PS X PT under the other two PS and PT, the conic section will become
a circle. And the same thing will happen if the four lines are drawn in
any angles, and the rectangle PQ X PR, under one pair of the lines drawn,
is to the rectangle PS X PT under the other pair as the rectangle under
the sines of the angles S, T, in which the two last lines PS, PT are drawn
to the rectangle under the sines of the angles Q, R, in which the first tw
134 THE MATHEMATICAL PRINCIPLES [BOOK 1.
PQ, PR are drawn. In all other cases the locus of the point P will be
one of the three figures which pass commonly by the name of the conic
sections. But in room of the trapezium A BCD, we may substitute a
quadrilateral figure whose two opposite sides cross one another like diago
nals. And one or two of the four points A, B, C, D may be supposed to
be removed to an infinite distance, by which means the sides of the figure
which converge to those points, will become parallel ; and in this case the
conic section will pass through the other points, and will go the same way
as the parallels in, infinitum.
LEMMA XIX.
To find a point P from which if four right lines PQ, PR, PS, PT an
drawn to as many other right lines AB, CD, AC, BD, given by posi
tion, each to each, at given angles, the rectangle PQ X PR, under any
two of the lines drawn, shall be to the rectangle PS X PT, under the
other tivo. in a given ratio.
Suppose the lines AB, CD, to which the two
right lines PQ, PR, containing one of the rect
angles, are drawn to meet two other lines, given
by position, in the points A, B, C, D. From one
of those, as A, draw any right line AH, in which
you would find the point P. Let this cut the
opposite lines BD, CD, in H and I
; and, because
all the angles of the figure are given, the ratio of
PQ to PA, and PA to PS, and therefore of PQ
to PS, will be also given. Subducting this ratio from the given ratio oi
PQ X PR to PS X PT, the ratio of PR to PT will be given ; and ad
ding the given ratios of PI to PR, and PT to PH, the ratio of PI to PH.
and therefore the point P will be given. Q.E.I.
COR. 1. Hence also a tangent may be drawn to any point D of the
locus of all the points P. For the chord PD, where the points P and D
meet, that is, where AH is drawn through the point D, becomes a tangent.
In which case the ultimate ratio of the evanescent lines IP and PH will
be found as above. Therefore draw CF parallel to AD, meeting BD in
F, and cut it in E in the same ultimate ratio, then DE will be the tan
gent ;
because CF and the evanescent IH are parallel, and similarly cut in
E and P.
COR. 2. Hence also the locus of all the points P may be determined.
Through any of the points A, B, C, D, as A, draw AE touching the locus,
and through any other point B parallel to the tangent, draw BF meeting
the locus in F ; and find the point F by this Lemma. Bisect BF in G,
and, drawing the indefinite line AG, this will be the position of the dia
meter to which BG and FG are ordinates. Let this AG meet the locus
SEC. V.J OF NATURAL PHILOSOPHY.
in H, and AH will be its diameter or latus transversum.
to which the latus rectum will be as BG2
to AG X GH. If AG nowhere meets the locus,
the line AH being infinite, the locus will be a par
abola ; and its latus rectum corresponding to the
diameter AG will be -.-7^ AC*
But if it does meet it
anywhere, the locus will be an hyperbola, when
the points A and H are placed on the same side the point G ; and an
ellipsis, if the point G falls between the points A and H ; unless, perhaps,
the angle AGB is a right angle, and at the same time BG2
equal to the
rectangle AGH, in which case the locus will be a circle.
And so we have given in this Corollary a solution of that famous Prob
lem of the ancients concerning four lines, begun by Euclid, and carried on
by Apollonius ; and this not an analytical calculus, but a geometrical com
position, such as the ancients required.
LEMMA XX.
