Principia Mathematica (Motte Translation, 1848) — part 21 of 45

being analogous. And if the quantity GD reciprocally proportional to p^, be augmented by the given quantity CG ; the sum CD, the time ABED uniformly increasing, will increase in a geometrical progression..

aE.D,

280 THE MATHEMATICAL PRINCIPLES [BoOK II,

Cor. 1. Therefore, if, having the points A and G given, the time be expounded by the hyperbolic area ABED, the velocity may be expounded

by p-pT the reciprocal of GD.

Cor. 2. And by taking GA to GD as the reciprocal of the velocity at the beginning to the reciprocal of the velocity at the end of any time ABED, the point G will be found. And that point being found the ve- locity may be found from any other time given.

PROPOSITION XII. THEOREM IX.

The same things being supposed, I say, that if the spaces described are . taken in arithmetical progression, the velocities augmented by a cer- tain given quantity will be in geometrical progressioii.

In the asymptote CD let there be given the point R, and, erecting the perpendicular RS meeting the hyperbola in S, let the space de- scribed be expounded by the hyperbolic area RSED ; and the velocity will be as the length GD, which, together with the given line CG, composes a length CD decreasing in a geo- metrical progression, while the space RSED increases in an arithmetical progression.

For, because the increment EDc?e of the space is given, the lineola Dc?, which is the decrement of GD, will be reciprocally as ED, and therefore directly as CD ; that is, as the sum of the same GD and the given length CG. But the decrement, of the velocity, in a time reciprocally propor- tional thereto, in which the given particle of space Dc/eE is described, is as the resistance and the time conjunctly, that is, directly as the sum of two quantities, whereof one is as the velocity, the other as the square of the velocity, and inversely as the velocity ; and therefore directly as the sum of two quantities, one of which is given, the other is as the velocity. Therefore the decrement both of the velocity and the line GD is as a given quantity and a decreasing quantity conjunctly ; and, because the decre- ments are analogous, the decreasing quantities will always be analogous; viz., the velocity, and the line GD. Q.E.D.

Cor. 1. If the velocity be expounded by the length GD, the space de- scribed will be as the hyperbolic area DESR.

Cor. 2. And if the point K be assumed any how, the point G will be found, by taking GR to GD as the velocity at the beginning to the velo- city after any space RSED is described. The point G being given, the space is given from the given velocity : and the contrary.

Cor. 3. Whence since (by Prop. XI) the velocity is given from the given

OF NATURAL PHILOSOPHY.

281

Sec. IIL]

time, and (by this Prop.) the space is given from the given velocity ; the space will be given from the given time : and the contrary.

PROPOSITION XIEI. THEOREM X.

Supposing that a body attracted doimiwards by an uniform gravity as- cends or descends in a right line; and that the same is resisted partly in the ratio of its velocity, and partly in the duplicate ratio thereof: I say, that, if right lines parallel to the diameters of a circle and an hyperbola, be drawn through the ends of the conjugate diame- ters, and the velocities be as some segments of those parallels drawn frcm a given point, the times will be as the sectors of the areas cut off by right lines draionfrom the ce7itre to the ends of the segmejits ; and the contrary.

Case V Suppose first that the body is ascending, and from the centre D, with any semi-diameter DB, describe a quadrant BETF of a circle, and through the end B of the semi-diameter DB draw the indefi- nite line BAP, parallel to the semi-diameter DF. In that line let there be given the point A, and take the segment AP proportional to the velocity. And since one part of the resistance is as the velocity, and ^ another part as the square of the velocity, let the whole resistance be as AP^ + 2BAP. Join DA, DP, cutting the circle in E and T, and let the gravity be expounded by DA^, so that the gravity shall be to the resistance in P as DA^to AP2-f2BAP; and the time of the whole ascent will be as the sector EDT of the circle.

