Principia Mathematica (Motte Translation, 1848) — part 20 of 45

as ABMI; IMNK, KNOL, &c., and the absolute forces AC, IC, KC, LC, &.C., will be in a geometrical progression. Q.E.D. And by a like rea- soning, in the ascent of the body, taking, on the contrary side of the point A, the equal areas ABmi, imnk, knol, &c., it will appear that the absolute forces AC, iC, kC, IG, &c,, are continually proportional. Therefore if all the spaces in the ascent and descent are taken equal, all the absolute forces IC, kC, iC, AC, IC, KC, LC, (fee, will be continually proportional. Q.E.D.

Cor. 1. Hence if the space described be expounded by the hyperbolic area ABNK, the force of gravity, the velocity of the body, and the resist- ance of the medium, may be expounded by the lines AC, AP, and AK re- spectively ; and vice versa.

Cor. 2. And the greatest velocity which the body can ever acquire in an infinite descent will be expounded by the line AC.

Cor. 3. Therefore if the resistance of the medium answering to any given velocity be known, the greatest velocity will be found, by taking it to that given velocity in a ratio subduplicate of the ratio which the force of gravity bears to that known resistance of the medium.

PROPOSITIOxN IX. THEOREM YII.

Supposing what is above demonstrated, I say, that if the tangents of the angles of the sector of a circle, and of an hyperbola, be taken propor- tional to the velocities, the radius being of a fit magnitude, all the time of the ascent to the highest place loill be as the sector of the circle, and all the time of descending from the highest place as tJie sector of the hyperbola.

To the right line AC, which ex- presses the force of gravity, let AD be \ drawn perpendicular and equal. From the centre D -with the semi-diameter AD describe as well the quadrant A/E of a circle, as the rectangular hyper- bola AYZ, whose axis is AK, principal vertex A, and asymptote DC. Let Dp, DP be drawn ; and the circular sector A^D will be as all the time of the as- cent to the highest place ; and the hy- perbolic sector ATD as all the time of descent from the highest place; if so be that the tangents Kp, AP of those sectors be as the velocities.

Case 1. Draw )vq cutting oif the moments or least particles tX^v and q\yp, described in the same time, of the sector AD/ and of the triangle ADjO. Since those particles (because of the common angle D) are in a du- plicate ratio of the sides, the particle iDv will be as ?2^1^ that is

265

THE MATHEMATICAL PRINCIPLES

[Book II.

(because tD is given), as

qDp

But^D^ is AD2 + A;?2, that is, AD^ +

particle of the sector, is as p, ,

AD X Ak, or AD X Ck ; and qDp is I AD X pq. Therefore tBv, the

pq

■ that is, as the least decrement pq of the

velocity directly, and the force Ck which diminishes the velocity, inversely ; and therefore as the particle of time answering to the decrement of the ve- locity. And, by composition, the sum of all the particles tDv in the sector AD^ will be as the sum of the particles of time answering to each of the lost particles pq of the decreasing velocity Ap, till that velocity, being di- minished into nothing, vanishes ; that is, the whole sector AD^ is as the whole time of ascent to the highest place. Q.E.D.

Case 2. Draw DQV cutting off the least particles TDV and PDQ of the sector DAY, and of the triangle DAQ, ; and these particles will be to each other as DT^ to DP% that is (if TX and AP are parallel), as DX^ to DA2 or TX2 to AP- ; and, by division, as DX^ — TX^ to DA^ — AP^ But, from the nature of the hyperbola, DX^ — TX^ isAD^ ; and, by the supposition, AP^ is AD X AK. Therefore the particles are to each other as AD-' toAD^— ADx AK; that is, as AD to AD — AK or AC

PDQ, X AC

to CK : and therefore the particle TDV of the sector is

CK

and

Pa

therefore (because AC and AD are given) as ^:^ • that is, as the increment

of the velocity directly, and as the force generating the increment inverse- ly ; and therefore as the particle of the time answering to the increment. And, by composition, the sum of the particles of time, in which all the par- ticles PQ, of the velocity AP are generated, will be as the sum of the par- ticles of the sector ATD ; that is, the whole time will be as the whole sector. Q.E.D.

