, and their sum ms, are reciprocally as the al- titude pC, and therefore the rectangle mk X ms is reciprocally as the square of the altitude pO. But, moreover, int is directly as ^mt, that is, as the altitude pG. These are the first ratios of the nascent lines ; and hence

that is, the nascent lineola mn, and the difference of the forces

mt

proportional thereto, are reciprocally as the cube of the altitude pC

aE.D.

Cor. 1. Hence the difference of the forces in the places P and jo, or K and k, is to the force with which a body may revolve with a circular motion from R to K, in the same time that the body P in an immovable orb de- scribes the arc PK, as the nascent line m.n to the versed sine of the nascent

mk X ms rk^ . , ,,

arc RK, that is, as — • to -^j^, or as Quk X ms to the square of

rk ; that is, if we take given quantities F and G in the same ratio to one another as the angle YCP bears to the angle YC;?, as GG — FF to FF. And, therefore, if from the centre C, with any distance CP or Cp, there be described a circular sector equal to the whole area YPC, which the body

Sec. IX.]

OF NATURAL PHILOSOPHY.

175

revolving in an immovable orbit has by a radius drawn to the centre de- scribed in any certain tiniCj the difference of the forces, with which the body P revolves in an immovable orbit, and the body p in a movable or- bit, will be to the centripetal force, with which another body by a radius drawn to the centre can uniformly describe that sector in the same time as the area YPC is described, as GG — FF to FF. For that sector and the area pCk are to one another as the times in which they are described.

Cor. 2. If the orbit YPK be an ellipsis, having its focus C, and its highest apsis V, and we suppose the the ellipsis upk similar and equal to / it, so that pG may be always equal / to PC, and the angle YC^ be to the j angle YCP in the given ratio of G \ to F ; and for the altitude PC or pG \ we put A; and 2R for the latus rec- /t' tum of the ellipsis, the force with ^" which a body may be made to re- volve in a movable ellipsis will be as

FF , RGG — RFF , .

-r-r + T-^^ ,2^-n.diViceversa.

AA A^ '

Let the force with which a body may

. FF

revolve in an immovable ellipsis be expressed by the quantity -t-t-, and the

force in Y will

FF

CY2-

But the force with which a body may revolve in

a circle at the distance CY, with the same velocity as a body revolving in an ellipsis has in Y, is to the force with which a body revolving in an ellip- sis is acted upon in the apsis Y, as half the latus rectum of the ellipsis to the

KFF

semi-diameter CY of the circle, and therefore is as ^^^^ ; and the force

which is to this, as GG — FF to FF, is as

R-GG

CY^ -RFF

cya

and this force

(by Cor. 1 of this Prop.) is the difference of the forces in Y, with which the body P revolves in the immovable ellipsis YPK, and the body p in the movable ellipsis upk. Therefore since by this Prop, that difference at

any other altitude A is to itself at the altitude CY as -r^ to /;p^, the same

A^ CY^'

A-^ ' 1.'. , ^ ■^^l. ^^^ " ^^F Qinerence m every altitude A will be as -r^ .

Therefore to the

FF

force — , by which the body may revolve in an immovable ellipsis YPKj

AA^

176 THE MATHEMATICAL PRINCIPLES [BoOK I.

RGG— RFF FF

add the excess -.-^ , and the sum will be the whole force -r-r 4-

A^ ' AA

RGG— RFF ,,.,,, . . , . . , .-5 by which a body may revolve m the same time m the mov- able ellipsis ttpk.

Cor. 3. In the same manner it will be foimdj that, if the immovable or- bit yPK be an ellipsis having its centre in the centre of the forces C, and there be supposed a movable ellipsis upk, similar, equal, and concentrical to it ; and 2R be the principal latus rectum of that ellipsis, and 2T the latus transversum, or greater axis; and the angle VC^ be continually to the angle TCP as G to F; the forces with which bodies may revolve in the im-

FFA FFA

movable and moveable ellipsis, in equal times, will be as ■ ^^3 and ^3

RGG— RFF . ,

• -p respectively.

