need only transfer the intersections of the right lines of which the first
figure consists, and through the transferred intersections to draw right lines
in the new figure. But if a curvilinear figure is to be transformed, we
must transfer the points, the tangents, and other right lines, by means of
which the curve line is denned. This Lemma is of use in the solution of
the more difficult Problems ; for thereby we maj transform the proposed
figures, if they are intricate, into others that are more simple. Thus any
right lines converging to a point are transformed into parallels, by taking
for the first ordinate radius any right line that passes through the point
of concourse of the converging lines, and that because their point of con
SEC. V.] OF NATURAL PHILOSOPHY. 143
course is by this means made to go off in infinitum ; and parallel lines
are such as tend to a point infinitely remote. And after the problem is
solved in the new figure, if by the inverse operations we transform the
new into the first figure, we shall have the solution required.
This Lemma is also of use in the solution of solid problems. For as
often as two conic sections occur, by the intersection of which a problem
may be solved, any one of them may be transformed, if it is an hyperbola
or a parabola, into an ellipsis, and then this ellipsis may be easily changed
into a circle. So also a right line and a conic section, in the construc
tion of plane problems, may be transformed into a right line and a circle
PROPOSITION XXV. PROBLEM XVII.
To describe a trajectory that shall pass through two given points, and
touch three right lines given by position.
Through the concourse of any two of the tangents one with the other,
and the concourse of the third tangent with the right line which passes
through the two given points, draw an indefinite right line ; and, taking
this line for the first ordinate radius, transform the figure by the preceding
Lemma into a new figure. In this figure those two tangents will become
parallel to each other, and the third tangent will be parallel to the right
line that passes through the two given points. Suppose hi, kl to be those
two parallel tangents, ik the third tangent, and hi a right line parallel
thereto, passing through those points a, b,
through which the conic section ought to pass
in this new figure; and completing the parallelogra
n fiikl, let the right lines hi, ik, kl be
BO cut in c, d, e, that he may be to the square
root of the rectangle ahb, ic, to id, and ke to
kd. as the sum of the right lines hi and kl is
to the sum of the three lines, the first whereof
is the right line ik, and the other two are the
square roots of the rectangles ahb and alb ; and c, d, e, will be the points
of contact. For by the properties of the conic sections, he2 to the rectan
gle ahb, and ic2 to id2
, and ke2 to kd2
, and el2 to the rectangle alb, are all
in the same ratio ; and therefore he to the square root of ahb, ic to id, ke
to kdj and el to the square root of alb, are in the subduplicate of that
ratio
; and by composition, in the given ratio of the sum of all the ante
cedents hi + kly to the sum of all the consequents ^/ahb -\- ik : *Jalb,
Wherefore from that given ratio we have the points of contact c, d, e, in
the new figure. By the inverted operations of the last Lemma, let those
points be transferred into the first figure, and the trajectory will be there
described by Prob. XIV. Q.E.F. But according as the points a, b, fall
between the points //, /, or without taem, the points c, d, e, must be taken
144 THE MATHEMATICAL PRINCIPLES BOOK I.J
Cither between the points, h, i, k, /, or without them. If one of the points
a, b, falls between the points h, i, and the other xvithout the points h, I,
the Problem is impossible.
PROPOSITION XXVI. PROBLEM XVIII.
To describe a trajectory that shall pass through a given point, and touch
four right lines given by position.
From the common intersections, of any two
of the tangents to the common intersection of
the other two, draw an indefinite right line ; and
taking this line for the first ordinate radius;
/ xs o
transform the figure (by Lem. XXII) into a new
figure, and the two pairs of tangents, each of
which before concurred in the first ordinate radius,
will now become parallel. Let hi and kl, Al l\
ik and hi, be those pairs of parallels completing the parallelogram hikl.
And let p be the point in this new figure corresponding to the given point
in the first figure. Through O the centre of the figure draw pq.: and O?
being equal to Op, q will be the other point through which the conic sec
tion must pass in this new figure. Let this point be transferred, by the
inverse operation of Lem. XXII into the first figure, and there we shall
have the two points through which the trajectory is to be described. But
through those points that trajectory may be described by Prop. XVII.