Cor. 1. Hence if AB be equal to a fourth part of AC, the space which a body will describe by falling in any time will be to the space which the body could de- scribe, by moving uniformly on in the same time with its greatest velocity AC, as the area ABNK, which ex- presses the space described in falling to the area ATD, which expresses the time. For since AC is to AP as AP _ to AK, then (by Cor. 1, Lem. II, of this

Book) LK is to PQ as 2AK to AP, that is, as 2AP 'to AC, and thence LK is to iPQ as AP to ^AC or AB ; and KN is to AC or AD as AB to

Sec. II.] OF NATURAL PHILOSOPHY. 267

, CK ; and therefore, ex cequo, LKNO to DPGl as AP to CK. But DPQ was to DTY as CK to AC. Therefore, ex cequo, LKNO is to DTV as

AP to AC ; that is, as the velocity of the falling body to the greatest velocity which the body by falling can acquire. Since, therefore, the moments LKNO and DTV of the areas ABNK and ATD are as the ve- locities, all the parts of those areas generated in the same time will be as the spaces described in the same time ; and therefore the whole areas ABNK and ADT, generated from the beginning, will be as the whole spaces de- scribed from the beginning of the descent. Q,.E.D.

Cor. 2. The same is true also of the space described in the ascent. That is to say, that all that space is to the space described in the same time, with the uniform velocity AC, as the area ABnk is to the sector ADif.

Cor. 3. The velocity of tlie body, falling in the time ATD, is to the velocity which it would acquire in the same time in a non-resisting space, as the triangle APD to the hyperbolic sector ATD. For the velocity in a non-resisting medium would be as the time ATD, and in a resisting me- dium is as AP, that is, as the triangle APD. And those velocities, at the beginning of the descent, are equal among themselves, as well as those areas ATD, APD.

Cor. 4. By the same argument, the velocity in the ascent is to the ve- locity with which the body in the same time, in a non-resisting space, would lose all its motion of ascent, as the triangle Aj!9D to the circular sector A^D ; or as the right line A.p to the arc At.

Cor. 5. Therefore the time in which a body, by falling in a resisting medium, would acquire the velocity AP, is to the time in which it would acquire its greatest velocity AC, by falling in a non-resisting space, as the sector ADT to the triangle ADC : and the time in which it would lose its velocity A^, by ascending in a resisting medium, is to the time in which it would lose the same velocity by ascending in a non-resisting space, as the arc At to its tangent A.p.

Cor. 6. Hence from the given time there is given the space described in the ascent or descent. For the greatest velocity of a body descending in infinitum is given (by Corel. 2 and 3, Theor. YI, of this Book) ; and thence the time is given in which a body would acquire that velocity by falling in a non-resisting space. And taking the sector ADT or AD^ to the tri- angle ADC in the ratio of the given time to the time just now found, there will be given both the velocity AP or Ap, and the area ABNK or ABn^, which is to the sector ADT, or AD^, as the space sought to the space which would, in the given time, be uniformly described with that greatest velocity found just before.

Cor. 7. And by going backward, from the given space of ascent or de- scent ABnIc or ABNK, there will be given the time AD^ or ADT.

268

1

THE MATHEMATICAL PRINCIPLES

[Book IL

PROPOSITION X. PROBLEM III.

Suppose the uniform force of gravity to tend directly to the plane of the horizon, aiid the resistance to be as the density of the medium and the square of the velocity conjunctly : it is proposed to find the density of the Qnedivm in each place, which shall make the body move in any given curve line ; the velocity of the body and the resistance of the inedium, in each vlace.