If the two opposite angularpoints A and P of any parallelogram ASPQ
touch any conic section in the points A and P ; and the sides AQ, AS
of one of those angles, indefinitely produced, meet the same conic section
in B and C ; and from the points of concourse, B and C to any fifth
point D of the conic section, two right lines BD, CD are drawn meeting
tlie two other sides PS, PQ of the parallelogram, indefinitely pro
duced in T and R ; the parts PR and PT, cut off from the sides, will
always be one to the other in a given ratio. And vice versa, if those
parts cut off are one to the other in a given ratio, the locus of the point
D will be a conic section passing through the four points A, B, C, F
CASE 1. Join BP, CP, and from the point
D draw the two right lines DG, DE, of which
the first DG shall be parallel to AB, and
meet PB, PQ, CA in H, I, G ; and the other
DE shall be parallel to AC, and meet PC,
PS, AB, in F, K, E ; and (by Lem. XVII)
the rectangle DE X DF will be to the rect
angle DG X DH in a given ratio. But
PQ is to DE (or IQ) as PB to HB, and con
sequently as PT to DH ; and by permutation PQ, is to PT as DE to
DH. Likewise PR is to DF as RC to DC, and therefore as (IG or) PS
to DG ; and by permutation PR is to PS as DF to DG ; and, by com
pounding those ratios, the rectangle PQ X PR will be to the rectangle
PS X PT as the rectangle DE X DF is to the rectangle DG X DH.
and consequently in "a given ratio. But PQ and PS are given, and there
fore the ratio of PR to PT is given. Q.E.D.
136 THE MATHEMATICAL PRINCIPLES
CASE 2. But if PR and PT are supposed to be in a given ratio one to
the other, then by going back again, by a like reasoning, it will follow
that the rectangle DE X DF is to the rectangle DG X DH in a given
rati) ; and so the point D (by Lem. XVIII) will lie in a conic section pass
ing through the points A., B, C, P, as its locus. Q.E.I).
COR. 1. Hence if we draw BC cutting PQ in r and in PT take Pt to
Pr in the same ratio which PT has to PR ; then Et will touch the conic
section in the point B. For suppose the point D to coalesce with the point
B, so that the chord BD vanishing, BT shall become a tangent, and CD
and BT will coincide with CB and Bt.
COR. 2. And, vice versa, if Bt is a tangent, and the lines BD, CD meet
in any point D of a conic section, PR will be to PT as Pr to Pt. And,
on the contrary, if PR is to PT as Pr to Pt, then BD and CD will meet
in some point D of a conic section.
COR. 3. One conic section cannot cut another conic section in more than
four points. For, if it is possible, let two conic sections pass through the
hve points A, B, C, P, O ; and let the right line BD cut them in the
points D, d, and the right line Cd cut the right line PQ, in q. Therefore
PR is to PT as Pq to PT : whence PR and Pq are equal one to the other,
against the supposition.
LEMMA XXI.
If two moveable and indefinite right lines BM, CM drawn through given
points B, C, as poles, do by their point of concourse M describe a third
right line MN given by position ; and other two indefinite right lines
BD,CD are drawn, making with the former two at those given points
B, C, given angles, MBD, MCD : I say, that those two right lines BD,
CD will by their point of concourse D describe a conic section passing
through the points B, C. And, vice versa, if the right lints BD, CD
do by their point of concourse D describe a conic section passing
through the given points B, C, A, and the angle DBM is always
equal to the giren angle ABC, as well as the angle DCM always
equal to the given angle ACB, the point M will lie in a right line
given by position, as its locus.
For in the right line MN let a point
N be given, and when the moveable point
M falls on the immoveable point N. let
the moveable point D fall on an immo
vable point P. Join ON, BN, CP, BP,
and from the point P draw the right lines
PT, PR meeting BD, CD in T and R, C
and making the angle BPT c jual to the
given angle BNM, and the angle CPR
SEC. V.J OF NATURAL PHILOSOPHY. 137
equal to the given angle CNM. Wherefore since (by supposition) the an
gles MBD, NBP are equal, as also the angles MOD, NCP, take away the
angles NBD and NOD that are common, and there will remain the angles
NBM and PBT, NCM and PCR equal; and therefore the triangles NBM,
PBT are similar, as also the triangles NCM, PCR. Wherefore PT is to
NM as PB to NB ; and PR to NM as PC to NC. But the points, B, C,
N, P are immovable: wheiefore PT and PR have a given ratio to NM,
and consequently a given ratio between themselves; and therefore, (by
Lemma XX) the point D wherein the moveable right lines BT and CR
perpetually concur, will be placed in a conic section passing through the
points B. C, P. Q.E.D.