LEMMA XXIII.
If two right lines, as AC, BD given by position, and terminating in
given points A, B, are in a given ratio one to the other, and the right
line CD, by which the, indetermined points C, D are joined is cut in
K in a given ratio ; I say, that the point K will be placed in a right
line given by position.
For let the right lines AC, BD meet in
E, and in BE take BG to AE as BD is to
AC, and let FD be always equal to the given
line EG ; and, by construction, EC will be
to GD, that is, to EF, as AC to BD, and
therefore in a given ratio ; and therefore the %- ,.--- I \
triangle EFC will be given in kind. Let E K cT^"^
CF be cut in L so as CL may be to CF in the ratio of CK to CD ; and
because that is a given ratio, the triangle EFL will be given in kind, and
therefore the point L will be placed in the right line EL given by position.
Join LK, and the triangles CLK, CFD will be similar ; and because FD
is a given line, and LK is to FD in a given ratio, LK will be also given
SEC. V.] OF NATURAL PHILOSOPHY. 145
To this let EH be taken equal, and ELKH will be always a parallelogram.
And therefore the point K is always placed in the side HK (given by po
tiition) of that parallelogram. Q.E.D.
COR. Because the figure EFLC is given in kind, the three right lines
EF, EL, and EC, that is, GD, HK, and EC, will have given ratios to
each other.
LEMMA XXIV.
If three right lines, two whereof are parallel, and given by position, touch
any conic section ; I say, that the semi-diameter of the section wkiJt
is parallel to those two is a mean proportional between the segments
of those two that are intercepted between the points of contact and the.
third tangent.
Let AF, GB be the two parallels touch
ing the conic section ADB in A and B ;
EF the third right line touching the conic
section in I, and meeting the two former
tangents in F and G, and let CD be the
semi-diameter of the figure parallel to
those tangents ;
I say. that AF, CD, BG
are continually proportional.
For if the conjugate diameters AB, DM G Q
meet the tangent FG in E and H, and cut one the other in C; and the
parallelogram IKCL be completed ; from the nature of the conic sections,
EC will be to CA as CA to CL ; and so by division, EC CA to CA -
CL, orEAto AL; and by composition, EA to EA + AL or EL, as EC to
EC + CA or EB ; and therefore (because of the similitude of the triangles
EAF, ELI, ECH, EBG) AF is to LI as CH to BG. Likewise, from tli?
nature of the conic sections, LI (or CK) is to CD as CD to CH ; and
therefore (ex aquo pertnrhatfy AF is to CD as CD to BG. Q.E.D.
COR. 1. Hence if two tangents FG, PQ, meet two parallel tangents AF,
BG in F and G, P and Q,, and cut one the other in O; AF (ex cequo pertnrbot,
) will be to BQ as AP to BG, and by division, as FP to GQ, and
therefore as FO to OG.
COR. 2. Whence also the two right lines PG, FQ, drawn through the
points P and G, F and Q, will meet in the right line ACB passing through
the centre of the figure and the points of contact A, B.
LEMMA XXV.
Iffour sides of a parallelogram indefinitely produced touch any conic
section, and are cut by a fifth tangent ; I say, that, taking those seg
ments of any two conterminous sides that terminate in opposite angles
10
146 THE MATHEMATICAL PRINCIPLES [BooK 1.
of the parallelogram, either segment is to the side from which it is
cut off as that part of the other conterminous side which is intercepted
between the point of contact and the third side is to Uie other segment,
Let the four sides ML, IK, KL, MI,
of the parallelogram MLJK touch the
conic section in A, B, C, I)
; and let the
fifth tangent FQ cut those sides in F,
Q, H, and E : and taking the segments
ME, KQ of the sides Ml, KJ, or the
segments KH, MF of the sides KL,
ML, 1 s/.y, that ME is to MI as BK to
KQ; and KH to KL as AM to MF.