Let PQ be a plane perpendicular to the plane of the scheme itself ; PFHQ, a curve line meeting tliat plane in the points P and Q : G, H, I, K four places of the body going on in this curve from F to Q, ; and GB, HC, ID, KE four parallel ordinates let fall P .^ Bcx) jt; q from these points to the horizon, and

standing on the horizontal line PQ. at the points B, C, D, E ; and let the distances BC, CD, DE, of the ordinates be equal among themselves. From the points G and H let the right lines GL. HN, be drawn touching the curve in G and H, and meeting the ordinates CH, DI, produced upwards, in L and N : and complete the parallelogram HO DM. And the times in which the body describes the arcs GH, HI, will be in a subduplicate ratio of the altitudes LH, NL which the bodies would describe in those times, by falling from the tangents ; and the velocities will be as the lengths de- Bcribed GH, HI directly, and the times inversely. Let the times be ex-

r^TT TTT

pounded by T and t, and the velocities by -^ and — -; and the decrement

of the velocity produced in the time t will be expounded by 7^^ ^.

T — t

_ JHI

T t

This decrement arises from the resistance which retards the body, and from the gravity which accelerates it. Gravity, in a falling body, which in its fall describes the space NI, produces a velocity with which it would be able to describe twice that space in the same time, as Galileo has demonstrated ;

2NI

that is, the velocity —j- : but if the body describes the arc HI, it augments

t

that arc only by the length HI — HN or 2M1 X Nl

MI X NI HI

and therefore generates

only the velocity

Let this velocity be added to the before-

^XHI

mentioned decrement, and we shall have the decrement of the velocity

■ , . GH HI 2MIXNI

arising from the resistance alone, that is, -^ :: — h

t

txm

Sec. II.]

OF NATURAL PHILOSOPHY.

269

Therefore sincCj in the same time, the action of gravity generates, in a fall-

, . 2NI , . .,, , , . GH

lug body, the velocity — — , the resistance will be to the gravity as -7^

HI 2MI X NI 2NI t X GH

2MI X NI

to 2NI.

Now for the abscissas CB, CD, OE, put — 0, 0, 2o, For the ordinate CH put P ; and for MI put any series Qo + Ro^ + So^ +, &c. And all the terms of the series after the first, that is, Ro^ + So^ +, (fee, will be NI ; and the ordinates DI, EK, and

BGwillbeP — Gio — Ro^ — So^ — ,-p ^. b c d js q

(fee, P — 2ao — 4Ro2— SSo^^— , (fee, and P + Qo — Ro^ + So»—, (fee, respectively. And by squaring the differences of the ordinates BG — CH and CH — DI, and to the squares thence produced adding the squares of BC and CD themselves, you will have 00 + QQoo — 2Q,Ro^ +, (fee, and 00 + dQoo + 2Q.Y,o^ +, (fee, the squares of the arcs GH, HI ; whose

Q,Roo QRoo

roots 0 v/ ■ ^=., ando ^/1 _u qo j- -^r are the

arcs GH and HI. Moreover, if from the ordinate CH there be subducted half the sum of the ordinates BG and DI, and from the ordinate DI there be subducted half the sum of the ordinates CH and EK, there will remain Roo and Roo + 3So^, the versed sines of the arcs GI and HK. And these are proportional to the lineolse LH and NI, and therefore in the duplicate

ratio of the infinitely small times T and t : and thence the ratio jy, is V

R + 3So R + lSo or

R

the values of

R

and

^XGH T

HI +

2MI X NI HI '

by substituting

GH, HI, MI and NI just found, becomes

3Soo 2R

V

1 + 0,0-. And since 2NI is 2Roo, the resistance will be now to the gravity as -^^ ^1 -f~QQ^ to 2Roo, that is, as 3S ^i^^TQQ to 4RR.

And the velocity will be such, that a body going off therewith from any place H, in the direction of the tangent HN, would describe, in vacuo, a

parabola, whose diameter is HC, and its latus rectum ^^ or — ^ .

And the resistance is as the density of the medium and the square of the velocity conjunctly ; and therefore the density of the medium is as the resistance directly, and the square of the velocity inversely ; that is, as

270 THE MATHEMATICAL PRINCIPLES [BoOK 11.

s

3S_v^^-f QA ^ij,g^ti ^^^ 1+39: inYersely : that is, as -

4RR R ^' Rv/l + QQ,-

GI.E.L

Cor. 1. If the tangent HN be produced both ways, so as to meet any HT ordinate AF in T -^ will be equal to ^ i + qq, and therefore in what

has gone before may be put for x/ 1 + Q.Q,. By this means the resistance will be to the gravity as 3S X HT to 4RR X AC ; the velocity will be as

-r-7q — 7^, and the density of the medium will be as .5 ^y^.