Cor. 4. And universally, if the greatest altitude CY of the body be called

T, and the radius of the curvature which the orbit VPK has in V, that is,

the radius of a circle equally curve, be called R, and the centripetal force

with which a body may revolve in any immovable trajectory VPK at the place

VFF Y be called 7^^-, and in other places P be indefinitely styled X ; and the

altitude CP be called A, and G be taken to F in the given ratio of the angle YGp to the angle YCP ; the centripetal force with which the sam^e body will perform the same motions in the same time, in the same trajectory ■upk revolving with a circular motion, will be as the sum of the forces X + VKGG — YRFF Jl •

Cor. 5. Therefore the motion of a body in an immovable orbit being given, its angular motion round the centre of the forces may be increased or diminished in a given ratio; and thence new immovable orbits may be found in which bodies may revolve with new centripetal forces.

Cor. 6. Therefore if there be erected the line YP of an indeterminate length, perpendicular to the line CY given by po- sition, and CP be drawn, and Cp equal to it, mak- ing the angle YCp having a given ratio to the an- gle YCP, the force with which a body may revolve in the curve line Ypk, which the point p is con- tinually describing, will be reciprocally as the cube of the altitude Cp. For the body P, by its vis in- ertice alone, no other force impelling it, will proceed uniformly in the right line YP. Add, then, a force tending to the centre C reciprocally as the cube of the altitude CP or Cp, and (by what was just demonstrated) the

Sec. IX.] OF natural philosophy. 177

body will deflect from the rectilinear motion into the curve line V^/j. But this curve Ipk is the same with the curve VPQ, found in Cor. 3, Prop. XLI, in which, I said, bodies attracted with such forces would ascend

oblicjuely.

PROPOSITION XLY. PROBLEM XXXL

To find the motio7i of the apsides in orbits approaching very near to

circles. This problem is solved arithmetically by reducing the orbit, which a body revolving in a movable ellipsis (as in Cor. 2 and 3 of the above Prop.) describes in an immovable plane, to the figure of the orbit whose apsides are required ; and then seeking the apsides of the orbit which that body describes in an immovable plane. But orbits acquire the same figure, if the centripetal forces with which they are described, compared between themselves, arc made proportional at equal altitudes. Let the point Y be the highest apsis, and write T for the greatest altitude CY, A for any other altitude CP or Cjo, and X for the difference of the altitudes CY — CP ; and the force with which a body moves in an ellipsis revolving about its

FF RGG RFF

focus C (as in Cor. 2), and which in Cor. 2 was as j-r -\ -r^ ,

, . FFA + RGG — RFF , , . . _

that IS as, p , by substituting T — X for A, will be-

RGG — RFF + TFF — FFX ^ ^.^ come as p . In like manner any other cen- tripetal force is to be reduced to a fraction whose denominator is A^, and the numerators are to be made analogous by collating together the homo- logous terms. This will be made plainer by Examples.

Example L Let us suppose the centripetal force to be uniform,

A^ and therefore as ^ or, writing T — X for A in the numerator, as

T^ _ 3TTX + 3TXX _ X^ ^^

== — =:=r. Then collating together the correspon-

dent terms of the numerators, that is, those that consist of given quantities,, with those of given quantities, and those of quantities not given with those of quantities not given, it will become RGG — RFF -f- TFF to T^ as — FFX to 3TTX + 3TXX — X^ or as — FF to — 3TT -{- 3TX — XX:. Now since the orbit is supposed extremely near to a circle, let it coincide with a circle ; and because in that case R and T become equal, and X is infinitely diminished, the last ratios will be, as RGG to T^, so — FF to — •3TT, or as GG to TT, so FF to 3TT; and again, as GG to FF, so TT to 3TT, that is, as 1 to 3 ; and therefore G is to F, that is, the angle YC^ to the angle YCP, as 1 to v/3. Therefore since the body, in an immovable?