For, by Cor. 1 of the preceding Lemma, ME is to El as (AM or) BK to
BQ ; and, by composition, ME is to MI as BK to KQ. Q.E.D. Also
KH is to HL as (BK or) AM to AF ; and by division, KH to KL as AM
to MF. Q.E.D.
COR. 1. Hence if a parallelogram IKLM described about a given conic
section is given, the rectangle KQ X ME, as also the rectangle KH X ME
equal thereto, will be given. For, by reason of the similar triangles KQH
MFE, those rectangles are equal.
COR. 2. And if a sixth tangent eq is drawn meeting the tangents Kl.
MI in q and e, the rectangle KQ X ME will be equal to the rectangle
K</ X Me, and KQ will be to Me as Kq to ME, and by division ns
Q? to Ee.
COR. 3. Hence, also, if
E<?, eQ, are joined and bisected, and a right line
is drawn through the points of bisection, this right line will pass through
the centre of the conic section. For since
Q</
is to Ee as KQ to Me, the
same right line will pass through the middle of all the lines Eq, eQ, MK
(by Lem. XXIII), and the middle point of the right line MK is the
centre of the section.
PROPOSITION XXVII. PROBLEM XIX.
To describe a trajectory that may touch jive right lines given by position.
Supposing ABG; BCF,
GCD, FDE, EA to be the
tangents given by position.
Bisect in M and N, AF, BE,
the diagonals of the quadri
lateral
tained
figure
under
ABFE conany
four of
them ; and (by Cor. 3, Lem.
XXV) the right line MN
draAvn through the points (,f
SEC. V.] OF NATURAL PHILOSOPHY. 147
bisection will pass through the centre of the trajectory. Again, bisect in
P and Q, the diagonals (if I may so call them) Bl), GF of the quadrila
teral figure EC OF contained under any other four tangents, and the right
line PQ, drawn through the points of bisection will pass through the cen
tre of the trajectory ; and therefore the centre will be given in the con
course of the bisecting lines. Suppose it to be O. Parallel to any tan
gent BG draw KL at such distance that the centre O may be placed in the
middle between the parallels; this KL will touch the trajectory to be de
scribed. Let this cut any other two tangents GCD, FJ)E, in L and K.
Through the points G and K, F and L, where the tangents not parallel,
CL, FK meet the parallel tangents CF, KL, draw GK, FL meeting in
K ; and the right line OR drawn and produced, will cut the parallel tan
gents GF, KL, in the points of contact. This appears from Gor. 2, Lem.
XXIV. And by the same method the other points of contact may be
found, and then the trajectory may be described by Prob. XIV. Q.E.F.
SCPIOLTUM.
Under the preceding Propositions are comprehended those Problems
wherein either the centres or asymptotes of the trajectories are given. For
when points and tangents and the centre are given, as many other points
and as many other tangents are given at an equal distance on the other
side of the centre. And an asymptote is to be considered as a tangent, ami
its infinitely remote extremity (if we may say so) is a point of contact.
Conceive the point of contact of any tangent removed in infinitum, and
the tangent will degenerate into an asymptote, and the constructions of
the preceding Problems will be changed into the constructions of those
Problems wherein the asymptote is given.
After the trajectory is described, we may
find its axes and foci in this manmr. In the
construction and figure of Lem. XXI, let those ,
legs BP, CP, of the moveable angles PEN, ^
PCN, by the concourse of which the trajec- \
tory was described, be made parallel one to
the other : and retaining that position, let
them revolve about their poles I
, C, in that
figure. In the mean while let the other legs
GN, BN, of those angles, by their concourse
K or k, describe the circle BKGC. Let O be the centre of this circle;
and from this centre upon the ruler MN, wherein those legs CN, BN did
concur while the trajectory was described, let fall the perpendicular OH
meeting the circle in K and L. And when those other legs CK, BK meet
in the point K that is nearest to the ruler, the first legs CP, BP will be
parallel to the greater axis, and perpendicular on the lesser ; and the con
148 THE MATHEMATICAL PRINCIPLES [Book I
trary will hajpen if those legs meet in the remotest point L. Whence ii
the centre of the trajectory is given, the axes will be given ; and those being
given, the foci will be readily found.