For draw DVQ, cutting oiF the moment PQ of the velocity AP, and the moment DTV of the sector DET answering to a given moment of time ; and that decrement PQ. of the velocity will be as the sum of the forces of gravity DA- and of resistance AP- + 2BAP, that is (by Prop. XII, Book II, Elem.), as DP - . Then the area DPQ, which is proportional to PQ, is as DP2, and the area DTV, which is to the area DPQ.as DT- to DP-, is as the given quantity DT^. Therefore the area EDT decreases uniformly according to the rate of the future time, by subduction of given particles DTV, and is therefore proportional to the time of the whole ascent. Q.E.D.

Case 2. If the velocity in the ascent of the body be expounded by the length AP as before, and the resistance be made as AP2 + 2BAP, and if the force of grav- ity be less than can be expressed by DA^ ; take BD of such a length, that AB^ — BD^ may be proportional to the gravity, and let T)Y be perpendicular and equal

r .-A

a p

/ /

o4

//

^

ra

2S2 THE MATHEMATICAL PRINCIPLES [BoOK 11.

1=0 DB, and througli the vertex F describe the hyperbola FTYE, whose con- jugate semi -diameters are DB and DP, and which cuts DA in E, and DP, DO, in T and Y ; and the time of the whole ascent will be as the hyper- bolic sector TDE.

For the decrement PQ, of the velocity, produced in a given particle of time, is as the sum of the resistance AP^ -f 2BAP and of the gravity AB^ — BD2, that is, as BP^ — BD^. But the area DTV is to the area DPQ, as DT^ to DP^ ; and, therefore, if GT be drawn perpendicular to DF, as GT^ or GD^ — DF^ to BD^, and as GD^ to BP% and, by di- vision, as DF 2 to BP 2 — BD 2. Therefore since the area DPQ, is as PQ, that is, as BP^ — BD^, the area DTV will be as the given quantity DF 2. Therefore the area EDT decreases uniformly in each of the equal particles of time, by the subduction of so many given particles DTV, and therefore is proportional to the time. Q-E.D.

Case 3. Let AP be the velocity in the descent of the body, and AP=^ + 2BAP the force of resistance, and BD 2 — AB ^ the force of gravity, the angle DBA being a right one. And if with the centre D, and the principal vertex B, there be described a rectangular hyperbola BETV cutting DA, DP, and DQ, produced in E, T, and V ; the sector DET of this hyperbola will be as the whole time of descent. For the increment PQ. of the velocity, and the area DPQ proportional to it, is as the excess of the gravity above the resistance, that is, as BD^ — AB^ _2BAP — AP2 or BD^— BP^ And the area DTV is to the area DPQ as DTMo DP^ ; and therefore as GT^ or GD- — BD^ to BP2, and as GD^ to BD^ and, by division, as BD^ to BD^ — BP2. Therefore since the area DPQ is as BD^ — BP% the area DTV will be as the given quantity BD^ Therefore the area EDT increases uniformly in the several equal particles of time by the addition of as many given particles DTV, and therefore is proportional to the time of the descent. Q.E.D.

Cor. If with the centre D and the semi-diameter DA there be drawn through the vertex A an arc At similar to the arc ET, and similarly sub- tending the angle A DT, the velocity AP will be to the velocity which the body in the time EDT, in a non-resisting space, can lose in its ascent, or acquire in its descent, as the area of the triangle DAP to the area of the sector D A^ ; and therefore is given from the time given. For the velocity in a non-resisting medium is proportional to the time, and therefore to this sector : in a resisting medium, it is as the triangle ; and in both mediums, where it is least, it approaches to the ratio of equality, as the sector and triangle do.

Sec. Ill]

OF NATURAL PHILOSOPHY.

SCHOLIUM.

283

One may demonstrate also that case in the ascent of the bodj; where the force of gravity is less than can be expressed by DA^ or AB^ + BD^, and greater than can be expressed by AB^ — DB^^ and must be expressed by AB^ But I hasten to other things.

