ARTES
1817
VERITAS
LIBRARY
SCIENTIA
OF THE
UNIVERSITY OF MICHIGAN
TUEBOR
QUÆRIS PENINSULAM AMŒNAN
CIRCUMSPICE
1.
!
A
GEOMETRICAL TREATISE
OF THE
CONIC SECTIONS.
2
A
GEOMETRICAL TREATISE
OF THE
CONIC SECTIONS,
IN WHICH
The PROPERTIES of the SECTIONS are derived
from the Nature of the CONE,
In an EASY MANNER, and by a NEW METHOD.
By HUGH HAMILTON, A. M.
Fellow of TRINITY COLLEGE, DUBLIN, and of the ROYAL SOCIETY, LONDON,
Now Dean of ARMAGH.
Tranflated from the LATIN ORIGINAL into ENGLISH.
LONDON,
Printed for J. NOURSE, in the Strand, Bookfeller to HIS MAJESTY.
MDCCLXXIII.
13903
THE
TRANSLATOR's PREFACE.
T
HE following Treatife, written in Latin, and pub-
liſhed in the year 1758, has been fo well received
by the learned, that the profeffors in the ſeveral
univerſities have uſed and recommended it in preference to
all others on the fame fubject; and therefore the tranſlator
thought that an Engliſh edition would be an acceptable pre-
fent to the publick, in a country where fo many, who had
not much cultivated the learned languages, had yet diftin-
guiſhed themſelves by their great proficiency in mathemati-
cal ftudies.
As the author in his preface has given an account of the
method in which he has deduced the properties of the Conic
Sections, the tranflator means only to add fome obfervations
here, which may fhew wherein this method is preferable to
any other that has been hitherto uſed.
Apollonius, and the other writers who have deduced the
properties of the fections from the cone, enquired but little
further
vi
The TRANSLATOR's PREFACE.
further into the nature of the cone than was neceſſary to
fhew that three different fections, befides the triangle and
circle, might be obtained by cutting its furface with a plane,
and to demonftrate in one particular inftance that leading
property of each Section, which is ufually called its equation.
This was much too narrow a foundation on which to raiſe,
with eafe and elegance, fo large a fuperftructure as the
doctrine of Conic Sections. Some of the modern writers
have betaken themfelves to a very different method, and
founded this doctrine entirely on thofe properties in each
fection, by which they may be defcribed mechanically on a
plane. But as theſe properties have a very remote relation
to any of the general and fundamental properties of the
fections, their method is ftill more prolix, as well as lefs
natural, than the former one.
De la Hire, the famous French mathematician, who has
written largely on this fubject, deduces the properties of the
fections from the cone, and that he might have more extenſive
principles to proceed on, he has premiſed one entire book of
lemmas, containing the properties of lines cut in harmonic
proportion, particularly of fuch as meet a circle, which is
the baſe of a cone. Yet this expedient has rather embar-
raſſed the doctrine of Conic Sections, by making it depend
fo much on another that is really foreign to it. For there
are but very few properties of theſe ſections that have any
immediate connection with thofe of lines cut harmonically;
and fuch the reader will find compriſed in a few propofitions
in our author's fifth book.
Now in all their reſearches on this ſubject, none of theſe
learned writers feem to have fufpected that any of the ge-
neral
The TRANSLATOR's PREFACE.
vii
•
neral properties of the fections, which they do not treat of
till towards the conclufion of their works, might have been
demonftrated in the beginning, and would have been an
ample fource from whence to derive all the other proper-
ties. And yet nothing can appear more eafy and natural
than the method of doing this, now that our author has
diſcovered it, and applied it ſo ſucceſsfully in the following
Treatife. And, as we may collect from his preface, he
feems to have been led to this difcovery by an obfervation,
very ingenious indeed, but fuch as one would think might
have been fufficiently obvious, viz. that whatever property
agrees to all the ſections in general, muſt be a property of
the conic furface on which they are placed. Such a pro-
perty as this he has demonftrated of the cone in the ele-
venth propofition of his firft book, and deduced from it
three other propofitions relative to the cone, which contain
very material properties of the fections. He then proceeds
to define the three conic fections, and the feveral lines be-
longing to them; and from the definitions he draws fome
corollaries, to fhew that the properties before demonftrated
of the cone may be equally affirmed of its ſections.
Having thus laid down the principal properties of the
fections in the beginning, he is enabled to deduce from
thence the other properties with conciſenefs and elegance, to
arrange the ſeveral parts of his fubject in their moſt natu-
ral order, and always to give general demonftrations for
every property that is in itſelf a general one: by which
means his work is freed from the intricacy and prolixity
that have been complained of in other writers. We may
therefore venture to ſay that this work has juftly mérited
the
viii
The TRANSLATOR's PREFACE.
the applauſe it has met with; and doubtleſs it will continue
to be admired while ever any tafte prevails for geometrical
learning.
Nothing but an ardent defire of promoting this uſeful
branch of ſcience among his countrymen could have en-
gaged the tranflator in this undertaking; and he flatters
himſelf he has fucceeded fo far as to give the juſt ſenſe of
the original, eſpecially as he has been affifted in carefully
revifing the whole by fome friends of acknowledged abi-
lities.
London, March 21, 1773.
THE
AUTHOR's PREFACE.
treats of the
Tis not neceffary to say any thing here of the importance
of that branch of geometry which treats of the properties
of the Conic Sections. It has been very much cultivated
by mathematicians both ancient and modern, and the learned
well know its great use in aftronomy, natural philoſophy, and
Jome of the practical arts.
Among the ancients who wrote on this fubject, Apollonius
was the moſt eminent, and on account of his excellent difcoveries,
and the accuracy of his demonftrations, obtained the name of
the Great Geometrician. Yet the learned and ingenious Doc-
tor Wallis complains, that the doctrine of Conic Sections, as de-
livered by him, is perplexed and difficult, and that neither My-
dorgius, nor others, had then removed this difficulty. Nor in-
deed do we find that any of the fubfequent writers have treated
this fubject with fufficient clearness. The reafon of which I
take to be; that thofe among the moderns who defined the Conic
Sections by their defcriptions on a plane, were thereby obliged to
deduce all their properties from a fingle one in each fection,
which is very remote from any of their general or fundamental
properties. So that in this method many propofitions must be
a
demon-
AUTHOR's PREFACE.
demonftrated, and in a very operofe manner, before thofe funda
mental properties can be treated of. The ancients indeed, and
they who followed their method, judged rightly that the proper-
ties of the Sections were beft derived from the Cone. And yet
their works are difficult and prolix; because they fet about de-
monftrating the properties of the Sections before they had fuffi-
ciently confidered the nature of the Cone. From whence it came
to pass that they demonſtrated with much labour many proper-
ties of each of the Sections Separately, which may, in a ſhort
and clear manner, be proved to belong to the Cone itself, and
may from thence be very easily transferred to all its Sections.
What has been here faid will, I believe, not appear ill founded
to those who compare the methods I have mentioned with that
which is uſed in the following Treatise. What that method is,
and the advantages arifing from it, may be hewn in a few
words.
In the elements of Geometry the properties of the triangle
and circle (which are the two moft fimple of the Gonic Sections)
are easily deduced from their defcriptions on a plane. But fince
the properties of the three others, which are to be the ſubject of
the following Treatise, cannot be eafily and concisely deduced
from their defcriptions on a plane; I propofe to retain their old
definitions, and to explain the nature of the conic furface in
fuch a manner, that as foon as thefe curves are defined by the
interfection of a plane with this furface, it may appear what
their principal properties are. Accordingly, before the Sections
are defined, certain properties of the conic furfaces are demon-
ftrated, as in Prop. XI. XII. XIII. XIV. Book I. which con-
tain in them most of the fundamental properties of the Sections.
And then fince it appears from their definitions, that thefe Sec-
tions
AUTHOR's PREFACE.
xi
tions are curves, all the points of which are placed on a conic
furface, it is manifest that every right line which any way
meets thefe Sections, muft in the fame manner, and in the fame
points, meet the conic furface; and therefore all the properties
which are proved to agree to right lines meeting the conic fur-
face, are immediately transferred to thofe which meet the Conic
Sections. And thus the principal and most general properties
of the Sections are laid down in the beginning, from whence we
may reaſonably expect their particular properties will be more
eafily deduced.
}
Several advantages arife from treating the fubject in this me-
thod. For thus every property which agrees to the three Sec-
tions may be always demonftrated of them all jointly. Then by
deducing particular properties from general ones the demonftra-
tions will be rendered fhorter. Befides, we shall have no occa-
fion to make uſe of Lemmas, or thofe Propofitions which are
brought in only to demonftrate others, with which works of this
kind do too much abound; ſo that ſuch Propofitions only will be
inferted as are of use in other ſciences, or feem to contain fome-
thing worthy of Speculation. By the help of this method, we
fhall be able to difpofe the Propofitions in fuch an order, that the
properties, which are of the fame kind, or relate to the fame
part of the ſubject, may be demonftrated together. And lastly,
we fhall perceive from hence the reaſon why theſe curves, which
differ fo much in fome properties, agree entirely in others.
The demonftrations of the theorems, and folutions of the pro-
blems, are moſtly new; as a new method and a new order of
the Propofitions neceffarily require. And that the demonftra-
tions, like thofe of the ancient geometricians, may be perfectly
accurate, Evaneſcent quantities and points fuppofed to be re-
a 2
moved
sii
AUTHOR's PREFACE.
:
moved to an infinite diftance, which are now made ufe of by
fome writers, find no place here.
In the following Treatife I have inferted whatever I found
in the works of others that feemed ufeful or proper for a work
of this kind, and have introduced fome Propofitions, which, as
far as I know, have not been delivered by others. Among
which may be reckoned those before mentioned, which relate to
the conic furfaces; as alfo Prop. LIV. Book I. Prop. XXIX.
XXXVI. XXXVII. Book II. Prop. III. V. Book III. Prop.
VI. IX. XI. XII. XIII. Book IV. Prop. XIV. Book V. be-
fides fome new Corollaries in different places.
The reader will find, in fome places, references to fuch `Pro-
pofitions of Sir Ifaac Newton's Mathematical Principles of
Natural Philofophy, as are more fully demonftrated in the fol-
lowing Work; which is chiefly defigned as an Introduction to
the Newtonian Philofophy, the mathematical part of which is
built on the doctrine of Conic Sections.
Trinity College, Dublin,
March 20, 1758.
CONTENTS of the BOOK S.
воок I.
THE first nine Propofitions contain ſuch properties of the conic fur-
faces as have been delivered by others. Then certain properties are
proved to belong to thefe furfaces, which are alſo principal and funda-
mental properties of the Sections, as appears afterwards from their de-
finitions. The centers of the ellipfe and hyperbola are found, and the
properties which agree to all the diameters of the Conic Sections in ge-
neral are demonftrated, and the proportion between the fquares of their
ordinates is fhewn. The generation and properties of thofe lines that are
called afymptotes of the hyperbola are clearly fet forth, and the nature
of conjugate hyperbolas explained. Several methods are fhewn of draw-
ing tangents to the Conic Sections. This Book alſo treats of the propor-
tions between the Squares and rectangles contained under the fegments of
lines which meet each other, and either touch or cut the Conic Sections:
and alſo of the rectangles contained under lines drawn from points in the
Sections to the fides of an infcribed trapezium, and which make given
angles with thoſe fides. It is proved that two Conic Sellions cannot meet
each other in more than four points.
BOOK
IT.
THE parameters of the Sections are defined, and thoſe properties of
the Sections demonftrated, from whence they have their denomina-
tions of ellipfe, parabola, and hyperbola. The axes of the Sections are
found,
[ xiv]
found, and thofe of the ellipfe and hyperbola are compared with their
other diameters. The foci and directrices of the sections are defined,
and the proportion is fhewn between a line drawn from any point in a
Section, and a perpendicular drawn from the fame point to the direc-
trix. Theſe lines are proved to be equal when the Section is a parabola,
and when from a point in an hyperbola a line is drawn to the focus, and
another to the directrix, and parallel to the afymptote, thefe lines are
also proved to be equal. It is proved, that if two lines be drawn from
a point in the ellipſe or hyperbola to the foci, their fum in the former,
and difference in the latter, will be equal to the tranfverfe axis of the
Section: that a tangent to the parabola makes .equal angles with two
lines drawn from the point of contact, one to the focus, and the other a
perpendicular to the directrix: and that a tangent to an ellipfe or hyper-
bola makes equal angles with lines drawn from the point of contact to the
foci. Various properties are demonftrated of lines paſſing through the
foci of the Sellions. A method is fhewn of defcribing the Conic Sec-
tions on a plane by means of inftruments. At the end of this Book it is
demonſtrated, that when a Conic Section is placed on the ſurface of a
right Cone, if a plane paſſing through its directrix forms a circle on the
Conic furface, the fide of the Cone, intercepted between this circle, and
any point in the Section, is equal to a line drawn from that point to
the focus.
THI
BOOK
III.
HIS Book treats of the proportion of the parameters of the feveral
diameters of a parabola, and of fuch properties of the parabola as
are less analogous to thofe of the other fections; and alfo of the quadra-
ture of the parabola, and the proportion of parabolic fegments to each
other.
BOOK
[ xv ]
воок
IV.
IN
'N this Book fuch properties are treated of as belong only to the ellipfe
and hyperbola. The parallelograms circumfcribed about their conju-
gate diameters, and about certain other diameters, are proved equal. It
is demonftrated that in the ellipfe the fum, and in the hyperbola the dif-
ference of the ſquares of any two conjugate diameters is equal to the fum
or the difference respectively of the fquares of their axes.
The equal con-
jugate diameters of the ellipfe are treated of. A new method is fhewn
for finding two conjugate diameters of the ellipfe or hyperbola, which may
contain given angles, by finding out the interfections of those curves with
a circle, whofe center lies in one of their axes. The interfection of an
ellipſe or hyperbola with a right line given in pofition is also found. The
proportion of hyperbolic sectors and trapezia to each other is fhewn, and
the ellipfe is compared with a circle defcribed about its axis.
воо к
V.
THIS Book treats of Conic Sections that are fimilar to each other, and
of fuch properties of the Sections as relate to lines cut in harmonic
proportion. It is proved that the concourfe of any two tangents of a
Conic Section will lie in a right line given in pofition, when the line join-
ing their points of contact paffes through a given point. This Book like-
wife treats of many other general properties of the Conic Sections, and
of the feveral points in which they may cut or touch each other of the
greatest and leaft lines that can be drawn from a point in the axis to the
Jection of circles that have the fame curvature with the Conic Sections:
and of the manner of defcribing Conic Sections, which may pass through
given points, and touch right lines given in poſition.
i
OF THE
CONIC SECTIONS.
BOOK THE FIRST.
Of the Nature of the Cone, and of its Section.
I.
"I
DEFINITION S.
F through the point A, without the plane of the circle BDZC, FIG., 1.
a right line AZ be extended indefinitely both ways, and if
(while the point A remains fixt) this line be carried round
the whole periphery of the circle; each of the furfaces generated by
this motion is called a conical furface, and both together are called
oppofite Surfaces.
II. The folid contained by the conical furface and the circle BDZ
is called a Cone.
III. The point A is called the vertex of the cone.
IV. The circle BDZ, the bafe of the Cone.
V. A right line AZ paffing through the vertex of the cone and
any point in the periphery of the bafe, is called a fide of the Cone.
A
VI. The
2
Book I.
Conic Sections.
VI. The right line AC, paffing through the vertex and center C
of the baſe and produced indefinitely, is called the axis.
VII. A right Cone is that whofe axis is perpendicular to the baſe.
VIII. A Scalene Cone is that whofe axis is inclined to the baſe.
IX. A right line meeting a conical furface in one point and
produced both ways falling without the furface, is called a Tan-
gent; but a right line meeting the furface, or the oppofite furfaces
in two points, is called a Secant.
COR. A right line paffing through any point in either furface
and the vertex of the cone, is wholly in that furface; and being
produced beyond the vertex, is in the oppofite ſurface.
X. The common interfection of a plane with a conical furface is-
called a Conic Section.
PROPOSITION
I.
Any plane paffing through the vertex of a cone and cut-
ting the periphery of the bafe, will cut the oppofite fur-
faces in two right lines and in them only.
F
OR if from the points, in which this plane meets the peri-
phery of the baſe, two right lines be drawn to the vertex; they
will be in this interfecting plane, and (by Cor. preceding) in the
furface of the cone, and being produced beyond the vertex, in the
oppofite furface; that is, the plane will cut the oppofite furfaces in
theſe two right lines; and as the interfection of this plane with
the plane of the bafe, cannot cut the periphery of the baſe in three
points, it is evident that the plane cannot cut the oppofite furfaces
in three right lines; therefore the plane will cut the furfaces in two
right lines only.
COR. The common interfection of any plane drawn through the
vertex with the conical furface and baſe, will be a triangle.
PROP.
Book I.
3
Conic Sections.
II.
PROP.
A right line ED joining any two points E, D, in a coni- Fig. 2.
cal ſurface, provided they be not in the fame right line
paffing through the vertex, falls wholly within the fur-
face, but produced both ways without it, and does not
meet again either of the furfaces. On the contrary, a
right line ED joining two points E, D, in the oppofite FIG. 3.
furfaces, provided they be not in the fame right line.
paffing through the vertex, falls without both furfaces,
and produced both ways it falls within them, and meets
neither of them again.
FOR
OR the fides AEC and ADB of the cone being drawn
through the points E and D, a plane paffing through theſe
lines does not meet the furface or oppofite furfaces, but in the
lines AEC, ADB; therefore the line DE does not meet them, but
in the points D, E.
In the firſt caſe the plane of the angle DAE and confequently FIG. 2.
the line DE, is within the conical furface; and therefore pro-
duced both ways falls without it.
In the ſecond cafe the plane of the angle DAE and confe- FIG. 3.
quently the line DE is without both the oppofite furfaces; and
therefore DE produced both ways falls within the oppofite furfaces.
If the points D, E, be in the fame right line paffing through
the vertex, the line joining theſe points (Cor. to Definitions) will be
wholly in the conical furfaces.
PRO P.
III.
If a right line DE touches the periphery of the bafe of FIG. 4.
the cone in the point D, and from A the vertex of the
cone, the right line AD be drawn to the point of con-
tact; the plane ADE paffing through both thefe lines
touches
A 2
4
Book I.
Conic Sections.
The
touches the conical furface in the line AD only.
meeting of this plane with the furface is called the line
of Contact.
OR the line AD is in the conical furface, and in the plane ADE,
Fo
and as every other point befides D, as F in the periphery of
the bafe, is without the line DE; it follows that every other line
befides AD, as FA in the conical furface, is without the plane-
ADE, and hence the propofition is manifeft.
COR. 1. Hence, and from Defin. I. it appears that the plane ADE
produced touches the oppofite furfaces in the line AD produced.
COR. 2. Hence appears a method of drawing a plane which ſhall
touch a conical furface in a given right line, as AD; for the right
line DE being drawn in the plane of the bafe, touching the periphery
in D, the plane ADE drawn through AD, DE touches the ſurface
in the line AD.
COR. 3. If any right line ED touches the conical furface in the
point D, and AD be drawn from the vertex; the plane paffing
through AD, DE touches the furface. For through the vertex A
in the plane ADE, draw AG parallel to ED, this right line meets
all the fides of the cone in the vertex; therefore if any other fide of
the cone beſides AD were in the plane ADE, the line ED parallel to
AG would neceffarily meet that fide, and then ED would meet the
conical furface in two fides, contrary to the hypothefis; wherefore
the plane ADE does not meet the conical furface, but in the line AD.
COR. 4. No other plane befides ADE touches the conical furface
in the line AD; for its interfection with the plane of the baſe will
cut its periphery. Hence it follows that this plane cuts the conical
furface in two right lines (by Prop. I.) and therefore does not touch it.
PROP. IV.
IV. PROB L. I.
FIG. 5, 6. Through a right line AC, paffing through the vertex of
the cone, but not in its furface, to draw a plane which
fhall touch the conical furface.
THROUGH
Book I.
5
Conic Sections.
T
HROUGH the line AC let any plane be drawn, cutting
the baſe in GH, which interfection the line AC (being in the
fame plane) will either meet, or be parallel to: firſt, let them meet
in E; and from this point on either fide of EHG let a right line FIG. 5.
be drawn touching the periphery of the bafe; and join AD, and
ADE will be the plane required; for it paffes through AC, and (by FIG. 6.
the preced.) touches the furface. Secondly, let AC be parallel to
GH; and let a diameter MD of the bafe be drawn perpendicular to
GH and meeting the periphery in D, and through the point D
draw DE touching the bafe; it will be parallel to GH and confe-
quently to AC; then joining AD, ADE will be the plane required;
for becauſe AC and DE are parallel, AC will be in the plane ADE
and this plane touches the conical furface (by the preced.)
COR. It is evident that only two planes paffing through AC can
touch the conical furface, viz. one on each fide of the plane AGH.
PROP. V.
Any right line ED parallel to a fide AB of the cone (pro- Fic. 7,
vided it be not in the plane touching the furface in AB)
meets one of the oppofite furfaces and that in one point
E only; that is, on one fide it is wholly without both
furfaces, and on the other wholly within that where
the point E is taken.
L'
ET a plane paffing through the parallels AB, ED, cut the fur-
faces in CAC; this plane does not meet the furfaces but in
the lines AB and CAC; and the line ED in this plane does not
meet AB (for they are parallel) but neceffarily meets CAC fome-
where, as in the point E, and on one fide of the point E is within
the angle BAC, and on the other part within the angle adjacent to
it. Whence the propofition is evident.
COR. It appears that the plane paffing through DE and pa-
rallel to the plane touching the conical furface in the line AB,
does not meet the oppofite furface.
PROP.
6
Book I.
Conic Sections.
PROP.
VI.
FIG. 8.
FIG. 9.
Every right line EF parallel to a right line AD, paffing
through the vertex A, and falling within the conical fur-
faces, meets each furface once as in the points E, F; and
every right line EF drawn through any point P within
either furface, and parallel to any right line AD paffing
through the vertex and falling without the furfaces,
meets the fame furface fomewhere, as in the points E, F.
L
ET the plane paffing through AD and EF cut the conical
furfaces in the lines BAb, CAc, which it is manifeſt may be
always done; fince in the firſt caſe AD falls within the angle BAC,
the line EF, which is parallel to AD, and in the fame plane with the
angle BAC, neceffarily meets the fides AC, AB, viz. AC in E, and AB
produced in the point F of the oppofite furface: and in the fecond
cafe, fince AD falls without the angle BAC, the line EF parallel to
it, drawn through the point P within this angle, neceffarily meets
AC, AB fomewhere as in E, F, that is, meets the furface in the
points E, F, within which the point P is taken.
COR. 1. Hence, and from Prop. II. it appears that a right line
does not meet a conical furface or oppofite furfaces in three points.
COR. 2. If a right line be drawn parallel to any right line EF
meeting the oppofite furfaces; this line will likewife meet the op-
pofite furfaces. For if AD be drawn through the vertex A parallel
to EF it is evident that it falls within the conical furfaces; there-
fore any right line drawn parallel to EF will be parallel to a right
line AD paffing through the vertex and falling within the furfaces,
and confequently it meets the oppofite ſurfaces.
COR. 3. Since the line EF joins two points in the fame furface,
the line AD drawn through the vertex parallel to it falls without
the ſurface; for it is manifeft that it falls without the angle BAC,
the fides of which EF meets.
COR?
Book I.
7
Conic Sections.
COR. 4. Every plane paffing through this line EF, in the firſt
cafe, will cut both furfaces; and in the fecond cafe, if a plane
falling without the conical furfaces, paffes through AD; the plane
drawn through EPF parallel to it, will cut the conical furface
within which the point P is taken throughout its whole circumfer-
ence, and will not meet the oppofite furface.
PROP. VII.
If either of the oppofite furfaces be cut by a plane GHF FIG. 10.
parallel to the baſe, the ſection EGFL will be the cir-
cumference of a circle whofe center is in the axis of
the cone.
L
ET C be the center of the bafe and AC the axis of the cone,
and let the cone be cut in any manner through the axis by
two planes making the triangles ABD, AIK, which (produced if
neceffary) may meet the plane GHF in the lines GHL, EHF; the
planes being parallel, the triangles AHF, ACK, and likewiſe AHL,
ACD, are fimilar: wherefore HL is to CD as AH to AC, and
likewife HF to CK will be in the fame ratio; whence by petmu-
tation, HL is to HF as CD to CK: but CD and CK are equal;
therefore HL and HF are equal. In like manner it may be de-
monftrated, that any other two lines, as HG, HE, drawn from
the point H to the fection EGFL, are equal; therefore that fection
is the circumference of a circle (by 9. 3.) and its center H is in the
axis of the cone as is evident.
COR. 1. Hence it appears that if a diameter of fuch a fection
be drawn, it will cut the axis of the cone and converſely, if a
right line drawn in the plane of this fection cuts the axis of the
cone, it will be a diameter of the fection.
COR. 2. It likewife appears that any circle parallel to the bafe
be taken for the baſe of the cone.
may
!
PROP.
8
Book I.
Conic Sections.
FIG. II
PROP. VIII.
If a ſcalene cone ABCL be cut through the axis by a plane
perpendicular to the bafe making the triangle ABL, and
from the vertical angle BAL another triangle ADI be cut
fimilar to ABL, but in a fubcontrary pofition, that is,
ſo that the angle ATD be equal to the angle ABL; and
if the cone be cut again by a plane paffing through DT
and perpendicular to the plane of the triangle ABL:
the fection DKTF in the conical furface will be the cir-
cumference of a circle.
L
ET the right line MN be the common interfection of the plane
DKTF with the plane of the baſe: from any point K, in the
fection DKTF, draw KGF parallel to MN, and meeting DT in G,
through G draw EH in the plane ABL and parallel to BL; then the
plane paffing through KGF, EGH will be parallel to the baſe (15.
11.) and therefore perpendicular to the plane ABL, and its interfec-
tion with the ſurface of the cone will be a circle, of which EGH is
a diameter (by the preced. and Cor. 1.) and becauſe both theſe
planes are perpendicular to the plane of the triangle ABL, their com-
mon interfection KGF will be perpendicular to the plane ABL (19.
11.) and therefore to both theſe right lines EH, DT; wherefore
the fquare of KG (by a property of the circle) is equal to the rect-
angle EGH; but the angles ATD and (ABL or) AEH are equal
(by hypothefis) and the vertical angles at G are equal; therefore the
remaining angles EDG, GHT are equal, and therefore the triangles
EDG, GHT are fimilar; wherefore DG is to EG as GH to GT;
and therefore the rectangle DGT is equal to the rectangle EGH,
that is, to the ſquare of KG. Therefore the point K is in the circum-
ference of the circle deſcribed about the diameter DT in the plane
DGK for if it were not, the rectangle DGT would be equal to the
fquare of a line greater or leſs than KG, contrary to what has been
proved in like manner the fame may be fhewn of any other point
in the fection DKTF, which is therefore the circumference of a circle.
This fection is called a fubcontrary fetion.
PROP.
Book I.
9
Conic Sections.
PROP.
IX.
If in the conical furface the fection DKTF made by a plane FIG, 11.
not parallel to the baſe be the periphery of a circle, it
will be a ſubcontrary ſection.
