proportion among themselves. Let the angles FGH, GHI, be so far in
creased that the right lines FG, GH, HI, may lie in directum ; and by
constructing the Problem in this case, a right line fghi will be drawn,
whose parts fg, gh, hi, intercepted between the four right lines given by
position, AB and AD, AD and BD, BD and CE, will be one to another
as the lines FG, GH, HI, and will observe the same order among them
selves. But the same thing may be more readily done in this manner.
Produce AB to K and BD to L,
so as BK may be to AB as HI to
GH ; and DL to BD as GI to FG;
and join KL meeting the right line
CE in i. Produce iL to M, so as
LM may be to iL as GH to HI ;
then draw MQ, parallel to LB, and
meeting the right line AD in g, and
join gi cutting AB, BD in f, h ;
I M*
say, the thing is done.
For let MO- cut the right line AB in Q, and AD the right line KL iu
II
^52 THE MATHEMATICAL PRINCIPLES [BOOK I.
S, arid draw AP parallel to BD, and meeting iL in P, and -M to Lh (g\
to hi, Mi to Li, GI to HI, AK to BK) and AP to BL, will be in the same
ratio. Cut DL in 11, so as DL to RL may be in that same ratio; and be
cause ffS to g~M, AS to AP. and DS to DL are proportional; therefore
(ex ceqit.o) as gS to LA, so will AS be to BL, and DS to RL ; and mixtly.
BL RL to Lh BL, as AS DS to gS AS. That is, BR is to
Eh as AD is to Ag, and therefore as BD to gQ. And alternately BR is
to BD as 13/i to g-Q,, or asfh to fg. But by construction the line BL
was cut in D and R in the same ratio as the line FI in G and H ; and
therefore BR is to BD as FH to FG. Wherefore fh is to fg as FH to
FG. Since, therefore, gi to hi likewise is as Mi to Li, that is, as GI to
HI, it is manifest that the lines FI, fi, are similarly cut in G and H, g
and //.. Q.E.F.
In the construction of this Corollary, after the line LK is drawn cutting
CE in i, we may produce iE to V, so as EV may be to Ei as FH to HI,
arid then draw V/~ parallel to BD. It will come to the same, if about the
centre i with an interval IH, we describe a circle cutting BD in X, and
produce iX to Y so as iY may be equal to IF, and then draw Yf parallel
to BO.
Sir Christopher Wren and Dr. Wallis have long ago given other solu
tions of this Problem.
PROPOSITION XXIX. PROBLEM XXI.
To describe a trajectory given in kind, that may be cut by four right
lines given by position, into parts given in order, kind, and proportion.
Suppose a trajectory is to be described that may be
similar to the curve line FGHI, and whose parts,
similar and proportional to the parts FG, GH, HI of
the other, may be intercepted between the right lines
AB and AD, AD, and BD, BD and CE given by po
sition, viz., the first between the first pair of those lines,
the second between the second, and the third between
the third. Draw the right lines FG, GH, HI, FI;
and (by Lem. XXVII) describe a trapezium fghi that
may be similar to the trapezium FGHI, and whose an
gles/, g, h, i, may touch the right lines given by posi
tion AB, AD, BD, CE, severally according to their order. And then about
bins trapezium describe a trajectory, that trajectory will be similar to the
curve line FGHI.
SCHOLIUM.
This problem may be likewise constructed in the following manner.
Joining FG, GH, HI, FI, produce GF to Y, and join FH, IG, and make
SEC. VI OF NATURAL PHILOSOPHY. 153
El
the angles CAK. DAL equal to
the angles PGH, VFH. Let
AK, AL meet the right line
BD in K and L, and thence
draw KM, LN, of which let
KM make the angle AKM equal
to the angle CHI, and be itself
to AK as HI is to GH ; and let
LN make the angle ALN equal to the angle FHI, and be itself
to AL as HI to FH. But AK, KM. AL, LN are to be drawn
towards those sides of the lines AD, AK, AL, that the letters
OA.KMC, ALKA, DALND may be carried round in the same
order as the letters FGHIF ; and draw MN meeting the right v
line CE in L Make the angle iEP equal to the angle IGF,
and let PE be to Ei as FG to GI ; and through P draw PQ/ that may
with the right line ADE contain an angle PQE equal to the angle FIG,
and may meet the right line AB in /, and join fi. But PE and PQ arcto
be drawn towards those sides of the lines CE, PE, that the circular
order of the letters PEtP and PEQP may be the same as of the letters
FGHIF ; and if upon the line/i, in the same order of letters, and similar
to the trapezium FGHI, a trapezium /^//.i is constructed, and a trajectory
given in kind is circumscribed about it, the Problem will be solved.