PROPOSITION XIV. THEOREM XL

The same things being supposed, 1 say, that the space described in the ascent or descent is as the difference of the area by which the time is expressed, and of so?ne other area lohich is augmented or diminished in an arithmetical progression ; if the forces com,pounded of the re- sistance and the gravity be taken in a geometrical progression. Take AC (in these three figures) proportional to the gravity, and AK

to the resistance ; but take them on the same side of the point A, if the

H

M

\

6

i-

B

c

A

/

^%J^

x^

^ -TiK

/^^^^1

D^

^^ 1

^.,„.

^F

body is descending, otherwise on the contrary. Erect Kb, which make to DB as DB^ to 4BAC : and to the rectangular asymptotes CK, CH, de- scribe the hyperbola 6N ; and, erecting KN perpendicular to CK, the area A6NK will be augmented or diminished in an arithmetical progression, while the forces CK are taken in a geometrical progression. I say, there- fore, that the distance of the body from its greatest altitude is as the excess of the area A6NK above the area DET.

For since AK is as the resistance, that is, as AP^ X 2BAP ; assume

^pa I 2BAP any given quantity Z, and put AK equal to ^ ; then (by Lem.

284

THE MATHEMATICAL PRINCIPLES [BoOK II.

2APa + 2BAxPQ

II of this Book) the moment KL of AK will be equal to

2RPQ

or — j^ — , and the moment KLON of the area A6NK will be equal to

/-A

BPa X BD^'

or

2BPa X LP

Z ^^ 2Z X CK X AB'

Case 1. Now if the body ascends, and the gravity be as AB^ + BD^,

BET being a circlej the line AC, which is proportional to the gravity,

AB^ 4- BD^ will be ^ , and DP^ or AP^ + 2BAP + AB^ + BD^ will be

AK X Z + AC X Z or CK X Z ; and therefore the area DTV will be to the area DPa as DT^ or DB^ to CK X Z.

Case 2. If the body ascends, and the gravity be as AB^ — BD^, the

A g2 BD^

line AC will be t-. , and DT^ will be to DP^ as DF^ or DB^.

Za

to BP=^ --BD2 or AP2 + 2BAP + AB^ — BD^, that is, to AK X Z +

Jbl

V

6

K

^

3

C

A

\ /.

^^ff

/^

;^ I.i

/^^

D^

^ /

^

J-F

AC X Z or CK X Z. And therefore the area DTV will be to the area

DPa as DB 2 toCKxZ.

Case 3. And by the same reasoning, if the body descends, and therefore

the gravity is as BD^ — AB^, and the line AC becomes equal to

BD2— AB^

7, ; the area DTV will be to the area DPQ as DB^ to CK X

Z : as above.

Since, therefore, these areas are always in this ratio, if for the area

Sec. Ill] OF NATURAL PHILOSOPHY. 285

DTV, by which the moment of the time, always equal to itself, is express- ed, there be put any determinate rectangle, as BD X m, the area DPQ., that is, iBD X PQ, will be to BD X m as CK X Z to BD^ And thence PQ X BD 3 becomes equal to 2BD X m X CK X Z, and the moment KLON

of the area A6NK, found before, becomes .-5 — . From the area

DET subduct its moment DTV or BD X m, and there will remain

AP X BD X m

-Tjy . Therefore the difference of the moments, that is, the

moment of the difference of the areas, is equal to T-5 ; and

therefore (because of the given quantity — t-:^ — ) as the velocity AP ;

that is, as the moment of the space which the body describes in its ascent or descent. And therefore the difference of the areas, and that space, in- creasing or decreasing by proportional moments, and beginning together or vanishing together, are proportional. Q,.E.D.