12

178 THE MATHEMATICAL PRINCIPLES [BoOK I.

ellipsis, in descending from the upper to the lower apsis, describes an angle, if I may so speak, of ISO deg., the other body in a movable ellipsis, and there- fore in the immovable orbit we are treating of, will in its descent from

180 the Tipper to the lower apsis, describe an angle YCp of — ^ deg. , And this

comes to pass by reason of the likeness of this orbit which a body acted upon by an uniform centripetal force describes, and of that orbit which a body performing its circuits in a revolving ellipsis will describe in a quies- cent plane. By this collation of the terms, these orbits are made similar ; not universally, indeed, but then only when they approach very near to a circular figure. A body, therefore revolving with an uniform centripetal

ISO force in an orbit nearly circular, will always describe an angle of — ^ deg., or

103 deg., 55 m., 23 sec, at the centre ; moving from the upper apsis to the lower apsis when it has once described that angle, and thence returning to the upper apsis when it has described that angle again ; and so on i/i in- finitum.

Exam. 2. Suppose the centripetal force to be as any power of the alti-

A° tnde A, as, for example. A" — ^, or -r-^ ; where n — 3 and n signify any in-

dices of powers whatever, whether integers or fractions, rational or surd,

affirmative or negative. That numerator A"" or T — X|" being reduced to an indeterminate series by my method of converging series, will become

T" — 77XT"— ^ -f — - — XXT"— 2, &c. And conferring these terms

with the terms of the other numerator RGG — RFF + TFF — FFX, it

becomes as RGG — RFF + TFF to T°, so — FF to — 7zT"— ^ + ^m — n

XT" — 2, &c. And taking the last ratios where the orbits approach to circles, it becomes as RGG to T", so — FF to — uIl"^ — \ or as GG to T'l— 1, so FF to ?iT"— ' ] and again, GG to FF, so T"— ^ to v!V''~-\ that is, as 1 to n ; and therefore G is to F, that is the angle YCp to the angle YCP, as 1 to ./». Therefore since the angle YCP, described in the de- scent of the body from the upper apsis to the lower apsis in an ellipsis, is of ISO deg., the angle YC/?, described in the descent of the body from the upper apsis to the lower apsis in an orbit nearly circular which a body de- scribes with a centripetal force proportional to the power A" — ^, will be equal

ISO to an angle of — ~ deg., and this angle being repeated, the body will re- ■s/ n

turn from the lower to the upper apsis, and so on in infinitum. As if the

centripetal force be as the distance of the body from the centre, that is, as A,

A^ .or -ri, n will be equal to 4, and •s/n equal to 2 : and therefore the angle

Sec. IX.] OF natural philosophy. i7'9

180

between the upper and the lower apsis will be equal to — deg., or 90 deg.

Therefore the body having performed a fourth part of one revolution, v/ill arrive at the lower apsis, and having performed another fourth part, will arrive at the upper apsis, and so on by turns in infinitum. This appears also from Prop. X. For a body acted on by this centripetal force will re- volve in an immovable ellipsis, whose centre is the centre of force. If the

1 A" centripetal force is reciprocally as the distance, that is, directly as -r- or — ^

n wiirbe equal to 2; and therefore the angle between the upper and lower

ISO apsis will be — - deg., or 127 deg., 16 min., 45 sec. ; and therefore a body re-

volving with such a force, will by a perpetual repetition of this angle, move alternately from the upper to the lower and from the lower to the upper apsis for ever. So, also, if the centripetal force be reciprocally as the biquadrate root of the eleventh power of the altitude, that is, reciprocally

as A -T-, and, therefore, directly as 77^, or as ~, n will be equal to , and

4 ISO ■ — deg. will be equal to 360 deg. ; and therefore the body parting from

the upper apsis, and from thence perpetually descending, will arrive at the lower apsis when it has completed one entire revolution ] and thence as- cending perpetually, when it has completed another, entire revolution, it will arrive again at the upper apsis ; and so alternately for ever.

Exam. 3. Taking m and n for any indices of the powers of the alti- tude, and h and c for any given numbers, suppose the centripetal force

6A^ + cA" , . b into T — X> -f- c into T ~ Xl" to be as 77 , that is, as tt, —

or (by the method of converging series above-mentioned) as

6T°^ + cT« — w&XT'^^-i wcXT"— 1 mm—m ___ nu — n

• -^ b XXT'" ~2

2

cXXT"— 2, &c. ,

— T--§ and comparing the terms of the numerators, there will

arise RGG — RFF -f TFF to ^T"^ + cT" as — FF to — m6T"^ — i —

.^cT"-^ + -J^!L_^ ,XT^-.+ -L^ ,XTn-^, &c. Andtak-

ing the last ratios that arise when the orbits come to a circular form, there .will come forth GG to ^T"^ — ' + cT" — ^ as FF to mbT"^ — ^ + ?icT" — ^ ; and again, GG to FF as 6T"^ — ^ + cT" — ^ to mbT"" — ^ + wcT" ~\ This proportion, by expressing the greatest altitude CY or T arithmeti- cally by unity, becomes, GG to FF as 6 -f- c to mb + nc, and therefore as 1