So far concerning the finding of the orbits. It remains that we deter
mine the motions of bodies in the orbits so found.
SECTION VI.
How the motions are to be found in given, orbits.
PROPOSITION XXX. PROBLEM XXII.
To find at any assigned time the place of a body moving in, a given
parabolic trajectory.
Let S be the focus, and A the principal vertex of
the parabola; and suppose 4AS X M equal to the
parabolic area to be cut off APS, which either was
described by the radius SP, since the body s departure
from the vertex, or is to be described thereby before
its arrival there. Now the quantity of that area to
be cut off is known from the time which is propor
tional to it. Bisect AS in G, and erect the perpendicular GH equal to
3M, and a circle described about th j centre H, with the interval HS, will
cut the parabola in the place P required. For letting fall PO perpendic
ular on the axis, and drawing PH, there will be AG2
-f- GH2
(=.= HP2 -_
AO^TAGJ* + PO GH|2
) = AO2 + PO2 2CA > ?G!I f PO
A G S
154 THE MATHEMATICAL PRINCIPLES [BOOK I
AG* + GH2
. Whence 2GH X PO ( AO2 + PO2 2GAO) = AOJ
PO2
-f | PO2
. For AO2 write AO X ; then dividing all the terms by
2PO; and multiplying them by 2AS, we shall have ^GH X AS (= IAO
the area APO SPO)| = to the area APS. But GH was 3M, and
therefore ^GH X AS is 4AS X M. Wherefore the area cut off APS is
equal to the area that was to be cut off 4AS X M. Q.E.D.
Con. 1. Hence GH is to AS as the time in which the body described
the arc AP to the time in which the body described the arc between the
vertex A and the perpendicular erected from the focus S upon the axis.
COR. 2. And supposing a circle ASP perpetually to pass through the
moving body P, the xelocity of the point H is to the velocity which the
body had in the vertex A as 3 to 8; and therefore in the same ratio is
the line GH to the right line which the body, in the time of its moving
from A to P, would describe with that velocity which it had in the ver
tex A.
COR. 3. Hence, also, on the other hand, the time may be found in which
the body has described any assigned arc AP. Join AP, and on its middle
point erect a perpendicular meeting the right line GH in H,
LEMMA XXVIII.
There is no oval figure whose area, cut off by right lines at pleasure, can,
be universally found by means of equations of any number of finite
terms and dimensions.
Suppose that within the oval any point is given, about which as a pole
a right line is perpetually revolving with an uniform motion, while in
that right line a mov cable point going out from the pole moves always
forward with a velocity proportional to the square of that right line with
in the oval. By this motion that point will describe a spiral with infinite
circumgyrations. Now if a portion of the area of the oval cut off by that
right line could be found by a finite equation, the distance of the point
from the pole, which is proportional to this area, might be found by the
same equation, and therefore all the points of the spiral might be found
by a finite equation also ; and therefore the intersection of a right line
given in position with the spiral might also be found by a finite equation.