Cor. If the length, which arises by applying the area DET to the line BD, be called M ; and another length V be taken in that ratio to the length M, which the line DA has to the line DE ; the space which a body, in a resisting medium, describes in its whole ascent or descent, will be to the space which a body, in a non-resisting medium, falling from rest, can de- scribe in the same time, as the difference of the aforesaid areas to

BD X V^

— yr- — ; and therefore is given from the time given. For the space in a

non-resisting medium is in a duplicate ratio of the time, or as V^ ; and,

BD X y^ .

because BD and AB are given, as r-^- — • This area is equal to the

DA^ X BD X M^ . . , ^ w 1 .1. - ^ .T,

area =-^^7 r^ — - and the moment 01 M is m : and thereiore the

DE2 X AB

. r -L- . DA^ X BD X 2M X m „ , ,, . , . ^

moment of this area is df^ v ar * moment is to

ihe moment of the difference of the aforesaid areas DET and A6NK, viz., to

APxBDXm DA^ XBDXM^ ^^^ ^ ^^' • . T^i^rp j^ , as j^p^ to |BD X AP, or as ^ypi mto DET

to DAP ; and, therefore, when the areas DET and DAP are least, in the

BD X V^^ ratio of equality. Therefore the area — -^ and the difference of the

areas DET and A6NK, when all these areas are least, have equal moments ; and are therefore equal. Therefore since the velocities, and therefore also the spaces in both mediums described together, in the beginning of the de- scent; or the end of the ascent, approach to equality, and therefore are then

286 THE MATHEMATICAL PRINCIPLES [BoOK II.

BD X Y^ one to another as the area Tr""; ^^^ *^® difference of the areas DET

and A6NK ; and moreover since the space, in a non-resisting medium, is

BD X V^ perpetually as r-^ — , and the space, in a resisting medium, is perpetu- ally as the difference of the areas DET and A^NK ; it necessarily follows, that the spaces, in both mediums, described in any equal times, are one to

BD X Y^

another as that area -.-p^ — , and the difference of the areas DET and

AB '

A6NK. aE.D.

SCHOLIUM.

The resistance of spherical bodies in fluids arises partly from the tena- city, partly from the attrition, and partly from the density of the medium. And that part of the resistance which arises from the density of the fluid is, as I s?id, in a duplicate ratio of the velocity ; the other part, which arises from the tenacity of the fluid, is uniform, or as the moment of the time ; and, therefore, we might now proceed to the motion of bodies, which are resisted partly by an uniform force, or in the ratio of the moments of the time, and partly in the duplicate ratio of the velocity. But it is suf- ficient to have cleared the way to this speculation in Prop. YIII and IX foregoing, and their Corollaries. For in those Propositions, instead of the uniform resistance made to an ascending, body arising from its gravity, one may substitute the uniform resistance which arises from the tenacity of the medium, when the body moves by its vis insita alone ; and when the body ascends in a right line, add this uniform resistance to the force of gravity, and subduct it when the body descends in a right line. One might also go on to the motion of bodies which are resisted in part uni- formly, in part in the ratio of the velocity, and in part in the duplicate ratio of the same velocity. And I have opened a way to this in Prop. XIII and XIY foregoing, in which the uniform resistance arising from the tenacity of the medium may be substituted for the force of gravity, or be_ compounded with it as before. But I hasten to other things.

Sec. IV.]

OF NATURAL PHILOSOPHY.

2S7

SECTION lY.

Of the circular motion of bodies in resisting mediums.

LEMMA III.

Let PQR be a spiral cutting all the radii SP, SQ, SR, t^^c., 171 equal angles. Draw the right line PT touching the spiral in any point P, and cutting the radius SQ, in T ; draio PO, QO perpendicular to the spiral, and meeting in O, and join SO. I say, that if the points P and Q approach and coincide, the angle PSO loill become a right angle, and the ultimate ratio of the rectangle TQ, X 2PS to PQ,^ will be the ratio of equality. For from the right angles OPQ, OQ,R, sub- duct the equal angles SPQ, SQ,R, and there will remain the equal angles OPS, OQ,S.