L
ET a conical furface be cut by two planes parallel to the baſe
making the circles EKHF, OPQR and cutting the plane
DKTF in the lines KF, PR, which will be parallel (by 16. 11.)
let the diameters EH, OQ of theſe circles be drawn perpendicular
to the lines KF, PR and meet them in the points G, S; the lines
EH, OQ will be parallel (Converf. 10. 11.) and will bifect the lines.
KF, PR in G and S (3. 3.) and will paſs through the axis of the
cone (by 1. Cor. 7. of this Book) therefore the plane AOEBLHQ
paffes through the axis of the cone; let the line DT be its inter-
fection with the plane of the circle DKTF; fince DT paffes
through the points G and S and confequently bifects the lines KF,
PR terminated by the circle DKTF, it is a diameter of that circle
and is perpendicular to theſe lines as appears from the (3. 3.) Then
becauſe the line KF is perpendicular to the lines DGT, EGH, it
will be perpendicular to the plane ABL paffing through theſe lines;
therefore (by 18. 11.) the planes of the circles DKTF, EKHF and
confequently the plane of the bafe will be perpendicular to the plane
ABL paffing through the axis of the cone; but (by a property of the
circle) each of the rectangles EGH, DGT is equal to the fquare of
KG and confequently are equal to each other; therefore DG will be
to EG as GH to GT, and the vertical angles at G are equal;
therefore the triangles DGE, HGT (by 6. 6.) are fimilar, and
therefore the angle ATD is equal to the angle AEH, that is to the
angle ABL, wherefore by the preced. Prop. the ſection DKTF is a
fubcontrary fection.
COR. Hence a conic fection, which is not a fubcontrary fection,
nor parallel to the bafe, will not be the circumference of a circle.
B
PROP.
10
Book I.
Conic Sections.
FIG. JZ,
13.
PROP. X.
Let two right lines meet each other in the point P, and let
one of them be parallel to the baſe of a cone, and the other
parallel to a right line given in pofition cutting the plane
of the baſe according as both touch or cut, or one of
them touches, and the other cuts a conical furface, or op-
pofite furfaces; the fquares of the fegments of the tan-
gents, or the rectangles under the fegments of the fecants,
between their point of concourfe, and the points where
they meet the conical furface or ſurfaces, will be in a con-
ſtant ratio to each other, wherefoever P the point of con-
courſe of theſe two lines is taken. I call a right line pa-
rallel to a plane, when in that plane a right line can be
drawn parallel to it.
L
}
ET the circle ASC be the bafe of the cone ABC, and BM the
right line given in pofition drawn from the vertex B to any
given point M in the plane of the baſe (but not in its periphery) and
from the point M let any right line be drawn cutting the periphery
of the baſe in the points K, L; let two right lines meet each other
in any point P, and let one of them POG or PT be parallel to
the baſe of the cone and cut the furface in the points O, G, or
touch it in T, and let the other line PEF be parallel to BM; firſt
let this line PEF cut the fame furface or oppofite furfaces in E, F;
the rectangle EPF will be to the rectangle OPG or to the fquare of
PT, as the fquare of BM to the rectangle KML, wherefoever P
the point of concourſe of the right lines may be taken.
For through the line POG let a plane be drawn parallel to the
bafe making the circle DOHG, and through the parallels BM and
PEF let another plane pafs meeting the furfaces in the lines QEA,
RFC, the plane of the baſe in the line MCA, and the plane of the
circle DOHG in the line DPH; MCA, DPH are parallel (16. 11.)
therefore the triangles EPD, BMA, are fimilar, and alſo the tri-
angles
}
}
FIG•I•
B
C
FIG.2.A
F
FIG. 3.
B
B
Ъ
A
FIG.7.
FIG.8.
B
FIG.5.
A
D
D
E
D
}
E
D
B
FIG.10.
E
A
G
FIG.4.
B
H
G
E
F
F
D
E
K
L
D
FIG.9
F
FIG. 6.
TAB I
A
c Page-10.
G
M
P
E
E
H
4
E
D
FIG.II.
D
R
O
Q
B
F
G
E
H
K
D
BA
UNIV
OF
J
M
7
Book I.
I I
Conic Sections.
angles FPH, BMC, and confequently EP is to DP as BM to AM,
and PF to PH as BM to MC; therefore (by taking the rectangles
under the antecedents of one order of proportionals and the antece-
dents of the other, and the rectangles under the confequents of one,
and the confequents of the other) the rectangle EPF is to the rect-
angle DPH as the fquare of BM to the rectangle AMC by (23. 6.)
but becauſe DOHG and ASC are circles, the rectangle DPH is
equal to the rectangle OPG or to the fquare of PT, and the rect-
angle AMC is equal to the rectangle KML by (35 and 36. 3.)
therefore the rectangle EPF is to the rectangle OPG or the fquare
of PT, as the fquare of BM to the rectangle KML, wherefoever
P the point of concourſe of the lines EF, OG is taken.
But if the line EF parallel to BM meets in P the line PB pa-
rallel to the baſe of the cone and paffing through the vertex; then
in like manner it may be demonſtrated (from the fimilar triangles
EPB, BMA; and FPB, BMC) that the rectangle EPF is to the
fquare of PB as the fquare of BM to the rectangle AMC, that is,
to KML.
In the fecond place, the reft remaining as before, let the right
line PF touch the furface in F, and through the point of contact F FIG. 14.
let the fide of the cone be drawn meeting the baſe in N; the plane
paffing through the parallels PF, BM touches the furface in the
line BFN (3. Cor. 3. of this Book) and its interfection MN with the
plane of the baſe, touches the baſe in N; let the line PT be the
interfection of the fame plane with the plane of the circle OTG in
which the point P is; this line touches the circle OTG in the point
T, viz. in the line BFN, and will be parallel to MN (16. 11.)
therefore, from the fimilar triangles FPT, BMN, the ſquare of PF
will be to the fquare of PT or to the rectangle OPG, as the fquare
of BM to the fquare of MN, or to the rectangle KML. Q. E. D.
B 2
1
PRO P.
;
I 2
Book I.
Conic Sections.
އ
PRO P. XI.
If two right lines meeting each other be parallel to two right
lines given in pofition; according as both touch or cut, or
one of them touches, and the other cuts a conical furface
or oppofite furfaces; the fquares of the tangents, or the
rectangles under the fegments of the fecants between
their point of concourſe and the points wherein they meet
the furface or oppofite furfaces, will be in the fame ratio
to each other wherefoever the concourſe of the two right
lines falls.
Cafe 1.
WH
HEN both the lines are parallel to the baſe of the
cone, a plane paffing through them will be parallel
to the baſe (15. 11.) and therefore will cut one of the furfaces in
the circumference of a circle; and therefore both the lines will either
touch or cut, or one of them will touch and the other cut the fame
circle; and therefore the fquares of the fegments of the tangents,
or the rectangles under the fegments of the fecants between the
point of concourſe of the two lines, and the points wherein they
meet the ſurface will be to each other in a ratio of equality by the
35 and 36. 3.
Cafe 2. When one of the lines meeting each other is parallel to a
line given in pofition cutting the plane of the bafe, and the other
parallel to a line given in pofition drawn in the plane of the baſe;
this line will be parallel to the plane of the baſe, and therefore this
cafe is manifeſt from the preceding Propofition.
Cafe 3. When neither of the lines meeting each other is parallel to
the plane of the baſe; through their point of concourſe let a right
line be drawn parallel to the plane of the bafe, and cutting either
furface, or perhaps paffing through the vertex of the cone; then
according as either of the lines not parallel to the bafe touch or cut
the furface or oppofite furfaces, the fquare of the fegment or the
rectangle under the fegments of the fecant, will be in a conftant ratio
to the fquare of the fegment, or to the rectangle under the ſegments
of the line parallel to the plane of the bafe (by preceding Prop.)
and
着
​Book I.
13
Conic Sections.
and confequently thefe fquares or rectangles under the fegments of
the lines which are not parallel to the plane of the baſe, will be in
a conftant ratio to each other as appears from (22. 5.) *
In this and the preceding propofition when one of the right lines
which meet each other paffes through the vertex of the cone, the
fquare of the fegment between the point of concourfe and the vertex,
muſt be taken as if the right line was a tangent, although it may
happen to fall within the conical furface.
PROP. XII.
If two parallel right lines meet a right line parallel to a fide
of a cone; according as both touch or cut, or one of them
touches, and the other cuts a conical furface or oppofite
furfaces; the fquares of the fegments of the tangents, or
the rectangles under the fegments of the fecants between
the furface or oppofite furfaces, and the right line which
they meet, will be to each other as the fegments of that
right line between the parallels, and the point wherein it
meets the conical furface.
L
ET MBC be a cone, and the right lines EF, LN parallel to FIG. 15.
each other, and let them cut in any manner the fame furface
or oppofite furfaces in E, F and L, N, and in the points P, Q meet the
line ST parallel to cBC the fide of the cone, and meeting the furface in
V; the rectangle EPF will be to the rectangle LQN as PV to QV.
For let the plane paffing through the parallels cBC and ST meet
again the ſurfaces in the line mBM, and through the points P, Q
draw in this plane the lines MPC, DQH parallel to each other, and
meeting the furface in M, C and D, H; then (by the preceding) the
rectangle DPF is to the rectangle MPC as the rectangle LQN to
The 22d Propofition of the fifth book of Euclid here referred to is that by which
the way of reafoning in proportions called, ex æquo ordinatim is proved, and from
thence the above conclufion will follow. It is alfo the Propofition by which Euclid
proves the eighth of his Data, which is, that two quantities which have a given ratio to
a third, have alſo a given ratio to each other; therefore the eighth of Euclid's Data
might have been more properly referred to on this occafion.
the
FIG. 16.
1 4
Conic Sections.
Book I.
the rectangle DQH; therefore by permutation the rectangle EPF
is to LQN as the rectangle MPC to DQH, that is, becauſe PC and
QH are equal (being oppofite fides of a parallelogram) as MP is to
DQ, or from the fimilarity of triangles as PV to QV; the fame
may be fhewn concerning the fquares of the fegments of the pa-
rallels between the line ST and the point of contact, if one or both
be tangents.
PROP.
XIII.
Q. E. D.
Let two parallel lines touch a furface, or oppofite furfaces,
and two points be taken in them equidiftant from their
points of contact; if the right line joining theſe points
meet the ſurface or oppoſite ſurfaces; its fegments be-
tween the tangents and furface or furfaces, will be equal.
L
ET the right lines EP, RQ be parallel, and touch the fame
furface or oppofite furfaces in E and R, and let the points P, Q
be equidiſtant from the points E, R, and let PQ be drawn cutting
the furface or oppofite furfaces in the points F, G; the fegments
PF, QG will be equal.
For becauſe the lines EP, RQ are parallel, and the fame right
line PQ meets them; the fquare of EP will be to the rectangle
FPG as the fquare of RQ to the rectangle GQF (by the 11. of
this Book) but the fquares of EP, RQ are equal (by hypothefis)
therefore the rectangles FPG and GQF are equal; let FG be bi-
fected in O; and when the point F, G are in the fame ſurface, let
the equal ſquares of OF and OG be added to the equal rectangles
FPG, GQF and the fums that is the fquares of OP, OQ (by 6. 2.)
are equal; and when the points F, G are in oppofite furfaces, let
the equal rectangles FPG, GQF be taken from the equal fquares
of OF, OG and the remainders, that is (by 5. 2.) the fquares of
OP, OQ will be equal; therefore the right lines PO, OQ_ in both
cafes are equal; but FO, GO are equal, therefore the fegments PF,
QG will be equal.
But
Book I.
Conic Sections.
15
But if the line PQ touches the furface in the point V; then, as
before, (by 11. of this Book) the fquare of EP is to the fquare of
VP as the fquare of RQ to the fquare of VQ; therefore the
fquares of VP, VQ, and confequently the fegments VP, VQ are
Q. E. D.
equal.
COR. Let the parallel lines PF, QH touch the conical furface in FIG. 17.
the points E, R, draw ER, and through the point O taken in it
draw a right line parallel to PF, QH meeting the furface in the
points M, N : if through the points M, N two lines be drawn pa-
rallel to ER and meeting PF, QH in the points F, H and P, Q;
theſe lines either at the fame time cut the furface again in the points
G, L., or at the fame time touch it in the points M, N. By con-
ſtruction and (34. 1.) the tangents EF, RH and likewiſe EP, RQ
are equal; then if the line FMH meets the furface in the points
M, G, FM, GH will be equal (by this Prop.) and if the line PNQ
be ſuppoſed to touch the furface in N, PN, NQ will be equal;
therefore, becauſe PF, NM, QH, are parallel, FM, MH will be
equal, which is abfurd; therefore when the line FMH meets the
furface in two points, PNQ will not touch the furface in N; con-
ſequently it meets it again; wherefore FMH, PNQ will either
at the fame time cut the conical furface, or at the fame time
touch it.
PROP. XIV.
The right line joining the two points of contact of parallel
tangents bifects all right lines which it meets parallel to
thoſe tangents, and terminated both ways by either of the
furfaces.
L
18.
ET the lines EP, RQ be parallel, and touch the fame furface FIG. 17,
or oppoſite ſurfaces, and draw ER joining the points of con-
tact; then let the right line MN be parallel to EP, RQ, and meet
ER (produced if neceffary) in the point O, and let it be termi-
nated
16
Book I.
Conic Sections.
3
FIG. 18.
nated by the conical furface in the points M, N; the fegments MO,
ON will be equal. For draw through M, N two right lines pa-
rallel to ER, and first, both meeting (by Cor. preced.) the fame or
oppofite furfaces again in G, L, and the parallel tangents in the
points F, H and P, Q; then the lines MF, NP, being oppofite
fides of a parallelogram, will be equal, and likewife FH, PQ and
the tangents EP, RQ; therefore (by the preced.) the lines NP,
QL will be equal: and in like manner it may be fhewn that MF,
HG are equal; therefore PL, FG will be equal and confequently
the rectangles NPL, MFG are equal; but (by II of this Book) the
rectangle NPL is to the fquare of PE as the rectangle MFG to
the fquare of EF; therefore the fquares of EP and EF are equal
and confequently the lines EP, EF, and therefore the fegments MO,
ON which are parallel to them, will be equal.
When the points E, R. are in oppofite furfaces, the lines drawn
through M, N parallel to ER neceffarily meet the oppofite furfaces
(by 2. Cor. 6. of this Book) but when the points E, R are in the fame
furface, if one of the lines which are drawn through M, N parallel
to ER, touches this furface in the point M, the other line touches.
it in N (by Cor. preced.) let thefe lines meet the tangent EPF
in the points X, V; then the tangents MX, NV, being oppofite
fides of a parallelogram, will be equal; therefore (by the preced.)
the tangents XE, EV will be equal, and therefore the fegments
MO, ON being parallel to them, will be equal.
Q. E. D.
FIG. 19.
DEFINITION S.
XI. If a plane ADE touches a conical furface in the right line
AD, and the furface be cut by a plane parallel to ADE; the fec-
tion MVN is called a Parabola.
XII
R
P
D
T
B
Q
FIG. 15.
B
Q
·
TAB 2:
m
R
G
Page·16.
D
N
D
FIG·13 P
B
7 17
D
FIG.12
H
F
G
H
D
H
F
K
M
A
M
L
T
E
S
•
C
M
K
L
S
B
R
Q
P
T
FIG.14
2
F
A
I.
FIG.16.B
R
G
E
PP
C
K
M
F
N
S
UNIK
OF
th
Book I.
17
Conic Sections.
XII. If through the vertex and without the conical furfaces a
plane DAE paffes, which is neither parallel to the plane of the FIG. 20.
bafe nor to the plane of fubcontrary fection, and if either furface
be cut by a plane XVLT parallel to DAE; the fection MVNL is
called an Ellipfe.
XIII. If through A the vertex of the cone BAG a plane paffes FIG. 21.
cutting the furfaces in the right lines dAD, eAE, and the conical
furface be cut by a plane XVM parallel to EAD; the ſection MVN
is called an Hyperbola. And becauſe this interfecting plane neceffa-
rily meets the oppoſite ſurface, and there makes a fection mLn of
the fame kind and denomination with the former, theſe two ſections
together are called oppofite fections.
XIV. The plane ADE paffing through the vertex of the cone
and parallel to the plane of any ſection, I call the vertical plane of
that fection.
XV. Any right line LV drawn in the plane of a parabola, and FIG. 19.
parallel to the right line AD in which the vertical plane touches the
furface, is called a diameter of the parabola, and the point V where
it meets the ſection is called its vertex.
XVI. The point within an ellipfe, or between the oppofite fec-
tions, in which is bifected every right line drawn through it, and
terminated both ways by the ellipfe or by the oppofite fections, I
call the center of the ellipfe, or of the hyperbola, or of the oppofite
fections.
XVII. Any right line paffing through the center of an ellipfe or
hyperbola, and meeting the ellipfe or oppofite fections in two
points, I call a diameter of the ellipfe, or a tranfverfe diameter of the
hyperbola or of the oppofite fections; and the points in which the
diameter meets the ellipfe, or oppofite fections, are called its ver-
tices.
XVIII. Any right line paffing through the center of an hyper-
bola, and bifecting a right line not paffing through the center and
terminated by the oppofite fections, is called a fecond diameter of the
hyperbolas.
XIX. Two diameters, each of which bifects all right lines ter-
minated by the ellipfe, hyperbola, or oppofite hyperbolas and pa-
C
rallel
18
Book I.
Conic Sections.
rallel to the other diameter, are called conjugate diameters of the el-
lipfe or hyperbola.
XX. Any right line, not paffing through the center, terminated
both ways by a conic fection or the oppofite fections, and bifected
by a diameter, is called an ordinate applied to that diameter, or more
fimply an ordinate: and more frequently the half of this right line
is called an ordinate.
XXI. The fegments of a diameter between its ordinate and ver-
tices are called abfciffes.
XXII. A diameter of a conic fection, which cuts the ordinates
at right angles, is called an axis.
XXIII. A right line drawn in the plane of a conic fection, and
meeting the ſection in one point, and produced both ways falling
without it, is faid to touch the ſection in this point, and this line is
called a tangent; but if a right line meets the fection or oppofite
fections in two points, it is called a fecant.
Corollaries to the preceding Definitions.
COR. 1. As every point in a conic fection is in the conical fur-
face, and any two points in the oppofite fections are in the oppofite
furfaces, it is evident that every right line drawn in the plane of a
conic ſection, and meeting in any manner the conical furface or op-
pofite furfaces, will in the fame manner and in the fame point or
the fame points, meet the conic fection or oppofite ſections: and
converſely it is evident that every right line meeting in any manner
a conic fection or oppofite fections, will in the fame manner and in
the fame point or the fame points, meet the conical furface or oppo-
fite furfaces.
COR. 2. A right line joining any two points in a conic fection falls
wholly within the ſection, and produced both ways falls without it, and
does not meet either it or the oppofite fection again; and converfely
a right line joining two points in the oppofite fections falls without
both fections, but produced both ways falls within each fection and
meets
Book I.
19
Conic Sections.
meets neither of them again. This corollary is manifeft from the
preceding and (Prop. II. of this Book). Hence or from (Cor. 1.
6. of this Book) it appears that a right line cannot meet a conic
fection or oppofite fections in three points.
COR. 3. Hence the fame right line cannot touch the oppofite fec-
tions.
COR. 4. Hence likewife if in the plane of the oppofite fections a
right line be drawn parallel to the line joining two points in the op-
pofite ſections, it will neceffarily meet the oppofite fections, as ap-
pears from Cor. 2. Prop. VI.
COR. 5. All the diameters of a parabola are parallel to each
other, and any line drawn in the plane of a parabola and parallel to
the diameters, will be a diameter as appears from Definition
15. and
each diameter meets the parabola in one point only and is on one
fide wholly without the parabola, and on the other within it; as ap-
pears from Prop. V. and Cor. 1. to the Definitions.
COR. 6. It is manifeft fince the planes of the parabola and hy-
perbola cut the conical furface (though produced indefinitely) that
theſe ſections may be extended ad infinitum, and never return again
or include a ſpace; but the ellipfe furrounds the cone, and return-
ing into itſelf includes a ſpace.
COR. 7. Every conic fection furrounding the cone is either a
circle or an ellipfe; for if the plane of the fection be parallel to
the baſe, or in fubcontrary pofition, it will be a circle, but if not,
it will be an ellipfe by Definition 12. and Cor. Prop. IX. of this
Book.
PROP.
C 2
20
Book I.
Conic Sections.
FIG. 19.
FIG. 19.
PROP. XV.
If through a point within a parabola a right line be drawn
meeting the diameters, it neceffarily meets the parabola in
two points.
ET BAD be a cone, and the ſection MVN in its furface
parabola, and the plane ADE its vertical plane touching the
furface in the line AD, and let LV be a diameter; and through the
point P within the parabola draw the right line MN cutting the dia-
meters, it will meet the parabola in two points.
For draw in the vertical plane, through the vertex A of the cone,
the right line AF parallel to MN; then becauſe the lines PN, FA,
and PL, AD are parallel, the angles NPL, FAD will be equal
(10. 11.) therefore AF will cut the line AD in which line only the
vertical plane meets the ſurface: therefore AF falls wholly without
the furface: and therefore (by 6. of this Book) the line MN pa-
rallel to AF meets the furface (within which the point P is taken)
in two points, fuppofe in M, N; confequently in thefe points it
meets the parabola by Cor. 1. to the Definitions.
PROP. XVI.
Through any point of a conic fection one right line only
can be drawn, which will touch the fection in that point.
ET BAD be a cone, and MVN any conic fection in its fur-
face; let any point V be taken in it, and through V draw the
fide AVB of the cone, and through the point B where the fide
AVB meets the circumference of the bafe draw BK touching the
baſe; then the plane ABK touches the furface (Cor. 2. 3. of this
Book) and the common interfection VX of the plane ABK, with
the plane of the fection, touches the ſection in the point V.
For the line VX is in the plane ABK touching the furface, and
meets the line of contact in the point V; therefore in this point it
touches
Book I.
21
Conic Sections.
touches the ſurface; but VX is likewife in the plane of the fection
MVN which it therefore touches in the fame point V (by Cor. 1.
to Definitions). Then if another right line, in the plane of the
fection, befides VX, be fuppofed to touch the fection MVN in the
point V, it would likewiſe touch the conical furface in this point :
and fince this line is not in the plane ABK, the plane paffing through
it and the line AB will not be the fame with the plane ABK, and
will touch the ſurface in the right line AB (by Cor. 3. 3. of this
Book) and then two planes would touch the furface in the ſame
right line (contrary to Cor. 4. 3. of this Book) wherefore one right
line only can touch a conic ſection in the fame point.
COR. Hence any right line MN, drawn through a point P within
a conic fection, and parallel to any tangent VX, meets the fection
in two points.
For let AF be the common interfection of the tangent plane
ABK, in which the tangent VX is drawn, with the vertical plane
ADE, it will be parallel to the tangent VX (by 16. 11.) and con-
fequently to the line MN drawn through P: but the line AF, be-
cauſe it is in the tangent plane and not the fame with AB the line
of contact, falls wholly without the furfaces; therefore the line
drawn through P meets the furface (within which the point P is
taken) in two points, fuppofe in M, N, (by the 6. of this Book)
and therefore meets the ſection in the fame points (by Cor. 1. to
the Definitions.)
PROP. XVII.
20,
21.
Let MN be a right line terminated by a conic fection; one FIG. 19.
right line only parallel to this line can touch the fection,
if it be a parabola; and if it be an hyperbola, one right
line only parallel to MN, can touch it, and likewiſe one
only the oppofite hyperbola; and if the ſection be an el-
lipfe, there are but two right lines parallel to MN which
can touch it.
OR let the ſection MVN be a parabola, by which the right FIG. 19.
line MN is terminated; and through A the vertex of the
FOR
cone,
22
Book I.
Conic Sections.
FIG. 20,
21.
cone, let the right line AF be drawn in the vertical plane and pa-
rallel to MN, it will (by Cor. 3. 6. of this Book) fall without the
conical furfaces, and becauſe the line AF is in the plane ADE
touching the furface, through AF another plane ABK may be
drawn (by the 4. of this Book) which will touch the furface in a
right line as AB: this plane will neceffarily cut the plane of the pa-
rabola, ſuppoſe in the line VX; this line touches the parabola in V
(by the preced.) and will be parallel (16. 11.) to the line AF, and
confequently to MN terminated by the parabola.
But if (the reft remaining as above) any other right line YZ,
parallel to MN, be fuppofed to touch the parabola; through the
vertex A and the point of contact Y draw a fide of the cone: the
tangent YZ (by hypothefis) is parallel to AF, and (by Cor. 3. 3.
of this Book) the plane paffing through thefe parallels touches the
furface in the line AY; but this plane will neceffarily be different
from the tangent planes ADE, ABK; then three planes touching
the furface would pafs through the line AF contrary to (Cor. 4.
this Book).
of
Now let the fection MVN be an ellipfe or hyperbola, and mLn
the oppoſite hyperbola, and let MN be the right line terminated by
the ellipfe or hyperbola: through the vertex A of the cone let the
line AF be drawn in the vertical plane, and parallel to MN, falling
(by Cor. 4. 6. of this Book) without the furface, and through AF
let two planes ABK, AGH be drawn touching the furfaces in the
lines AVB, ALG (by 4. of this Book). Since theſe planes cut the
vertical plane ADE in the line AF, they will likewife cut the plane
of the ellipſe, or of the oppofite ſections, in two lines VX, LT pa-
rallel to AF (16. 11.) but thefe lines VX, LT touch the ellipfe, or
oppofite hyperbolas (by the preced.) and becauſe they are parallel to
AF they will be parallel to MN.
If then (the reft remaining as before) any third line parallel to
MN be fuppofed to touch the ellipfe or hyperbola, in the fame man-
ner, as in the former cafe, it may be fhewn that three planes touch-
ing the furface might be drawn through AF, which is abfurd
(Cor. Prop. IV. of this Book).
COR.
Book I.
23
Conic Sections.
COR. 1. Hence it appears that if any right line touches an el-
lipfe or hyperbola, another right line may be drawn which fhall
touch the fame ellipfe, or oppofite hyperbola, and be parallel to the
other tangent.
COR. 2. Hence it likewife appears, that if from a point in a pa-
rabola a right line be drawn parallel to a tangent, it meets the para-
bola again; for it cannot touch it (by the firſt part of this Prop.)
therefore it falls on one fide within the parabola, and therefore (by
Cor. to preced. Prop.) it meets it again. In like manner it
may be
fhewn that a right line drawn from a point in an ellipfe, or hyper-
bola, and parallel to two tangents, meets the ellipfe, or hyperbola
again.
THE reader is defired to obferve that the circle is reckoned among the
Conic Sections, for whatſoever is proved univerfaliy in what follows
of the three ſections already defined, obtains alſo in the circle, as gene-
rally appears from the elements of plain Geometry.
PROP. XVIII.
If two right lines meeting each other, be always parallel to
two right lines given in pofition; according as they both
touch or cut, or one of them touches and the other cuts
a conic fection or oppofite fections; the fquares of the feg-
ments of the tangents, or the rectangles under the feg-
ments of the fecants between the point of concourſe of
the two lines, and the fection, or fections, will be in a
conſtant ratio to each other, wherefoever the point of
concourſe of the right lines be taken.
HIS Propofition is evident from (Prop. XI. of this Book) and
Cor. 1. to the Definitions.
TH
+
COR.
24
Book I.
Conic Sections.
FIG. 22,
23.
