Page:Cyclopaedia, Chambers - Volume 2.djvu/238

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Cor. 3. Again, if T =(, then will S ==/. Two moving Bodies therefore, whofe Mafles and Momenta are equal, defcribe equal Spaces in equal Times.

tor. 4. It' befides I = i, S =/; then will i»T = Mt; and therefore M : m : : T : r; that is, if two moving Bo- dies, whofe Momenta are equal, pafs over equal Spaces, their Maffes are proportionable to their Times.

Cor. 5. Further, if T = r, then will M = >«; and there- fore Bodies, whofe Momenta are equal, and which moving equably, defcribe equal Spaces in equal Times, have their Maffes equal.

Cor. 6. If befides I = 1, T=tj then willMS = >»/; and therefore S:/: : ib: M; that is, the Spaces pafs 'd o- ver in the fame time, by two moving Bodies, whofe Mo- menta ate equal, are in a reciprocal Ratio of their Mafles.

Tieor. VII. In an equable Mot/on, the Spaces S and/ are in a Ratio compounded of the direct Ratio's of the Mo- menta 1 and 1, and the Times T and t; and the reciprocal one of the Mafles M and m.

Dem. Becaufe 1 : i : : M S t : mfT,

I mfT: :,,:: IT: if, T : t: iiM:IiB. If two Bodies therefore move equably over equal Spaces, 1. Their Momenta will be in a Ratio compounded of the direcf Ra- tio of the Mafles, and the reciprocal one of the Times, a. Their Mafles will be in a Ratio compounded of the Momenta and the Times. 5. The Times will be in a Ra- tio compounded of the direcl Ratio of the Mafles, and the reciprocal one of the Momenta.

Cor.2. If befide S =/, M = m ; then will IT = 1 t ; and therefore I : < : : t : T. That is, Bodies whofe Mafles are equal, have their Momenta reciprocally proportionable to the Times in which they move over equal Spaces.

Cor. 3. If befide S =/, T = f; then will i M = I m ; and therefore two Bodies moving equably, and thro equal Spaces in equal Times, have their Momenta proportion- able to their Maffes.

Tieor.VIII. Two Bodies moving equably, have their Mafles M and m, in a Ratio compounded of the direef Ra- tio's of the Momenta I and i, and the Times T and t, and the reciprocal one of the Spaces f and S.

Dem. Becaufe I: ; :: M Sr: mfT, lmfT = iUSt. Wherefore M : m : : I T / : i t S.

Q.E.D. In Numbers 7:5:13. 28, 16: 8.10. 11 : : 3. 7.1 : I. IO. 3 : : 7 : 5.

Again I : i : : M St : mft.

In Numbers 28 : JO: : 7. 12. 8. : 5. itf. 3 : : 7. 4. I : 5.2. I : : 28 : IO.

Corol. 1. If M = m, then will IT/=irS; and there- fore I : i : : t S : T/, S :/: : I T : i t, and I : t : : i S : If. That is, in two moving Bodies, whofe Mafles are equal ;

1. The Momenta are in a Ratio compounded of the direct Ratio of the Spaces, and the reciprocal one of the Times.

2. The Spaces are in a Ratio compounded of the Momenta and the Times. 5 . The Times are in a Ratio compounded of the direct Ratio of the Spaces, and the reciprocal one of the Momenta.

Cor.2. If befide M — m, T = r, then will/S= If; and therefore I : i ; :S:f. That is, the Momenta of two Bodies, whofe Maffes are equal, are proportional to the Spaces pafs'd over in equal Times.

Tieor. IX. In equable Motions, the Times T and t are in a. Ratio compounded of the direfl Ratio's of the Mafles M and >n, and the Spaces S and/, and the reciprocal one of the Momenta I andi.

Dem. Becaufe I : i : : M S t : m/T, 1 111/T = i M S I. Wherefore T : 1 : : i M S : I mf. CL E. D.

Cor. If T = r, i M S = I mf; and therefore I : i : : M S
 * mf, M :m: : I S ; i S j and S :/: : I m : i M. That is, if

two Bodies, moving equably, defcribe equal Spaces in e- qual Times ; 1. Their Momenta will be in a Ratio com- pounded of the Mafles and the Spaces. 2. Their Mafles will be in a Ratio compounded of the diredl Ratio of the Momenta, and the reciprocal one of the Spaces. 3. The Spaces will be in a Ratio compounded of the direct Ratio of the Momenta, and the reciprocal one of the Mafles.

The Laws 0/ Motions uniformly accelerated and retarded. . Def. By an accelerated Motion, we mean fuch a one as continually receives frefh Acccffions of Velocity ; and it is faid to be uniformly accelerated, when in equal Times its Acceflions of Velocity arc equal. See Acceleration.

By a retarded Motion, is understood fuch a one, whofe Velocity continually decreafes; and it is faid to be uniform-

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ly retarded, wheilfts Decreafe is continually proportional la- the Time. See Retardation.

Axiom A Body once at reft, will never move, unlefs lome other Body put it in Motion ; and when once in Mo- tion, it will continue for ever to move, with the fame Velo- city, and in the fame Direclion, unlefs it be forced from its State by force other Caufe : This is evident from that fundamental Axiom in Philofophy, That there is nothing without a fufheient Caufe.