180 THE MATHEMATICAL PRINCIPLES [BoOK I

to — -, . Whence G becomes to F, that is, the an^le YCp to the aD-

^22,^ -j— 'jiQ

e-le YCP, as 1 to ^/—, — ; . And therefore since the anoie VCP between

^ ' 6 + c °

the upper and the lower apsis, in an immovable ellipsis, is of 180 deg., the

angle YCp between the same apsides in an orbit which a body describes

wdth a centripetal force, that is, as — — , will be equal to an angle of

180 V-~j-T — ^^S- ^^^ ^^y ^^^ ^^™^® reasoning, if the centripetal force

be as -p ■, the angle between the apsides will be found equal to

^ c

180 v''^i cleo\ After the same manner the Problem is solved in

mo — nc °

more difficult cases. The quantity to which the centripetal force is pro- portional must always be resolved into a converging series whose denomi- nator is Al Then the given part of the numerator arising from that operation is to be supposed in the same ratio to that part of it which is not given, as the given part of this numerator RGG — RFF + TFF — FFX is to that part of the same numerator which is not given. And taking away the superfluous quantities, and writing unity for T, the proportion of G to F is obtained.

Cor. 1. Hence if the centripetal force be as any power of the altitude, that power may be found from the motion of the apsides ; and so contra- riwise. That is, if the whole angular motion, with which the body returns to the same apsis, be to the angular motion of one revolution, or 360 deg., as» any number as m to another as ?i, and the altitude called A ; the force

nn

will be as the power A i^ — ^ of the altitude A; the index of which power is

— 3. This appears by the second example. Hence it is plain that

mm i-i. ^ i -L

the force in its recess from the centre cannot decrease in a greater than a triplicate ratio of the altitude. A body revolving with such a force^. and parting from the apsis, if it once begins to descend, can never arrive at the lower apsis or least altitude, but will descend to the centre, describing the curve line treated of in Cor. 3, Prop. XLI. But if it should, at its part- ing from the lower apsis, begin to ascend never so little, it will ascend in ivjlnitum, and never come to the upper apsis ; but will describe the curve line spoken of in the same Cor., and Cor. 6, Prop. XLIY. So that where the force in its recess from the centre decreases in a greater than a tripli- ca^te ratio of the altitude, the body at its parting from the apsis, will either descend to the centre, or ascend in i?ijinitu7n, according as it descends or ascends at the beginning of its motion. But if the force in its recess from

Sec. IX.] OF natural philosophy. ISl

the centre either decreases in a less than a triplicate ratio of the altitude,

or increases in any ratio of the altitude whatsoever, the body will never

descend to the centre, but Avill at some time arrive at the lower apsis ; and,

on the contrary, if the body alternately ascending and descending from one

apsis to another never comes to the centre, then either the force increases

in the recess from the centre, or it decreases in a less than a triplicate ratio

of the altitude; and the sooner the body returns from one apsis to another,

the farther is the ratio of the forces from the triplicate ratio. As if the

body should return to and from the upper apsis by an alternate descent and

ascent in 8 revolutions, or in 4, or 2, or 1^ ; that is, if m should be to n as S,

nil or 4, or 2, or 1^ to 1, and therefore 3, be -^V — 3,or A — ^, or | — 3, or

A _ 3 ; then the force will be as A«'* "" '^ or A^'^"~ "' or A^" '^ or A" ~ ' '

that is, it will be reciprocally as A "^ *' or A ^ ^' or A^ "^- or A •' If the body after each revolution returns to the same apsis, and the apsis