COR. 1. Hence, if any right line LCV cutting a conic fection,
or oppofite fections, in the points L, V, meets two parallel lines in
the points O, S cutting the fame fection, or oppofite fections, or
both the oppofite fections, in the points A, B and H, G; the rect-
angles BOA, HSG under the fegments of the parallel lines between
the ſection or fections, and the line LCV, will be to each other as
the rectangles VOL, VSL under the fegments of the line LCV be-
tween the parallel lines and ſection or fections.
For becauſe BA, LVO are parallel to the fame lines to which
HG, LVO are parallel; in whatfoever ratio the rectangle BOA is
to the rectangle VOL, in the fame ratio the rectangle HSG will
be to the rectangle VSL, by this Propofition: therefore by permu-
tation, BOA is to HSG as VOL to VSL.
COR. 2. Or if any right line cutting a fection, or oppofite fec-
tions, meets two parallel lines touching the ſame ſection, or oppoſite
fections, in the fame manner it may be fhewn that the fquares of
the fegments of the paralel tangents between their points of con-
tact and the ſecant which they meet, are to one another as the rect-
angles under the fegments of that fecant between the parallel lines
and fection, or fections.
COR. 3. Or if any right line cutting a fection, or oppofite fec-
tions, meets two parallel right lines, one of them a tangent, and the
other a fecant; the fquare of the fegment of the tangent, and the
rectangle under the fegments of the fecant between the fection and
the fecant which they meet, will be to each other as the rectangles
under the fegments of that fecant between the parallel lines and fec-
tion, or fections.
COR. 4. Or if any right line touching a conic fection meets two
parallel right lines cutting the ſection, or ſections; the rectangles
under the ſegments of the fecants between the fection, or fections,
and the tangent which they meet, will be to each other as the
fquares of the fegments of the tangent between the parallel lines,
and the point of its contact.
COR. 5. Or if any right line touching a conic ſection meets two
parallel right lines touching the fame fection, or oppofite fections;
the fquares of the fegments, of the parallel tangents between their
points
Book I.
25
Conic Sections.
points of contact and the tangent which they meet, will be to each
other as the fquares of the fegments of that tangent between the
parallels and its point of contact.
COR. 6. Or if any right line touching a conic fection, meets two
parallel lines, one of them a tangent and the other a fecant; the
fquare of the ſegment of the tangent, and the rectangle under the
fegments of the fecant between the fection and tangent which they
meet, will be to each other as the fquares of the fegments of that
tangent between the parallels and its point of contact.
PROP. XIX.
If two right lines parallel to each other meet a right line
parallel to any fide of a cone, and if both touch or cut,
or if one of them touches, and the other cuts a conic
fection or oppofite fections; the fquares of the fegments
of the tangents, or the rectangles under the ſegments of
the fecants between the right line, which they meet, and
the ſection or ſections, will be to each other as the feg-
ments of that line between the parallels, and the point
in which it meets the conic fection.
F
OR fince this line, which the parallels meet, is parallel to a
fide of the cone, it will meet with the furface of the cone, and
confequently with the conic fection (in the plane of which it is) in
one point only (by Prop. V. and Cor. 1. Def.) and therefore this
propofition follows from Prop. XII. and Cor. 1. to the Definitions.
COR. Hence if MVN be a parabola in the furface of a cone,
and the parallel right lines MN and AB cut it, meeting its diameter
LV in D, C; the rectangles MDN, ACB under the fegments of
the parallels between the diameter and parabola, will be to each
other as the fegments DV, CV of the diameter between the pa-
rallels and the vertex V: or if the right line XT parallel to MN
D
touches
FIG. 21.
26
Book I.
Conic Sections.
FIG. 22,
23.
1
touches the parabola in T, and meets the diameter LV in X; the
rectangle MDN will be to the fquare of TX as DV to XV.
For by Definition 15, every diameter of a parabola is parallel to
a fide of the cone, viz. to that in which the vertical plane touches
the furface of which the parabola is a fection. Whence the Co-
rollary is evident by this Propofition.
PROP. XX.
Let two right lines parallel to each other, touch an ellipfe,
or oppofite hyperbolas, and in thofe tangents let two
points be taken equidiftant from their points of contact.
If the right line joining theſe points cuts the ellipfe, or
oppoſite hyperbolas in two points; its fegments between
theſe points and the tangents will be equal: or if it
touches an ellipfe, the fegments between its point of con-
tact and the tangents, will be equal.
T
HIS Propofition is evident from Prop. XIII. and Cor. 1.
Def.
PROP. XXI.
Let two right lines, parallel to each other, touch an ellipfe,
or oppoſite hyperbolas; a right line joining their points
of contact will bifect all right lines which are parallel to
the tangents, and terminated by the ellipfe, or either of
the hyperbolas.
FOF
OR all right lines parallel to the tangents, neceffarily meet the
The
line which joins their points of contact; therefore this Propo-
fition follows from Prop. XIV. and Cor. 1. to Def.
COR. 1. Hence if two right lines AB, GH parallel to each other,
be terminated both ways by an ellipfe, hyperbola, or different and
oppofite hyperbolas; the right line OS bifecting both theſe, meets
the
Book I.
27
Conic Sections.
the ellipfe, or hyperbolas in two points; through which points, if
tangents be drawn, they will be parallel to the lines biſected.
For let two right lines VX, LT parallel to AB, GH be con-
ceived to touch the ellipfe, or oppofite hyperbolas (by Prop. XVII.)
in the points V, L; then the right line L.V joining their points of
contact, will biſect the lines AB, GH and confequently will coin-
cide with OS: therefore the Corollary is evident.
COR. 2. Hence if the right lines AB, GH parallel to each other,
and terminated by an ellipfe, hyperbola, or different and oppofite
hyperbolas, be biſected by the right line OS; and if through the
points V and L in which the bifecting line meets the ellipfe or hy-
perbolas, two right lines VX, LT be drawn parallel to the bifected
lines, they will be tangents; for if not (by preced. Cor.) tangents
drawn through theſe points will be likewife parallel to the lines AB,
GH, which is abfurd.
COR. 3. Hence appears a method of drawing two right lines,
which fhall touch an ellipfe or oppofite hyperbolas, and be parallel
to a right line given in pofition, cutting the ellipfe or hyperbola:
for let another fecant be drawn parallel to the given fecant, and bi-
fect theſe parallels, and through the points in which the bifecting
right line meets the ellipfe or hyperbolas, draw two right lines pa-
rallel to the line given in pofition; thefe lines will be tangents (by
Cor. preced.)
PROP. XXII. PROBL. II.
To find the center of an ellipfe, or oppofite hyperbolas,
given in pofition; and to demonftrate that there can be
but one center only of an ellipfe or hyperbola.
L
23.
ET VAGHB be an ellipfe or hyperbola, and m Ln its oppofite FIG. 22,
hyperbola, and let two right lines be drawn parallel to each
other, cutting the ellipfe or hyperbola in the points A, B and G, H,
and let the right line OS be drawn bifecting the parallels AB, GH,
and meeting (by Cor. 1. preced.) the ellipfe or oppofite hyperbolas
D 2
in
28
Book I.
Conic Sections.
FIG. 23.
in two points, fuppofe in V, L, and bifect the line VL in C; that
point will be the center. For through the points V, L draw two
lines VX, LT parallel to AB, GH, they will be tangents (by Cor.
2. preced.) therefore draw through the point C any line meeting the
ellipfe or oppofite hyperbolas in the points E, F and the tangents
VX, LT in Q and P; then becauſe the triangles CVQ, CLP are
fimilar, and the lines CV, CL equal; the lines CQ, CP will be
equal, and likewife the tangents VQ, LP; therefore QE, PF are
equal (by 20. of this Book) and therefore CE, CF are equal. But
if the line drawn through C, and meeting the ellipfe in two points
M, N be parallel to the tangents VX, LT, it will be bifected in C
(by the preced. Prop.) therefore fince every right line drawn through
the point C, and terminated by the ellipſe or oppoſite hyperbolas is
bifected in C, this point C will be the center of the ellipfe or oppo-
fite hyperbolas by Definition XVI. Q, E. I.
If (the above remaining as premiſed) another point D be fup-
pofed to be the center of the hyperbolas, draw through this point a
line parallel to VL, it will meet the oppofite hyperbolas in two
points A, G (Cor. 4. Def.) let it meet the tangents VX, LT in
Q, R; VQ, LR will be equal (being oppofite fides of a parallelo-
gram) and therefore QA, RG are equal (by Prop. XX. of this
Book) but becauſe D is fuppofed to be the center, DA, DG are
equal, therefore DQ, DR are equal: draw any other line through
D, meeting the hyperbolas in the points Y, Z, and the tangents in
I, K; becauſe D is a center (by hypoth.) DY, DZ are equal, and
becauſe the triangles DQI, DRK are fimilar, and the lines DQ,
DR equal, DI, DK will be equal, and confequently (5. 2.) the
rectangles YIZ, YKZ are equal: but thefe rectangles are as the
fquares of the tangents VI, LK (by Cor. 2. Prop. XVIII.) there-
fore theſe fquares, and confequently the lines VI, LK are equal;
therefore the line IK is parallel to VL (33. 1.) that is to AG,
which is abfurd; and therefore no point befide C can be the center
of the hyperbolas; which may in like manner be demonſtrated of
the ellipfe; but this is felf-evident, for if there were two centers in
an ellipfe, a line terminated by the ellipfe and paffing through
the
Book I.
29
Conic Sections.
the centers, would be biſected in two different points; which is
abfurd.
COR. 1. All right lines joining the points of contact of two pa-
rallel right lines touching an ellipfe or oppofite hyperbolas, bifect
each other in one and the fame point, viz. in the center. For if
they could be bifected in different points, there would be ſeveral
centers of the ellipfe or oppofite hyperbolas, by the firſt part of the
preceding demonftration; the contrary to what is fhewn in the fe-
cond part.
COR. 2. Any right line as LV joining the points of contact of
two parallel tangents LT, VX paffes through the center, and con-
ſequently is a diameter of an ellipfe, or a tranſverſe diameter of an
hyperbola, by Def. XVII.
COR. 3. Hence two tangents LT, VX, drawn through the two
vertices of the fame diameter LV of an ellipfe or hyperbolas are
parallel; for if not, another tangent may be drawn parallel to one
of them (Cor. 1. 17. of this Book) and the right line joining the
points of contact of theſe parallels would alſo pafs through the cen-
ter (Cor. 2. of this Prop.) which is abfurd.
Cor. 4. No right line touching an hyperbola paffes through the
center; for if a right line touches an hyperbola, another right line
parallel to it may be drawn to touch the oppofite hyperbola (Cor. 1.
17. of this Book) and the right line joining their points of contact
paffes through the center, and confequently neither of the tangents
can paſs through the center.
COR. 5. Hence any right line, drawn from the center of the hy-
perbola to any point V in it, will be a tranfverfe femidiameter; for
it does not touch the hyperbola in V by the preceding Corollary;
therefore through the point V let VX be drawn touching the hyper-
bola, and LT parallel to it touching the oppofite hyperbola; it ap-
pears that the line joining the points L, V is a tranfverfe diameter
(by Cor. 2.) and confequently coincides with the line CV, therefore
this line is a tranfverfe femidiameter.
PROP.
30
Book I.
Conic Sections.
FIG. 23.
FIG. 19.
PROP. XXIII.
Every right line drawn through the center of an hyperbola,
and parallel to a right line touching the hyperbola, or cut-
ting it in two points, is a fecond diameter.
HROUGH C the center of the hyperbolas BVA, mLn let the
TH
right line MN be drawn parallel to any tangent VX, and
draw through the point of contact the tranfverfe diameter VCL, and
through the other vertex L draw LT, touching the oppofite hyper-
bola, and parallel to VX (by Cor. 3. preced.) and let any right
line be drawn parallel to VCL, and meeting the oppofite hyperbo-
las (by Cor. 3. Def.) in two points A, G, the tangents VX, LR
in Q, R, and the line MN in D; then the tangents VQ, LR are
equal, being oppofite fides of a parallelogram, and therefore the
lines QA, RG are equal (Prop. XX.) and becauſe VC, CL are
equal, and CD parallel to VX, LT; the lines QD, DR will be
equal; therefore the line AG will be bifected in D by the line
MN and therefore this right line is a fecond diameter (by Def.
18.) 2. E. D.
+
PROP. XXIV.
A diameter of any conic fection bifects all right lines termi-
nated by the ſection which are parallel to the tangent
drawn through its vertex.
IRST let BAD be a cone, and the fection MVN in its furface
F"
a parabola, ADE its vertical plane, and LV a diameter, and
let the line VX touch the parabola in the vertex V of the diameter
LN, and let MN be parallel to VX, and terminated by the para-
bola, and meet the diameter LV in P; it will be bifected in this
point.
For draw the fide of the cone AV, and let the plane ABK pafs
through AV and the tangent VX; this plane touches the conical
furface in the fide AVB (Cor. 3. 3. of this Book) and becauſe AV
cuts
Book I.
3 L
Conic Sections.
I
cuts the parallels LV, AD, it will be in the fame plane with them
(7. 11.) therefore the line LV is in the plane of the triangle BAD:
draw in this plane through the point P any line meeting AD in any
point Q and AB in O; draw through theſe points in the tangent
planes ADE, ABK, the lines QR, OS parallel to AF, the inter-
fection of the tangent plane ABK with the vertical plane ADE,
theſe lines touch the furface in the points Q, O, and becauſe they
are parallel to AF they will be parallel to the tangent VX (by 16.
11.) and likewiſe to MN (9. 11.) but the line MN is bifected in
P by the line joining the points of contact Q, O (by Prop. XIV.)
and confequently by the diameter LV.
I
In the fecond place, let the fection be an ellipfe or hyperbola,
and becauſe the tangents drawn through the two vertices of any dia-
meter are parallel to each other (Cor. 3. 22. of this Book) this dia-
meter will bifect all right lines parallel to thofe tangents, and ter-
minated by the ellipfe or either of the hyperbolas, as appears from
Prop. XXI.
PROP.
XXV.
A fecond diameter MN of an hyperbola bifects all right
lines terminated by the hyperbolas, and parallel to the
tranſverſe diameter LV joining the points of contact of
the tangents VX, LT, which are parallel to that fecond
diameter MN and on the contrary, if a right line AG
be bifected by a fecond diameter MN, it will be parallel
to the tranſverſe diameter LV joining the points of con-
tact of the tangents VX, LT, which are parallel to that
fecond diameter.
Part I.
L
ET the line AG be terminated by the hyperbolas FIG. 23.
BVA, mLn and be parallel to the diameter LV join-
ing the points of contact of the tangents VX, LT which are pa-
rallel to MN, and let AG meet the diameter MN in D. It will
be bifected in this point.
For
32
Eook I.
Conic Sections.
For let AG meet the tangents in the points Q, R; then the
tangents VQ, LR will be equal, being the oppofite fides of the
parallelogram LVQR; therefore the lines AQ, RG are equal (by
Prop. XX. and becauſe VC, CL are equal, and the line CD is
parallel to VQ, LR; the lines QD, DR will be equal; therefore
AD, DG are equal; and therefore AG is bifected by the fecond
diameter MN.
Part 2. And if (the reft remaining as above) the line AG be
bifected by the diameter MN, it will be parallel to the diameter
LV joining the points of contact of the tangents VX, LT which
are parallel to MN; for becaufe VC, CL are equal, and the lines
VQ, CD, LR parallel; QD, DR will be equal; but, by hypo-
thefis, AD, DG are equal; therefore AQ, RG are equal, and
therefore the rectangles AQG, ARG are equal; but thefe rect-
angles are to each other as the fquares of the tangents VQ, LR
(by Cor. 2. Prop. XVIII.) therefore theſe fquares and conſe-
quently the lines VQ, LR are equal wherefore (33. 1.) the lines
AQRG and LV are parallel. 2. E. D.
Corollaries to the two preceding Propofitions.
COR. 1. Every right line terminated by a conic fection, and bi-
fected by a diameter, is parallel to the tangent paffing through the
vertex of that diameter (provided when the fection is an ellipfe,
the bifected line be not a diameter) for if it be not parallel to this
tangent, let it be parallel (by 17. of this Book) to a tangent drawn
through the vertex of another diameter; then by Prop. 24. it
would be alfo bifected by this other diameter, which is abfurd.
COR. 2. All right lines ordinately applied to the fame diameter
of a conic fection are parallel to each other; for when they are ap-
plied to a diameter of a parabola, ellipfe, or a tranfverfe diameter of
an hyperbola, they will all be parallel to the tangent paffing through
the vertex of that diameter (by preced. Cor.) and when they are
applied to a fecond diameter of the hyperbola, they will be pa-
rallel
Book I.
33
Conic Sections.
rallel to one and the fame tranfverfe diameter, by Cafe 2. of the pre-
ceding Propofition.
COR. Hence
3.
any right line terminated by a conic fection or
by oppofite hyperbolas, and parallel to the ordinates of any dia-
meter, will be ordinately applied to the fame diameter.
COR. 4. Hence alfo, two right lines terminated by a conic fec-
tion or oppofite fections, cannot bifect each other unleſs they meet
in the center of the ellipfe or of the oppofite fections; for if they
could, a diameter of the ſection may be drawn bifecting both of
them, and confequently both would be ordinately applied to the
ſame diameter, and therefore (by Cor. 2.) parallel to each other,
which is abfurd.
COR. 5. Hence alſo it appears that if two right lines parallel to
each other be terminated by a conic fection or oppofite fections,
the diameter which biſects one will biſect the other.
COR. 6. A right line bifecting two parallel lines terminated by a
conic ſection or oppofite fections is a diameter; for a diameter bi-
fecting one of them will bifect the other.
COR. 7. Hence, if a fegment of a conic fection be given in
po-
fition, the diameters and center of the fection may be found for
let two parallel right lines be drawn terminated by the fegment,
the right line bifecting thefe will be a diameter; in like man-
ner another diameter may be found, and if it be parallel to the
former, the fection will be a parabcla; but if the diameters cut
each other the point of interfection will be the center of an ellipfe,
if the fegment be concave towards the center, and if convex, the
fection will be an hyperbola.
COR. 8. A right line drawn through the vertex of a diameter pa-
rallel to its ordinates, touches the fection; for if not, let a tangent
E
be
34
Book I.
Conic Sections.
FIG. 25,
26.
be drawn through its vertex, this likewife will be parallel to the or-
dinates of that diameter (by Cor. 1.) which is abfurd.
COR. 9. Hence appears a method of drawing a right line which
fhall touch a conic fection given in pofition, and be parallel to any
right line cutting the fection in two points; for let a line be drawn
parallel to this fecant and terminated by the fection, the right line
biſecting theſe parallels will be a diameter, and the right line drawn
through the vertex of this diameter and parallel to the bifected
lines touches the ſection by the preced. Cor.
1
PROP.
XXVI.
If two right lines touching a conic fection or oppofite fec-
tions meet each other; a right line paffing through their
point of concourſe and biſecting the right line joining
their points of contact, will be a diameter of the fection.
F%
OR let the right lines PM, PN touch the conic fection or
oppofite fections in M, N, and join the points M, N, and
draw PO bifecting MN in O; PO will be a diameter: for if not,
through the point O let a diameter of the fection be drawn meeting
the tangent PM in Q, and let NQ be drawn meeting the fection
again in A; through the point A draw the right line AD parallel
to MN, this meets the oppofite ſection, ſuppoſe in B (Cor. 4. Def.)
when the points M, N are in oppofite fections; and when thefe
points are in the fame fection, becauſe A is not the vertex of the
diameter OQ, the line AD parallel to MN does not touch the
fection (by Cor. 1. preced. and Cor. 2. 17. of this Book) let it then
meet the fection again in B and OQ in X and PM in D; fince
the line AB, is parallel to MN, which is bifected by the diameter
OQ, it will be bifected by the fame diameter (by Cor. 5. preced.)
therefore the fegment AX will be equal to BX; and becauſe in the
triangle MON the line OQ drawn from the vertex, bifects the
baſe in O, the fame line will bifect AD, parallel to the bafe in X;
there-
!
Book I.
35
Conic Sections.
therefore the fegment DX is equal to the fegment AX, that is, to
BX, which is impoffible; and therefore no line drawn through O
befide PO will be a diameter of the fection. 2. E. D.
COR. I. On the contrary, if two right lines touching a conic
fection or oppofite fections meet each other; the diameter bifecting
the right line joining the points of contact, will pafs through the
point where the tangents meet; for one diameter only can bifect
the line joining the points of contact, and a right line bifecting this
line joining the contacts and paffing through the concourfe of the
tangents is a diameter; therefore the Corollary is manifeſt.
COR. 2. Hence it appears that if two right lines touching a conic
fection or oppofite fections meet each other, their point of con-
courſe will be in the diameter which bifects the line that joins their
points of contact.
Cor. 3. Hence, three right lines touching a fection or oppofite
fections cannot meet in one point. For if it were poffible, draw
two right lines joining one of the points of contact with the other
two; then (by Cor. 1.) two diameters bifecting theſe lines would
meet each other in the point of concourfe of the tangents, which is
impoffible; for if the fection be a parabola, the diameters are pa-
rallel, and the diameters of an ellipfe or hyperbola meet only in the
center, through which no tangent paffes (Cor. 4. 22. of this
Book.)
PROP. XXVII.
Two diameters of an ellipfe or hyperbola, of which each is
parallel to the ordinates of the other, are conjugate dia-
meters.
L'
23.
ET MN be a diameter of an ellipfe or a fecond diameter of FIG. 22,
an hyperbola, to which let the right lines AG be ordinately
applied meeting it in the point D, and let LV be the diameter pa-
E 2
rallel
36
Book I.
Conic Sections.
rallel to AG; theſe two diameters MN, LV will be conjugate di-
ameters.
For let the tangents VX, LT be drawn through the two vertices
of the diameter LV; they will be parallel (by Cor. 3. Prop. XXII.)
and let them meet AG (if neceffary produced) in the points Q, R;
then the tangents VQ, LR will be equal, being oppofite fides of a
parallelogram, and therefore (by Prop. XX.) the right lines AQ,
RG will be equal; but AD, DG (by hypothefis) are equal; there-
fore the lines QD, DR are equal, and likewiſe VC, CL are equal;
therefore CD, or MN is parallel to VQ, LT; but the diameter
LV bifects (by 24. of this Book) all right lines which are parallel
to VQ, LT, that is to the diameter MN; therefore the diameters.
MN, LV are conjugate diameters (by Def. XIX.)
Now let the right line AB be ordinately applied to the tranfverfe
diameter LV of an hyperbola, and draw the fecond diameter
MN parallel to AB; LV, MN will be conjugate diameters. For
through the two vertices of the diameter LV let the lines VX, LT
be drawn parallel to AB, that is to the diameter MN; they will
be tangents (Cor. 8. 25. of this Book) therefore by Prop. XXV.
the diameter MN bifects all right lines terminated by the oppofite
hyperbolas and parallel to the diameters LV; and therefore LV, MN
are conjugate diameters. 2. E. D.
COR. It appears from hence that an ordinate to any diameter is
parallel to its conjugate diameter, and that two diameters are con-
jugate, if one of them be parallel to the tangent drawn through the
vertex of the other.
And on the contrary a tangent drawn through the vertex of a
diameter is parallel to its conjugate.
PROP..
Book I.
37
Conic Sections.
PROP.
XXVIII.
PROB. III
A conic ſection and its diameter being given in pofition ;
through a given point in the fection to draw a right line
which fhall be ordinately applied to that diameter.
IRST let the fection be a parabola or an hyperbola and MN FIG. 27.
FIR
the diameter given in pofition (which in the hyperbola muft
be a tranfverfe diameter) and let A be the point given in the fec-
tion, to the diameter MN let any right line AN be drawn and
produced to H, ſo that AN, NH be equal; through H let a right
line be drawn parallel to MN which (by Cor. 4, 5. to Def.) will
meet the fection in fome point as G; the right line joining the
points A, G will be ordinately applied to the diameter MN.
For becauſe AN, NH are equal and DN, GH parallel; AD,
DG will be equal.
Secondly, let the ſection be an ellipfe or hyperbola, and MN any
diameter of the ellipſe or a conjugate diameter of the hyperbola, to
which, from the point A in the fection, an ordinate is to be ap-
plied find the center of the fection (22. of this Book) and draw
through the point A the diameter ACH; if the right line drawn
through its vertex H or A parallel to MN touches the fection, then
the diameter ACH will be conjugate to MN, and theſe diameters
are faid to be ordinately applied to each other: but if the line drawn
through H and parallel to MN meets the ſection again in any point
G; the line joining the points A, G will be an ordinate to the dia-
meter MN; for let it meet it in D; becauſe AC, CH are equal and
HG, CD parallel; AD, DG will be equal; therefore the line AG
is ordinately applied to the diameter MN. 2. E. F.
7.
COR. Hence, a line may be drawn through a given point in a
conic fection given in pofition to touch the fection: for (by Cor.
Prop. XXV.) let two diameters of the fection be found, and
through the given point draw a diameter to which an ordinate may
be applied (by this Prop.) the line parallel to this ordinate drawn
through the given point will touch the ſection (by Cor. 8. Prop. XXV.)
PRO P.
FIG. 22,
23.
38
Book I.
Conic Sections.
FIG. 28,
FIG. 22,
23.
PROP. XXIX.
The fquares of right lines which are ordinately applied to
the fame diameter of a parabola, are as the abfciffes of
that diameter between the ordinates and vertex.
L
ET LV be a diameter of a parabola and L its vertex, and let
it meet the lines BA, HG ordinately applied to it in the
points C, D.
Becauſe theſe lines are parallel to each other (by Cor. 2. Prop.
XXV.) the rectangles BCA, HDG, that is the ſquares of the or-
dinates BC, HD are (by Cor. Prop. XIX.) to each other as CL,
DL, viz. the abfciffes of the diameter between the ordinates and
the vertex L (Def. XXI.) 2. E. D.
PROP.
XXX.
The ſquares of right lines, ordinately applied to any diameter
of an ellipfe or to a tranfverſe diameter of an hyperbola,
are to each other as the rectangles under the abfciffes of
the diameter between the two vertices and ordinates.
L
ET LV be a diameter of an ellipfe or a tranfverfe diameter of
an hyperbola and the points L, V its two vertices, and let it
meet in the points O, S two ordinates BA, HG terminated by the
ellipfe, hyperbola or oppofite hyperbolas.
Becauſe theſe right lines are parallel (Cor. 2. Prop. XXV.) the
rectangles BOA, HSG (by Cor. 1. Prop. XVIII.) that is the
fquares of the ordinates BO, HS, will be to each other as the
rectangles LOV, LSV under the abfciffes of the diameter LV be-
tween the ordinates and the two vertices. 2. E. D.
COR. 1. As it is manifeft by this Propofition, that in the ellipfe or
oppoſite hyperbolas the ordinates applied to the fame diameter and
at equal diſtances from the center, are equal; it is likewife evident,
that if the oppofite fections be ſo placed that the line VO may coin-
cide with LS, and the angle VOB with its alternate angle LSG, the
whole
Book I.
39
Conic Sections.
:
whole fection will be congruous and alfo the two fegments of an
ellipſe into which it is divided by its diameter, may be ſo placed as
wholly to coincide.