Therefore a circle which passes through the 1 v -<^^ ^X-~-^:>\Dy

points OSP will pass also through the point \ XT^X ^~^P

Q. Let the points P and Q coincidcj and \ \ o*^T

this circle will touch the spiral in the place ^^ J^^

of coincidence PQ,, and will therefore cut the right line OP perpendicularly. Therefore OP will become a diameter of this circle, and the angle OSP, being in a semi-circle, becomes a right one. aE.D.

Draw Qjy, SE perpendicular to OP, and the ultimate ratios of the lines will be as follows : TQ to PD as TS or PS to PE, or 2P0 to 2PS ; and PD to PQ as PQ. to 2P0 ; and, ex mquo perturbatp^, to TQ, to PQ as PQ to 2PS. Whence PQ^ becomes equal to TQ X 2PS. Q.E.D.

PROPOSITION XY. THEOREM XII.

If the density of a mediiun in each place thereof be reciprocally as the distance of the places from an immovable centre, and the centripetal force be in the duplicate ratio of the density ; / say, that a body may revolve in a spiral which cuts all the radii drawn from, that centre ill a given angle.

Suppose every thing to be as in the forego- ing Lemma, and produce SQ to Y so that SY may be equal to SP. In any time let a body, in a resisting medium, describe the least arc PQ, and in double the time the least arc PR : and the decrements of those arcs arising from the resistance, or their differences from the arcs which would be described in a non-resist- ing medium in the same times, will be to each other as the squares of the times in which they are generated ; therefore the decrement of the

288 THE MATHEMATICAL PRINCIPLES * [BoOK II.

arc PQ, is the fourth part of the decrement of the arc PR. Whence also if the area Q.Sr be taken equal to the area PSQ, the decrement of the arc PQ, will be equal to half the lineola Rr / and therefore the force of resist- ance and the centripetal force are to each other as the lineola ^RrandTQ, which they generate in the same time. Because the centripetal force with which the body is urged in P is reciprocally as SP^, and (by Lem. X, Book I) the lineola TQ, which is generated by that force, is in a ratio compounded of the ratio of this force and the duplicate ratio of the time in which the arc PQ, is described (for in this case I neglect the resistance, as being infinitely less than the centripetal force), it follows that TQ X SP-, that is (by the last Lemma), ^PQ^ X SP, will be in a duplicate ra- tio of the time, and therefore the time is as PQ X /SP ; and the velo- city of the body, with which the arc PQ is described in that time, as

PQ 1

^j57^ ^^ or — ^-5, that is, in the subduplicate ratio of SP reciprocally.

And, by a like reasoning, the velocity with which the arc QR is described, is in the subduplicate ratio of SQ reciprocally. Now those arcs PQ and QR are as the describing velocities to each other ; that is, in the subdu- plicate ratio of SQ to SP, or as SQ to ^/SF X SQ; and, because of the equal angles SPQ, SQr, and the equal areas PSQ, QSr, the arc PQ is to the arc Qr as SQ to SP. Take the differences of the proportional conse- quents, and the arc PQ will be to the arc Rr as SQ to SP — v^SP X SQ, or Y^Q. For the points P and Q coinciding, the ultimate ratio of SP — VSP X SQ' to ^YQ is the ratio of equality. Because the decrement of the arc PQ arising from the resistance, or its double Rr, is as the resistance

and the square of the time conjunctly, the resistance will be aSp^— — <cp.

But PQ was to Rr as SQ to |YQ, and thence p^-^ ^ becomes as

^YQ ^OS

pa X SP X Sa' °' ''' OF^SP^- ^°' ^^' P°^"*' ^ '^"'^ ^ coinciding,

SP and SQ, coincide also, and the angle PYQ becomes a right one ; and,

because of the similar triangles PYQ, PSO, PQ. becomes to ^YQ as OP

OS to I^OS. Therefore ^^b — opi ^^ ^^ *^® resistance, that is, in the ratio of

the density of the medium in P and the duplicate ratio of the velocity conjunctly. Subduct the duplicate ratio of the velocity, namely, the ratio

1 * OS

^, and there will remain the density of the medium in P, as qp w gp-

Let the spiral be given, and, because of the given ratio of OS to OP, the density of the medium in P will be as ^p. Therefore in a medium whose

Sec. IY.] of natural philosophy. 289

density is reciprocally as SP the distance from the centre^ a body will re- volve in this spiral. Q.E.D.