Cor. 2. And hence, if the curve line PFHQ, be defined by the relation between the base or abscissa AC and the ordinate CH, as is usual, and the value of the ordinate be resolved into a converging series, the Problem will be expeditiously solved by the first terms of the series ; as in the fol- lowing examples.

Example 1. Let the line PFHQ. be a semi-circle described upon the diameter PQ,, to find the density of the medium that shall make a projec- tile move in that line.

Bisect the diameter PQ in A ; and call AQ, n ; AC, a ; CH, e ; and CD, 0 ; then DI^ or AQ,^ — AD ^ = vn — aa — 2ao — 00, or ee — 2ao

— 00 ; and the root being extracted by our method, will give DI = e — ao 00 aaoo ao^ a^ 0^

• — -^- — -^ ~2^ — -^5 > <^c- Here put nn for ee -f aa, and

__. .,, , ao nnoo anno^ ^

Dl will become = e 77— 7^- , &c.

Such series I distinguish into successive terms after this manner : I call that the first term in which the infinitely small quantity 0 is not found ; the second, in which that quantity is of one dimension only ; the third, in which it arises to two dimensions ; the fourth, in which it is of three ; and so ad wfinitum. And the first term, which here is e, will always denote the length of the ordinate CH, standing at the beginning of the indefinite

quantity 0. The second term, which here is — , will denote the difiierence

between CH and DN ; that is, the lineola MN which is cut off by com- pleting the parallelogram HC DM; and therefore always determines the

position of the tangent HN ; as, in this case, by taking MN to HM as —

to 0, or a to e. The third term, which here is -g-^, will represent the li- neola IN, which lies between the tangent and the curve ; and therefore determines the angle of contact IHN, or the curyature which the curve line

Sec. ILJ

OF NATURAL PHILOSOPHY.

271

has in H. If that lineola IN is of a finite magnitude, it will be expressed by the third term, together with those that follow in infinitum. But if that lineola be diminished in ivfini- tu?n, the terms following become in- finitely less than the third term, and therefore may be neglected. The fourth term determines the variation of ihe curvature ; the fifth, the varia- tion • of the variation ; and so on.

Whence, by the way, appears no con- p" ^ b c d e Q

temptible use of these series in the solution of problems that depend upon tano^ents, and the curvature of curves.

■ . ao ?inoo a7ino^ • i i

Now compare the series e — ^-^ ^-^ occ, with the

cii mi &c., and for P, Q., R and S, put e, -, ^-^

series P — Q,o — Roo — So '

ann

and ^ , and for ^ 1 + Q,Q, put

V

aa

1 H or -I; and the density of

ee

the medium will come out as — : that is (because n is ffiven), as - or

pPPj, that is, as that length of the tangent HT, which is terminated at the

semi-diameter AF standing perpendicularly on PQ, : and the resistance will be to the gravity as 3a to 2?i, that is, as 3AC to the diameter PQ, of the circle ; and the velocity will be as v/ CH. Therefore if the body goes from the place F, with a due velocity, in the direction of a line parallel to PQ, and the density of the medium in each of the places H is as the length of the tangent HT, and the resistance also in any place H is to the force of gravity as 3AC to PQ,, that body will describe the quadrant FHQ, of a circle. Q.E.I.

But if the same body should go from the place P, in the direction of a line perpendicular to PQ., and should begin to move in an arc of the semi- circle PFQ,, we must take AC or a on the contrary side of the centre A ; and therefore its sign must be changed, and we must put — a for + a.

a

Then the density of the medium would come out as . But nature

does not admit of a negative density, that is, a density which accelerates the motion of bodies ; and therefore it cannot naturally come to pass that a body by ascending from P should describe the quadrant PF of a circle. To produce such an effect, a body ought to be accelerated by an impelling medium, and not impeded by a resisting one. Example 2. Let the line PFQ, be a parabola, having its axi3 AF per-

272

THE MATHEMATICAL PRINCIPLES

[Book 11.

pendicular to the horizon PQ,, to find the density of the medium, which will make a projectile move in that line.