nn ^

remains unmoved, then m will be to n as 1 to 1, and therefore A^i

will be equal to A -, or -r--r ; and therefore the decrease of the forces vrill

be in a duplicate ratio of the altitude ; as was demonstrated above. If the body in three fourth parts, or two thirds, or one third, or one fourth part of an entire revolution, return to the same apsis ; m Avill be to n as | or |

nn ^ 1_6 3 9 3 9

or ^ or ^ to 1, and therefore Amm ^ is equal to A ^' ^or A* or A

3 1 6 3 ^ ^ LI

' ' or A ; and therefore the force is either reciprocally as A ^ or

^ . 6 13

A*^ or directly as A or A . Lastly if the body in its progress from the upper apsis to the same upper apsis again, goes over one entire revolution and three deg. more, and therefore that apsis in each revolution of the body moves three deg. in conseqiieiitia ; then 7n will be to n as 363 deg. to

n n

360 deg. or as 121 to 120, and therefore Amm will be equal to

2.9.5.21

A 14 6 4 17 ^^^ therefore the centripetal force will be reciprocally as

29.5.23 o—A^

^1 4 6 4 ij Qj. reciprocally as A" " * ^ very nearly. Therefore the centripetal force decreases in a ratio something greater than the duplicate ; but ap- proaching 59| times nearer to the duplicate than the triplicate.

Cor. 2. Hence also if a body, urged by a centripetal force which is re- ciprocally as the square of the altitude, revolves in an ellipsis whose focus is in the centre of the forces ; and a new a-nd foreign force should be added to or subducted from this centripetal force, the motion of the apsides arising from that foreign force may (by the third Exeanple) be known ; and so on the contrary. As if the force with which the body revolves in the ellipsis

182 THE MATHEMATICAL PRINCIPLES [BoOK L

be as -j-r- ; and the foreign force subducted as cA, and therefore the remain-

ing force as r^ ; then (by the third Example) b will be equal to 1,

m equal to 1, and n equal to 4 ; and therefore the angle of revolution be-

1 — c

tween the apsides is equal to ISO V- y deg. Suppose that foreign force

to be 357.45 parts less than the other force with which the body revolves

in the ellipsis : that is^ c to be itItIs ; A or T being equal to 1 ; and then

1-c 1^0^/:^--^ will be ISOv^lffff or 180.7623, that is, 180 deg., 45 min.,

44 sec. Therefore the body, parting from the upper apsis, will arrive at the lower apsis with an angular motion of 180 deg., 45 min., 44 sec , and this angular motion being repeated, will return to the upper apsis ; and therefore the upper apsis in each revolution will go forward 1 deg., 31 min., 28 sec. The apsis of the moon is about twice as swift.

So much for the motion of bodies in orbits whose planes pass through the centre of force. It now remains to determine those motions in eccen- trical planes. For those authors who treat of the motion of he?tvy bodies used to consider the ascent and descent of such bodies, not only in a per- pendicular direction, but at all degrees of obliquity upon any given planes ; and for the same reason we are to consider in this place the motions of bodies tending to centres by means of any forces whatsoever, when those bodies move in eccentrical planes. These planes are supposed to be perfectly smooth and polished, so as not to retard the motion of the bodies in the least. Moreover, in these demonstrations, instead of the planes upon which those bodies roll or slide,, and which are therefore tangent planes to the bodies, I shall use planes parallel to them, in which the centres of the bodies move, and by that motion describe orbits. And by the same method I afterwards determine the motions of bodies performed in curve superficies.

SECTION X:

Of the motion of bodies in given superjicies.andof th-e reciprocal motion of fimependuloiis bodies.

PROPOSITION XLYI. PROBLEM XXXII.

Any kind of centripetal force being supposed^ and the centre of for ce^ and any plane whatsoever in which the body revolves, being given, and the qiiadratnres of cnrvilinear figures being alloived ; it is required tods- termine the motion of a body going off from a given place, ivith a p^iven velocity, in the direction of a given right line in that plane»

Sec. X.l

OF NATURAL PHILOSOPHY.