COR. 2. Hence, if a circle be deſcribed upon a diameter of an
ellipfe and from two points in this diameter ordinates applied to this
diameter be drawn to the ellipfe, and at the fame points two per-
pendiculars be erected meeting the circumference of the circle:
the ordinates will be to each other as theſe perpendiculars: for by
this Prop. the fquares of the ordinates are to each other as the
rectangles under the abfciffes of the diameter between thefe ordi-
nates and its two vertices, that is (from a property of the circle) as
the ſquares of theſe perpendiculars; therefore thefe ordinates are as
the perpendiculars.
1
PROP. XXXI.
If two right lines meeting each other touch or cut, or one
of them touches and the other cuts an ellipfe; the ſquares
of the fegments of the tangents, or the rectangles under
the ſegments of the fecants between the point where the
right lines meet and the ellipſe, will be to each other as the
fquares of the femidiameters to which they are parallel.
FOR
OR the diameters of an ellipfe are fecants which meet each
other in the center, and the rectangles under their fegments
between the center and the ellipfe are the fquares of the femidiame-
ters; therefore (by 18. of this Book) the ſquares of the tangents,
or the rectangles under the fegments of the fecants, are to each
other as the fquares of the femidiameters to which they are parallel.
COR. 1. Hence, if in an ellipfe a right line AD be ordinately ap- FIG. 22.
plied to the diameter MN; the rectangle MDN under the abfciffes
of the diameter MN will be to the fquare of the ordinate AD as
the fquare of the diameter MN to the fquare of its conjugate LV:
for let AD produced meet the ellipfe in G, the fquare of AD is
equal.
4.0
Book I.
Conic Sections.
equal to the rectangle ADG, and (by Cor. 27. of this Book) the
diameter LV and right line AG are parallel; wherefore the Corol-
lary appears from this Propofition.
COR. 2. If two conjugate diameters MN, LV of an ellipſe are
perpendicular to each other, they will be unequal; for if they were
equal, from any point A of the ellipfe, let the right line AD be
ordinately applied to the diameter MN, it will be parallel to LV
(Cor. 27. of this Book) and from hence, MDA is a right angle (by
hypotheſis) let C be the center and join C, A; the ſquare of
CA (by 47. 1.) is equal to the fquares of CD, DA, that is to the
fquare of CD, together with the rectangle MDN (for the rectangle
MDN and the fquare of DA are equal, by the preced. Cor. be-
cauſe the diameters MN, LV are equal by hypothefis) or to the
fquare of the femidiameter CN (5. 2.); therefore the femidiame-
ters CA, CN are equal; and in like manner it may be fhewn that
all the other femidiameters would be equal to each other and then
the ellipfe would be a circle, which is abfurd; and therefore this
Corollary is manifeſt.
COR. 3. Hence, it appears, that if two right lines touching an
ellipſe meet each other; the ſegments of the tangents between their
point of concourſe and the points of contact, will be to each other
as the femidiameters to which they are parallel.
FIG. 23.
In
DEFINITION XXIV.
F in a fecond diameter of an hyperbola, a fegment MN bifected
in the center be taken fo that the fquare of the fegment MN,
be to the fquare of the diameter LV conjugate to this fecond di-
ameter, as the ſquares of the ordinates applied to the diameter LV,
to the rectangles under the abfciffes; the points M, N the ex-
tremities of this fegment are called the two vertices of that fecond
diameter and the magnitude of any fecond diameter is determined
by thefe points, as the magnitude of a tranfverfe diameter is deter-
mined by its two vertices.
PROP.
Book I.
4I
Conic Sections.
PROP. XXXII.
If from a point in an hyperbola a right line be ordinately
applied to a fecond diameter; the fquare of this fecond
diameter is to the fquare of the tranfverfe diameter con-
jugate to it, as the ſum of the ſquares of the ſecond ſemi-
diameter and of its fegment, between the center and ordi-
nate, will be to the ſquare of the ordinate.
FRO
ROM the point A in an hyperbola, let AD be ordinately ap-
plied to the ſecond diameter MN, and let the diameter LV be
conjugate to MN; the fquare of CN is to the fquare of CV, as
the fquares of CN, CD together to the fquare of AD.
For let AO be an ordinate to LV, COAD will be a parallelogram
(by Cor. Prop. XXVII.) and confequently AD, OC and likewiſe
AO, DC are equal.
By the preceding Definition, the fquare of CN is to the fquare
of CV as the fquare of AO or CD to the rectangle VOL; there-
fore (by 12. 5.) the fquare of CN is to the ſquare of CV as the
fum of the fquares of CN, CD to the fquare of CV and the rect-
angle VOL together, that is (6. 2.) to the fquare of CO or AD.
Q, E. D.
COR. Hence, if from two points in an hyperbola or oppoſite hy-
perbolas, two right lines be drawn ordinately applied to a fecond
diameter; the fquare of the firſt line drawn will be to the fquare of
the other as the fum of the fquares of the fecond femidiameter and
of the diſtance of the firſt line from the center to the fum of the
fquares of the fame fecond femidiameter and of the diftance of the
other line from the center.
FIG. 23.
F
PROP.
42
Book I.
Conic Sections.
FIG. 29.
PROP.
XXXIII. -
Any indefinite right line VL being given in poſition, and a
point V given in it; and likewiſe the right line MN given,
bifected by VL in the point P; a parabola may be de-
fcribed, of which the line VL fhall be a diameter, V its
a¸
vertex, and to which MN fhall be ordinately applied.
FO
OR let any plane different from the plane MPV paſs through
the line MN, and in this plane defcribe a circle about the dia-
meter MN and draw the diameter BPD perpendicular to MN,
meeting the circle in the points B, D; let B, V be joined, and
through the point D in the plane VBD draw the right line DA pa-
rallel to VL which may meet fomewhere BV, ſuppoſe in the point
A; whilft this point remains fixt, let the line AD be moved round
the periphery of the circle BNDM; the interfection of the conical
furface, generated by this motion, with the plane MPV, will be a
parabola.
For through the point D let DE be drawn parallel to MN, it
will be perpendicular to the diameter BD (by conftruction) and
therefore touches the circle in D, and hence the plane ADE touches
the furface (3. of this Book) and will be parallel to the plane of the
fection MVN (15. 11.) this fection is therefore a parabola and the
line VL parallel to AD a diameter whofe vertex is the point V,
and the line MN an ordinate applied to this diameter; all which are
evident from the 11th and 15th Definitions. 2; E. D.
COR. If two parallel right lines NP, YZ meet any right line VL,
and a point V be taken in it (but not between P and Z) ſo that the
ſquare of NP be to the fquare of YZ as PV to ZV; the points
N, Y will be in a parabola to which LV is a diameter, and V its
vertex, and to which NP, YZ, are ordinately applied; for if (by
this Prop.) a parabola be deſcribed about the diameter VL whoſe
vertex is V, and the line NPM be ordinately applied to that di-
ameter; then becauſe the line YZ cuts the diameter below the ver-
tex,
B
M
H
R
I
n
F'
L
TAB.3.
Page · 42.
L
2
H
R
Q
FIG. 19.
Z
X
FIG. 18.
N
P
di
R
M
B
E
FIG. 17.
T
P
FIG.21.
B
F
K
V
F
E
X
D
B
N
FIG. 20.
K
V
X
T
K
B
H
M
E
D
H
E
E
P
O
M
m
F
H
S
Z
-T'
R K
P
T
F
H
R
-T
G
M
N
D
FIG. 23.
D
1
M
FIG. 22.
I Q
O
x
B
I
B
X
X
Q
II
IS
برااد
OF
H
F
1
Book I.
43
Conic Sections.
tex, produced both ways, it will cut the parabola (15. of this Book)
in two points, and will be ordinately applied to the diameter LV
(Cor. 3. Prop. XXV.) and therefore the point Y will be in this pa-
rabola; for if the parabola meets the line ZY produced, or between
the points Z and Y (by Prop. XXIX.) the fquare of NP would be
to the fquare of a line either greater or leſs than YZ, as PV to ZV,
which is contrary to the hypothefis.
I
LEMMA
I.
F in two planes interfecting each other two right lines be parallel;
they will be parallel to the common interfection of the planes.
For, if not, they will meet the interfection of the planes fome-
where, and then the interfection of the planes would be in the fame
plane with the two parallel right lines (by 7. 11.) contrary to the
hypothefis.
PROP. XXXIV.
Two right lines biſecting each other being given; an ellipſe
may be deſcribed in which the given lines fhall be conju-
gate diameters; but the lines muſt not be equal, if they
be at right angles to each other.
L
ET VL, MN be the two given lines, and C the point of in- FIG. 30.
terſection, and let any plane different from the plane MCV
paſs through the line MN, and through the point C, in this plane
draw the right line BD perpendicular to MN, and in this line let
any other point P, befide C, be taken as a center, and a circle de-
fcribed paffing through the points M, N, meeting BD fomewhere
as in the points B, D. Let the lines joining the points B, V and
L, D meet each other in A (for they are in the fame plane with the
lines BD, VL (2. 11.) and will not be parallel, for if they were,
the triangles VCB, LCD would be fimilar, and therefore becauſe
CV, CL are equal, BC, CD would be equal contrary to the hypo-
thefis.) Let the line AD, whilft the point A remains fixt, be carried
round the periphery of the circle BNDM: the interſection of the
conical furface, generated by this motion, with the plane MVNL,
will
F &
і
?
44
Book I.
Conic Sections.
will be an ellipfe, of which the lines VL, MN are conjugate dia-
meters.
Through the point D, the extremity of the diameter BD draw
DE a tangent to the circle, it will be parallel to MN (by conftruc-
tion, and 16. 3.) and the plane ADE touches the ſurface in the line
ADL, and the line LT the interfection of this tangent plane with
the plane of the fection VNLM, touches the fection in L; and
fince the lines MN, DE in theſe interfecting planes are parallel to
each other, they will (by Lemma 1.) be parallel to the common
interfection LT; but fince the lines VL, MN are terminated by the
fection MVNL and bifect each other in the point C; they will be
diameters of this fection, whether it be an ellipfe (Cor. 4. 25. of
this Book) or a circle. If the diameters VL, MN be not perpen-
dicular to each other, the tangent LT parallel to MN will not be
perpendicular to the diameter LV drawn through the point of con-
tact; therefore the ſection will not be a circle (16. 3.) and if the
diameters VL, MN be perpendicular to each other, by hypotheſis,
they are unequal; therefore in this cafe the fection MVNL will not
be a circle, and therefore the ſection furrounding the cone is an el-
lipfe (by Cor. 7. to the Defin.) and the diameters VL, MN are
conjugate (Cor. 27. of this Book) becauſe MN is parallel to the tan-
gent LT drawn through the vertex of the diameter VL. 2, E. D.
COR. Let the given right lines VL, MN bifect each other in C,
as in the propofition: if YZ be drawn from any point Y to the line
VL, and parallel to MN, and if the rectangle LZV be to the
fquare of YZ as the fquare of LC to the fquare of MC; the point
Y will be in the ellipfe deſcribed about the conjugate diameters
VL, MN. For if not, the rectangle LZV would be to the fquare
of a line greater or leſs than YZ, as the fquare of LC to the fquare
of MC (Cor. 1. Prop. XXXI.) contrary to the hypothefis.
PROP.
Book I.
45
Conic Sections.
XXXV.
PROP.
Oppoſite hyperbolas may be defcribed about two given right
lines bifecting each other, one of which lines fhall be a
tranſverſe diameter, and the other a fecond diameter con-
jugate to it.
L
ET VL, MN be the given lines interfecting each other in C; FIG. 31.
let any point O be taken in the line LV produced, and through
this point draw RQ parallel to MN, and take on each fide of it
RO equal to OQ, fo that the fquare of RO or OQ may be to
the rectangle VOL, as the fquare of MN to the fquare of LV;
let any plane, different from the plane ROV, pafs through the line
RQ, and in this plane draw through the point O the right line BD
perpendicular to RQ, and take in this line any point P for a center,
and defcribe a circle paffing through the points R, Q which may
meet the line BD fomewhere as in the points B, D, and join BV
and DL: thefe lines BV, DL meet fomewhere, fuppofe in A (for
they are in the fame plane with BD, VL and are not parallel) let a
line drawn through the point A and produced both ways, revolve
about the periphery of the circle BQDR; the interfections of the
oppofite furfaces, which are generated by the motion of this line,
with the plane MVNL, will be oppofite hyperbolas by Def. XIII.
Through the points B, D, let BK, DE be drawn touching the.
circle, they will be parallel to RQ which is perpendicular to the
diameter BD; then VX the interfection of the tangent plane ABK
with the plane of the hyperbolas touches the hyperbola RVQ in V,
and (by Lemma 1.) it will be parallel to OR, BK; and LT the
interfection of the tangent plane ADE with the plane of the hyper-
bolas, touches the oppofite hyperbola in L, and will be parallel to
OR, DE (by Lemma 1.) therefore the tangents VX, LT are pa-
rallel to each other and to the line MN, and therefore the line VL
is a tranſverſe diameter of the hyperbolas (by Cor. 2. Prop. XXII.)
and
46
Book I.
Conic Sections.
FIG. 32.
and MN a fecond diameter conjugate to VL (Cor. Prop. XXVII.)
and the points M, N, are its two vertices, by Conſtruction and De-
finition XXIV. In the fame manner, two other oppofite hyperbo-
las may be deſcribed, of which MN fhall be a tranfverfe diameter
and VL a fecond diameter conjugate to it. 2. E. D.
COR. Let the given lines VL, MN bifect each other, and to the
line VL produced draw RO parallel to MN, fo that the fquare of
MN will be to the fquare of VL as the fquare of RO to the rect-
angle VOL; the point R will be in the hyperbola of which VL is
a tranſverſe diameter to which RO is ordinately applied, and MN
will be its conjugate, as is evident.
L
DEFINITION XXV.
ET BAF be a cone, and the ſections aVt, mL.n oppofite hyper-
bolas, and ADE the vertical plane meeting the oppofite fur-
faces in the right lines DAd, EAe: draw the planes ADGC, AEHC
touching the conical furfaces in the right lines DAd, EAe, and let
GCg, HCb be the interfections of thefe planes with the plane of
the hyperbolas; theſe lines are called the Afymptotes of the hyper-
bolas.
COR. 1. The afymptotes GCg, HCb of the hyperbolas are pa-
rallel to the fides of the cone AD, AE, in which lines the vertical
plane meets the furface; for the vertical plane is parallel to the
plane of the hyperbolas; therefore the Corollary is evident from
this Definition and (16. 11.)
COR. 2. The aſymptotes GCg, HCh do not meet the hyperbolas ;
for the aſymptotes are in the planes touching the conical furface,
and (by Cor. preced.) are parallel to the lines DAd, EAe in which
lines only theſe planes meet the furface; therefore the afymptotes
do not meet the conical furfaces and therefore do not meet the hy-
perbolas which are in thofe furfaces.
COR.
Book I.
47
Conic Sections.
COR. 3. Every right line as NV drawn in the plane of the hy-
perbolas parallel to either of the afymptotes HCh meets one of the
hyperbolas in one point only, and will be wholly on one fide within
this hyperbola, and on the other part wholly without both of them.
For (by Cor. 1.) the line NV is parallel to the fide AE of the
cone; therefore (by Prop. V.) it meets one of the oppofite furfaces
in one point only, and on one part falls wholly within this furface,
and on the other part wholly without both furfaces; therefore be-
cauſe the line NV is in the plane of the hyperbolas, this Corollary
from Cor. 1. to the Definitions of the Sections.
appears
COR. 4. If a right line PQ cuts both the fides which contain
the angle GCh adjacent to the angle GCH, containing the hyper-
bola, it will meet both the hyperbolas. For if CV be drawn from
the point of interſection C of the afymptotes parallel to PQ, it falls
within the angle GCH, and therefore becauſe the angles GCH,
DAE are equal (10. 11.) if a line be drawn from the vertex A of
the cone parallel to CV or PQ, it falls within the angle DAE, and
confequently within the conical furfaces; therefore the right line
PQ meets the oppofite furfaces (by Cor. 2. Prop. VI.) and there-
fore meets both the hyperbolas, becauſe it is in the fame plane with
them.
COR. 5. If a right line touches an hyperbola in any point V, it
meets both the afymptotes HC, GC which contain that hyperbola;
for if it were parallel to either of the afymptotes it would be on one
fide of the point V within the hyperbola (by Cor. 3.) contrary to
the hypothefis; or if it could meet the fides containing the angle
adjacent to HCG, it would meet both the hyperbolas (by Cor. 4.)..
and therefore would not be a tangent (by Cor. 2. to the Def. of
the Sections) therefore it meets the fides of the angles HCG con-
taining the hyperbola.
COR. 6. It is evident that a right line RS cutting the hyperbola
or oppofite hyperbolas meets the afymptotes.
PROP.
48
Book I.
Conic Sections.
FIG. 32.
PROP. XXXVI.
If a right line cutting an hyperbola or oppofite hyperbolas
meets the afymptotes in two points; the fegments be-
tween the hyperbola or hyperbolas and afymptotes will
be equal: Or if a right line touches an hyperbola ;
the fegments between the point of contact and the afymp-
totes, will be equal: Or if a right line meets the afymp-
totes and the fegments between the afymptotes and a
point in the hyperbola be equal; it will touch the hyper-
bola in that point.
Part I.
L'
ET the line RS cut the hyperbola or oppofite hyperbo-
las in the points R, S and meet the afymptotes in
P, Q; the fegments PR, QS will be equal.
For let AC be the common interfection of the tangent planes
ADGC, AEHC, and draw through the points P, Q, in theſe planes,
the lines PK, QI parallel to AC, meeting the fides of the cone dAD,
eAE in the points K, I, it is manifeft that theſe lines touch the
fame or oppofite conical furfaces in the points K, L, and being op-
pofite fides of parallelograms are equal to AC, and confequently to
each other; and therefore fince the right line joining the points
P, Q cuts the fame or oppofite furfaces in R, S; the fegments RP,
SQ will be equal (by Prop. XIII.)
Part 2. Let any right line as TX touch an hyperbola in V and
meet the afymptotes in the points X and T; the fegments VX, VT
will be equal; for from the points X, T, draw in the tangent planes
two lines XY, TZ, parallel to AC, meeting the fides of the cone
dAD, eAE in the points Y, Z. It is evident that theſe lines touch
the conical furface BAF in thofe points and are equal to AC, being
oppofite fides of parallelograms, and confequently are equal to each
other; therefore fince the line joining the points X, T touches the
conical furface in V, the fegments VX, VT will be equal by Prop.
XIII.
Part
:
Book I.
49
Conic Sections.
Part 3. Or if the right line XT meeting the afymptotes in X, T
meets the hyperbola in V, and the fegments VX, VT be equal;
this line touches the hyperbola in the point V.
For if it could meet the fection in any other point, fuppofe in p,
the fegment Tp would be equal to the fegment VX, by the first
part, that is to VT, which is abfurd: therefore it does not meet the
hyperbola but in the point V and therefore it touches it in this
point; for it does not fall on one part wholly within the hyperbola,
becauſe by hypotheſis it meets both the afymptotes. Q. E. D.
COR. 1. The afymptotes meet in the center of the hyperbolas
for draw two parallel right lines QP, HG cutting the hyperbola in
the points S, R and a, t and meeting the afymptotes in the points
Q, P and H, G, and through C the interfection of the afymptotes
draw CV bifecting the line QP in O; this will likewife bifect the
line HG (4. 6.) in the point q; but as the fegments RP, SQ₂
and likewiſe tG, aH are equal, the line CV will bifect the lines
SR, at, terminated by the hyperbola; therefore this line is a tranf-
verſe diameter of the hyperbola: and in like manner it may be
fhewn that any other line drawn through the interfection of the
afymptotes is a tranfverfe diameter; and therefore the afymptotes
meet in the center of the hyperbolas.
COR. 2. Every right line LCV drawn through the center and
within the angle HCG containing the hyperbola is a tranfverfe dia-
meter. For any line drawn parallel to LCV, and in the plane of the
hyperbolas, neceffarily meets the fides containing the angle adjacent
to HCG, and therefore meets the oppofite hyperbolas (Cor. 4. to
Def. XXV.) therefore LCV parallel to this line meets (Cor. 3. to
Def. of the Sections) the oppofite hyperbolas; but it paffes through
the center and therefore is a tranfverfe diameter by Def. XVII.
COR. 3. Every right line drawn through the center as MN, and
within the angle adjacent to the angle HCG containing the hyper-
bola is a fecond diameter. For through any point O within the hy-
perbola draw a line parallel to MN; it is evident that it meets the
fides containing the angle HCG, and confequently the hyperbola
G
in
50
Book I.
Conic Sections.
FIG. 32.
in two points; therefore the line MN is a fecond diameter, by
Prop. XXIII.
may
COR. 4. Hence, the afymptotes being given in pofition, a line
be drawn which fhall touch an hyperbola in a given point; for
let the afymptotes HC, GC be given in pofition, and the point V
given in the hyperbola; through V let the right line Vr be drawn
parallel to HC, one of the afymptotes, and meeting the other CG
in the point r, and let rX be taken in the afymptote CG equal to
Cr, and join VX, this line will be a tangent; for let it, produced,
meet HC in T; then becauſe CT, rV are parallel and Cr, rX equal,
TV, VX will be equal; therefore the line TX touches the hyper-
bola in V, by Cafe 3. of this Prop.
COR. 5. Hence, oppofite hyperbolas and their aſymptotes being
given in pofition; a line may be drawn from a given point. X in the
afymptote CG, which fhall touch the adjacent hyperbola: for let
CX be bifected in r, and through the point r draw a right line pa-
rallel to the other aſymptote CH, meeting the adjacent hyperbola
in V, the right line joining the points X, V will be a tangent; for
let it, produced, meet the afymptote CH in T, then becaufe Xr,
rC are equal, and Vr, TC are parallel, XV, VT will be equal
(2. 6.) therefore the line XVT is a tangent.
PROP. XXXVII.
If a right line cutting an hyperbola meets an aſymptote;
the rectangle under the fegments of the fecant between
the afymptote and hyperbola will be equal to the ſquare
of the fegment of the tangent parallel to this fecant, in-
tercepted between the point of contact, and the aſymptote.
L
ET a right line cut the hyperbola in the points S, R and meet
the afymptote CG in P, and let the right line XT parallel to
SR touch the hyperbola in V, and meet the afymptote CG in X;
the rectangle RPS will be equal to the fquare of VX.
<
Through
Book I.
band
Conic Sections.
5
Through the points P, X let two parallel right lines be drawn in
the tangent plane ADGC, and meet the fide dAD of the cone in
the points K, Y; they will touch the conical furface in thoſe
points and will be equal, becauſe XP, YK are parallel (Cor. 1.
Def. XXV.) then becauſe the line PRS cuts and PK touches the
conical furface, and thefe lines are parallel to the two lines XV, XY
touching the fame furface; the rectangle RPS (by Prop. XI.) will
be to the fquare of PK as the fquare of XV to the fquare of XY:
but the fquares of PK, XY are equal; therefore the rectangle RPS
is equal to the fquare of VX (14. 5.) 2. E. D.
COR. Hence, if through a point R in an hyperbola a right line
be drawn meeting the afymptotes in the points P, Q and the fame
hyperbola again in S; the rectangle PRQ will be equal to the
fquare of the fegment of the tangent VX, parallel to PQ, inter-
cepted between the point of contact and afymptote. For becauſe
RP, SQ are equal, PS, RQ will be equal; therefore the rectangle
PRQ will be equal to the rectangle RPS, that is to the fquare of
XV, by this Prop.
PROP.
XXXVIII.
Any right line terminated by the afymptotes and touching
an hyperbola, is equal to the conjugate diameter which
is parallel to it.
L
7-
ET a line touch an hyperbola in the point V, and meet its FIG. 32.
afymptotes in the points T, X; the line TX will be equal to
the conjugate diameter MN parallel to TX.
For through the point of contact draw the tranfverfe diameter
LVO, and through a point O taken in it, draw the right line SR
ordinately applied to this diameter; this line will be parallel to the
tangent VX; let SR be produced both ways that it may meet the
afymptotes in the points Q, P; the fegments OQ, OP will be
equal and becaufe (by preced. Cor.) the rectangle PRQ is equal
G2.
to
52
Book I.
Conic Sections.
to the fquare of VX, the fquare of OR is the difference of the
fquares of OP and VX (5. 2.) and the rectangle VOL (6. 2.) is
the difference of the fquares of CO, CV; then becauſe the trian-
gles CVX, COP are fimilar, the fquare of OP will be to the
fquare of VX as the ſquare of CO to the fquare of CV; therefore,
by divifion, the fquare of OR will be to the fquare of VX as the
rectangle VOL to the fquare of CV, and therefore, by permutation,
the fquare of OR is to the rectangle VOL as the fquare of VX to
the fquare of CV, that is as the fquare of TX to the ſquare of
LV: therefore TX is equal to the fecond diameter conjugate to
LV (by Def. XXIV.) and conſequently is parallel to TX.
COR. 1. The right line VN joining the vertices of the conjugate
diameters LV, MN, is parallel to one of the afymptotes and bi-
fected by the other. Let the afymptote CG meet VN in r, and let
CH be the other afymptote; through the vertex V of the diameter
VL draw a right line touching the hyperbola and meeting the
afymptotes in the points T, X: (by Cor. 27. of this Book) VT will
be parallel to CN and will be equal to it (by this Prop.) therefore
(33. 1.) the lines VN, TC are parallel and equal; but becauſe XV
is half of XT, Vr will be (2. 6.) half of TC or VN; therefore
VN is bifected in the point r, and from hence the Corollary is
manifeft.
' COR. 2. Hence an hyperbola being given in pofition, the afymp-
totes are eafily found; for let any tranfverfe diameter LV and the
center C be found (by Cor. 7. Prop. XXV.) and (by Prop.
XXVIII.) draw RO ordinately applied to the diameter LV, and
(by Def. XXIV.) let the fecond diameter MN conjugate to LV be
found, and join the vertices V, N of the conjugate diameters, and
from the center draw the right lines CG bifecting VN, and CH pa-
rallel to VN; the lines CG, CH will be the afymptotes, as is
evident,
}
!
.
PRO Pa
Book I.
53
Conic Sections.
!
PROP. XXXIX.
If through a point a right line be drawn taken in an aſymp-
tote cutting an hyperbola or oppofite hyperbolas; the
rectangle under the fegments of the fecant between the
afymptote and hyperbola or oppofite hyperbolas, will be
equal to the fquare of the femidiameter parallel to that
fecant.
F
IRST draw from the point P in the afymptote CG a right FIG. 32.
line cutting the hyperbola in the points R, S, and let CN be
the ſecond femidiameter parallel to PRS; the rectangle RPS will
be equal to the fquare of the fegment VX of the tangent (by 37.
of this Book) that is to the fquare of CN by the preced. Prop.
Secondly draw a right line through the fame point P cutting
the oppofite hyperbolas in the points R, S, and let LCV be the
tranfverfe diameter parallel to RPS, and draw in the tangent plane
ADG the line PK, parallel to the interſection CA of the tangent
planes ADG, AEH and meeting the fide AD of the cone in K;
it is manifeft that PK touches the conical furface BAF in K, and
is equal to CA; then becauſe the line PK touches the conical fur-
face and the line PRS cuts both furfaces and thefe lines are, pa-
rallel to the two lines CA, LV meeting in the point C and one of
them CA paffes through the vertex of the cone and confequently
touches the furface in A, and the other cuts both the furfaces; the
rectangle (by x1. of this Book) RPS will be to the fquare of PK
as the rectangle VCL to the fquare of CA: but the ſquares of PK,
and CA are equal; therefore the rectangle RPS (14. 5.) is equal to
the rectangle VCL, that is to the fquare of CV or CL the tranf-
verſe femidiameter parallel to RPS. 2. E. D.