Cor. 1. The velocity in any place P, is always the same wherewith a body in a non-resisting medium with the same centripetal force would re- volve in a circle, at the same distance SP from the centre.

Cor. 2. The density of the medium, if the distance SP be given, is as

OS OS

^rrTp, but if that distance is not given, as vr^ ^. And thence a spiral

may be fitted to any density of the medium.

Cor. 3, The force of the resistance in any place P is to the centripetal

force in the same place as |0S to OP. For those forces are to each other

^YQ X PQ iPQ-2 as |Rr and TQ., or as - — ^7^ and ^^7^-, that is, as iYQ and PQ,

or I OS and OP. The spiral therefore being given, there is given the pro- portion of the resistance to the centripetal force ; and, vice versa, from that proportion given the spiral is given.

Cor. 4. Therefore the body cannot revolve in this spiral, except where the force of resistance is less than half the centripetal force. Let the re- sistance be made equal to half the centripetal force, and the spiral will co- incide with the right line PS, and in that right line the body will descend to the centre with a velocity that is to the velocity, with which it was proved before, in the case of the parabola (Theor. X, Book I), the descent ■would be made in a non-resisting medium, in the subduplicate ratio of unity to the number two. And the times of the descent will be here recip- rocally as the velocities, and therefore given.

Cor. 5. And because at equal distances from the centre the velocity is the same in the spiral PQ.R as it is in the right line SP, and the length of the spiral is to the length of the right line PS in a given ratio, namely, in the ratio of OP to OS ; the time of the descent in the spiral will be to the time of the descent in the right line SP in the same given ratio, and therefore given.

Cor. 6. If from the centre S, with any two

given intervals, two circles are described ; and

these circles remaining, the angle which the spiral makes with the radius

PS be any how changed ; the number of revolutions which the body can

complete in the space between the circumferences of those circles, going

PS round in the spiral from one circumference to another, will be as 7^, or as

Ob

the tangent of the angle which the spiral makes with the radius PS ; and

19

290 THE MATHEMATICAL PRINCIPLES [BoOK II.

OP

the time of the same revolutions will be as -^-^, that is, as the secant of the

same angle, or reciprocally as the density of the medium.

Cor. 7. If a body, in a medium whose density is reciprocally as the dis- tances of places from the centre, revolves in any curve AEB about that centre, and cuts the first radius AS in the same angle in B as it did before in A, and that with a velocity that shall be to its first velocity in A re- ciprocally in a subduplicate ratio of the distances from the centre (that is, as AS to a mean propor- tional between AS and BS) that body will con- tinue to describe innumerable similar revolutions BFC, CGD, (fee, and by its intersections will distinguish the radius AS into parts AS, BS, CS, DS, (fee, that are con- tinually proportional. But the times of the revolutions will be as the perimeters of the orbits AEB, BFC, CGD, (fee, directly, and the velocities

at the beginnings A, B, C of those orbits inversely ; that is as AS-. BS^,

3

CS^. And the whole time in which the body will arrive at the centre, will be to the time of the first revolution as the sum of all the continued

3. 3_ 3_ ,

proportionals AS^, BS^, CS^ , gomg on ad wfinitum, to the first term

11 3.

AS 2 ; that is, as the first term AS - to the difference of the two first AS ^

— BS% or as f AS to AB very nearly. Whence the whole time may be easily found.