But the squares of the axes are one to the
other as KH to LH, and thence it is easy to
describe a trajectory given in kind through
fmr given points. For if two of the given
points are made the poles C, 13, the third will
give the moveable angles PCK, PBK ; but
those being given, the circle BGKC may be
described. Then, because the trajectory is
given in kind, the ratio of OH to OK, and
and therefore OH itself, will be given. About
the centre O, with the interval OH, describe another circle, and the right
line that touches this circle, and passes through the concourse of the legs
CK, BK, when the first legs CP; BP meet in the fourth given point, will
be the ruler MN, by means of which the trajectory may be described
Whence also on the other hand a trapezium given in kind (excepting a
few cases that are impossible) may be inscribed in a given conic section.
There are also other Lemmas, by the help of which trajectories given in
kind may be described through given points, and touching given lines.
Of such a sort is this, that if a right line is drawn through any point
given by position, that may cut a given conic section in two points, and
the distance of the intersections is bisected, the point of bisection will
to ich ano her conic section of the same kind with the former, arid havin^
its axes parallel to the axes of the former. But I hasten to things of
greater use.
LEMMA XXVI.
To place 1ht lit rev angles of a triangle, given both in kind and magni
tude, in, respect of as many rigid lines given by position, -provided th\]
are not all parallel among themselves, in such manner tfia t jic spiral
angles may touch the several lines.
Three indefinite right lines AB, AC, BC, are
given by position, and it is required so to place
the triangle DEF that its angle 1) may touch
the line AB, its angle E the line AC, and
its angle F the line BC. Upon DE, DF, and
EF, describe three segments of circles DRE,
DGF. EMF, capable of angles equal to the
Rubles BAG, ABC, ACB respectively. But those segments are to be de
scribed t wards such sides of the lines DE, DF; EF; that the letters
3 EC. V.I OF NATURAL PHILOSOPHY. 1411
DRED may turn round about in the same order with the letters I1ACB :
the letters DGFD in the same order with the letters ABCA ; and the
letters EMFE in the same order with the letters ACBA ; then, completing
th se segmerts into entire circles let the two former circles cut one the
other in G, and suppose P and Q to be their centres. Then joining GP,
PQ, take Ga to AB as GP is to PQ ; and about the centre G, with the
interval Ga, describe a circle that may cut the first circle DGE in a.
Join aD cutting the second circle DFG in b, as well as aE cutting the
third circle EMF in c. Complete the figure ABCdef similar and equal
to the figure a&cDEF : I say, the thing is done.
For drawing Fc meeting D in n,
and joining aG; bG, QG, QD. PD, by
construction the angle EaD is equal to
the angle CAB, and the angle acF equal
to the angle ACB; and therefore the
triangle aiic equiangular to the triangle
ABC. Wherefore the angle anc or FnD
is equal to the angle ABC, and conse-
< uently to the angle F/>D
; and there
fore the point n falls on the point b,
Moreover the angle GPQ, which is half
the angle GPD at the centre, is equal
to the angle GaD at the circumference \
and the angle GQP, which is half the angle GQD at the centre, is equal
to the complement to two right angles of the angle GbD at the circum
ference, and therefore equal to the angle Gba. Upon which account the
triangles GPQ, Gab, are similar, and Ga is to ab as GP to PQ. ; that is
(by construction), as Ga to AB. Wherefore ab and AB are equal; and
consequently the triangles abc, ABC, which we have now proved to be
similar, are also equal. And therefore since the angles I), E, F, of the
triangle DEF do respectively touch the sides ab, ar, be of the triangle
afjc
/ the figure AECdef may be completed similar and equal to the figure
afrcDEFj and by completing it the Problem will be solved. Q.E.F.