And, vice versa, if the moveable point
D lies in a conic section passing through
the given points B, C, A ; and the angle
DBM is always equal to the given an
gle ABC, and the angle DCM always
equal to the given angle ACB, and when
the point D falls successively on any
two immovable points p, P, of the conic
section, the moveable point M falls suc
cessively on two immovable points /?, N.
Through these points ??, N, draw the right line nN : this line nN will be
the perpetual locus of that moveable point M. For, if possible, let the
point M be placed in any curve line. Therefore the point D will be placed
in a conic section passing through the five points B, C, A, p, P, when the
point M is perpetually placed in a curve line. But from what was de
monstrated before, the point D will be also placed in a conic section pass
ing through the same five points B, C, A, p, P, when the point M is per
petually placed in a right line. Wherefore the two conic sections will both
pass through the same five points, against Corol. 3, Lem. XX. It is
therefore absurd to suppose that the point M is placed in a curve line.
QE.D.
PROPOSITION XXII. PROBLEM XIV.
To describe a trajectory that shall pass through Jive given points.
Let the five given points be A, B, C, P, D. c
From any one of them, as A, to any other sv
two as B, C, which may be called the poles,
draw the right lines AB, AC, and parallel to
those the lines TPS, PRO, through the fourth
point P. Then from the two poles B, C,
draw through the fifth point D two indefinite
lines BDT, CRD, meeting with the last drawn lines TPS, PRQ (the
138 THE MATHEMATICAL PRINCIPLES IBOOK L
former with the former, and the latter with the latter) in T and R. Then
drawing the right line tr parallel to TR, cutting off from the right lines
PT, PR, any segments Pt, Pr, proportional to PT, PR ; and if through
their extremities, t, r, and the poles B, C, the right lines lit, Cr are drawn,
meeting in d, that point d will be placed in the trajectory required. For
(by Lena. XX) that point d is placed in a conic section passing through
the four points A, B, C, P ; and the lines R/ , TV vanishing, the point d
comes to coincide with the point D. Wherefore the conic section passes
through the five points A, B, C, P, D. Q.E.D.
The same otherwise.
Of the given points join any three, as A, B,
C ; and about two of them 15, C, as poles,
making the angles ABC, ACB of a given
magnitude to revolve, apply the legs BA,
CA, first to the point D, then to the point P,
and mark the points M, N, in which the other
legs BL, CL intersect each other in both cases. C
Draw the indefinite right line MN, and let
those moveable angles revolve about their
poles B, C, in such manner that the intersection, which is now supposed to
be ???, of the legs BL, CL; or BM7 CM, may always fall in that indefinite
right line MN ; and the intersection, which is now supposed to be d, of the
legs BA ^A, or BD; CD, will describe the trajectory required, PADc/B.
For (by Lem. XXI) the point d will be placed in a conic section passingthrough
the points B, C ; and when the point m comes to coincide with
the points L, M, N, the point d will (by construction) come to coin
cide with the points A, D, P. Wherefore a conic section will be described
that shall pass through the five points A, B. C, P, D. Q,.E.F.
COR. 1. Hence a right line may be readily drawn which shall be a tan
gent to the trajectory in any given point B. Let the point d come to co
incide with the point B, arid the right line Bt/ Avill become the tangent
required.
COR. 2. Hence also may be found the centres, diameters, and latera recta
of the trajectories, as in Cor. 2, Lem. XIX.
SCHOLIUM.
The former of these constructions will be- c
come something more simple by joining ,
and in that line, produced, if need be, aking
Bp to BP as PR is to PT ; and t rough p
draw the indefinite right inc j0e parallel to S
PT, and in that line pe taking always pe
equal to Pi , and draw the right lines Be, Cr
SEC. Y.J OF NATURAL PHILOSOPHY. 139
to meet in d. For since Pr to Pt, PR to PT, pB to PB, pe to Pt, are all in
the same ratio, pe and Pr will be always equal. After this manner the
points of the trajectory are most readily found, unless you would rather
describe the curve mechanically, as in the second construction.