COR. 1. Or if through the point R in an hyperbola a right line
be drawn meeting the afymptotes in two points P, Q; the rect-
angle PRQ under the fegments between the hyperbola and afymp-
totes,
1
54
Book I.
Conic Sections.
totes, is equal to the fquare of the femidiameter which is parallel
to PQ: for let the line PQ be parallel to the fecond femidiameter
CN; then the rectangle PRQ will be equal to the rectangle RPS (by
Cor. 37. of this Book) that is to the fquare of CN, by the firſt
cafe of this Prop.
Now let the line PQ be parallel to the tranfverſe femidiameter
CV, and being produced meet the oppofite hyperbola in S; then
becauſe the ſegments PR, QS are equal, the rectangle PRQ is
equal to the rectangle RPS, that is to the fquare of CV, by the
fecond cafe of this Prop.
COR. 2. Hence, if a right line as RS terminated by the oppofite
hyperbolas be cut in the points P, Q, ſo that each of the rect-
angles RPS, RQS be equal to the fquare of the femidiameter CV
which is parallel to RS; the points P, Qwill be in the afymptotes,
as is evident.
COR. 3. Hence, likewiſe, a tranfverfe diameter is less than any
line parallel to it, and terminated by the oppofite hyperbolas.
COR. 4. Since any right line touching an hyperbola SVR meets
both its afymptotes CG, CH, any two lines touching the fame hy-
perbola, meet each other within the angle GCH; and if two lines
not parallel to each other touch the oppofite hyperbolas, they ne-
ceffarily meet within the angle adjacent to the angle GCH. For
two tangents cannot meet each other in an afymptote; for if they
could, the diameter, viz. the line drawn from the point of concourſe
of the tangents, and bifecting the line joining their points of con-
tact (26. of this Book) would meet the afymptoté in another point
befide the center, which is abfurd.
PROP.
Book I.
55
Conic Sections.
,
PRO P. XL.
If two right lines meet each other, and according as both
touch or cut, or one of them touches and the other cuts.
an hyperbola or oppofite hyperbolas; the fquares of the
fegments of the tangents, or the rectangles contained by
the ſegments of the fecants between the point of concourfe
of the right lines and hyperbola or oppofite hyperbolas,
will be to each other as the fquares of the femidiameters.
to which the right lines are parallel.
FOF
OR if the lines RS, VT meet each other in the point Q of the FIG. 33
afymptote CY, and the hyperbola or oppofite hyperbolas in
the points R, S and V, T, and if they be parallel to the two femi-
diameters CA, CB; by Prop. XXXVIII. XXXIX. the rectangle
RQS will be equal to the fquare of the femidiameter CA, and the
rectangle VQT, or the fquare of the tangent QVT will be equal to
the fquare of the femidiameter CB: therefore the rectangle RQS is
to the rectangle VQT or to the fquare of the tangent QVT as the
fquare of CA to the fquare of CB. Then if two lines parallel to
RS, VT meet each other in a point not in the afymptote; the fquares
of thefe lines, or the rectangles under their fegments, will be to
each other as the fquares or rectangles under the fegments of RS,
VT (by 18. of this Book) and therefore are as the fquares of the
femidiameters CA, CB to which theſe lines are parallel.
COR. 1. Hence, if from any point P two right lines PO, PX be FrG. 34
drawn touching the fame hyperbola or oppofite hyperbolas; the
lines PO, PX will be to each other as the femidiameters to which
they are parallel; for their fquares are as the fquares of thefe femi-
diameters.
COR. 2. If two right lines MN, TR touching an hyperbola or
oppofite hyperbolas in the points O, X meet the afymptotes in M, N
and T, R; the right lines MR, TN which join the points of con-
courfe
56
Book I.
Conic Sections.
?
FIG. 35.
courfe will be parallel to each other, and to the line OX joining the
points of contact. For (by Prop. XXXVIII. and preced. Cor.)
PO will be to PX as OM to XR; therefore, OX is parallel to MR
(2. 6.) and in like manner becauſe PO is to PX as NO is to TX,
by permutation, PO is to NO as PX is to TX, and, by conver-
fion, PO is to PN as PX to PT; therefore the triangles OPX,
NPT, having the fides about equal angles proportional, are equi-
angular (6. 6.) confequently the angle TNP is equal to the alter-
nate angle XOP, and therefore the line XO is parallel to TN.
COR. 3. Hence (the reft remaining as premifed) the fegment NR
of the afymptote CZ intercepted by two right lines touching the
hyperbola or oppofite hyperbolas, will be bifected by the right line
OX (produced if neceffary) joining the points of contact; for OX
is parallel to MR (by preced. Cor.) let it meet then (if neceffary
produced) the afymptote CZ in B, and becauſe MO, ON are equal
(36. of this Book) the lines RB, BN will be equal.
PROP. XLI.
If a right line cutting an hyperbola or oppofite hyperbolas
in the points E, F, or touching the hyperbola in O, meets
a fecond diameter MN in the point P; the rectangle EPF
under the ſegments of the fecant, or the fquare of the
tangent PO, will be to the fquare of the ſegment PC be-
tween the right line and center, together with the fquare
of the femidiameter MC, as the fquare of the femidia-
meter CA, to which the fecant or tangent is parallel, to
the fquare of the femidiameter MC which the fecant or
tangent meets.
TH
HROUGH the point P let the right line QR be drawn or-
dinately applied to MN, and let LV be the conjugate dia-
meter to MN and therefore parallel to the line QR; by the pre-
ceding Prop. the rectangle FPE will be to the rectangle (QPR or)
to
TAB · 4 ·Page · 56.
L
M
P
FIG.28-
B
·FIG.27%
D
A
G
H
FIG.25%
A
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FIG.24
D
D
H
BX
L
V
M
W
B
MD
FIG. 30.
&
V
A
n\
R
I
M
FIG. 32.
T
Q
m
In
T
A B
Y
/R
B
E
P
B
K R
F
FIG-33.
VT
T
M
Y
A
Z
A
FIG-31.
•
E
B
P
E
M
T
Y
L
ľ
X
FIG. 29.
Y
B
Z
P
K
N
P
R
M
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E
Р
D
G
B
D
M
FIG.26.
NIK
C
H
O
X
N
A
ļ
Book I.
57
Conic Sections.
to the fquare of PQ as the fquare of CA to the fquare of CL, and
by Prop. XXXII. the fquare of QP is to the fquares of PC, MC
together, as the fquare of CL to the fquare of CM: therefore, ex
æquo, the rectangle FPE will be to the fquares of PC, CM toge-
ther, as the fquare of CA to the fquare of CM. In like manner
the ſquare of the tangent PO will be to the fquares of PC, CM to-
gether, as the fquare of CA to the fquare of CM. 2, E. D.
PROP.
XLII.
If two right lines be parallel to each other, and if both
touch or cut, or one of them touches and the other cuts
an hyperbola or oppofite hyperbolas, and meet a right line
parallel to one of the afymptotes; the fquares of their
ſegments between the line parallel to the afymptote and
the points of contact, or the rectangles under their feg-
ments between this line and the hyperbola or oppoſite hy-
perbolas, will be to each other as the fegments of that
line parallel to the afymptote, intercepted between the pa-
rallels, and the point in which that line meets one of the
hyperbolas.
FO
AOR (by Cor. 1. and 3. Def. XXV.) a line parallel to an afymp-
tote is parallel to a fide of the conical furfaces in which the
hyperbolas are, and meets one of the hyperbolas in one point only;
therefore this Prop. is evident, from Prop. XIX. and Cor. 1. to
Def. of the Sections.
COR. 1. Hence if the tranfverfe diameter LV of an hyperbola FIG. 36.
meets a line KH in O, parallel to the afymptote CY, meeting the
other afymptote CZ in P, and the hyperbola in E; the rectangle
VOL under the fegments of the diameter between its two vertices.
and the line KH will be to the fquare of the femidiameter CV as
the ſegment OE of the line KH between the diameter and hyper-
bola to the fegment EP of the fame line between the hyperbola and
afymptote CZ; for through P draw a line parallel to the diameter.
H
LV,
58
Book I.
Conic Sections.
FIG. 37.
LV, meeting the hyperbolas in the points R, S: then by this Prop.
the rectangle VOL will be to the rectangle RPS or (Prop. XXXIX.)
to the fquare of CV as OE to EP.
COR. 2. Or (the reft remaining the fame) if the ſecond diameter
MN meets KH in B; the fquare of the fegment CB of the dia-
meter between the center and the line KH, together with the
fquare of the femidiameter CN, will be to the fquare of the femi-
diameter CN as the fegment BE of the line KH between the dia-
meter and hyperbola, to the fegment PE of the fame line between
the afymptote CZ and hyperbola.
For through the points B, P draw the right lines AD, RS ordi-
nates to the diameter MN, and let LV be the diameter conjugate
to MN, and confequently parallel to its ordinates; then (by Prop.
XXXII.) the fquare of BC together with the fquare of CN, will
be to the ſquare of CN as the fquare of AB to the fquare of CV;
that is as the rectangle ABD to the rectangle RPS (39. of this Book)
that is (by this Prop.) as BE to PE.
PROP.
XLII.
If any two right lines be drawn from a point in an hyper-
bola to the afymptotes, and two other right lines be drawn
from any other point in the fame or oppofite hyperbola
to the afymptotes and parallel to the two firſt drawn lines;
the rectangle under the two firſt lines will be equal to the
rectangle under the other two lines.
ET A, B be the points in the hyperbola or oppofite hyperbo-
las, and through A draw the right lines AD, AE to the afymp-
totes, and through B the lines BG, BF parallel to them; the rectangle
under AD, AE will be equal to the rectangle under BG, BF.
For draw the right line AB meeting the afymptotes in the points-
M, N, and (by Prop. XXXVI.) the fegments AM, BN are equal,.
and therefore AN, BM will be equal; and becaufe the triangles
ADM, BGM are equiangular, AD is to BG as AM to BM, or as
BN to AN, that is, becauſe the triangles BNF, ANE are equian-
angular
Book I.
Conic Sections.
59
angular as BF to AE; therefore AD is to BG as
therefore the rectangles DAE, FBG will be equal.
BF to AE, and
Q. E. D.
COR. 1. Hence if two right lines AD, BG be drawn from two
points A, B in an hyperbola or oppofite hyperbolas, to the fame or
different afymptote and parallel to the other aſymptote; the rectan-
gle under the line AD and the abfcifs DC between this line and the
center C, will be equal to the rectangle under the line BG and ab-
fcifs GC; for complete the parallelograms CDAE, CGBF, the
rectangles DAE, FBG, that is, the rectangles ADC and BGC will
be equal.
COR. 2. And (the fame thing being fuppofed) becauſe the rect-
angles DAE, FBG are equal; DA is to BG as BF to AE; but the
parallelograms DAEC, FBGC are equiangular, and therefore they
will be equal (14. 6.)
COR. 3. If a line AD be drawn from any point A in an hyper-
bola AVB to one of the afymptotes and parallel to the other, and
if a point P in either afymptote be taken, from which a line PO is
drawn towards the adjacent hyperbola and parallel to the other
afymptote, and if the rectangle CPO be equal to the rectangle
CDA; the point O will be in the hyperbola towards which the line
PO was drawn: for PO meets fomewhere the adjacent hyperbola
(Cor. 3. to Def. XXV.) and if it does not meet it in O, a rectangle
(Cor. 1.) greater or lefs than CPO would be equal to the rectangle
CDA, contrary to the hypothefis.
L
DEFINITION XXVI.
ET LV be a tranfverfe diameter of two oppofite hyperbolas, FIG. 38.
and MN the ſecond diameter conjugate to LV; and by Prop.
XXXV. let KMS, HNX be two oppofite hyperbolas, of which let
MN be a tranfverfe diameter and LV the fecond diameter conjugate
to MN; theſe four hyperbolas are called conjugate hyperbolas.
H 2
Сека
бо
Book I.
Conic Sections.
FIG. 38.
COR. 1. It is evident that the point C, the interſection of the
diameters LV and MN, is the common center of the conjugate hy-
perbolas, and that the afymptotes are the fame (for the things re-
maining which were affumed in the Definition) let VM be joined,
and draw CY bifecting VM in D, and CZ parallel to VM; becauſe
LV, MN are conjugate diameters of both hyperbolas TVO, KMS,
the lines CY, CZ are afymptotes of the hyperbola TVO its oppo-
fite hyperbola, and likewiſe of the hyperbola KMS conjugate to
TVO, and its oppofite HNX (by Cor. 2. 38. of this Book.)
COR. 2. Becauſe conjugate hyperbolas have the fame afymptotes;
it is evident from Prop. XXXVI. that if a right line terminated by
oppofite hyperbolas, touches an hyperbola conjugate to them; it
will be bifected in the point of contact; or if it meets a conjugate
hyperbola in two points; the fegments between this hyperbola and
the oppofite hyperbolas will be equal.
PRO P. XLIV.
If a ſecond diameter of any hyperbola be drawn; its two
vertices will be in the hyperbolas conjugate to that hy-
perbola.
ET LV, MN be the common conjugate diameters of any con-
jugate hyperbolas, as in the preceding Definition, and CY, CZ
their common afymptotes, and join VM meeting the afymptote
CY in D, and draw a ſecond diameter BCb of the hyperbola TVO;
and the two vertices in B, b, will be in the hyperbolas conjugate
to TVO.
For draw the ſecond diameter CA conjugate to CB, and join AB
meeting the afymptote CY in E; it will be bifected in E, and be
parallel to the other aſymptote CZ (Cor. 1. Prop. XXXVIII.) and
confequently parallel to MV; becaufe BE, EA are equal, the rect-
angle BEC will be equal to the rectangle AEC, that is to VDC
(by
Book I.
6I
Conic Sections.
(by Cor. 1. Prop. preced.) or to the rectangle MDC, becauſe
VD, DM are equal; therefore fince the point M is in the hyper-
bola KMS and the lines MD, BE are parallel to the afymptote CZ,
and the rectangles MDC, BEC are equal; the point B will be in the
fame hyperbola KMS (Cor. 3. Prop. preced.) and its other vertex
b in the hyperbola HNX oppofite to KMS.
COR. 1. Hence, if right lines be drawn through the vertices of
two conjugate diameters CA, CB and touch conjugate hyperbolas,
they will meet each other in the afymptote CY, which the right line
joining the vertices A, B meets; for through the point A draw a
right line touching the hyperbola TVO, and meeting the afymp-
tote CZ in R, and CY in P, and join PB meeting the afymptote
CZ in Q; then becauſe PAR touches the hyperbola, the lines
PA, AR are equal (36. of this Book) therefore becauſe RQ, AB
(Cor. 1. Prop. XXXVIII.) are parallel, PB, BQ will be equal;
therefore the line PBQ_touches the hyperbola in B (by Prop.
XXXVI.) and confequently the tangents drawn through A, B, meet
in the point P in the afymptote CY.
COR. 2. Hence likewife, any fecond diameter of an hyperbola is
the tranfverfe diameter of its conjugate hyperbola; and the contrary.
PROP.
XLV.
If two right lines be drawn from two points in conjugate
hyperbolas to the fame, or different afymptotes, and pa-
rallel to the afymptotes; the rectangles contained by thefe
lines and the abfciffes, between theſe lines and the cén-
ter, will be equal.
L
ET CZ, CY be the afymptotes to the conjugate hyperbolas, FIG. 38.
and draw, from a point A in one of them, AE to the afymp
tote CY, and parallel to CZ, and let FG be drawn from a point F
in a conjugate hyperbola to one afymptote and parallel to the other;
the rectangles AEC, FGC will be equal.
Let
62
Book I.
Conic Sections.
Let the vertices V, M of the conjugate diameters be joined, the
right line joining theſe points will be parallel to one of the afymp-
totes and will be bifected by the other in D, and (by Cor. 1.
Prop. XLIII.) the rectangles AEC, VDC will be equal, and like-
wife the rectangles FGC, MDC; but the rectangles VDC, MDC
are equal, and therefore the rectangles AEC, FGC are equal.
Q. E. D.
COR. 1. Hence, the parallelograms AECe, FGCg will be equal,
as in Cor. 2. Prop. XLIII.
COR. 2. Hence alfo if any two right lines PR, pr touch the
fame hyperbola, or oppofite, or adjacent hyperbolas, and meet the
afymptotes in the points P, R and p, r; the triangles PCR, pCr
contained between the tangents and the fegments of the afymptotes
will be equal. For from the points A, F of contact, draw AE,
Ae and FG, Fg parallel to the afymptotes; then becaufe PA, AR
are equal, PE, EC, and Ce, eR will be equal: therefore the tri-
angle PCR is double the parallelogram AECe: in the fame man-
ner it may be ſhewn that the triangle pCr is double the parallelo-
gram FGCg: but thefe parallelograms are equal (Cor. preced.)
therefore the triangles are equal.
COR. 3. Hence alſo, if a right line terminated by adjacent hyper-
bolas be parallel to one of the afymptotes, it will be bifected by the
other afymptote. For let the line AB be terminated by the adja-
cent hyperbolas and be parallel to the afymptote CZ, and let it meet
CY in E; becaufe by this Prop. the rectangles AEC, BEC are
equal, the lines AE, EB will be equal, as is evident.
COR. 4. And on the contrary, if AB, terminated by the conju-
gate hyperbolas, be bifected by the afymptote CY; it will be pa-
rallel to the other afymptote CZ; for, if not, a right line may be
drawn from the point A to another point K in the adjacent hyper-
bola, parallel to CZ: this line AK will alſo be bifected by the
afymptote CY (by preced. Cor.) therefore the afymptote CY will
be parallel to the line joining the two points B, K in the hyperbola
(2. 6.) which is impoffible (Cor. 3. to Def. XXV.)
COR.
Book I.
63
Conic Sections.
COR. 5. It appears from this Prop. and Cor. 2. Prop. XLIII.
that if two right lines be drawn from points in the fame hyperbola,
or from points in any two of the four conjugate hyperbolas to one
afymptote and parallel to the other; thefe lines will be reciprocally
as the fegments of the afymptotes between theſe lines and the center;
and confequently the hyperbola and afymptotes produced indefi-
nitely continually approach each other, and arrive at an interval lefs
than any given diſtance.
COR. 6. If through the point A in the hyperbola TAO, a right
line be drawn meeting the conjugate hyperbolas BMS, HNX in the
points K and H; the rectangle KAH will be double the fquare of
the femidiameter CM which is parallel to KH; for through the
point A draw the femidiameter CA, it will be a fecond femidiameter
of the hyperbolas BMS, HNX (Cor. 2. 44. of this Book) there-
fore by Prop. XLI. the rectangle KAH will be to twice the ſquare
of CA as the fquare of CM to the fquare of CA; therefore the
rectangle KAH is double the fquare of CM.
PROP.
XLVI.
If through a point in an hyperbola two indefinite right lines
be drawn parallel to the afymptotes and meeting any dia-
meter; the femidiameter will be a mean proportional be-
tween the ſegments of this diameter intercepted between
the center and the lines which it meets.
DR
RAW through the point A of the hyperbola TVO, the right FIG. 39-
lines DE, FG parallel to the afymptote CY, CZ, and first let
them meet the tranſverſe diameter LCV in the points P, Q; CP,
CV, CQ will be proportional.
For let FG meet the afymptote CY in X, and from the points
V, P draw to CY the lines Vr, PS parallel to FG or CZ; PS will
be equal to AX; and becauſe Vr, PS are parallel, CS will be to
Cr, as (SP or) XA to rV, that is (by Cor. 5. preced.) as Cr to
CX; therefore becauſe CS, Cr, CX are proportional, CP, CV,
will be proportional, for the lines SP, rV, XQ are parallel.
се
Now
64
Book I.
Conic Sections.
'A
1
Now let the lines DE, FG meet the fecond diameter MCN in
the points K, H, and let its vertex M be in the hyperbola conjugate
to TVO, which let the line FG meet in the point B, through which
point draw BR parallel to DE or CY, meeting the diameter MN
in R; by the firft cafe, CR, CM, CH are proportional: but be-
cauſe BR, XC, DE are parallel, and the lines AX, XB are equal
(by Cor. 3. preced.) the lines KC, CR will be equal; therefore
CK, CM, CH are proportional. 2. E. D.
FIG. 39.
SCHOLIU M.
IF two indefinite right lines DE, FG cut each other in the point A, and
another indefinite line revolves about any given point C taken without
the lines DE, FG, and in this revolving line, whilſt it meets (as in the
points P, 2) the fides of the angle EAF (without which is affumed
the point C) or the fides of its vertical angle GAD, let a point V be
taken on the fame fide of the point C with the points P, Q, fo that the
Segments CP, CV, CQ be always proportional; the Locus of all the
points V will be the hyperbola TVO paffing through the point A, whofe
center is the given point C, and its afymptotes will be parallel to DE,
FG, and the revolving line will at length coincide with the afymptotes;
and if on the contrary fide of the point C, the point L be fo taken that
CL be always equal to CV; the hyperbola oppofite to TVO will be the
Locus of the points L. But if in the revolving line, whilst it meets
(as in the points K, H) the fides of the angle EAG within which the
point C is placed, two points M, N be fo taken, that the fegments CK,
CM or CN, and CH be proportional, the Loci of all the points M, N
will be the oppofite hyperbolas conjugate to the former.
:
PROP.
Book I.
65
Conic Sections.
XLVII.
PROP.
If from a point F in an hyperbola a right line FE be drawn FIG. 47.
ordinately applied to a diameter BA, and through the
ſame point F a tangent be drawn meeting this diameter in
D; the ſegment of the diameter between the ordinate and
tangent will be bifected in its vertex B.
FOR
the
OR let the ordinate FE produced meet the parabola again in
L, and draw through the point F the diameter FG, and the
lines LG, EC, BH parallel to the tangent FD; LG, BH will be
ordinates to the diameter FG (Cor. 27. of this Book) and EC will
be equal to BH; then becauſe LF is double EF, GF will be double
CF and LG double EC or BH (2. 6.) and therefore the fquare of
LG is quadruple the ſquare of BH (4. 2.) and therefore the ab-
fcifs GF is quadruple the abfcifs HF (29. of this Book) and there-
fore CF is double HF, and confequently ED double BD; and
therefore the fegment ED of the diameter is bifected at the ver-
tex B. 2. E. D.
PROP.
XLVIII.
If from a point F in an ellipfe an ordinate FE be drawn to FIG. 41.
the diameter AB, and alfo a right line touching the el-
lipfe and meeting the fame diameter in D; the femidia-
meter CB will be a mean proportional between the feg-
ments CE, CD of the diameter between the center and
ordinate, and between the center and tangent.
FOF
OR draw through the two vertices of the diameter AB right
lines touching the ellipfe, and meeting the tangent drawn
through F in the points H, G; thefe tangents will be parallel to
the ordinate FE (Cor. 1. 25. of this Book); take CT in the dia-
meter AB equal to CE; AT, EB will be alfo equal: but (by Cor.
5. Prop. XVIII. of this Book, and 22. 6.) AH is to BG as HF
I
to
1
66
Book I.
Conic Sections.
to FG or (becauſe HA, FE, GB are parallel lines) as AE to EB;
therefore AE is to EB as AD to BD; and therefore by divifion,
TE will be to EB as AB to BD, and by taking the halves of the
antecedents CE will be to EB as CB to BD, and therefore by com-
pofition and converfion, CE is to CB as CB to CD. 2, E. D.
PROP. XLIX.
If from a point in an hyperbola an ordinate be drawn to a
diameter, and alfo a right line touching the hyperbola
and meeting the fame diameter; the femidiameter will be
a mean proportional between the fegments of the diame-
ter between the center and ordinate and between the
center and tangent.
FIG. 42. Cafe 1.
W
HEN an ordinate is drawn to a tranfverfe diameter
let AB be the tranfverfe diameter of the hyperbola,
C the center, and from the point F in it draw an ordinate FE ap-
plied to the diameter AB, and a tangent meeting the fame diameter
in D; CE will be to CB as CB to CD.
Draw through the two vertices of the diameter AB two tangents
meeting the tangent drawn through F in the points H, G, which
will be parallel to the ordinate FE (Cor. 1. Prop. XXV. of this
Book) and take CT equal to CE, and AT, BE will be alfo equal:
but (by Cor. 5. Prop. XVIII. of this Book, and 22. 6.) AH is to
BG as HF to GF, that is (becaufe AH, GB, FE are parallel) as.
AE to BE, and as AH is to BG fo is AD to BD; therefore AE
will be to BE as AD to DB, and by compofition TE will be to BE
as AB to DB, and by taking the halves of the antecedents, CE is
to BE as CB to DB, and therefore by converfion CE is to CB as
CB to CD.
Cafe 2. When an ordinate is drawn to a fecond diameter. From
the point F in an hyperbola draw FO an ordinate to the fecond dia-
meter MN, which the tangent drawn through F meets in P, and let
AB be the tranfverfe diameter conjugate to MN, meeting the tan-
gent
Book I.
67
Conic Sections.
gent FP in D, and draw FE ordinately applied to AB; it will be
parallel and equal to CO.
Becauſe, by the firft cafe, EC, BC, DC are proportional, the
fquare of EC will be to the fquare of BC as EC to DC (Cor. 20. 6.)
therefore by divifion the rectangle BEA is to the fquare of BC as
ED to DC; but the fquare of the ordinate FE is to the fquare of
the ſecond femidiameter CM as the rectangle BEA to the fquare of
BC, as appears from Def. XXIV. therefore the fquare of FE is to
the fquare of CM (as ED to DC) that is as FE to CP; therefore
FE, CM, CP are proportional (by the converſe of Cor. 20. 6.) and
therefore CO, CM, CP are proportional. Q. E. D.
Corollaries to the two preceding Propofitions.
COR. I. If a right line touches an ellipfe or hyperbola in any
point F, and meets the diameter AB (which in the hyperbola ſhould
be a tranfverfe diameter) in the point D, and if an ordinate FE be
drawn from the point of contact to the diameter; the rectangle
CED under the fegments of the diameter, between the ordinate and
center, and between the fame and the tangent, will be equal to the
rectangle AEB under the fegments between the ordinate and the two
vertices of the diameter.
For becauſe (by the preced. Propofitions, and 17. 6.) the rect-
angle ECD is equal to the fquare of CB, if in the ellipfe the ſquare
of CE be taken from each of them; the remainders, viz. the rect-
angles CED, and AEB (by the 3. and 5. 2.) will be equal, and in
the hyperbola theſe equals, viz. the rectangle ECD and the fquare
of BC being taken from the fquare of EC; the remainders viz.
the rectangles CED and AEB (by 2. and 6. 2.) will be equal.
FIG. 41,
42€
COR. 2. Let the tangent and ordinate drawn from the point F, FIG. 42
meet the ſecond diameter CM in P and O, and becauſe the rect-
angle PCO is equal to the fquare of CM; let the fquare of CO
be added to each, and the rectangle POC (3. 5.) will be equal to
the fquares of CM, CO together.
I 2
COR.
68
Book I.
Conic Sections.
FIG. 41,
42.
FIG. 4',
42.