^ ^ From the nature of the parabola, the rectangle PDQ,

^^ is equal to the rectangle under the ordinate DI and some

given right line ; that is, if that right line be called b ;

PC, a; PQ,, c; CH, e; and CD, o ; the rectangle a

P A. CD Q + 0 into c — a — o or ac — aa — 2ao + co — oo, is

equal to the rectangle b into DI, and therefore DI is equal to c

ac

aa.

-2a 00 , c — 2a . , . . . ,

7 — ^ A • ^^ second term — r — o of this series is to be put

00

for 0,0, and the third term j- for Roo.

But since there are no more

terms, the co-efficient S of the fourth term will vanish ; and therefore the

S

quantity z=z _, to which the density of the medium is propor-

R v' 1 + 0,0.

tional, will be nothing. Therefore, where the medium is of no density, the projectile will move in a parabola ; as Galileo hath heretofore demon- strated. O.E.L

Example 3. Let the line AGK be an hyperbola, having its asymptote NX perpendicular to the horizontal plane AK, to find the density of the medium that will make a projectile move in that line.

Let MX be the other asymptote, meeting the ordinate DG produced in Y ; and from the nature of the hyperbola, the rectangle of XV into VG will be given. There is also given the ratio of DN to VX, and therefore the rectangle of DN into VG is given. Let that be bb: and, completing the parallelo- gram DNXZ, let BN be called a; BD, o; NX, c; and let the given ratio of VZ to

ZX or DN be -. Then DN will be equal

to a — 0, VG equal to

n m

a — 0'

7Jl 771

•VZ—VG equal to c a H o —

VZ equal to — X a

Let the term

n bb

0, and GD or NX bb

be

71 71

. bb bb bb bb ^ .

resolved into the converffinor series 1 o -] — - oo -\ — - o^, &c., and

GD will become equal to c

m

a aa a^ ^

bb m hh bb bb

771

-+-0 a 71

aa

a-

Sec. II.]

OF NATURAL PHILOSOPHY.

273

&c. The second term — o — — o of this series is to be used for 0,0 ; the n aa

third — o^j with its sign changed for Ro^ ; and the fourth — o'j with its

m bh bb bb sign changed also for So^, and their coemcients , — and — are to

be put for Q,, R, and S in the former rule. Which being done, the den-

bb

sity of the medium will come out as

bh ^

mm nn

2mbb

naa

'2:^^ ^~, that is, if in VZ you take VY equal to

■ +

2mhb ft 4

-1 are the squares of XZ

V' mm

aa -\ aa

nn n aa

VG, as r^T^. For aa and -— - «^ — ' XY n^

and ZY. But the ratio of the resistance to gravity is found to be that of

3XY to 2YG ; and the velocity is that with which the body would de-

XY^

scribe a parabola, whose vertex is G, diameter DG, latus rectum ^„ . Sup- pose, therefore, that the densities of the medium in each of the places G are reciprocally as the distances XY, and that the resistance in any place G is to the gravity as 3XY to 2YG ; and a body let go from the place A, with a due velocity, will describe that hyperbola AGK. Q.E.I.

Example 4. Suppose, indefinitely, the line AGK to be an hyperbola described with the centre X, and the asymptotes MX, NX, so that, having constructed the rectangle XZDN, whose side ZD cuts the hyperbola in G and its asymptote in V, YG may be reciprocally as any power DN" of the line ZX or DN, whose index is the number n : to find the density of the medium in which a projected body will describe this curve.