PROPOSITION XXIII. PROBLEM XV.
To describe a trajectory that shall pass through four given points, and
touch a right line given by position.
CASE 1. Suppose that HB is the
given tangent, B the point of contact,
and C, 1., P, the three other given
points. Jo n BC. and draw IS paral
lel to BH, and PQ parallel to BC ;
complete the parallelogram BSPQ.
Draw BD cutting SP in T, and CD
cutting PQ, in R. Lastly, draw any
line tr parallel to TR, cutting off
from PQ, PS, the segments Pr, Pt proportional to PR, PT respectively ;
and draw Cr, Bt their point of concourse d will (by Lem. XX) always fall
on the trajectory to be described.
The same otherwise.
1 et tl e angle CBH of a given magnitude re
volve about the pole B; as also the rectilinear rad
: us 1C, both ways produced, about the pole C.
Mark the points M, N, on which the leg BC of
the angle cuts that radius when BH; the other
leg thereof, meets the same radius in the points
P and D. Then drawing the indefinite line MN,
let that radius CP or CD and the leg BC of the
angle perpetually meet in this Ikie; and the
point of concourse of the other leg BH with the
radius will delineate the trajectory required.
For if in the constructions of the preceding Problem the point A comes
to a coincidence with the point B, the lines CA and CB will coincide, and
the line AB, in its last situation, will become the tangent BH ; and there
fore the constructions there set down will become the same with the con
structions here described. Wherefore the concourse of the leg BH with
the radius will describe a conic section passing through the points C, D,
P, and touching the line BH in the point B. Q.E.F.
CASE 2. Suppose the four points B, C, D, P, given, being situated withont
the tangent HI. Join each two by the lines BD, CP meeting in G,
and cutting the tangent in H and I. Cut the tangent in A in such mannr:
140 THE MATHEMATICAL PRINCIPLES [BOOK I
X IT
that HA may be to IA as the rectangle un
der a mean proportional between CG and
GP, and a mean proportional between BH
and HD is to a rectangle under a mean pro
portional between GD and GB, and a mean
proportional betweeen PI and 1C, and A will
be the point of contact. For if HX, a par
allel to the right line PI, cuts the trajectory
in any points X and Y, the point A (by the
properties of the conic sections) will come to be so placed, that HA2 will
become to AP in a ratio that is compounded out of the ratio of the rec
tangle XHY to the rectangle BHD, or of the rectangle CGP to the rec
tangle DGB; and the ratio of the rectangle BHD to the rectangle PIC.
But after the point of contac.t A is found, the trajectory will be described as
in the first Case. Q.E.F. But the point A may be taken either between
or without the points H and I, upon which account a twofold trajectory
may be described.
PROPOSITION XXIV. PROBLEM XVI.
To describe a trajectory that shall pass through three given points, and
touch two right lines given by position.
Suppose HI, KL to be the given tangents
and B, C, D, the given points. Through any
two of those points, as B, D, draw the indefi
nite right line BD meeting the tangents in
the points H, K. Then likewise through
any other two of these points, as C, D, draw
the indefinite right line CD meeting the tan
gents in the points I, L. Cut the lines drawn
in R and S, so that HR may be to KR as
the mean proportional between BH and HD is to the mean proportional
between BK and KD ; and IS to LS as the mean pioportional between
CI and ID is to the mean proportional between CL and LD. But you
may cut, at pleasure, either within or between the points K and H, I and
L, or without them ; then draw RS cutting the tangents in A and P, and
A and P will be the points of contact. For if A and P are supposed to
be the points of contact, situated anywhere else in the tangents, and through
any of the points H, I, K, L, as I, situated in either tangent HI, a right
line IY is drawn parallel to the other tangent KL, and meeting the curve
in X and Y, and in that right line there be taken IZ equal to a mean pro
portional between IX and IY, the rectangle XIY or IZ2
, will (by the pro
perties of the conic sections) be to LP2 as the rectangle CID is to the rect
angle CLD, that is (by the construction), as SI is to SL2
; and therefore
SEC. V.] OF NATUKAL PHILOSOPHY. 141
IZ is to LP as SI to SL. Wherefore the points S, P, Z. are in one right
line. Moreover, since the tangents meet in G, the rectangle XIY or IZ2
will (by the properties of the conic sections) be to IA2 as GP2
is to GA2
,
and consequently IZ will be to IA as GP to GA. Wherefore the points
P, Z, A, lie in one right line, and therefore the points S, P, and A are in
one right line. And the same argument will prove that the points R, P,
and A are in one right line. Wherefore the points of contact A and P lie
in the right line RS. But after these points are found, the trajectory may
be described, as in the first Case of the preceding Problem. Q,.E.F.