COR. 3. And the rectangle ADB, under the fegments between
the tangent and the two vertices of the diameter (which in the hy-
perbola fhould be the tranfverfe) will be equal to the rectangle CDE
under the fegments between the tangent and center and between
the fame tangent and ordinate: for in the ellipfe, equals being fub-
ducted, viz. the fquare of CB and the rectangle ECD from the
fquare of CD, the remainders, viz. ADB and CDE (by 6. and
2. 2.) will be equal, and in the hyperbola, if from equals viz.
from the fquare of CB and from the rectangle ECD, the fquare of
CD be taken, the remainders ADB and CDE will be equal (5-
and 3. 2.)
PRO P. L.
If two parallel right lines touching an ellipſe or oppofite
hyperbolas, meet any other tangent; the rectangle under
the fegments of the parallel lines between the points of
contact and the tangent which they meet, will be equal to
the fquare of the femidiameter, to which the tangents are
parallel and the rectangle under the fegments of the
tangent which the parallels meet, viz. between its point
of contact and the parallel lines, will be equal to the
ſquare of the femidiameter parallel to this tangent.
L
ET the parallel lines AH, BG touch an ellipfe or oppofite hy-
perbolas in the points A, B, and let them meet a tangent
drawn through any point F in H and G; the rectangle under AH,
BG will be equal to the fquare of the femidiameter CM parallel to
AH, BG, and the rectangle HFG will be equal to the fquare of
the femidiameter CK parallel to HG.
For the line joining the points A, B will be a diameter (Cor. 2.
22. of this Book) and is conjugate to CM (Cor. 27. of this Book) if
the tangent HG in the ellipfe be parallel to AB, the propofition is
manifeft; but if it be not parallel to AB, let it meet it in D and CM
produced in P, and from the point F draw FE, FO ordinately ap-
plied to the diameters AB, CM.
Part
Book I.
69
Conic Sections.
!
Part 1. Becauſe by Cor. 3. preced. the rectangles ADB and CDE
are equal; AD will be to CD as ED to BD (16. 6.) therefore
(being parallel lines) AH is to CP as EF to BG (2. 6.) and there-
fore the rectangle under AH, BG is equal to the rectangle under
CP, EF or CO, that is, to the fquare of the femidiameter CM:
for by the two preceding Propofitions, CO, CM, CP are propor-
tional.
Part 2. Becaufe AH is to BG as HF to FG (Cor. 5. 18. of
this Book, and 22. 6.) the rectangle HFG is fimilar to the rectangle
under AH, BG and therefore is to the fame rectangle AH, BG as
the fquare of HF to the fquare of AH, that is, as the fquare of
CK to the fquare of CM (by 31. and 40. of this Book) therefore
by permutation, the rectangle HFG is to the fquare of CK as the
rectangle AH, BG to the fquare of CM, that is, in a ratio of
equality by the first part. 2. E. D.
PRO P.
LI.
42.
If a right line touching an ellipfe or hyperbola in any point FIG. 41,
F, meets two conjugate diameters AB, CM in the points
D, P; the rectangle PFD under the fegments of the tan-
gent between the point of contact and the diameters will
be equal to the fquare of the femidiameter CK parallel to
the tangent.
OR draw from the point of contact F a right line FE ordi-
dinately applied to the diameter AB (which in the hyperbola
ſhould be the tranfverfe diameter) and draw through the two ver-
tices of this diameter two tangents, and let them meet the tangent
drawn through F in the points H, G; they will be parallel to the
ordinate FE and to the diameter CM; then becauſe (by Cor. 1.49.
of this Book) the rectangle CED is equal to the rectangle AEB,
CE is to EB as AE to ED; therefore (becauſe of the parallel lines).
PF is to FG as HF to FD, and therefore the rectangle PFD is
equal
70
Book I.
Conic Sections.
FIG. 40,
41,
42,
43.
equal to the rectangle HFG, that is, to the fquare of the femidia-
meter CK, by the fecond part of the preced. Prop. 2. E. D.
COR. 1. It is manifeft, if the right line PFD touching the el-
lipfe or hyperbola, meets two diameters CB, CM, and if the rect-
angle PFD be equal to the ſquare of the femidiameter CK conju-
gate to that which paffes through the point of contact F, that CB,
CM are conjugate diameters.
PROP. LII. PROB L. IV.
A conic fection being given in pofition, and a point given
without the ſection; to draw two right lines which fhall
touch the ſection or oppoſite ſections, provided the given
point be not in an aſymptote of an hyperbola.
L
ET FL be the given fection, and D the point given; draw
through the point D a diameter meeting the fection in B, or
if it be a fecond diameter of an hyperbola let its vertex or extre-
mity be B: or if the fection be a parabola take BE equal to BD,
and through E draw FEL parallel to the ordinates applied to the
diameter BE, meeting the parabola in the points L, F; the lines.
joining the points, DL, DF touch the fection; for if either of them,
fuppofe DL, does not touch the fection, let LR be a tangent meet-
ing the diameter BE in R, (by the 47. of this Book) BR is equal
to BE, that is to BD, which is abfurd.
But if the fection be an ellipfe or an hyperbola; let C be its
center, and let CE be a third proportional to CD, and CB (which
muſt be taken on the fide of the center, oppofite to that on which
CD is placed, if CD be a ſecond diameter of the hyperbola) but
if not, take it on the fame fide, and through the point E draw FEL
parallel to the ordinates applied to the diameter CD, meeting the
fection in the points L, F; the lines joining theſe points DL, DF
touch the ſection or oppofite fections.
For if either of them, fuppofe DL does not touch the fection,
let
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Page.70.
TAB.5.
FIG. 35°
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FIG. 37.
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Book I.
zi
Conic Sections.
let LR be a tangent meeting the diameter CD in R; the lines CE,
CB, CR will be proportional (by the 48. 49. of this Book) which
is abfurd.
If the given point D be in one of the afymptotes of an hyper-
bola, two tangents cannot meet in this point (Cor. 4. 39. of this
Book) but a method of drawing a right line which fhall touch
the adjacent hyperbola was fhewn in Cor. 5. Prop. XXXVI.
PROP. LIII.
45.
If two right lines touching a conic fection or oppofite fec- FIG. 44,
tions meet each other, and through a point in one tan-
gent a right line be drawn parallel to the other, and meet-
ing the right line joining their points of contact and alſo
the conic ſection in two points; the fquare of its ſegment
between the tangent and the line joining the points of
contact will be equal to the rectangle under the ſegments
intercepted between the tangent and fection.
L
ET the lines AB, AC touch a conic fection or oppofite fections
in the points B, C, and join thefe points and draw a right
line through the point D in the tangent AC parallel to the other
tangent AB, meeting BC in G and the ſection in the points E, F;
the fquare of DG will be equal to the rectangle EDF.
For becauſe the tangent AC meets the parallels FD, BA, the
rectangle EDF under the fegments of the fecant is to the ſquare of
the tangent BA as the fquare of DC to the fquare of AC (by Cor.
6. 18. of this Book) that is (from fimilar triangles) as the fquare
of DG to the fquare of BA; therefore the rectangle EDF is equal
to the fquare of DG. 2. E. D.
COR. Let two right lines touching a conic fection, or oppofite
fections meet each other in the point A, if through a point P in
the tangent AC a right line be drawn parallel to the other tangent,
and meeting the right line joining their points of contact in L, and
any other line be drawn from the fame point P cutting the fec-
if
tion
72
Book I.
Conic Sections.
FIG. 46.
tion or oppofite fections in the points R, Q; the rectangle RPQ
and the ſquare of PL will be to each other as the fquares of the
fegments of the tangents, which are parallel to PRQ, and PL, in-
tercepted between their points of concourfe and contact; or as the
ſquares of the femidiameters parallel to PRQ and PL.
For through the point D in the tangent AC draw two right lines.
parallel to PRQ, PL, cutting the ſection or oppofite fections in the
points O, N and E, F, and let the right line DEF parallel to PL
meet the right line joining the points of contact in G: then be-
cauſe the tangent AC meets two parallel fecants (Cor. 4. 18. of this
Book) the rectangle RPQ will be to the rectangle ODN as the
fquare of PC to the fquare of DC, that is from fimilar triangles as
the fquare of PL to the fquare of DG or to the rectangle EDF
(by this Prop.) therefore by permutation the rectangle RPQ is to
the fquare of PL as the rectangle ODN to the rectangle EDF,
that is (by the 18. of this Book) as the fquares of the fegments of
the tangents which are parallel to PRQ, PL, intercepted between
their points of concourfe and contact: or as the fquares of the fe-
midiameters of the ellipfe or hyperbola which are parallel to PRQ,
PL, by (31. and 40. of this Book).
LEMMA
11.
FR
ROM a point E in any indefinite right line take the lines EA,
ED on the fame fide of the point E, and from the fame point
E take the lines EB, that its extremity B be either between the
points A, D, or on the contrary fide of the point E; if the fquare
of AB be to the fquare of DB as AE to DE, EB will be a mean
proportional between the lines AE and DE.
For if not, take from the point E towards B a line EF (greater
or leſs than EB) a mean proportional between AE and DE; then
becauſe AE, FE, DE are proportional, by converfion and permu-
tation, as AE the firft is to FE the fecond term, fo will AF the
fum or difference of the firſt and ſecond terms be to DF the fum or
differ-
Book I.
73
Conic Sections.
difference of the fecond and third; therefore the fquare of AF will
be to the fquare of DF as the fquare of AE to the fquare of FE,
that is as AE to the third DE, or, by conftruction, as the fquare of
AB to the fquare of DB; therefore AF is to DF as AB to DB,
which is abfurd; as is felf-manifeft when the points F, B are between
A, D, and if the points F and B be on contrary fides of the
point E, it is evident alfo that the ratio of the unequal lines AF,
DF will not remain the fame, if to theſe unequal lines, the fame
line FB be added or taken from them; therefore no line either
greater or leſs than EB is a mean proportional between AD, DE;
and therefore EB is a mean proportional between theſe right lines.
PRO P. LIV.
If a diameter of a parabola, or a right line parallel to one
of the afymptotes of an hyperbola, meets two tangents
and the right line joining the points of contact; the ſquare
of its fegment between the fection and the line joining the
points of contact, will be equal to the rectangle under its
other fegments, viz. between the fection and tangents.
L
48.
ET two lines MN, MO touch a parabola, an hyperbola or op- FIG. 47.
pofite hyperbolas in the points N, O; draw the line TX, which
may either be a diameter of the parabola or be parallel to one of
the afymptotes of the hyperbola, meeting the parabola or either of
the hyperbolas in E, the tangents in A, D and the line joining the
points of contact in B; the fquare of EB will be equal to the rect-
angle AED.
Cafe 1. If the lines touching the parabola or hyperbola meet Fic. 47•
each other in M, through the point D, in which the line TX meets
one of the tangents, let a line be drawn parallel to the other tan-
gent, cutting the ſection in the points P, L and the line joining the
points of contact in K, which may be always done; then by the
preced. Prop. the fquare of DK is equal to the rectangle PDL,
and becauſe AO is parallel to DK, the fquare of AB will be to the
fquare
K
74
Book I.
Conic Sections.
FIG. 48.
ſquare of DB as the fquare of AO to the fquare of DK, or
to the rectangle PDL, that is (Prop. XIX.) as AE to DE;
but the lines AE, BE, DE are taken from the point E as in
the preceding Lemma; therefore they are proportional by the fame
Lemma, and confequently the fquare of BE is equal to the rect-
angle AED.
Cafe 2. If the right lines touching the oppofite hyperbolas meet
each other in M; through either of the points A or D, ſuppoſe D,
draw a right line parallel to the other tangent AO and meeting the
line joining the points of contact in K, and let two right lines be
drawn through the points A, D parallel to each other, cutting the
oppofite hyperbolas in the points G, H and P, L: by Prop. XL.
the rectangle GAH and the fquare of AO will be to each other as the
ſquares of the femidiameters parallel to the lines GH, AO, and by Cor.
preced. the rectangle PDL will be to the ſquare of DK as the fquares
of the fame femidiameters; therefore, by permutation, the rectangle
GAH will be to the rectangle PDL as the ſquare of AO to the
fquare of DK or (becauſe AO, DK are parallel) as the fquare of
AB to the ſquare of DB; but (Prop. XIX.) the rectangle GAH
is to the rectangle PDL, as AE to DE and confequently the fquare
of AB is to the fquare of DB as AE to DE; therefore the lines.
AE, BE, DE (by Lemma preced.) are proportional, and therefore
the fquare of BE is equal to the rectangle AED.
Cafe 3. But if the lines AO, DN touching the oppofite hyper-
bolas ſhould be parallel to each other, then their fquares would be
to each other as the fquares of AB, DB, becauſe the triangles ABO,
DBN are fimilar, and they would be likewiſe as AE, DE by Prop.
XIX. therefore the fquare of AB would be to the ſquare of DB
as AE to DE, and therefore the lines AE, BE, DE would be pro-
portional and the fquare of BE equal to the rectangle, AED, as
before. 2. E. D.
COR. 1. If the right line TX be a diameter of a parabola, or pa-
rallel to one of the afymptotes of an hyperbola, and meets the fec-
tion in E, and in the point A the line touching the ſection in O,
and in B a line cutting the fection or oppofite fections in the points
O, N;
*
:
Book I.
75
Conic Sections.
O, N; and if from the point E in TX without the ſection DE
be taken a third proportional to AE, BE; the line joining the points
D, N will be a tangent. For if not, a line, touching the ſection in
N would meet TX without the fection in fome point C different
from the point D, and therefore by this Prop. CE would be a third
proportional to AE, BE, contrary to the hypothefis; therefore the
line DN touches the fection in N.
COR. 2. Hence, if, from M the point of concourfe of two right
lines touching a parabola, an hyperbola, or oppofite hyperbolas in
the points N, O, a line MF be drawn to the line joining the points
of contact, which may be a diameter of the parabola, or be pa-
rallel to an afymptote of the hyperbola, it will be bifected by the
fection in S; for if MS, FS be not equal, let SR in the line MF
be taken without the fection from the point.S, a third proportional
to MS, FS and (by Cor. preced.) the line RN or RO joining the
points touches the ſection in N or O (contrary to 16. of this Book)
therefore MS, FS are equal.
Prop. XXXV. Book I. and Prop. XXX. XXXI. Book III. of
Apollonius, are contained in this Corollary.
SCHOLIU M.
48.
LET the fides of a triangle MNO be indefinitely produced both ways, FIG. 47,
and let the right line TX cutting all the fides of that triangle be
moved in fuch manner as to be always parallel to itſelf, and let a point
E be fo taken in it, that the fquare of the fegment EB between the point
E and the given fide NO of the triangle be always equal to the rectangle
AED under the fegments of the line TX between the point E and the
other fides of the triangle. If the line TX, when it bifects in its motion
the given fide NO of the triangle, paffes through the oppofite angle M;
the Locus of all the points E will be a parabola and its diameters will
be parallel to TX, and the lines MN, MO will touch this parabola in.
the points N, O. But if the line TX in its motion bifecting the fide NO,
does not at the fame time pass through the angle at M; the Locus of
all the points will be the oppofite hyperbolas and one of its afymptotes
K 2
will
ti
76
Book I.
Conic Sections.
FIG. 49)
59, 51.
will be parallel to TX; and that afymptote is eaſily found, viz. by draw-
ing parallel to TX a right line whofe fegment intercepted by the fides
MN and MO of the triangle would be bifected by the other fide NO, as
is evident (from Cor. 3. Prop. XL.) when this afymptote paſſes between
the points N, O, the lines MN, MO will touch in the points N, O op-
pofite hyperbolas generated by the motion of the point E. If it does not.
pass between theſe points they will touch in those points one of the oppo-
fite hyperbolas, generated by the motion of this point E; the right line
drawn through M bifecting NO will (by Prop. XXVI.) be a diameter,
and its interfection with the afymptote will be the center of the hyper-
bolas; and from hence the other afymptote is found. Or if the moving
line TX meets two parallel lines MO, MN and any other line ON cut-
ting them; and a point E be taken in TX, fo that the fquare of the
fegment EB of the line TX between the point E and the line ON be
equal to the rectangle AED, under the fegments between the fame point
E and the parallels; the point E in its motion will defcribe the oppofite
hyperbolas, which the parallels touch in the points O, N.
PROP. LV.
If from two given points in a parabola, an hyperbola or op-
pofite hyperbolas two right lines be inflected to any third
point, in the fame parabola, or in either of the hyperbolas,
and meet a right line given in pofition, which may be
either a diameter of the parabola, or be parallel to an
afymptote of the hyperbola; the fegments of this line
between the inflected lines and the point in which it meets
the fection, will be to each other always in the fame ratio.
wherefoever in the ſection the point be taken to which
the lines are inflected.
ET N, M be the two points given in a parabola, or in an
hyperbola or oppofite hyperbolas, and let TX be a diameter of
the parabola, or let it be parallel to one of the afymptotes of the
hyperbola meeting the fection in the given point E, and from the
points N, M to any point O in the parabola, or in either of the hy-
perbolas,
Book I.
77
Conic Sections.
perbolas, inflect two right lines NO, MO meeting TX in the points
B, C; the abfciffes EB, EC will be to each other in the fame ratio
wherefoever the point O be taken in the fection.
For draw through the three points M, N, O, three tangents which
may meet TX in three points F, D, A, and let MN be joined meet-
ing TX in G; it appears from the preced. Prop. that ED, EG, EF
are proportional.
Becauſe the line TX meets the two tangents DN, AO in the
points D, A, and the line joining the points of contact in B; the
fquare of EB will be equal to the rectangle AED (by preced. Prop.)
and in like manner becauſe the fame line meets the tangents MF,
AO in the points F, A, and the line joining the points of contact
in C; the fquare of EC will be equal to the rectangle AEF;
therefore the fquare of EB is to the fquare of EC as the rectangle
AED to the rectangle AEF, that is, as ED to EF, or as the ſquare
of ED to the fquare of EG (a mean proportional between ED,
EF) therefore EB is to EC as ED to EG; but becauſe the points
M, N are given, and the line TX given in pofition; the fegments
ED, EG remain always the fame; therefore the point O moving
through the whole parabola, hyperbola, or the oppofite hyperbola,
the fegments EB, EC of TX between the inflected lines and ſection
will be always in the fame ratio, viz. as the lines ED, EG or EG,
EF. 2. E. D.
This Propofition (So far as it relates to the parabola, is the fame
which Fermat propofed in his letter to Sir Kenelm Digby to be demon-
Strated by Dr. Wallis (fee page 858. Tom. 2. Oper. Math. Wallifi)
but by the demonftration bere given, it appears that the fame property.
extends to the hyperbolas alfo.
PROP.
78
Book I.
Conic Sections.
PROP. LVI.
If a trapezium be inſcribed in a conic fection or oppofite
fections, and all its fides be indefinitely produced, and if
from any point E in the ſection two right lines be drawn
parallel to two adjacent fides of the trapezium; the rect-
angles under the fegments of thefe lines between the
point in the ſection and the oppoſite ſides of the trape-
zium will be to each other, as the fquares of the tan-
gents or of the femidiameters to which theſe lines are
parallel.
FIG. 52, Cafe 1.
53.
ET ABdC be a trapezium infcribed in the fection or
oppofite fections, and firft let the two fides AC, Bd be
parallel, and draw from any point E in the fection two right lines
parallel to the adjacent fides AB, AC meeting the oppofite fides of
the trapezium in the points Q, a and b, R; the rectangles QEn,
bER will be to each other as the fquares of the tangents or of the
femidiameters to which En, Eb are parallel. For let bER meet the
fection again in T, and draw the right line MK bifecting the pa-
rallels AC, Bd, it will be a diameter of the fection and will bisect
in the point V the line TE terminated by the ſection (Cor. 6. and
5. Prop. XXV.) and likewife the line bR, terminated by the lines
Cd, AB, and parallel to the bifected lines AC, Bd; therefore bE,
TR are equal, and therefore the rectangles ER, TRE are equal,
and (becauſe of the parallelograms) the rectangles QEn, ARB are
equal; therefore the rectangles bER, QEn are to each other as the
rectangles TRE, ARB, that is, as the fquares of the tangents or
of the femidiameters to which the lines Eb, En are parallel, by
Prop. XVIII. or XXXI. and XL.
Caſe 2. Let ABDC be a trapezium inſcribed, none of whofe fides
are parallel to each other, and through the point E in the ſection
draw two lines parallel to AB, AC meeting the oppofite fides of
the trapezium in the points Q, N and H, R; the rectangles HER,
QEN
Book I.
79
Conic Sections.
QEN will be to each other as the fquares of the tangents or of
the femidiameters to which HE, EN are parallel.
For draw through the point B a right line parallel to the fide
AC, and meeting the fection again in d and the line EN in n, let
the right line joining the points d, C meet EH in b, and draw
through the point D a right line parallel to the fide AC, meeting
the fection again in F, and the fide AB in G, and the line joining
the points d, C, in S; draw the right line KM bifecting the pa-
rallels AC, Bd; it will be a diameter of the ſection, and will bifect
in the point O the line DF terminated by the ſection (as in the
former cafe) and likewife SG parallel to AC, Bd; therefore SD and
FG are equal: and becauſe the triangles HCh, DCS are fimilar;
Hb will be to (SD or) FG as (Cb to CS, that is as RA or) EQ
to AG; and becauſe the triangles NBn, GDB are fimilar (zB or)
ER will be to DG as Nn to BG: therefore by taking the rectangles
under the correſponding antecedents of the first and fecond order of
the proportionals, and the rectangles under the confequents of the
fame orders, the rectangle Hb, ER is to the rectangle DGF, as
the rectangle EQ, Nn to the rectangle AGB; therefore, by permu-
tation, the rectangles under Hb, ER and EQ, Nn are to each
other as DGF and AGB, that is as the fquares of the tangents or
of the femidiameters to which the lines FG and AG, or HE and
EN are parallel (by Prop. XVIII. XXXI. and XL.) and by the
preceding cafe, the rectangles bE, ER, and nE, EQ are to each
other in this fame ratio; therefore the fums or differences of theſe
rectangles, that is, the rectangles HER, QEN will be to each
other in this fame ratio. 2. E. D.
This Propofition holds good, although in the firft cafe the line
¿ER drawn parallel to AC, Bd, touches the fection in E; for then
the points E, T coinciding, the point E will be at the vertex of
the diameter MK of the fection, and the rectangle ER or TRE
becomes the ſquare of hE or ER; and in like manner in the fe-
cond cafe, the points D, F coinciding, the rectangle DGF becomes
the fquare of the tangent DG or DS.
COR.
80
Book I.
Conic Sections.
FIG. 52.
FIG. 54.
COR. 1. Hence, the points A, B, D, C, remaining fixt; and the
point E moving through the fection or oppofite fection, and the
lines drawn through it being always parallel to AB, AC, the ratio
of the rectangles HER, QEN will be conftantly the fame.
COR. 2. Or the points A, B, C, E, and the lines drawn through
E remaining fixt as before; the point D (that is the interfection of
the lines CD, BD) moving through the fection or oppofite fection,
the ratio of the rectangles HER, QEN will continue the fame;
and therefore (becauſe their fides ER, QE are given) the ratio of
the lines HE, NE will always remain the fame.
COR. 3. A conic fection does not meet another conic fection or
oppofite fections in more than four points. For if it be poffible, let
two fections pafs through five points as A, B, C, D, E, and (every
thing remaining as affumed above) draw through the point B a right
line cutting both fections in the points d, P, join Cd meeting HE
in b, and join CP meeting the fame line HE in L; then becauſe
the points d, D are in the fame fection paffing, through the points.
A, B, C, E, the lines E, nE are to each other as HE, NË (by
Cor. preced.) and in like manner when the points P, D are in the
fame fection paffing through the points A, B, C, E; the lines LE,
nE will be to each other as HE, NE; therefore bE, LE are equal,
which is abfurd; for the points h and L are on the fame fide of the
point E; from hence the Corollary is manifeft.
T-
COR. 4. If a trapezium ABCD, whoſe fides are given in pofition,
be infcribed in a conic fection given in pofition, and if from any
point E in the ſection the right lines EF, EG, EH, EK be drawn
in given angles to the four fides of the trapezium, the rectangles
FEH, GEK under the lines drawn to the oppofite fides, will be in
a given ratio to each other.
For draw from the point E the right lines EL, EO parallel to
two adjacent fides of the trapezium, viz. to BC, BA; and let them
meet the oppofite fides in the points L, M, and N,.O: then be-
cauſe in the triangle ELF, the angle LFE is given by hypotheſis,
and
Book I.
81
Conic Sections.
and the angle ELF is equal to the given angle CBA, the triangle
ELF (by 40. of Euclid's Data) is given in fpecies; and therefore
the ratio of FE to EL is given. In like manner in the triangle
EMH the ratio of EH to EM will be given, and therefore the
ratio compounded of theſe ratios is given; that is (by 23. 6.) the
ratio of the rectangle FEH to the rectangle LEM.
It may be demonftrated in the fame manner that the ratio of the
rectangle OEN to the rectangle GEK is given. But the ratio of
the rectangle LEM to the rectangle OEN is given (by Cor. 1.)
therefore becauſe it has been fhewn that the ratio of the rectangle
FEH to LEM is given, and alſo the ratio of LEM to OEN, and
of OEN to the rectangle GEK; the ratio of the rectangle FEH
to the rectangle GEK will be given (by 8. Euclid. Dat.)
All the cafes of the 17. Lemma, Book I. of Newton's Math.
Principles of Nat. Philofophy, are fully demonftrated in the Pro-
pofition and this laft Corollary.
END OF THE FIRST BOOK.
L
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FIG.44.
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N
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FIG. 51 D
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FIG. 46.
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LINTY
OF
H
CONIC SECTIONS.
BOOK THE SECOND.
Concerning the Properties from which the Sections
derive their names; their parameters, axes, foci,
and defcriptions in plano.
İ.
I
DEFINITIONS I. and II.
F from a point in a parabola a right line be drawn ordinately
applied to a diameter; a right line which is a third propor-
tional to the abfcifs and this ordinate, is called the Latus
Rectum, or Parameter of that diameter.
II. A right line, which is a third proportional to two conjugate
diameters of an ellipfe or hyperbola, is called the Latus Rectum or
Parameter of that diameter, which is the firſt of the three pro-
portionals.
L 2
COR
84
Book II.
Conic Sections.
FIG. 1.
COR. 1. If an ordinate be drawn from a point in an ellipfe or
hyperbola to a diameter (which in the hyperbola is a tranfverfe dia-
meter) the rectangle under the abfciffes of the diameter between its
two vertices and the ordinate will be to the fquare of the ordinate
as the diameter to its parameter. For, by Cor. 1. 31. and Def.
XXIV. Book I. this rectangle is to the fquare of the ordinate as
the ſquare of the diameter to the fquare of its conjugate, that is as
the diameter to its parameter (by Def. preced. and Cor. 20. 6.)
COR. 2. If an ordinate be drawn from a point in an hyperbola to
a fecond diameter; the fum of the fquares of the fecond femidia-
meter and of its ſegment between the center and ordinate will be to
the fquare of the ordinate as (the fquare of that fecond diameter
to the ſquare of its conjugate, by Prop. XXXII. Book I. that is
as) the fecond diameter to its parameter, by Def. preced. and
Cor. 20. 6.
PROP. I.
The ſquares of right lines ordinately applied to a diameter
of a parabola are equal to the rectangles under the ab-
fciffes and its parameter.