For BN, BD, NX, put A, O, C, respec- tively, and let VZ be to XZ or DN as d to

bb e, and VG be equal to

DN'^'

be equal to A — O, VG ^=. - d

then DN will

bb ==, VZ =

Of

A — O, and GD or NX — VZ — VG equal

bb

toG-^A + ^0-.

e e A — 0|°

=— . Let the

18

274

term

THE MATHEMATICAL PRINCIPLES

hh 11. . ,. . . bb . nbb z=— be resolved into an infinite series -?- + -,

A — 01 A A° + ^

[Book II X O +

2An"jrT X ^^ 02 + ^^„ 3 X bb 03,(fec,,andGD will be equal

toC— -A-TT+-0- .„ . . O - v.., „ bb 02 -

A°

2A°+ 2

71 3 -f- 3/1», +2?^ 6A"+ 2

^

?i65

550 3, &c. The second term - O — . "'"," , O of this ' e A" + '

series is to be used for Q,o, the third ^ .„ . „ 550 ^ for Koo, the fourth

2A»+ 2

n^ + 3ra?. + 2n

550^ for So 3. And thence the density of the medium

71 -{-2

6A" + 3

S ^

^ V. r^r, in any place G, will be o ~

c?(Z 2dnbb nnb^

and therefore if in YZ you take Y Y equal to n X YG^ that density is re-

-. ^^^ _, . dd 2dnbb ^ nnh'^

ciprocally as XY. For A^ and — A^ —- A + - — are the

^ -^ ee eA" A^"

squares of XZ and ZY. But the resistance in the same place G is to the

XY 2nv + 2?! force of gravity as 3S X —r- to 4RR; that is, as XY to '- — YG.

And the velocity there is the same wherewith the projected body would

move in a parabokj whose vertex is G, diameter GD, and latus rectum

1 + aa 2XY2

or == . GI.E.I.

R

nn + nx YG

SCHOLIUM.

In the same manner that the den- sity of the medium comes out to be as

I^ -5 rrrp; ^"^ ^o^- Ij if ^^10 rcsistauce

' is put as any power Y° of the velocity

Y, the density of the medium will

S

come out to be as

B C D E Q

4-n

R-2—

And therefore if a curve can be found, such that the ratio of

R-

4 — D

-to

Sec. IL]

OF NATURAL PHILOSOPHY.

275

HT

AC

0^ of p4_ „ to ' ^^' may lo^ given ; the body, in an uni-

R

will

form medium, whose resistance is as the power V" of the velocity V, move in this curve. But let us return to more simple curves.

Because there can be no motion in a para- bola except in a non-resisting medium, but in the hyperbolas here described it is produced by a perpetual resistance ; it is evident that the line which a projectile describes in an uniformly resisting medium approaches nearer to these hyperbolas than to a parabola. That line is certainly of the hyperbolic kind, but about the vertex it is more distant from the asymptotes, and in the parts remote from the

vertex draws nearer to them than these hy- I^A. BDX~K

perbolas here described. The difference, however, is not so great between the one and the other but that these latter may be commodiously enough used in practice instead of the former. And perhaps these may prove more useful than an hyperbola that is more accurate, and at the same time more compounded. They may be made use of, then^ in this manner.

Complete the parallelogram XYGT, and the right line GT will touch the hyperbola in G, and therefore the density of the medium in G is re-

ciprocally as the tangent GT, and the velocity there as a/

GT^ GV

and the

resistance is to the force of gravity as GT to

Therefore if a body projected from the place A, in the direction of the right line AH, describes the hyperbola AGK and AH produced meets the asymptote NX in H, and AI drawn parallel to it meets the other asymptote MX in I ; the density of the medium in A will be reciprocally as AH, and the velocity of the body as ^/

AH^

'-^ , and the resistance there to the force AI

27m + 2n

n+2~

X GY.

of gravity as AH to

2nn + 2n

X AI. Hence the

following

rules are

n --2 deduced.

Rule 1. If the density of the medium at A, and the velocity with which the body is projected remain the same, and the angle NAH be changed ; the lengths AH, AI, HX will remain. Therefore if those lengths, in any

276 • THE MATHEMATICAL PRINCIPLES [BoOK 11.

one case, are found, the hyperbola may afterwards be easily determined from any given angle NAH.