Cor. 8. From hence also may be deduced, near enough, the motions of bodies in mediums whose density is either uniform, or observes any other assigned law. From the centre S, with intervals Sxi, SB, SC, (fee, con- tinually proportional, describe as many circles ; and suppose the time of the revolutions between the perimeters of any two of those circles, in the medium whereof we treated, to be to the time of the revolutions between the same in the medium proposed as the mean density of the proposed me- dium between those circles to the mean density of the medium whereof we treated, between the same circles, nearly : and that the secant of the angle in which the spiral above determined, in the medium whereof we treated, .cuts the radius AS, is in the same ratio to the secant of the angle in which the new spiral, in the proposed medium, cuts the same radius : and also that the number of all the revolutions between the same two circles is nearly as the tangents of those angles. If this be done every where between every two circles, the motion will be continued through all the circles. And by this means one may without difficulty conceive at what rate and in what time bodies ought to revolve in any regular medium.

Sec. IY.] of natural philosophy. 291

Cor. 9. And although these motions becoming eccentrical should be performed in spirals approaching to an oval figure, yet, conceiving the several revolutions of those spirals to be at the same distances from each other, and to approach to the centre by the same degrees as the spiral above described, we may also understand how the motions of bodies may be per- formed in spirals of that kind.

PROPOSITION XVI. THEOREM XIII.

If the density of the medium in each of the places be reciprocally as the distance of the places from the immoveable centric, and the centripetal force be reciprocally as any power of the same distance, I say, that the body may revolve in a spiral intersectitig all the radii drawn from that centre in a given angle. This is demonstrated in the same manner as the foregoing Proposition. For if the centri- petal force in P be reciprocally as any power SP°+^ of the distance SP whose index is w

• 1 ; it will be collected, as above, that the time in which the body describes any arc PQ,

will be as PQ, X PS^"^ ; and the resistance in Rr 1 _ ^^ X Va

^ ^'PQ^ X sp^^'^'^'pGTx^p^sa' ^""^

^, „ l — \nXO^.. ^. 1— iTiXOS .

therefore as ^^ o^^TTT' ^^^^ ^^? (because pr--5 is a given

vJJr X oJr -J- L/Jr

quantity), reciprocally as SP" + ^ . And therefore, since the velocity is recip-

rocally as SP^°, the density in P will be reciprocally as SP.

Cor. 1. The resistance is to the centripetal force as 1 — \n X OS to OP.

Cor. 2. If the centripetal force be reciprocally as SP^j \ — \7i will be = 0 ; and therefore the resistance and density of the medium will be nothing, as in Prop. IX, Book I.

Cor. 3. If the centripetal force be reciprocally as any power of the ra- dius SP, whose index is greater than the number 3, the aJOBrmative resist- ance will be chano;ed into a neo-ative.

SCHOLIUM.

This Proposition and the former, which relate to mediums of unequal density, are to be understood of the motion of bodies that are so small, that the greater density of the medium on one side of the body above that on the other is not to be considered. I suppose also the resistance, cceteris paribus^ to be proportional to its density. Whence, in mediums whose

292 THE MATHEMATICAL PRINCIPLES [BoOK IL

force of resistance is not as the density, the density must be so much aug- mented or diminished, that either the excess of the resistance may be taken away, or the defect supplied.

PROPOSITION XYII. PROBLEM IV,

To find the centripetal force and the resisting force of the medium, by which a body, the law of the velocity being given, shall revolve in a given spiral.

Let that spiral be PQR. From the velocity, with which the body goes over the very small arc PQ,, the time will be given ; and from the altitude TO,, which is as the centripetal force, and the square of the time, that force will be given. Then from the difference RSr of the areas PSQ, and Q,SR described in equal particles of time, the re- tardation of the body will be given ; and from the retardation will be found the resisting force and density of the medium.

PROPOSITION XVIIL PROBLEM V.

The law of centripetal force being given, to find the density of the me- dium^ in each of the places thereof, by which a body may describe a given spiral.

From the centripetal force the velocity in each place must be found ; then from the retardation of the velocity the density of the medium is found, as in the foregoing Proposition.