FR
ROM a point B in a parabola let the right line BG be ordi-
nately applied to the diameter FL whofe vertex is F, and let
the right line P be a third proportional to the abfcifs GF and ordi-
nate BG; it will be the parameter of the diameter FL, by Def. I.
of this Book, and the fquare of BG will be equal to the rectangle
under GF and the parameter P: then if any other right line as DL
be ordinately applied to the diameter FL; by Prop. XXIX. Book I.
the fquare of DL will be to the fquare of BG as LF to GF, that
is as the rectangle under LF and P, to the rectangle under GF and
P; therefore the fquare of DL is to the rectangle under LF and P
as the fquare of BG to the rectangle under GF and P, that is, in a
ratio of equality, and from hence Apollonius called this fection a
Parabola. 2. E. D.
COR.
Book II.
85
Conic Sections.
COR. 1. If from a point G in the diameter FL of a parabola, a
right line GB be drawn parallel to the ordinates applied to that dia-
meter, and the fquare of BG be equal to the rectangle under the
abfcifs GF (between GB and the vertex F of the diameter) and the
parameter of the fame diameter; the point B will be in the para-
bola, as is evident.
COR. 2. If the right line AB terminated by the parabola be ordi-
nately applied to the diameter FL, and if the abfcifs FG, between
the ordinate and the vertex of the diameter, be equal to a fourth
part of the parameter of the fame diameter FL; the line AB will
be equal to the parameter. For by this Prop. AG, viz. half of
AB is a mean proportional between the parameter and abfcifs GF,
viz. the fourth part of the parameter; therefore AG is the half of
the parameter, and therefore AB is equal to the parameter.
PROP. II.
If a right line touching a parabola meets a diameter; the
fquare of the fegment between the point of contact and
this diameter will be equal to the rectangle under the feg-
ment of this diameter, between its vertex and the tan-
gent, and the parameter of the diameter which paſſes
through the point of contact.
L'
ET the right line FR touch the parabola in F, and meet the FIG. 1.
diameter ED in the point R, and draw the diameter FG
through the point of contact, and let the right line P be its para-
meter; the fquare of FR will be equal to the rectangle under DR
and the line P.
For draw from the vertex of the diameter ED the line DL ordi-
nately applied to the diameter FG; it will be parallel and equal to
the tangent FR, and its abfcifs LF will be equal to DR : then be-
cauſe (by the preced. Prop.) the ſquare of DL is equal to the rect-
angle under the line LF and the line P; the fquare of FR will be
equal to the rectangle under DR and the line P. 2. E. D.
PROP.
86
Book II.
Conic Sections.
FIG. 1.
PRO P. III.
If a right line cutting a parabola meets a diameter; the
rectangle under its fegments between the diameter and
parabola will be equal to the rectangle under the ſegment
of the diameter, between its vertex and the fecant, and
the parameter of the diameter to which the fecant is or-
dinately applied.
L
ET a right line cut a parabola in the points A, B, and meet
the diameter ED in E, and let FG be the diameter to which
AB is ordinately applied, and the right line P its parameter; the
rectangle AEB will be equal to the rectangle under ED and the
line P.
For through the vertex of the diameter FG draw a right line
touching the parabola, and meeting the diameter in R, it will be
parallel to the fecant AB (by Cor. 1. 25. Book I.) and therefore
(by Cor. 19. Book I.) the rectangle AEB is to the fquare of FR as
ED to DR, or as the rectangle under ED and the line P to that
under DR and the line P; therefore, by permutation, the rectangle
AEB is to the rectangle under ED and the line P as the fquare of
FR to the rectangle under DR and the line P; that is in a ratio
of equality by the preced. Prop. Q. E. D.
Corollaries to the two preceding Propofitions.
COR. 1. If two right lines touching a parabola meet each other;
the fquares of their fegments, between the points of contact and con-
courſe, will be to each other as the parameters of the diameters which
pafs through their points of contact. For (a diameter being drawn
through the point of concourfe of the tangents) the fquare of either
of the tangents will be equal to the rectangle under the parameter
of the diameter which paffes through its point of contact, and the
fame right line, viz. the fegment of the diameter between its vertex
and the point of concourfe of the tangents.
CORS
Book II.
87
Conic Sections.
COR. 2. And in like manner, if two right lines cutting a parabola
meet each other; the rectangles under the fegments of the fecants
between their point of interſection and the parabola, will be to each
other as the parameters of the diameters to which the fecants are
ordinately applied, this appears from the preced. Prop.
COR. 3. Or if a right line touching a parabola meets a fecant, it
is manifeft from the two preceding Propofitions, that the fquare of
the ſegment of the tangent between the points of concourfe and
contact, will be to the rectangle under the fegments of the fecant
between the point of concourfe and the parabola as the parameter
of the diameter which paffes through the point of contact, to the
parameter of the diameter to which the ſecant is ordinately applied.
COR. 4. Becauſe the rectangle AEB under the fum and difference
of the ordinates BG, DL is equal to the rectangle under the para-
meter P and ED, the difference of the abfciffes; the parameter of
any diameter will be to the fum of the two ordinates, as their dif-
ference to the difference of the abfciffes.
PROP. IV.
If from a point in an ellipfe, an ordinate FG be applied to FIG. 2.
the diameter AB; the fquare of the ordinate will be equal
to the rectangle under one abfcifs GB and a part of the
parameter of the diameter AB, which is to the whole as
the other abfcifs AG to the diameter AB.
FR
ROM the vertex B draw BH perpendicular to the diameter and
equal to its parameter; let AH be joined, and draw from the
point G a line parallel to BH, meeting AH in K, and let the rect-
angle BGKL be completed; the fquare of FG will be equal to this
rectangle.
For by Cor. 1. to Def. II. the rectangle AGB is to the fquare of
FG as the diameter AB to its parameter BH, or (becauſe GK is
parallel to BH) as AG to GK, that is as the fame rectangle AGB
to the rectangle KGB; therefore the fquare of FG is equal to the
rectangle
88
Book II.
Conic Sections.
FIG. 3.
KGB under the abfcifs GB and the line KG, which is to the para-
meter BH as the other abfcifs AG to the diameter AB.
COR. Let the rectangles KLHM, ABHN be completed; it is
manifeft that the fquare of the ordinate FG is deficient from the
rectangle GH under the abfcifs GB and the parameter of the dia-
meter AB, by a rectangle KH fimilar and fimilarly placed to the
rectangle AH under the diameter AB and its parameter BH. From
this defect Apollonius called this fection an Ellipse.
PRO P. V.
If an ordinate FG be drawn from a point in an hyperbola to
the tranfverfe diameter AB; the fquare of the ordinate
will be equal to the rectangle under the leffer abfcifs GB
and a right line which is to the parameter of the diameter
AB, as the other abfcifs AG to the diameter AB.
FR
ROM the vertex B draw BH perpendicular to the diameter
and equal to its parameter; join AH, and draw through the
point G a line parallel to BH, meeting AH in K, and complete
the rectangle KGBL; the fquare of FG will be equal to this rect-
angle.
For by Cor. 1. to Def. II. the rectangle AGB is to the ſquare of
FG as the diameter AB to its parameter BH, or (becauſe GK is
parallel to BH) as AG to KG, that is as the fame rectangle AGB
to the rectangle KGB; therefore the fquare of FG is equal to the
rectangle KGB under the leffer abfcifs GB and the line KG, which
is to the parameter BH as the greater abfcifs AB to the diameter
AB. 2. E. D.
COR. Complete the rectangles KLHM, ABHN, and it is mani-
feſt that the ſquare of the ordinate FG exceeds the rectangle GH
under the abfcifs GB and the parameter of the diameter AB, by a
rectangle
Book II.
89
Conic Sections.
rectangle fimilar and fimilarly placed to the rectangle AH, which
is contained by the diameter AB and its parameter BH. From
this excefs, Apollonius called this fection an Hyperbola.
PROP. VI. PROBL. I.
A parabola being given in pofition; to find its axis and to
demonſtrate there can be one axis only.
L
ET FBL be the parabola, aad find any diameter VO, and FIG. 4
draw through a point X below the vertex a perpendicular to
VO meeting the parabola in F, L; if the line FL be bifected in
X, the diameter VO will be the axis, by Def. XXII. Book I. but if
it be not bisected in X, draw another diameter AB bifecting FL in
E, this diameter will be the axis; becauſe it cuts the ordinate FL
at right angles, for AB is parallel to VO.
But if two diameters could be axes, the line terminated by the
parabola and perpendicular to both of them, would be an ordinate
to each of them, which is abfurd: therefore there is one axis only
to the fame parabola. 2, E. D.
PROP. VII. PROBL. II.
Two conjugate diameters of an ellipfe or hyperbola being
given in poſition and magnitude; to find their axes, and
to demonſtrate that there can be two axes only in each
fection.
L
ET the right lines FL, GH, not at right angles to each other, FIG.
be conjugate diameters of an ellipfe or hyperbola, and C the
center through F the vertex of one of the diameters, which in an
hyperbola is a tranfverfe diameter, draw FV parallel to GH, and
take from the ſame point F in the diameter FL the fegment FK (to-
wards the center in the hyperbola, and the contrary way in the el-
lipfe) fo that the rectangle CFK be equal to the fquare of CG, and
KC being bifected in P, draw PQ perpendicular to KC, and meet-
M
ing
6.
90
Book II.
Conic Sections.
ing the line FV in Q; then deſcribe a circle from the center Q
paffing through the points C, K, and meeting the line FV in the
points D, O; the right lines CD, CO will be the two axes.
For by Cor. 27. Book I. the line FV touches the fection, of
which FL, GH are conjugate diameters, and by a property of the
circle, the rectangle DFO is equal to the rectangle CFK, that is to
the ſquare of the femidiameter CG (by conftruction) therefore
CD, CO are conjugate diameters (Cor. 51. Book I.) and cut each
other at right angles, becauſe the angle DCO is in a femicircle, and
therefore they are axes (Dcf. XXII. Book I.) for they cut their or-
dinates at right angles: the two vertices are thus found; draw
from the point. F the right line FE to the axis CD, and parallel to
the other axis, and let CD, CB, CE be proportional, and take CA
equal to CB; the points B, A will be the two vertices of the axis
CD, (by 48. 49. Book I.) in like manner the two vertices M, N of
the other axis are found. 2. E. I.
The above conftruction remaining, if two other conjugate dia-
meters as CX, CV could be axes, let them meet the tangent FD in
the points X, V; the rectangle XFV will be equal to the fquare of
CG (51. Book I.) that is to the rectangle CFK; therefore the
circle deſcribed through the points C, K, X will alfo paſs through
V, and becauſe, by hypothefis, XCV is a right angle, the center
of this circle will be in the line FXV, and becauſe the line PQ
bifects CK perpendicularly, the center of the circle will be likewiſe
in the line PQ, and therefore in the point Q, that is in the center
of the circle paffing through the points C, K, D; therefore two
circles, cutting each other in the points C, K, have the ſame center,
which is abfurd: therefore there can be but two axes in each fec-
tion. 2. E. D.
If in the hyperbola the diameters FL, GH were equal, the point
K would coincide with the point C, and the circle defcribed from
the center Q would touch FL in C, and the demonftration of this
cafe is the fame as the preceding, by only fubftituting the fquare of
FC inftead of the rectangle CFK.
If the given conjugate diameters are perpendicular to each other,
they are axes, as is evident.
COR
Book II.
91
Conic Sections.
COR. 1. Hence, an ellipfe or hyperbola being given in pofition
their axes may be found, for the center is found by Cor. 7. 25.
Book I. and two conjugate diameters by the 27. Book I. and the
magnitude of a ſecond diameter of the hyperbola is determined by
Def. XXIV. Book I. Or the fection being given in pofition and
its center being found, the axes may be more readily found: for if
a circle whofe center is the fame as the center of the fection, be de-
fcribed meeting the fection in two points, and thefe points be joined
by a right line, the diameter of the fection bifecting this line will
be an axis, as is evident.
COR. 2. It is manifeft from Cor. 8. 25. Book I. that a right line
drawn through the vertex of an axis of a conic fection and perpen-
dicular to it, touches the fection in that point; and on the contrary,
if a right line touches the fection at the vertex of the axis, it will
be perpendicular to it.
PROP.
VIII.
The axes of an ellipſe are unequal; and the greater axis is
the greateſt, and the leffer the leaſt of all the diameters
of the ellipſe; and the axes of an hyperbola are the leaft
of all its diameters.
Part 1.
ET AB, MN be the axes of an ellipfe, they will be FIG. 7.
unequal by Cor. 2. 31. Book I. Let AB be the
greater, which is called the tranfverfe axis, and MN the leffer,
which is called the fecond or conjugate axis: any femidiameter as
CF is less than CB, but greater than CM; for let an ordinate FE
be applied to AB; and becauſe CEF is a right angle, the fquare of
CF is equal to the fquares of CE, FE together; but the fquare of
FE is less than the rectangle AEB; for it is to that rectangle as the
fquare of CM to the fquare of CB (Cor. 1. 31. Book I.) therefore
the fquare of CF is lefs than the fquare of CE and the rectangle
AEB together, that is, than the fquare of CB (5. 2.) therefore CF
is less than CB; and in the fame manner, an ordinate FL being
drawn
MN be axes of an will
M 2
92
Book II.
Conic Sections.
FIG. 6.
FIG. 7+
FIG. 8.
drawn to the axis MN, it may be demonſtrated that the femidia-
meter CF is greater than CM.
Part 2. Let an ordinate FE be drawn from the vertex of the
diameter CF of the hyperbola to the tranfverfe axis AB; and be-
cauſe FEC is a right angle, the femiaxis CB is lefs than CF, and
în like manner it may be ſhewn that the fecond femiaxis CM is lefs
than any other fecond femidiameter; for CM is the tranfverfe axis.
of the hyperbola, conjugate to FB.
COR. 1. It is evident that the more remote any diameter is from
the axis of the hyperbola, the greater it is.
COR. 2. The diameter CF nearer to the greater axis of the ellipfe
is greater than the diameter CP more remote. For if CF, CP were
equal, the diameter bifecting the right line FP would cut it at right
angles and would be an axis, contrary to what has been proved by
the preceding; and if CF be lefs than CP, the circle defcribed
from the center C with the diftance CF would cut CP within the
ellipſe, but it meets the leffer axis without the ellipfe, and therefore
meets the ellipfe fomewhere between the points F, M, fuppofe in
Q; then CF, CQ would be equal, and the diameter bifecting FQ
would be an axis, as before (contrary to the preced.) Hence the
Corollary is evident.
COR. 3. Hence, it is manifeft that two diameters of an ellipfe
or hyperbola, equally diftant from the fame axis are equal; and
converfely.
COR. 4. The axes AB, MN of an hyperbola bifect the angles con--
tained by the afymptotes CY, CZ. For draw through the vertex
of the axis AB a line parallel to the other axis, it will touch the
hyperbola; let it meet the afymptotes in the points R, S; RB, BS
will be equal (36. Book I.) and ſince the angles at B are right an-
gles, the triangles RBC, SBC will be equiangular, and therefore:
the angles at C are equal.
DEFI
Book II.
93
Conic Sections.
DEFINITIONS III, IV, V.
vertex
III. F a point in the axis of a parabola be taken within the
fection, and its diftance from the vertex of the axis be equal
to a fourth part of its parameter; that point is called the Focus of
the parabola.
10.
IV. If two points F, O, in the tranfverfe axis of an ellipfe or hy- FIG. 9,
perbola, be taken within the ellipfe or oppofite hyperbolas, fo that
the rectangle AFB or AOB be equal to the fquare of the conjugate
femiaxis CM; each of theſe points F and O is called the Focus of
the ellipfe or oppofite hyperbolas, and when the two points are
mentioned together they are called the Foci.
V. Through the focus of a conic fection let an ordinate be drawn
to the axis, and through the point where it meets the fection, let a
tangent be drawn; this tangent is called the Focal tangent.
COR. 1. If from the vertex M of the conjugate axis of an el- FIG. 9.
lipfe a circle be defcribed with an interval equal to the tranfverfe fe-
miaxes CA; it will cut the tranfverfe axis in the foci: for let it
meet it in the points F, O; the fquare of MF or CA is equal to the
fquares of CF and CM together: but the fquare of CA is equal to
the ſquare of CF and the rectangle AFB together (5. 2.) therefore
the rectangle AFB is equal to the fquare of CM; and confequently
F is a focus; and in like manner O will be the other focus.
COR. 2. If from the center C, and in the tranfverfe axis of an FIG. 10,
hyperbola, a ſegment CF or CO be taken equal to the right line
AM joining two vertices of the axes; the extremity of this fegment
will be a focus.
For the fquare of AM or CF is equal to the fquares of CA, CM
together, and (by 6. 2.) the fame fquare of CF is equal to the
fquare of CA and rectangle AFB; therefore the rectangle AFB is
equal to the fquare of CM, and confequently F is a focus, and in
like manner O will be the other focus.
COR. 3. It is evident that the foci of an ellipfe, or of oppofite or
conjugate hyperbolas, are equally diftant from the center.
PROP.
92
Book II.
Conic Sections.
A
FIG. 9,
10,
11.
FIG. 11.
FIG. 9,
10.
PROP. IX.
Draw through the focus F of the conic fection RAP, FT an
ordinate to the axis AB, and draw the focal tangent TD;
if a tangent be drawn through the vertex of the axis AB,
the ſegment intercepted between the point of contact and
the focal tangent, will be equal to the fegment AF of the
axis between the fame point of contact and the focus F.
FIR
IRST, let the ſection be a parabola, and let the focal tangent
cut the axis in D, and meet the tangent AH, drawn through
the vertex of the axis, in the point H; AH, AF will be equal.
For (by Prop. XLVII. Book I.) AD is half of FD; therefore
AH will be the half of FT; but FT is half of the parameter of
the axis (Def. III. and Cor. 2. 1. of this Book) therefore AH is a
fourth part of the fame parameter, and confequently is equal to AF.
Secondly, let the fection be an ellipfe or hyperbola, and let the
focal tangent meet the axis AB in D; and the tangents drawn
through the vertices of the axis AB in H and G; AH, BG will be
equal to AF, BF; for the rectangle under AH, BG will be equal
to the ſquare of the conjugate femiaxis CM (by 50. Book I.) that
is to the rectangle AFB (by Def. IV.) and by (Cor. 5. 18. Book I.)
AH is to BG as (HT to TG, or on account of parallel lines as)
AF to FB; therefore theſe equal rectangles under AH, BG and AF,
FB are alfo fimilar, and confequently their homologous fides AH,
AF and BG, BF are equal. 2. E. D.
COR. The above conftruction remaining, if a right line be drawn
from any point P of a conic fection or oppofite fection to the focus
F, and the perpendicular PQ to the axis AB, meeting the focal
tangent in L, and the ſection again in R; the fquare of FQ will
be equal to the rectangle PLR; for becauſe the tangent AH is pa-
rallel to the fecant LPR, the fquare of AH will be to the rectangle
PLR as (the fquare of HT to the fquare of TL (Cor. 6. 18.
Book
+
Book II.
95
Conic Sections.
Book I.) or on account of parallel lines) as the fquare of AF to
the ſquare of FQ: but the fquare of AH is equal to the ſquare of
AF by this Prop. therefore the rectangle PLR is equal to the
fquare of FQ.
PROP. X.
10,
II.
Through the focus F of the conic fection RAP, draw FT FIC. 9,
an ordinate to the axis AB, and draw the focal tangent
TD; and a right line from any point P in this ſection or
in the oppoſite ſection, to the focus F, and the perpen-
dicular PQ to the axis AB, meeting the focal tangent TD
in L; PF, LQ will be equal.
L
ET the line PQ meet the fection again in R; then becauſe
FQP is a right angle, the fquare of FP is equal to the fquares
of PQ, FQ, that is (by Cor. preced.) to the fquare of PQ_ toge-
ther with the rectangle PLR, that is to the ſquare of LQ (6. 2.)
therefore the line FP is equal to LQ. 2. E. D.
L
DEFINITION VI.
10,
11.
ET the focal tangent TH meet AB the axis of a parabola, or FIG. 9,
the tranfverfe axis of an ellipfe or hyperbola in the point D,
and draw through the point D a perpendicular to AB: this right
line is called the Directrix of the conic fection.
COR. 1. Every right line perpendicular to the directrix of the
pa-
rabola is a diameter, and converfely, as is evident from Cor. 5. to
the Definitions of the fections, Book I.
COR. 2. The diſtance of the directrix from the vertex of the
parabola is equal to the diftance of the focus from the fame ver-
tex, by Prop. XLVII. Book I.
COR. 3. The femiaxis CA in the ellipfe or hyperbola is a mean
proportional between the diſtance CD of the directrix from the
center,
1
96
Book II.
Conic Sections.
}
FIG. 9,
10,
II.
center, and the diftance CF of the focus from the center, as is evi-
dent from Prop. XLVIII. and XLIX. Book I. It is likewife evi-
dent that there are two directrices in the ellipfe and hyperbola equally
diftant from the center.
1
PROP. XI.
If from any point P in a conic ſection RAP, a right line PE
be drawn perpendicular to the directrix DE, and the
right line PF to the focus nearer to the directrix; PF
drawn to the focus will be to the perpendicular PE as FA,
the diſtance of the focus F from the nearer vertex of the
axis, to DA, the diſtance of the directrix DE from the
fame vertex.
FOR
OR draw from the point P a right line PQ perpendicular to
the axis AB, and meeting the focal tangent DT in L; the
fegment QD of the axis between PQ and the directrix will be
equal to PE, and LQ is equal to PF (by Prop. preced.) draw
through the vertex A of the axis a tangent meeting DT in H;
then becauſe AH, LQ are parallel, LQ will be to QD as HA to
AD, that is as FA to AD (9. of this Book) therefore PF is to PE
as FA to AD. 2. E. D.
COR. 1. Hence, in the ellipfe and hyperbola, the right line PF
drawn to the focus is to PE perpendicular to the directrix as CF,
the diſtance of the focus from the center, to CA the tranfverfe femi-
axis. For by Cor. 3. preced. CF is to CA as CA to CD; there-
fore by converfion CF is to FA as CA to AD, and, by permutation,
CF is to CA as FA to AD, that is as PF to PE by this Prop.
therefore PF is to PE as CF to CA.
COR. 2. From the preceding demonftration, the diſtance CF of
the focus from the center is to the femiaxis CA as the diſtance FA
of the focus from the nearer vertex to the diftance DA of the di-
rectrix from the fame vertex.
COR.
1
Book II.
97
Conic Sections.
COR. 3. Right lines drawn from points in the fame fection or op-
pofite fections to a focus, are to each other as the perpendiculars
drawn from the fame points to the directrix nearer to that focus.
SCHOLI U U M.
LET DAH be a right angled triangle, and take in the bafe DA pro-
duced beyond the right angle A, a line FA equal to the perpendicular
HA, and let the line FA revolve about the point F, and increaſe in fuch
a manner, that it be always equal to the line drawn through its extremity
parallel to HA, and terminated by the lines DA, DH produced: the line
defcribed by the motion of the extremity of this revolving line will be a
conic fection, viz. an hyperbola, a parabola or an ellipfe, according as
the baſe LA of this triangle may be leſs, equal, or greater than the per-
pendicular HA; hence the point D, the interfection of the base with
the hypothenuje, receding ad infinitum, the hypothenufe will at last be
parallel to the bafe, and the ellipfe will be changed into a circle whofe
center is F; in the other cafes, the point F will be a focus of the ſection
defcribed, and the point A the vertex of its axis; and when the revolv-
ing line ſhall be parallel to HA, its extremity will be in the line DH,
which will therefore touch the fection. All which is manifeft from what
has been demonftrated above: from hence likewife the point B, the other
extremity of the tranſverſe axis, and center C may be found, when the
fection is an ellipfe or hyperbola.
PROP. XII.
FIG. 9:
M
10,
11.
A right line drawn from any point P in a parabola to the FIG. 11,
focus F, is equal to the perpendicular PE drawn from the
fame point to the directrix.
Fo
OR, by the preceding, PF is to PE as FA to AD, that is, in a
ratio of equality, by Cor. 2. Def. VI. of this Book. Q, E. D.
COR. 1. The diſtance of a point within a parabola from the focus
is lefs, and of a point without a parabola is greater, than the perpen-
dicular drawn from the fame point to the directrix. Let the point
M be within the parabola, and draw MF to the focus, and from
the
N
98
Conic Sections.
FIG. 10.
Book II.
the fame point a perpendicular to the directrix meeting it in E, and
the parabola in P, and join PF; becauſe MF is less than MP, PF
together, it will be leſs than MP, PE, that is, than ME. Now let
the point N be (without the parabola) in the line PE perpendicular
to the directrix, and join NF; NF and NP together are greater than
PF or PE; therefore NF is greater than NE.
COR. 2. Hence, if the diſtance of any point from the focus of a
parabola be equal, lefs, or greater than the perpendicular drawn
from the fame point to the directrix, that point, in the firſt caſe,
will be in the parabola, in the fecond cafe, within the parabola, and
in the third cafe, without it.
PROP. XIII.
If from a point P in the hyperbola RAP a right line PF be
drawn to the focus F, and the right line PN, parallel to
the afymptote CY, be drawn to the directrix DE, nearer
to the focus F; thefe lines PF, PN will be equal.
FOR
OR let the perpendicular PE be drawn from the point P to the
directrix DE, and let O be the other focus, and draw through
the vertex B of the tranfverfe axis a tangent meeting the afymptote
CY in the point Y, BY will be equal to the conjugate femiaxis CM
(XXXVIII. Book I.) and CY will be equal to CO the diſtance
of the focus from the center (by Cor. 2. Def. IV.) and (by Cor. 1.
XI. of this Book) PF is to PE as (CO, or) CY to CB: but becauſe
the triangles PEN, CBY are fimilar, PN is to PE as CY to CB;
therefore PF, PN are equal. Q. E. D.
COR. It may be fhewn in the fame manner as in the preceding
Cor. that the diftance of a point, within an hyperbola, from the fo-
cus, is lefs, and from a point taken without the hyperbola, greater
than the line drawn parallel to the afymptote from the fame point to
the directrix nearer to this focus; and from hence, if the diſtance of
any point from the focus of an hyperbola be equal to the line drawn
from the fame point to the directrix nearer to this focus and parallel
to the afymptote; that point will be in the hyperbola, to the focus
PROP.
of which the right line was drawn...
Book II.
99
Conic Sections.
PROP.
XIV.
If two right lines be drawn from a point in an ellipfe or hy-
perbola to the foci; the fum of theſe lines in the ellipſe,
and their difference in the hyperbola, is equal to the tranf-
verſe axis.
L
IQ.
ET AB be the tranfverfe axis of an ellipfe or hyperbola, and Fia. 9,
the points F, O the foci, and DE and ZX the directrices;
draw from a point P in the ſection PF and PO to the foci, and a
line parallel to AB, meeting the directrices DE, ZX in the points
E and X.
(By Cor. 1. Prop. XI. of this Book) PF is to PE as CF to CA,
that is, as CA to CD (Cor. 3. Def. VI.) or as AB to DZ; and in
like manner, PO is to PX, as AB to DZ; therefore the fum or
difference of the antecedents PF, PO is to the fum or difference of
the confequents PE, PX as AB to DZ; but in the ellipfe the fum.
and in the hyperbola the difference of the confequents PE, PX is
equal to DZ; therefore in the ellipfe the fum and in the hyperbola
the difference of the antecedents PF, PO is equal to the tranſverſe
axis AB. 2. E. D.