But every right line infinitely produced cuts a spiral in an infinite num
ber of points ; and the equation by which any one intersection of two lines
is found at the same time exhibits all their intersections by as many roots,
and therefore rises to as many dimensions as there are intersections. Be
cause two circles mutually cut one another in two points, one of those in
8FC. Vl.J OF NATURAL PHILOSOPHY. 155
terscctions is not to be found but by an equation of two dimensions, fo
which the other intersection may be also found. Because there may b(-
four intersections of two conic sections, any one of them is not to be found
universally, but by an equation of four dimensions, by which they may bi>
all found together. For if those intersections are severally sought, be
cause the law and condition of all is the same, the calculus will be the
same in every case, and therefore the conclusion always the same, which
must therefore comprehend all those intersections at once within itself, and
exhibit them all indifferently. Hence it is that the intersections of the
conic se"
f ions with the curves of the third order, because they may amount
to six, (\,me out together by equations of six dimensions ; and the inter
sections of two curves of the third order, because they may amount to nine,
come out together by equations of nine dimensions. If this did not ne
cessarily happen, we might reduce all solid to plane Problems, and those
higher than solid to solid Problems. But here i speak of curves irreduci
ble in power. For if the equation by which the curve is defined may bo
reduced to a lower power, the curve will not be one single curve, but com
posed of two, or more, whose intersections may be severally found by different
calculusses. After the same manner the two intersections of right lines
with the conic sections come out always by equations of two dimensions ; the
three intersections of right lines with the irreducible curves of the third
urder by equations of three dimensions ; the four intersections of right
lines with the irreducible curves of the fourth order, by equations of four
dimensions ; and so on in iitfinitum. Wherefore the innumerable inter
sections of a right line with a spiral, since this is but one simple curve
and not reducible to more curves, require equations infinite in r- .imber of
dimensions and roots, by which they may be all exhibited together. For
the law and calculus of all is the same. For if a perpendicular is let fall
from the pole upon that intersecting right line, and that perpendicular
together with the intersecting line revolves about the pole, the intersec
tions of the spiral will mutually pass the one into the other ; and that
which was first or nearest, after one revolution, will be the second ; after
two, the third ; and so on : nor will the equation in the mean time be
changed but as the magnitudes of those quantities are changed, by which
the position of the intersecting line is determined. Wherefore since those
quantities after every revolution return to their first magnitudes, the equa
tion will return to its first form ; and consequently one and the same
equation will exhibit all the intersections, and will therefore have an infi
nite number of roots, by which they may be all exhibited. And therefore
the intersection of a right line with a spiral cannot be universally found by
any finite equation ; and of consequence there is no oval figure whose area,
cut off by right lines at pleasure, can be universally exhibited by an^
such equation.
1 56 THE MATHEMATICAL PRINCIPLES [BOOK 1
By the same argument, if the interval of the pole and point by which
the spiral is described is taken proportional to that part of the perimeter
of the oval which is cut off, it may be proved that the length of the peri
meter cannot be universally exhibited by any finite equation. But here I
speak of ovals that are not touched by conjugate figures running out in
infinitvm.
COR. Hence the area of an ellipsis, described by a radius drawn from
the focus to the moving body, is not to be found from the time given by a
finite equation ; and therefore cannot be determined by the description ol
curves geometrically rational. Those curves I call geometrically rational,
all the points whereof may be determined by lengths that are definable
by equations ; that is, by the complicated ratios of lengths. Other curves
(such as spirals, quadratrixes, and cycloids) I call geometrically irrational.
For the lengths which are or are not as number to number (according to
the tenth Book of Elements) are arithmetically rational or irrational.
And therefore I cut off an area of an ellipsis proportional to the time in
which it is described by a curve geometrically irrational, in the following
manner.
PROPOSITION XXXI. PROBLEM XXIII.
Tofind the place of a body moving in a given elliptic trajectory at any
assigned time.
Suppose A to be
the principal vertex,
S the focus, and O
the centre of the
ellipsis A PB ; and
let P be the place of
the body to be found.
Produce OA to G so
as OG may be to OA
as OA to OS. Erect
the perpendicular GH; and about the centre O, with the interval OG, de
scribe the circle* GEF ; and on the ruler GH, as a base, suppose the wheel
GEF to move forwards, revolving about its axis, and in the mean time by
its point A describing the cycloid ALL Which done, take GK to the
perimeter GEFG of the wheel, in the ratio of the time in which the body
proceeding from A described the arc AP, to the time of a whole revolution
in the ellipsis. Erect the perpendicular KL meeting the cycloid in L ;
then LP drawn parallel to KG will meet the ellipsis in P, the required
place of the body.