Rule 2. If the angle NAH, and the density of the medium at A, re- main the same, and the velocity with which the body is projected be changed, the length AH will continue the same ; and AI will be changed in a duplicate ratio of the velocity reciprocally. -

Rule 3. If the angle NAH, the velocity of the body at A, and the ac- celera^tive gravity remain the same, and the proportion of the resistance at A to the motive gravity be augmented in any ratio ; the proportion of AH to A I will be augmented in the same ratio, the latus rectum of the above-

AH^

mentioned parabola remaining the same, and also the length propor-

tional to it ; and therefore AH will be diminished in the same ratio, and AI will be diminished in the duplicate of that ratio. But the proportion of the resistance to the weight is augmented, when either the specific grav- ity is made less, the magnitude remaining equal, or when the density of the medium is made greater, or when, by diminishing the magnitude, the resistance becomes diminished in a less ratio than the weight.

Rule 4. Because the density of the medium is greater near the vertex of the hyperbola than it is in the place A, that a mean density may be preserved, the ratio of the least of the tangents GT to the tangent AH ought to be found, and the density in A augmented in a ratio a little greater than that of half the sum of those tangents to the least of the tangents GT.

Rule 5. If the lengths AH, AI are given, and the figure AGK is to be described, produce HN to X, so that HX may be to AI as n + 1 to 1 ; and with the centre X, and the asymptotes MX, NX, describe an hyperbola through the point A, such that AI may be to any of the lines VG as XY° to XP.

Rule 6. By how much the greater the number n is, so much the more accurate are these hyperbolas in the ascent of the body from A, and less accurate in its descent to K ; and the contrary. The conic hyperbola keeps a mean ratio between these, and is more simple than the rest. There- fore if the hyperbola be of this kind, and you are to find the point K, where the projected body falls upon any right line AN passing through the point A, let AN produced meet the asymptotes MX, NX in M and N, and take NK equal to AM.

Rule 7. And hence appears an expeditious method of determining this hyperbola from the phenomena. Let two similar and equal bodies be pro- jected with the same velocity, in different angles HAK, hAk, and let them fall upon the plane of the horizon in K and k ; and note the proportion of AK to Kk, Let it be as d to e. Then erecting a perpendicular AI of any lengthy assume any how the length AH or AA, and thence graphically,

Sec. IL]

OF NATURAL PHILOSOPHY.

277

or by scale and compass, collect the lengths AK, Ak (by Rule 6). If the ratio of AK to Ak be the same with that of d to e, the length of AH was

^K nr

1

"t

l^-M^

rightly assumed. If not, take on the indefinite right line SM, the length SM equal to the assumed AH ; and erect a perpendicular MN equal to the

diiFerence -r-y of the ratios drawn into any given right line. By the

like method, from several assumed lengths AH, you may find several points N ; and draw through them all a regular curve NNXN, cutting the right line SMMM in X. Lastly, assume AH equal to the abscissa SX, and thence find again the length AK ; and the lengths, which are to the as- sumed length AI, and this last AH, as the length AK known by experi- ment, to the length AK last found, will be the true lengths AI and AH, which were to be found. But these being given, there will be given also the resisting force of the medium in the place A, it being to the force of gravity as AH to f AI. Let the density of the medium be increased by Rule 4, and if the resisting force just found be increased in the same ratio, it will become still more accurate.

Rule 8. The lengths AH, HX being found ; let there be now re- quired the position of the line AH, according to which a projectile thrown with that given velocity shall fall upon any point K. At the points A and K, erect the lines AC, KF perpendicular to the horizon ; whereof let AC be drawn downwards, and be equal to AI or JrHX. With the asymp- totes AK, KP, describe an hyperbola, whose conjugate shall pass through the point C ; and from the centre A, with the interval AH. describe a cir- cle cutting that hyperbola in the point H ; then the projectile thrown in th?e direction of the right line AH will fall upon the point K. Q,.E.I. For the point H, because of the given length AH, must be some^here in the circumference of the described circle. Draw CH meeting AK and KF in E and F ; and because CH, MX are parallel, and AC, AI equal, AE will be equal to AM, and therefore also equal to KN. But CE is to AE as FH to KN, and therefore CE and FH are equal. Therefore the point H falls upon the hyperbolic curve described with the asymptotes AK, KP, whose conjugate passes through the point C ; and is therefore found in the