183

Let S be the centre of force, SC the least distance of that centre from the given planC; P a body issuing from the place P in the direction of the right line FZ, Q, the same body revolving in its trajectory, and PQ,R the trajectory itself whicli is required to be found, described in that given plane. Join CQ,, Q.S, and if in Q,S we take SV proportional to the centripetal force with which the body is attracted to- wards the centre S, and draw VT parallel to CO., and meeting SC in T ; then will the force SV be resolved into two (by Cor. 2, of the Laws of Motion), the force ST, and the force TV ; of Tfhich ST attracting the body in the direction of a line perpendicular to that plane, does not at all change its motion in that plane. But the action of the other force TV, coinciding with the position of the plane itself, at- tracts the body directly towards the given point C in that plane; and therefore causes the body to move in this plane in the same manner as if the force ST were taken away, and the body were to revolve in free space about the centre C by means of the force TV alone. But there being given the centripetal force TV with which the body Q. revolves in free space about the given centre C, there is given (by Prop. XLII) the trajectory PQR which the body describes ; the place Q., in which the body will be found at any given time ; and, lastly, the velocity of the body in that place Q. And so e contra. Q,.E.L

PPvOPOSITION XL VII. THEOREM XV.

Supposing the centripetal force to be proportional to the distance of the body from the centre ; all bodies i^evolving in any planes tohatsoever will describe ellipses, and complete their revolutions in equal tim£s ; and those which move in right lines, rwining backwards and forwards alternately, will complete their several periods of going and returning in the same trmes.

For letting all things stand as in the foregoing Proposition, the force SV, with which the body Q, revolving in any plane PQR is attracted to- wards the centre S, is as the distance SQ ; and therefore because SV and SQ, TV and CQ, are proportional, the force TV with which the body is attracted towards the given point C in the plane of the orbit is as the dis- tance CGt. Therefore the forces with which bodies found in the plane PQ,R are attracted towards the point C, are in proportion to the distances equal to the forces with which the same bodies are attracted every way to- wards the centre S ; and therefore the bodies will move in the same times, and in the same figures, in any plane PQ.R about the point C, as they

IS4 THE MATHEMATICAL FrvlNCIPLES [BoOK I.

would do in free spaces about the centre S ; and therefore (by Cor. 2, Prop. X, ar.d Cor. 2, Prop. XXXYIII.) they will in equal times either describe ellipses in that plane about the centre C; or move to and fro in right lines passing through the centre C in that plane ; completing the same periods of time in all cases. Q,.E.D.

SCHOLIUM.

The ascent and descent of bodies in curve superficies has a near relation to these motions we have been speaking of. Imagine curve lines to be de- scribed on any plane, and to revolve about any given axes passing through the centre of force, and by that revolution to describe curve superficies ; and that the bodies move in such sort that their centres may be always found in those superficies. If those bodies reciprocate to and fro with an oblique ascent and descent, their motions will be performed in planes passing through the axis, and therefore in the curve lines, by whose revolution those curve superficies were generated. In those cases, therefore, it will be sufficient to consider the motion in those curve lines.

PROPOSITION XLYIII. THEOREM XYI.

If a loheel stands upon the outside of a globe cd right angles thereto, and revolving about its oivn axis goes forioard in a great circle, the length of the curvilinear path which any point, given in the pervmeter of the loheel, hath described since the time that it touched the globe {which curvilinear path we may call the cycloid or epicycloid), will be to double the versed sine of half the arc id hick since that time has touched the globe in passing over it, qs t/ie sum of the diameters of the globe and the wheel to the semi-diameter of the globe.

PROPOSITION XLIX. THEOREM XVII.

If a wheel stand upon the inside of a concave globe at right angles there- to, and revolviiig about its oion axis go forioard in one of the great circles of the globe, the length of the curvilinear path tohich any point, given in tJie perimeter of the wheel, hath described since it touched the globe, will be to the double of the versed sine of half the arc which in all that time has touched the globe in passing over it, as the difference of the diameters of the globe and the tvheel to the semi-diameter of the globe.

Let ABL be the globe, C its centre, BPY the wheel insisting thereon, E the centre of the wheel, B the point of contact, and P the given point in the perimeter of the wheel. Imagine this wheel to proceed in the great circle ABL from A through B towards L, and in its progress to revolve in such a manner that the arcs AB, PB may be always equal one to the other, . and the given point P in the perimeter of the wheel may describe in the

Sec. X.]