In this Proposition, and Case 2 of the foregoing, the constructions are
the same, whether the right line XY cut the trajectory in X and Y, or
not ; neither do they depend upon that section. But the constructions
being demonstrated where that right line does cut the trajectory, the con
structions where it does not are also known ; and therefore, for brevity s
sake, I omit any farther demonstration of them.
LEMMA XXII.
To transform figures into other figures of the same kind.
Suppose that any figure HGI is to be
transformed. Draw, at pleasure, two par
allel lines AO, BL, cutting any third line
AB, given by position, in A and B, and from
any point G of the figure, draw out any
right line GD, parallel to OA, till it meet
the right line AB. Then from any given
point in the line OA, draw to the point
D the right line OD, meeting BL in d ; and
from the point of concourse raise the right
line dg containing any given angle with the right line BL, and having
such ratio to Qd as DG has to OD ; and g will be the point in the new
figure hgi, corresponding to the point G. And in like manner the several
points of the first figure will give as many correspondent points of the new
figure. If we therefore conceive the point G to be carried along by a con
tinual motion through all the points of the first figure, the point g will
be likewise carried along by a continual motion through all the points of
the new figure, and describe the same. For distinction s sake, let us call
DG the first ordinate, dg the new ordinate, AD the first abscissa, ad the
new abscissa ; O the pole. OD the abscinding radius, OA the first ordinate
radius, and Oa (by which the parallelogram OABa is completed) the new
ordinate radius.
I say, then, that if the point G is placed in a right line given by posi
tion, the point g will be also placed in a right line given by position. If
the point G is placed in a conic section, the point g will be likewise placed
J42 THE MATHEMATICAL PRINCIPLES [BOOK 1.
in a conic section. And here I understand the circle as one of the conic
sections. But farther, if the point G is placed in a line of the third ana
lytical order, the point g will also be placed in a line of the third order,
and so on in curve lines of higher orders. The two lines in which the
points G, g, are placed, will be always of the same analytical order. For
as ad is to OA, so are Od to OD, dg to DG, and AB to AD ; and there-
OA X AB OA X dg fore AD is equal to , , and DG equal to 7 . Now if the ad ad
point G is placed in a right line, and therefore, in any equation by which
the relation between the abscissa AD and the ordinate GD is expressed,
those indetermined lines AD and DG rise no higher than to one dimenv
v xu- ,. OA X AB . OA X dg
sion, by writing this equation . m place of AD, and -. -
in place of DG, a new equation will be produced, in which the new ab
scissa ad and new ordinate dg rise only to one dimension ; and which
therefore must denote a right line. But if AD and DG (or either of
them) had risen to two dimensions in the first equation, ad and dg would
likewise have risen to tAvo dimensions in the second equation. And so on
in three or more dimensions. The indetermined lines, ad} dg in the
second equation, and AD, DG, in the first, will always rise to the same
number of dimensions ; and therefore the lines in which the points G, g,
are placed are of the same analytical order.
I say farther, that if any right line touches the curve line in the first
figure, the same right line transferred the same way with the curve into
the new figure will touch that curve line in the new figure, and vice versa.
For if any two points of the curve in the first figure are supposed to ap
proach one the other till they come to coincide, the same points transferred
will approach one the other till they come to coincide in the new figure ;
and therefore the right lines with which those points are joined will be
come together tangents of the curves in both figures. I might have given
demonstrations of these assertions in a more geometrical form ; but I study
to be brief.
Wherefore if one rectilinear figure is to be transformed into another, we