278

THE MATHEMATICAL PRINCIPLES

[Book II,

common intersection of this hyperbolic curve and the circumference of the de- scribed circle. Q.E.D. It is to be ob- served that this operation is the same, whether the right line AKN be parallel to the horizon, or inclined thereto in any an- o-le ; and that from two intersections H, h, there arise two angles NAH, NA^; and that in mechanical practiae it is suf- ficient once to describe a circle, then to apply a ruler CH, of an indeterminate lengch, so to the point C, that its part FH, intercepted between the circle and the right line FK, may be equal to its part CE placed between the point C and the right line AK.

"What ha^ been said of hyperbolas may be easily applied to parabolas. For if a parabola be re- presented by XAGK, touched by a right line XV in the vertex X, and the ordinates lA, YG be as any powers XI", XY", of the abscissas XI, XY ; draw XT, GT, AH, whereof let XT be parallel to YG, and let GT, AH touch the parabola in G and A : and a body projected from any place A, in the direction of the right line AH, Avith a due velocity, will describe this parabola, if the density of the medium in each of the places G be reciprocally as the tangent GT. In that case the velocity in G will be the same as would cause a body, moving in a non- resisting space, to describe a conic parabola, having G for its vertex, YG

produced downwards for its diameter, and for its latus

nil — n X YG

rectum. And the resisting force in G will be to the force of gravity as GT to

5— YG, Therefore if NAK represent an horizontal line, and both

the density of the medium at A, and the velocity with which the body is projected, remaining the same, the angle NAH be any how altered, the lengths AH, AI, HX will remain ; and thence will be given the vertex X of the parabola, and the position of the right line XI ; and by taking YG to lA as XY° to XI", there will be given all the points G of the parabola, through which the projectile will pass.

Sec. III.] OF NATURAL PHILOSOPHY. 279

SECTION III.

Of the motions of bodies which are resisted partly in the ratio of the ve- locities, and partly in the duplicate of the same ratio.

PROPOSITION XL THEOREM YIII.

If a body be resisted partly in the ratio and partly in the duplicate ratio of its velocity, and moves in a similar m^edium by its innate force only; and the times be taken in arithmetical progressioti ; then quantities reciprocally proportional to the velocities, incj^eased by a cer- tain given quantity, loill be in geometrical progression. With the centre 0, and the rectangular asymptotes

CADo? and CH, describe an hyperbola BEe, and let

AB, DE, de, be parallel to the asymptote CH. In

the asymptote CD let A, G be given points ; and if

the time be expounded by the hyperbolic area ABED

uniformly increasing, I say, that the velocity may iq-

be expressed by the length DP, whose reciprocal

GD, together with the given line CG, compose the ^

length CD increasing in a geometrical progression.

For let the areola DEe<i be the least given increment of the time, and

"Dd will be reciprocally as DE, and therefore directly as CD. Therefore

the decrement of ^^, which (by Lem. 11, Book II) is ttft^j will be also as

^-r^„ or — T^^- — J that is, as ;:;rFr + T^Ti^^. Therefore the time ABED GD^ GD^ GU GD^

uniformly increasing by the addition of the given particles EDc?^, it fol- lows that pYi ^iecreases in the same ratio with the velocity. For the de- crement of the velocity is- as the resistance, that is (by the supposition), as the sum of two quantities, whereof one is as the velocity, and the other as

the square of the velocity ; and the decrement of ^^s ^^ ^^ *^® sum of the

quantities ^TpT and ttt^j whereof the first is pj=r itself, and the last

CG 1 1

^T^„ is as 7z^i^„ : therefore t^^r is as the velocity, the decrements of both

GD2 GD^ GD -^'