But I have explained the method of managing these Problems in the tenth Proposition and second Lemma of this Book- and will no longer detain the reader in these perplexed disquisitions. I shall now add some things relating to the forces of progressive bodies, and to the density and resistance of those mediums in which the motions hitherto treated of, and those akin to them, are performed.

Sec. v.] of natural philosophy. 293

SECTION Y. Of the density and compression of fluids ; and of hydrostatics.

THE DEFINITION OF A FLUID.

A fluid is any body whose parts yield to any force impressed on it, and, hy yielding, are easily Tnoved among themselves.

PROPOSITION XIX. THEOREM XIV.

All the parts of a homogeneous and unmoved fluid included in any un- moved vessel, and compressed on every side [setting aside the consider- ation of conde7isation, gravity, and all centripetal forces), will he equally pressed on every side, and remain in their places without any •motion arising fi'om that pressure. Case 1. Let a fluid be included in the spherical vessel ABC, and uniformly compressed on every side : I say, that no part of it will be moved by that pressure. For if any part, as D, be moved, all such parts at the same distance from the centre on every side must necessarily be moved at the same time by a like motion ; because the pressure of them all is similar and equal ; and all other motion is excluded that does not arise from that pressure. But if these parts come all of them nearer to the centre, the fluid must be condensed towards the centre, contrary to the supposition. If they recede from it, the fluid must be condensed towards the circumfer- ence ; which is also contrary to the supposition. Neither can they move in any one direction retaining their distance from the centre, because for the same reason, they may move in a contrary direction ; but the same part cannot be moved contrary ways at the same time. Therefore no part of the fluid will be moved from its place. Q..E.D.

Case 2. I say now, that all the spherical parts of this fluid are equally pressed on every side. For let EF be a spherical part of the fluid ; if this be not pressed equally on every side, augment the lesser pressure till it be pressed equally on every side ; and its parts (by Case 1) will remain ia their places. But before the increase of the pressure, they would remain in their places (by Case 1) ; and by the addition of a new pressure they will be moved, by the definition of a fluid, from those places. Now these two conclusions contradict each other. Therefore it was false to say that the sphere EF was not pressed equally on every side. Q,.E.D.

Case 3. I say besides, that different spherical parts have equal pressures. For the contiguous spherical parts press each other mutually and equally in the point of contact (by Law III). But (by Case 2) they are pressed on every side with the same force. Therefore any two spherical parts not

294 THE MATHEMATICAL PRINCIPLES fBoOK 11.

contiguouSj since an intermediate spherical part can toucli both, will be pressed with the same force. Q,.E.D.

Case 4. I say now, that all the parts of the fluid are every where press- ed equally. For any two parts may be touched by spherical parts in any points whatever ; and there they will equally press those spherical parts (by Case 3), and are reciprocally equally pressed by them (by Law III). Q.E.D.

Case 5. Since, therefore, any part GHI of the fluid is inclosed by the Test of the fluid as in a vessel, and is equally pressed on every side ; and also its parts equally press one another, and are at rest among themselves ; it is manifest that all the parts of any fluid as GHI, which is pressed equally on every side, do press each other mutually and equally, and are at rest among themselves. Q.E.D.

Case 6. Therefore if that fluid be included in a vessel of a yielding substance, or that is not rigid, and be not equally pressed on every side, the same will give way to a stronger pressure, by the 'Definition of fluidity.

Case 7. And therefore, in an inflexible or rigid vessel, a fluid will not sustain a stronger pressure on one side than on the other, but will give way to it, and that in a moment of time ; because the rigid side of the vessel does not follow the yielding liquor. But the fluid, by thus yielding, will press against the opposite side, and so the pressure will tend on every side to equality. And because the fluid, as soon as it endeavours to recede from the part that is most pressed, is withstood by the resistance of the vessel on the opposite side, the pressure will on every side be reduced to equality, in a moment of time, without any local motion : and from thence the parts of the fluid (by Case 5) will press each other mutually and equal- ly, and be at rest among themselves. Q.E.D.