For about the centre O with the interval OA describe the semi-circle
AQB, and let LP, produced, if need be, meet the arc AQ, in Q, and join
SEC. VI. OF NATURAL PHILOSOPHY. 157
SQ, OQ. Let OQ meet the arc EFG in F, and upon OQ let fall the
perpendicular Sll. The area APS is as the area AQS, that is, as tlie
difference between the sector OQA and the triangle OQS, or as the difLience
of the rectangles *OQ, X AQ, and -J.OQ X SR, that is, because .
>,_
is given, as the difference between the arc AQ and the right line Sll : ai.;l
therefore (because of the equality of the given ratios SR to the sine of the
arc AQ,, OS to OA, OA to OG, AQ to GF; and by division, AQ Sii
to GF sine of the arc AQ) as GK, the difference between the arc C 1
and tlie sine of the arc AQ. Q.E.D.
SCHOLIUM.
But since the description of this curve
is difficult, a solution by approximation
will be preferable. First, then, let there
be found a certain angle B which may
be to an angle of 57,29578 degrees,
which an arc equal to the radius subtends,
as SH, the distance of the foci, to AB,
the diameter of the ellipsis. Secondly, a certain length L, which may be to
the radius in the same ratio inversely. And these being found, the Problem
may be solved by the following analysis. By any construction (or even
by conjecture), suppose we know P the place of the body near its true
place jo. Then letting fall on the axis of the ellipsis the ordinate PR
from the proportion of the diameters of the ellipsis, the ordinate RQ of
the circumscribed circle AQB will be given ; which ordinate is the sine of
the angle AOQ, supposing AO to be the radius, and also cuts the ellipsis
in P. It will .be sufficient if that angle is found by a rude calculus in
numbers near the truth. Suppose we also know the angle proportional to
the time, that is, which is to four right a iules as the time in which tlie
body described the arc A/?, to the time of one revolution in the ellipsis.
Let this angle be N. Then take an angle D, which may be to the angle
B as the sine of the angle AOQ to the radius ; and an angle E which
may be to the angle N AOQ -fD as the length L to the same length
L diminished by the cosine of the angle AOQ, when that angle is less
than a right angle, or increased thereby when greater. In the next
place, take an angle F that may be to the angle B as the sine of the angle
1OQ H- E to the radius, and an angle G, that may be to the angle NAOQ
E -f F as the length L to the same length L diminished by the
cosine of the angle AOQ + E, when that angle is less than a right angle,
or increased thereby when greater. For the third time take an angle H,
that may be to the angle B as the sine of the angle AOQ f- E 4- G to the
radius; and an angle I to the angle N AOQ E G -f- H, as the
58 THE MATHEMATICAL PRINCIPLES jB(OK 1.
length L is to the same length L diminished by the cosine of the angle
AOQ -f- E + G, when that angle is less than a right angle, or increased
thereby when greater. And so we may proceed in infinitum. Lastly,
take the angle AOy equal to the angle AOQ -f- E 4- G + I -\-} &c. and
from its cosine Or and the ordinatejor, which is to its sine qr as the lesser
axis of the ellipsis to the greater, \\ e shall have p the correct place of the
body. When the angle N AOQ, -f D happens to be negative, the
sign -|- of the angle E must be every where changed into , and the sign
into +. And the same thing is to be understood of the signs of the angles
G and I, when the angles N AOQ E -f F, and N AOQ E
G + H come out negative. But the infinite series AOQ -f- E -f- G -|- I +,
&c. converges so very fast, that it will be scarcely ever needful to pro
ceed beyond the second term E. And the calculus is founded upon
this Theorem, that the area APS is as the difference between the arc
AQ and the right line let fall from the focus S perpendicularly upon the
radius OQ.
And by a calculus not unlike, the Problem
is solved in the hyperbola. Let its centre be
O, its vertex A, its focus S, and asymptote
OK ; and suppose the quantity of the area to
be cut off is known, as being proportional to
the time. Let that be A, and by conjecture
suppose we know the position of a rij;ht i ne
SP, that cuts off an area APS near the truth.
Join OP, and from A and P to the asymptote
T A S
draw AI, PK parallel to the other asymptote ; and by the table of loga
rithms the area AIKP will be given, and equal thereto the area OPA,
which subducted from the triangle OPS, will leave the area cut off APS.