OF NATURAL PHILOSOPHY. S

185

mean time the curvilinear path AP. Let AP be the whole cnrvilinear path described since the wheel touched the globe in A, and the length of this path AP will be to twice the versed sine of the arc |-PB as 2CE to CB. For let the right line CE (produced if need be) meet the wheel in V, and join CP, BP, EP, YP ; produce CP, and let fall thereon the perpen- dicular YF. Let PH, YH, meeting in H, touch the circle in P and Y, and let PH cut YF in G, and to YP let fall the perpendiculars GI, HK. From the centre C with any interval let there be described the circle 7iom, cutting the right line CP in n, the perimeter of the wheel BP in o, and the curvilinear path AP in m ; and from the centre Y with the interval Yo let there be described a circle cutting YP produced in q.

Because the wheel in its progress always revolves about the point of con- tact B, it is manifest that the right line BP is perpendicular to that curve line AP which the point P of the wheel describes, and therefore that the right line YP will touch this curve in the point P. Let the radius of the circle noTn be gradually increased or diminished so that at last it become equal to the distance CP; and by reason of the similitude of the evanescent figure Fnomq, and the figure PFGYI, the ultimate ratio of the evanescent lineolae Pm, P/?, Po, Vq, that is, the ratio of the momentary mutations of the curve AP; the right line CP, the circular arc BP, and the right line YP, will be

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[Book L

the same as of the lines PY, PF, PG, PI, respectively. But since YF is perpendicular to CF, and YH to CY, and therefore the angles HYG, YCF equal ; and the angle YHG (because the angles of the quadrilateral figure HYEP are right in Y and P) is equal to the angle CEP, the triangles VHG, CEP will be similar; and thence it will come to pass that asEP is to CE so is HG to HY or HP, and so KI to KP, and by composition or division as CB to CE so is PI to PK, and doubling the consequents as CB to 2CE so PI to PY, and so is Vq to Vm. Therefore the decrement of the line YP, that is, the increment of the line BY— YP to the increment of the curve line AP is in a given ratio of CB to 2CE, and therefore (by Cor. Lem. lY) the lengths'BY— YP and AP, generated by those increments, are in the same ratio. But if BY be radius, YP is the cosine of the angle BYP or IBEP, and therefore BY— YP is the versed sine of the same angle, and therefore in this wheel, whose radius is |BY, BY— YP will be double the versed sine of the arc -^BP. Therefore AP is to double the versed sine of the arc iBP as 2CE to CB. Q.E.D.

The line AP in the former of these Propositions we shall name the cy- cloid without the globe, the other in the latter Proposition the cycloid within the globe, for distinction sake.

Cor. 1. Hence if there be described the entire cycloid ASL, and the same be bisected in S, the length of the part PS will be to the length PY (which is the double of the sine of the angle YBP, when EB is radius) as 2CE to CB, and therefore in a given ratio.

Cor. 2. And the length of the semi-perimeter of the cycloid AS will be equal to a right line which is to the diameter of the wheel BY as 2CE to CB.

PROPOSITION L. PROBLEM XXXIII.

To cause a pendulous body to oscillate in a given cycloid.

$ Let there be given within the globe QYS de- ^ scribed with the centre C, the cycloid QRS, bi- sected in R, and meeting the superficies of the lobe with its extreme points Q and S on either hand. Let there be drawn CR bisecting the arc QS in O, and let it be produced to A in such sort that CA may be to 'CO as CO to CR. About the centre C, with the interval CA, let there be described an exterior globe DAF ; and within this globe, by a wheel whose diameter is AO, let there be described two semi-cycloids AQ, AS, touching the interior globe in Q. and S, and meeting the exterior globe in A. From that point A, with a thread APT in length equal to the line AR, let the body T depend, and oscillate in such manner between the two

Sec. X.] OF natural philosophy. 187

Bemi-cycloids AQ, AS, that, as often as the penclalum parts from the per- pendicuhir AR, the upper part of the thread AP may be applied to that semi-cycloid APS towards which the motion tends, and fold itself round that curve line, as if it were some solid obstacle, the remaining part of the same thread PT which has not yet touched the semi-cycloid continuing straight. Then will the weight T oscillate in the given cycloid Q,RS.

aE.F.