COR. Hence a right line may be drawn whose parts given in length may
be intercepted between three right lines given by position. Suppose the
triangle DEF, by the access of its point D to the side EF, arid by having
the sides DE, DF placed i>t directum to be changed into a right line
whose given part DE is to be interposed between the right lines AB; AC
given by position; and its given part DF is to be interposed between the
right lines AB; BC, given by position; then, by applying the preceding
construction to this case, the Problem will be solved.
THE MATHEMATICAL PRINCIPLES [BOOK 1.
PROPOSITION XXVIII. PROBLEM XX.
To describe a trajectory giren both in kind and magnitude, given parts
of which shall be interposed between three right lines given by position.
Suppose a trajectory is to be described that
may be similar and equal to the curve line DEF,
-and may be cut by three right lines AB, AC,
BC, given by position, into parts DE and EF,
similar and equal to the given parts of this
curve line.
Draw the right lines DE, EF, DF: and
place the angles D, E, F, of this triangle DEF, so
as to touch those right lines given by position (by
Lem. XXVI). Then about the triangle describe
the trajectory, similar and equal to the curve DEF.
Q.E.F.
LEMMA XXVII.
To describe a trapezium given in kind, the angles whereof may be ,
placed, in respect offour right lines given by position, that are neither
all paralhl among themselves, nor converge to one common point, //////
the several angles may touch the several lines.
Let the four right lines ABC, AD, BD, CE, be
given by position ; the first cutting the second in A,
the third in B, and the fourth in C and suppose a
trapezium fghi is to be described that may be similar
to the trapezium FCHI, and whose angle /, equal to
the given angle F, may touch the right line ABC ; and
(lie other angles g, h, i, equal to the other given angles,
G, H, I, may touch the other lines AD, BD, CE, re
spectively. Join FH, and upon FG. FH, FI describe J%
as many segments of circles FSG, FTH, FVI, the first
of which FSG may be capable of an angle equal to
the angle BAD ; the second FTH capable of an angle
equal to the angle CBD ; and the third FVI of an angle equal to the angle
ACE. Bnrf>, the segments are to be described towards those sides of the
lines FG, FH, FI, that the circular order of the letters FSGF may be
the same as of the letters BADB, and that the letters FTHF may turn
.ibout in the same order as the letters CBDC and the letters FVIF in the
game order as the letters ACEA. Complete the segments into entire cir
cles, and let P be the centre of the first circle FSG, Q, the centre of the
second FTH. Join and produce both ways the line PQ,, and in it take
QR in the same ratio to PQ as BC has to AB. But QR is to be taken
towards that side of the point Q that the order of the letters P, Q,, R
SEC. V.J OF NATURAL PHILOSOPHY. 15]
may be the same as of the letters A, B, C ;
and about the centre R with the interval
RF describe a fourth circle FNc cutting
(lie third circle FVI in c. Join Fc1 cut
ting the first circle in a, and the second in
/ . Draw aG, &H, cl, and let the figure
ABC/4f/ii be made similar to the figure
w^cFGHI; and the trapezium fghi will
be that which was required to be de
scribed.
For let the two first circles FSG, FTH
cut one the other in K ; join PK, Q,K,
RK, "K, 6K, cK, and produce QP to L.
The angles FaK, F6K, FcK at the circumferences are the halves of the
angles FPK, FQK, FRK, at the centres, and therefore equal to LPK,
LQ.K, LRK, the halves of those angles. Wherefore the figure PQRK is
iquiangular and similar to the figure 6cK, and consequently ab is to be
res PQ, to Q,R, that is, as AB to BC. But by construction, the angles
Air, /B//,/C?, are equal to the angles FG, F&H, Fcl. And therefore
the figure ABCfghi may be completed similar to the figure abcFGHl.
vVliich done a trapezium fghi will be constructed similar to the trapezium
FGHI, and which by its angles/, g, h, i will touch the right lines ABC,
AD, BD, CE. Q.E.F.
COR. Hence a right line may be drawn whose parts intercepted in a
given order, between four right lines given by position, shall have a given