And by applying 2APS 2A, or 2A 2A PS, the double difference of
the area A that was to be cut off, and the area APS that is cut off, to the
line SN that is let fall from the focus S, perpendicular upon the tangent
TP, we shall have the length of the chord PQ. Which chord PQ is to
be inscribed between A and P, if the area APS that is cut off be greater
than the area A that was to be cut off, but towards the contrary side of the
point P, if otherwise : and the point Q will be the place of the body more
accurately. And by repeating the computation the place may be found
perpetually to greater and greater accuracy.
And by such computations we have a general
analytical resolution of the Problem. But the par
ticular calculus that follows is better fitted for as
tronomical purposes. Supposing AO, OB, OD, to
be the semi-axis of the ellipsis, and L its latus rec
tum, and D the difference betwixt the lesser semi
SEC. VII.] OF NATURAL PHILOSOPHY. J 59
axis OD, and -,L the half of the latus rectum : let an angle Y be found, whose
sine may be to the radius as the rectangle under that difference J), and
AO 4- OD the half sum of the axes to the square of the greater axis AB.
Find also an angle Z, whose sine may be to the radius as the double rec
tangle under the distance of the foci SH and that difference D to triple
the square of half the greater semi-axis AO. Those angles being once
found, the place of the body may be thus determined. Take the angle T
proportional to the time in which the arc BP was described, or equal to
what is called the mean motion ; and an angle V the first equation of thr
mean motion to the angle Y, the greatest first equation, as the sine of
double the angle T is to the radius ; and an angle X, the second equation,
to the angle Z, the second greatest equation, as the cube of the sine of the
angle T is to the cube of the radius. Then take the angle BHP the mean
motion equated equal to T + X + V, the sum of the angles T, V. X,
if the angle T is less than a right angle; or equal to T + X V, the
difference of the same, if that angle T is greater than one and less than
two right angles ; and if HP meets the ellipsis in P, draw SP, and it will
cut off the area BSP nearly proportional to the time.
This practice seems to be expeditious enough, because the angles V and
X, taken in second minutes, if you please, being very small, it will be suf
ficient to find two or three of their first figures. But it is likewise
sufficiently accurate to answer to the theory of the planet s motions.
For even in the orbit of Mars, where the greatest equation of the centre
amounts to ten degrees, the error will scarcely exceed one second. But
when the angle of the mean motion equated BHP is found, the angle oi
the true motion BSP, and the distance SP, are readily had by the known
methods.
And so far concerning the motion of bodies in curve lines. But it mav
also come to pass that a moving body shall ascend or descend in a right
line : and I shall now go on to explain what belongs to such kind of
motions.
SECTION VII.
Concerning the rectilinear ascent and descent of bodies,
PROPOSITION XXXII. PROBLEM XXIV.
Supposing that the centripetal force is reciprocally proportional to tht
square of tlie distance of the places from the centre ; it is required
to define the spaces which a body, falling directly, describes in given
times.
CASE 1. If the body does not fall perpendicularly, it will (by Cor. I
160 THE MATHEMATICAL PRINCIPLES [BOOK I
Prop. XIII) describe some conic section whose focus is A
placed in the centre of force. Suppose that conic sec
tion to be ARPB and its focus S. And, first, if the
figure be an ellipsis, upon the greater axis thereof AB
describe the semi-circle ADB, and let the right line
I) PC pass through the falling body, making right angles
with the axis; and drawing DS, PS, the area ASD will c
be proportional to the area ASP, and therefore also to
the time. The axis AB still reaiaining the same, let the
breadth of the ellipsis be perpetually diminished, and s
the area ASD will always remain proportional to the
time. Suppose that breadth to be diminished in, in fruitum ; and the orbit
APB in that case coinciding with the axis AB, and the focus S with the
extreme point of the axis B, the body will descend in the right line AC1
.
and the area ABD will become proportional to the time. Wherefore the
space AC will be given which the body describes in a given time by itsperpendicular
fall from the place A, if the area ABD is taken proportional
to the time, and from the point D the right line DC is let fall perpendic
ularly on the right line AB. Q,.E.I.
CASE 2. If the figure RPB is an hyperbola, on the
same principal diameter AB describe the rectangular
hyperbola BED ; and because the areas CSP, CB/P,