Cor. 1. A Body therefore moved by one only Impulfe, mult proceed in a right Line.

Cor. 2. It then it be carry'd in a Curve, it muff be acted on by a double Power ; one, whereby it would proceed in a right Line 5 another, whereby it is continually drawn out of it.

Axiom 1. If the Aflion and Re-aSion of two (unelaftic) Bodies be equal, there will no Motion enfue 5 but the Bo- dies after Collifion, will remain at reft by each other.

^iio»<2. If a moving Body be impell'd in the Direflion of its Motion, it will be accelerated 5 if by a refilling Force, it will beretatded. Heavy Bodies defcend with an acce- lerated Motion.

Tbeor. X. If a Body move with an uniform Velocity 5 the Spaces will be in a duplicate Ratio of the Times.

Dem. For let the Velocity acquir'd in the Time 1 be=a ■v, then will the Velocity a'cquir'd in the Time 2 r= 2 *>, in the Time 3 t sa 3 i, i£c. and the Spaces correfpohding to thofe Times, 1,21, 3 t, $gc. will be as r v, 421 r, a r, &*• (by Tbeor. 2.) The Spaces therefore are as r. 4.9. fi?c. And the Times as 1.2. 3. ti?c. that is, the Spaces are in a duplicate Ratio of the Times. Q. E. D.

Cor In a Motion uniformly accelerated, the Times are in a fub-duplicare Ratio of the Spaces.

Tieor. XI. The Spaces pafs'd over by a Body uniformly accelerated, increafe, in equal Times, according to the un- equal Numbers 1. 3. 5. 7, £c?c.

Dem. If the Times, wherein a moving Body equably ac- celerated, proceeds, be as 1. 2. 3. 4. 5, ££?c. the Space pafs'd over in one Moment, will be as 1, in 2 Moments as 4, in 3 as 9, in 4 as irj, in 5 as 25, iyc. (Tbeor. 10.) If there- fore you fubftraft the Space pafs'd over in one Moment, viz. 1. from that pafs'd over in two Moments, 4. there will remain the Space cbrrefpondihg to the fecond Minute, ■visa. 3. In the fame manner may be found the Space pafs'd' over in the third Minute, 9 — 4=5. The Space corre- fponding to the fourth Minute, 16 — 1; = 7 ; and fo of the reft. The Space of the firfl: Minute therefore is as 1, that of the fecond as 3, that of the third as 5, of the fourth as 7, of the fifth as 9, £s?c. Therefore the Spaces pafs'd over by a Body, moving with an uniformly accelerated Motion, in equal Times increafe according to the unequal Numbers, 1. 5. 5. 7. 9, S?c. Q. E. D.

Tieor. XII, The Spaces pafled over by a Body equably accelerated, are in a duplicate Ratio of the Velocities.

Dem. For fuppofing the Velocities to be V and v, the Times T and t, the Spaces S and s ; then will V : 11: : T: r. Wherefore, fince S : ; : : T* : >', f Tbeor. 10 } S:s:: V* : »'. '

Corol. Wherefore in a Motion uniformly accelerated, the Velocities are in a fubduplicate Ratio of the Spaces.

Tbeor. XIII. Heavy Bodies defcend with an uniformly accelerated Motion, in a Medium void of Reliftance ; if the Spaces be not very great.

Dem. Since heavy Bodies defcend with one accelerated Velocity, the Power of Gravity mufl continually impel them. But the Power of Gravity is found the fame at all Diftances from the Earth where the Experiment can be made. Therefore heavy Bodies muft be driven down- wards in the fame manner in equal times. If then, in thefirft Moment of Time, they be impell'd with the Velo- city o, they will be impell'd with the fame Velocity 21 in the fecond Moment, and with the fame in the 3d, 4th, Igc. Moments. Now the Medium being fuppofed void of all Refinance, (by Hyfoth.) they will ilill retain the Velocity they acquire ; and by rcafon of their equal frefh Acqui- sitions every Minute, 'they will defcend with a Motion uni- formly accelerated. 1^. E. D. See Gravity.

Cor. 1. The Spaces of Defcent therefore, are in a du- plicate Ratio of their Times, and alfo of their Velocities, (Tieor. io,i2.J and increafe according to the uneven Num- bers I, 3, 5, 7, 9, js?c. (Tbeor. 11.)

Cor. 2. The Times, and likewife the Velocities, are in a fubduplicate Ratio of the Spaces, (Tieor. 10, 13.)

Scbol. In fuppofing heavy Bodies to move thro' a Medium void of RefiHance, we exclude, at once, all manner of Im- pediments, under what Name foever they be call'd, or from whatfoever Caufe they proceed ; and among the reft, that Motion, wherewith the Earth revolving on its Axis, carries with it heavy Bodies during the time of their fall 5 tho* this is not fenfible at any moderate diftance.

Scbol. It was GaliUus who firft difcover'd the Law of

the Defcent of heavy Bodies j and that too by Re&foning ;

7 L tho"