COR. 1. If two right lines be drawn from a point without the
ellipfe to the foci; thefe two lines together will be greater than the
tranſverſe axis: but if they be drawn from a point within the ellipfe
they will together be less than the fame axis, as is evident from the
20, I.
COR. 2. And on the contrary, if the lines drawn from any point
to the foci, are together equal, greater, or lefs than the tranfverfe
axis; that point in the firſt caſe will be in the ellipfe, in the ſecond
cafe without it, and in the third caſe within the ellipfe.
COR. 3. If right lines MF, MO be drawn from a point M within FIG. 1.
the hyperbola LAP to the foci; their excefs will be greater than the
tranſverſe axis AB: but if they be drawn from a point N without
the hyperbola, their excefs will be lefs than the fame axis: for let
N 2
the
100
Book II.
Conic Sections.
FIG. 13.
the point M be within the hyperbola, whofe focus is F, and let the
line MO drawn to the other focus O, meet this hyperbola in L, and
join LF; the excefs of the line MO above MF is greater than its
excefs above LM, LF together, that is, than the excefs of the line
LO above LF, that is, than the tranfverfe axis by this Prop.
Now let the point N be without the hyperbola, and draw from
the point L, where the line NF meets the hyperbola, a right line to
the focus O; NO will be leſs than NL and LO together; there-
fore the excess of NO above NF is leſs than the exceſs of NL, LO
above NF, that is, than the exceſs of the line LO above LF, that
is, than the tranſverſe axis AB.
COR. 4. And on the contrary, if the excefs of the lines, which
are drawn from any point to the foci of the hyperbolas, be equal,
greater, or leſs than the tranfverfe axis; this point will be in one of
the hyperbolas, or within, or without the hyperbola.
SCHOLIU
M.
W
HILE I endeavoured to deduce from the nature of the cone,
the origin of those lines that are called the directrices of the ſec-
tions; a theorem occurred to me, which feemed not an inelegant one,
and in ſome meaſure answered my defign. But as it would here interrupt
the feries of the Propofitions, it is inferted at the end of this Book.
PROP. XV.
If a right line PF be drawn from any point P in a parabola
to the focus, and a perpendicular PE to the directrix; the
right line PT bifecting the angle FPE contained by theſe
lines will touch the parabola: and on the contrary, the
line PT touching the parabola in P will bifect the angle
contained by the lines PF, PE.
Part 1.
L
ET any point X be taken in the line. PT, and draw
the right lines XF, XE and EF, and draw the per-
pendicular XD to the directrix; then becauſe in the triangles PTF,
PTE,
C
RN
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TAB · 7.
B
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HI Page 100
FIG. 1.
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FIG.7.
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FIG. 4
FIG. 2:
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FIG.3.
E
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FIG.10
FIG. 5
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FIG.II.
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FIG.8.
FIG.6.
H
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I
FIG. 9°
G
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A F
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Y
Z
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RM
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NIV
CH
Q.
Book II.
IOI
Conic Sections.
PTE, PF, PE are equal (XII. of this Book) and PT common to
both, and the angles TPF, TPE equal, TF, TE will be equal, and
the angles at T right angles; therefore XF is equal to XE, and
confequently greater than XD (19. 1.) therefore the point X is with-
out the parabola (by Cor. 2. XII. of this Book) and therefore the
line PT touches the parabola in P.
Part 2. If now the line PT touches the parabola in P, it will
bifect the angle FPE. For if not, draw another line bifecting this
angle, this line likewiſe will touch the parabola in P, by the firſt
cafe, which is impoffible (XVI. Book I.) therefore the tangent PT
bifects the angle FPE. 2. E. D.
PROP.
XVI.
If two right lines PF, PO be drawn from any point P in an FIG. 14.
ellipſe to the foci; the right line PT bifecting the angle
FPE, which is adjacent to the angle FPO contained by
the lines drawn to the foci, will touch the ellipfe in the
point P: and on the contrary, if the line PT touches the
ellipfe in P, it will bifect the angle FPE adjacent to the
angle FPO contained by the lines drawn from the point P
to the two foci.
Part 1.
L'
ET OP be produced to E, fo that PE be equal to PF,
and let a point X be taken in the line PT, and draw
XO, XF, XE, and alfo FE meeting PT in T; then becauſe in the
triangles PTF, PTE, PF, PE are equal, and PT common to both,
and the angles TPF, TPE equal, TF, TE will be equal, and the
angles at T right angles; therefore XF, XE will be equal; but
XO, XE together are greater than OE; therefore XO, XF
toge-
ther will be greater than OE, that is, than OP, PF, that is, than
the tranfverfe axis of the ellipfe (XIV. of this Book) therefore the
point X is without the ellipfe (by Cor. 2. XIV. of this Book) and
conſequently the line PF touches the ellipſe in the point P. Q. E. D.
Part 2. May be demonſtrated in the fame words as the fecond
part of the preced. Prop.
PROP.
102
Book II.
Conic Sections.
FIG. 12.
PROP. XVII.
If two right lines PF, PO be drawn from a point P in an
hyperbola to the foci, the right line PT bifecting the
angle FPO contained by theſe lines, touches the hyperbola
in P and on the contrary, if the line PT touches the
hyperbola in P, it will bifect the angle FPO contained by
the fame lines drawn from the point P to the foci.
Part 1.
IN
[
N the greater right line PO take PE equal to PF; EO
will be equal to the tranſverſe axis (XIV. of this 'Book)
join FE meeting PT in T, and take any point X in the line PT,
and draw XO, XF, XE, and as in the preceding, XE, XF will be
equal and becauſe XO is leſs than XE, EO together, its exceſs
above XF will be leſs than the exceſs of XE, EO above XF, that
is than EO, which is equal to the tranſverſe axis; and confequently
the point X is without the hyperbola (Cor. 4. XIV. of this Book)
and therefore the line PT touches the hyperbola in the point P.
Part 2. Is demonſtrated in the fame words as the ſecond part of
Prop. XV. 2. E. D.
SCHOLIU M.
THE foci of the ellipſe and hyperbola were called by Apollonius, Puncta
ex comparatione facta.
The focus of the parabola he does not mention. Modern writers have
named theſe points Umbilici or Foci. Perhaps from hence; that the rays
of light falling on a concave speculum, formed by the revolution of a
conic fection round its principal axis, will all meet after reflection in the
focus of the fection, whofe revolution generates the fpeculum.
For if the rays fall on a parabolic fpeculum and are parallel to its axis,
they will meet in its focus after reflection: Or if they fall on an elliptic
Speculum, and are diverging from one focus, they will meet in the other
focus after reflection; and if they fall on an hyperbolic fpeculum, and are
converging towards the focus of the hyperbola oppofite to the one which
gene-
Book II.
103
Conic Sections.
generates the fpeculum, they will meet in its focus after reflection. All
which appears from the three foregoing Propofitions, and from hence that
the angle of reflection is equal to the angle of incidence.
PROP.
XVIII.
16.
Draw through the extremities of the tranfverfe axis AB of FIG. 15,
an ellipfe or hyperbola two tangents which may meet any
other tangent PT in the points H, G; a circle defcribed
about the diameter GH will pass through the foci F, O
of the fection.
ET CM be the conjugate femiaxis, and becauſe the tangents
AH, BG are parallel, and meet a third tangent PT in the
points H, G; the rectangle under AH, BG will be equal to the
fquare of CM (by L. Book I.) that is, to the rectangle AFB (by
Def. IV. of this Book) therefore AH is to AF as FB to BG, and
theſe lines are about the equal angles FAH, FBG; therefore the
triangles FAH, FBG are equiangular (6. 6.) and therefore the angles.
FHA, BFG are equal; but the angles FHA, HFA are equal to a
right angle (for FAH is a right angle); therefore the angles BFG,
HFA are equal to a right angle; and therefore GFH is a right
angle, and confequently the point F is in the periphery of the circle
deſcribed about GH (converſe. 31. 3.) in like manner it may be
ſhewn that the other focus O is in the periphery of the fame circle.
2. E. D.
PROP.
XIX.
16.
If two right lines PF, PO be drawn from a point P in an FIG. 15,
ellipfe or hyperbola to the foci; they will contain a rect-
angle equal to the fquare of the femidiameter CD parallel.
to the tangent drawn through the point P.
FOR
OR draw through the extremities of the tranfverfe axis the
tangents AH, BG meeting the tangent drawn through P in the
points H, G by the preceding, a circle defcribed about the dia-
meter.
104
Book II.
Conic Sections.
FIG. 15,
16.
meter GH will pafs through the foci; draw from either focus as F,
the perpendicular FT to GH, and let it meet OP in E: and be-
cauſe the angles FPT, EPT are equal (by Part 2. Prop. XVI.
XVII. of this Book) and PT common; FT will be equal to TE
and PF to PE; therefore becauſe the diameter GH of the circle
bifects FE and is perpendicular to it, and the point F is in its circum-
ference, the point E will be in the fame; and confequently by a pro-
perty of the circle, the rectangle OPE, that is, OPF is equal to
the rectangle GPH, that is, to the fquare of the femidiameter CD
parallel to GH (by Part 2. Prop. L. Book I.) 2. E. D.
PROP. XX.
If from the foci F, O of an ellipfe or hyperbola, the right
lines FT, OL be drawn perpendicular to the tangent GPH;
a circle deſcribed about the tranſverſe axis will paſs through
the points T, L, where the perpendiculars meet the tan-
gent.
L
ET AB be the tranfverfe axis and C the center, and draw PF,
PO from the point of contact P to the foci, and let the per-
pendicular FT meet PO in E, and join CT. Becauſe the angles FPT,
EPT are equal (XVI. and XVII. of this Book) and PT common,
FP will be equal to PE; therefore OE is equal to the tranfverfe
axis (XIV. of this Book) but FT, TE are equal, and likewiſe FC,
CO (Cor. 3. Def. V. of this Book) therefore OE, CT are pa-
rallel, and confequently becauſe CF is half of OF, CT will be the
half of OE, that is, half the tranfverfe axis AB: therefore the
point T is in the circumference of the circle defcribed about AB:
and in like manner the point L will be in the fame circumference.
2. E. D.
:
PROP.
Book II.
105
Conic Sections.
PROP. XXI.
16.
If the perpendiculars FT, OL be drawn from the two foci Fic. 15,
F, O to any tangent GPH; they will contain a rectangle
equal to the ſquare of the conjugate femiaxis.
OR let AB be the tranſverſe axis, and C the center, and join
FOR
CT, which may meet OL in N; becauſe ON, FT are pa-
rallel, the triangles CFT, CON will be equiangular, and therefore
becauſe CF, CO are equal, CT, CN, and likewiſe ON, FT will be
equal: but the circle defcribed about the axis AB will pafs through
the points F and L (by the preceding) and becauſe CT, CN are
equal, it will likewife pafs through the point N; and therefore by
a property of the circle, the rectangle under LO (ON, or) TF will
be equal to the rectangle BOA, that is, to the fquare of the con-
jugate femiaxis (by Def. IV. of this Book). 2. E. D.
PROP.
XXII.
15.
Let the right line PT touch an ellipſe or hyperbola, and at FIG. 17,
the point of contact P draw PN, meeting the tranfverfe
axis, and perpendicular to the tangent, and draw from the
center, CK perpendicular to the fame tangent; theſe per-
pendiculars will contain a rectangle equal to the fquare of
the conjugate femiaxis CM.
L
ET AB, Mm be the axes of the ellipfe or hyperbola, and C the
center. Draw from the point of contact P the right lines
PE, PQ perpendicular to the axes; QC will be equal to PE: let
CM meet PT in R, and on account of parallel lines, the angles
NPE, RCK will be equal; therefore the right angled triangles
CKR, PEN are fimilar; and CK will be to CR as PE to PN, and
con-
106
Book II.
Conic Sections.
FIG. 15,
15.
confequently the rectangle under CK and PN is equal to the rect-
angle under CR and PE or CQ, that is, to the ſquare of CM (by
XLVIII. XLIX. Book I). Q. E. D.
PROP.
XXIII.
Let PT touch an ellipfe or hyperbola, and draw from the
point of contact P, the right line PO to the focus, and
from the center, CT parallel to PO, meeting the tangent;
CT will be equal to half of the tranfverfe axis AB.
FO
OR draw PF to the other focus and join FT, and let it be pro-
duced that it may meet PO in E: then becaufe OE, CT are
parallel and OC, CF equal; ET, TF will be equal; and becauſe
the tangent PT bifects the angle FPE, FP will be to PE as FT to
TE (by 3. 6.) therefore FP, PE are equal, and therefore the line.
OE is equal to AB (by XIV. of this Book) but becaufe CF is half
of OF, CT will be the half of OE, or of the tranfverfe axis AB,
Q. E. D.
FIG. 17,
18,
1.9.
PRO P. XXIV.
Let the right line PT touch an ellipfe or hyperbola, and
draw from the point of contact P, two right lines to the
axis of the ſection, one of them PE, an ordinate to the
axis, and the other PN perpendicular to the tangent; the
fegment CE of the axis between the center and ordinate
will be to the fegment NE of the fame axis, between the
ordinate and perpendicular to the tangent, as the axis to
its parameter.
L
ET the tangent PT meet the axis of the fection AB in D,
and firft let AB be either of the axes of the ellipfe or the
tranfverfe axis of the hyperbola; CE is to NE as the rectangle
CED to the rectangle NED; but the rectangle CED is equal to
the
Book II.
107
Conic Sections.
the rectangle BEA (Cor. 1. XLIX. Book I.) and becauſe NPD is
a right angle, and PE perpendicular to AB, the rectangle NED is
equal to the fquare of PE; therefore CE is to NE, as the rectangle
BEA to the fquare of PE, that is, as the axis AB to its parameter
(by Cor. 1. Def. 2. of this Book).
Secondly, let AB be the conjugate axis of the hyperbola (byCor. FIG. 15.
2. XLIX. Book I.) the rectangle CED will be equal to the fquares
of CE and CB, and, as before, the rectangle NED will be equal
to the fquare of PE, and CE will be to NE, as the rectangle CED
to the rectangle NED, that is, as the fquares of CE, CB to the
fquare of the ordinate PE, that is, as the axis AB to its
parameter
(by Cor. II. Def. II. of this Book).
PROP. XXV.
Let PT touch a parabola, and meet the axis AB in T, and FIG. 20.
draw from the point of contact P two right lines to the
axis, one of them PE ordinately applied to the axis, and
the other PN perpendicular to the tangent; the fegment
EN of the axis intercepted by theſe lines will be half the
parameter of the axis: and the fegment TN between the
tangent and the perpendicular PN will be half the para-
meter of the diameter which paffes through the point of
contact P.
FOF
}
OR draw through the point of contact P the diameter DQ,
and by Prop. XLVII. Book I. the right line EA or TA is the
half of TE.
Part 1. Becauſe NPT is a right angle and PE is perpendicular
to TN, the rectangle under TE, EN is equal to the fquare of PE,
that is, to the rectangle under AE and the parameter of the axis
(I. of this Book) therefore as TE is to AE, ſo is the parameter of
the axis to EN, which is therefore the half of that parameters
This fegment EN of the axis is called the Subnormal.
02
Part
}
108
Conic Sections.
(
Book II.
Part 2. Becauſe of the fimilar rectangled triangles TPN, TEP,
the rectangle under TE, TN, will be equal to the ſquare of TP,
that is, to the rectangle under TA and the parameter of the dia-
meter (Prop. II. of this Book) therefore TE is to TA as the para-
meter of the diameter DQ to TN, which is therefore the half of
that parameter. 2. E. D.
COR. 1. Hence, if a right line PF be drawn from the vertex P
of any diameter DQ to the focus, it will be a fourth part of the
parameter of that diameter. For, the tangent PT being drawn,
becauſe the diameter DQ is perpendicular to the directrix of the
parabola, the angle FPT will be equal to the angle DPT (XV. of
this Book) that is, to the alternate angle, FTP; therefore FP, FT
are equal then if a circle be defcribed from the center F through
the points T, P, it will pafs through N, becauſe TPN is a right
angle; therefore the line PF is half of TN; and therefore a fourth
part of the parameter of the diameter DQ, by the ſecond part of
this Prop.
See Newton's Principia Math. Lemma 13. Book I.
COR. 2. From hence the fegment of any diameter between its
vertex and directrix is a fourth part of its parameter, as is evident
from (Cor. preced. and Prop. XII. of this Book) but if the point
P were the vertex of the axis, both the Corollaries are evident from
the Def. of the focus.
COR. • 3•
If the right line joining the vertices of two diameters
be an ordinate to the axis, their parameters will be equal. For
their fegments between the vertices and directrix are equal, being
the oppofite fides of a parallelogram.
PROP.
Book II.
109
Conic Sections.
PRO P. XXVI.
A right line terminated by a conic fection paffing through
the focus, and ordinately applied to the axis, is equal
to the parameter of the axis: but every right line ter-
minated by a parabola and paffing through the focus,
is equal to the parameter of that diameter to which the
line is ordinately applied.
Part 1.
18.
IRST let the ſection be an ellipfe or hyperbola, whofe FIG. 17
tranſverſe axis is AB and conjugate axis Mm, the cen-
ter C, and focus F, and through F let XY pafs ordinately applied
to the tranſverſe axis AB; XY will be equal to the parameter of
the axis AB.
For (by Cor. 1. Prop. XXXI. and Def. XXIV. Book I.) the
fquare of CA is to the fquare of CM as the rectangle BFA, that is,
the fquare of CM (Def. IV.) to the fquare of FY; therefore CA,
CM, FY are proportional; therefore the whole lines AB, Mm, XY
are proportional, and therefore XY is the parameter of the tranf-
verſe axis AB, by the definition of a parameter; fecondly, if the
fection be a parabola, the ordinate to the axis paffing through the
focus is equal to the parameter of the axis, by Cor. 2. Prop. I. of
this Book, and the Def. of the focus.
Part 2. Let XY terminated by a parabola pafs through the FIG. 20.
focus and be ordinately applied to the diameter DQ and meet it in
the point O; through the vertex P of this diameter draw PF to the
focus, and let the line PT touch the parabola in P, it will be pa-
rallel to the ordinate OX; therefore the angle POF is equal to the
angle DPT, or TPF (XV. of this Book) that is, to the alternate
angle PFO; therefore the abfcifs PO is equal to PF, that is, to a
fourth part of the
of the parameter of the diameter DQ, by Cor. 1. preced-
ing; therefore XY is equal to the parameter of the diameter DQ,
to which it is ordinately applied, by Cor. 2. Prop. I. of this Book.
COR.
110
Book II..
Conic Sections.
1
FIG. 18.
FIG. 17,
18,
20.
FIG. 20.
FIG. 17,
18.
COR. Hence, if from the focus F of an hyperbola a right line
FL be drawn parallel to an afymptote, till it meets the fection, it
will be a fourth part of the parameter to the tranfverfe axis.
Draw through the focus F an ordinate FX applied to the axis,
and meeting the hyperbola in X, and let dr be the directrix of the
hyperbola, let FL produced meet the directrix in d, and draw Xr
to the directrix and parallel to Fd; Xr will be equal to Fd (34. 1.)
and by Prop. XIII. of this Book, FL, Ld will be equal, as alfo
FX, Xr; therefore Fd is equal to FX, that is, to half the para-
meter of the tranfverfe axis (by Part 1. of this Prop.) and there-
fore FL the half of Fd is a fourth part of the parameter of the
tranfverfe axis of the hyperbola.
PRO P.
XXVII.
If a right line PT touches a conic fection, and a right line
PF be drawn from the point of contact P to the focus F,
alfo PN perpendicular to the tangent, and meeting in N
the axis AB, which paffes through the focus, and if the
perpendicular NV be drawn from the point N to PF;
this line cuts from PF the fegment VP equal to half the
parameter of the axis AB.
WH
HEN the fection is a parabola, draw PE perpendicular to
the axis, 'becauſe it has been fhewn in Cor. 1. XXV. of
this Book, that PF is equal to NF, the angle FPN is equal to the
angle FNP; wherefore the right angled triangles PEN, NVP are
equiangular, and have the fide PN common, therefore theſe tri-
angles are equal, and therefore PV is equal to NE, that is, to half
the parameter of the axis (Part 1. Prop. XXV. of this Book).
But if the ſection be an ellipfe or hyperbola, draw CK from the
center, perpendicular to the tangent, and CT parallel to PF, and
let CM be the conjugate femiaxis: and becaufe CK, CT are pa-
rallel to PN, PV, the angles KCT, NPV will be equal; and there-
fore the right angled triangles CKT, NVP are equiangular; there-
fore the rectangle under CT, PV is equal to the rectangle under
CK
1
Book II:
III
Conic Sections.
?
CK and NP, that is, to the fquare of CM (XXII. of this Book) and
therefore CT, CM, PV are proportional: but CT is equal to CA
(by XXIII. of this Book) and confequently PV is half the
of the axis AB, by the Def. of a parameter. 2. E. D.
parameter
PROP.
XXVIII.
18,
21,
If through the focus O of an ellipfe or hyperbola a right FIG. 17,
line be drawn meeting the ſection or oppofite hyperbolas
in the points H, I; the diameter UZ drawn parallel to
this line will be a mean proportional between the ſegment
HI of the line intercepted by the fection or hyperbolas,
and the tranſverſe axis AB.
L
ET C be the center of the fection, and draw a diameter biſect-
ing HI inG; it will be conjugate to UZ (by XXVII. Book I.)
through the point H draw HS an ordinate applied to the diameter
UZ, it will be parallel to CG (Cor. XXVII. Book I.) and draw the
tangent HT meeting UZ in T: then becaufe CT drawn from the
center to the tangent is parallel to HO joining the point of contact
and focus; CT will be equal to CA: but CS is equal to GH (34. I.)
and the femidiameter CZ is a mean proportional between CT, CS
(XLVIII. XLIX. Book I.) that is, between CA, GH; therefore the
whole diameter UZ is a mean proportional between AB, HI. Q. E. D.
COR. Hence, right lines terminated both ways by an ellipfe or
hyperbola (and when neceffary produced) paffing through the focus,
are to each other as the fquares of the diameters to which they are
parallel.
}
PROP.
112
Book II.
Conic Sections.
FIG. 13.
PROP. XXIX.
If right lines terminated both ways by a conic fection or op-
pofite fections pafs (produced when neceffary) through
the focus; the rectangles under their fegments, viz. be-
tween the focus and ſection or fections, will be to each
other as theſe right lines.
Fo
OR if the fection be an ellipfe or hyperbola, the rectangles
under the ſegments of theſe lines will be to each other as the
fquares of the diameters to which the lines are parallel (XXXI. and
XL. Book I.) that is, as theſe lines by Cor. preced.
If the ſection be a parabola, the rectangles under the fegments
of the lines are as the parameters of the diameters to which theſe
lines are ordinately applied (Cor. 2. Prop. III. of this Book) that
is, as the lines themſelves, by XXVI. of this Book. 2. E. D.
PROP. XXX.
If from the focus of a parabola a right line be drawn per-
pendicular to any tangent; the fquare of the perpendi-
cular will be equal to the rectangle under the diſtance of
the point of contact from the focus, and the diſtance of
the focus from the vertex of the axis.
L
ET AB be the axis of the parabola and F the focus, and draw
through the point P in the parabola a tangent meeting the axis
in N, draw the right lines PF to the focus, PB an ordinate to the
axis, and PE perpendicular to the directrix: let FT be drawn from
the focus perpendicular to the tangent, and join AT; becauſe the
angle NPF is equal to the angle NPE (by XV. of this Book) that
is, to the alternate angle PNF, the lines FP, FN will be equal;
therefore the perpendicular FT bifects PN in T, the fegment BN is
alfo bifected in the vertex A (by the XLVII. Book I.) therefore TA
iş
Book II.
113
Conic Sections.
is parallel to PB, and confequently is perpendicular to the axis, and
therefore, becauſe FTN is a right angle, the fquare of FT is equal
to the rectangle under FA, FN or FP. 2. E. D.
COR. Hence, the fquares of perpendiculars drawn from the focus
of a parabola to the tangents, are to each other as the diſtances of
their points of contact from the focus.
PROP. XXXI.
16.
If from the focus O of an ellipfe or hyperbola, a right line FIG. 15,
OL be drawn perpendicular to any tangent GPH, and OP
be drawn to the point of contact; OL will be to OP,
as the conjugate femiaxis CM to the femidiameter CD,
which is parallel to the tangent.
ROM the other focus F, draw FT perpendicular to the tan-
FR
gent GP, and PF to the point of contact; becaufe (from the
fecond parts of Prop. XVI. and XVII. of this Book) the angles
FPT, OPL are equal, and confequently the right angled triangles
FTP, OLP ſimilar, and therefore the rectangles under OL, FT
and OP, FP are fimilar; therefore the fquare of OL is to the fquare
of OP, as the rectangle under OL, FT to the rectangle under OP,
FP, that is, as the fquare of CM to the fquare of CD (by XXI.
and XIX. of this Book) therefore OL is to OP, as CM to CD.
2. E. D.
COR. 1. Hence, if from the center C of an ellipfe or hyperbola,
CK be drawn perpendicular to any tangent GH; CK will be to the
tranfverfe femiaxis CA, as the conjugate femiaxis
diameter CD parallel to the tangent.
CM to the femi-
For draw from the focus the right line OL perpendicular to GH,
and OP to the point of contact, and CT from the center to the
tangent, parallel to OP; CT will be equal to CA (XXIII. of this
P
Book)
114
Book II.
Conic Sections.
:
í
T
"
FIG. 22.
Book) and becauſe the triangles KCT, LOP are fimilar, CK is to
CT or CA, as OL to OP, that is, as CM to CD, by this Prop.
COR. 2. Becauſe in the ellipfe or hyperbola, the fquare of FT is
to the fquare of FP, as the fquare of CM to the fquare of CD, and
the fquare of the conjugate femiaxis CM is given; the fquare of
FT will be as that is (by XIX. of this Book) as
FP 9
CDq
FP q
FP XOP
or as
FP therefore the fquare of FT increaſes in the ratio in which FP is
OP
increaſed, and OP diminiſhed; but in the ellipfe, when FP is in-
creaſed, OP is diminiſhed, and the contrary: therefore in the ellipſe
the ſquare FT varies more than in the ratio of FP; but in the hy-
perbola FP, OP are increaſed together, or diminiſhed together;
therefore in the hyperbola the fquare of FT varies lèfs than in the
ratio of FP; and confequently the perpendicular FT drawn from
the focus to the tangent in the ellipfe varies more, and in the hy-
perbola leſs than in the ſubduplicate ratio of FP, viz. the diſtance
of the focus F from the point of contact P.
See Newt. Prin. Math. Cor. 6. Prop. XVI. Book I.
PROP.
XXXII.
Let AB be the tranfverfe, and MN the conjugate axis of an
ellipfe, and C the center: if from a point D in the con-
jugate axis a right line DG be placed, meeting the tranſ-
verfe axis in G, and equal to the fum or difference of
the femiaxes CA, CM, and if in DG (produced beyond
the point G, when DG is the difference of the femiaxes)
GP be taken equal to CM; the point P will be in the
ellipfe.
FOF
OR draw PE through the point P ordinately applied to the
axis AB, and from the center draw a right line parallel to DP,
and meeting PE produced in F; becauſe CD, PF are parallel, CF
is
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FIG.20.
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FIG.19.
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FIG. I.
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FIG.18.
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1
Book II.
115
Conic Sections.
*
is equal to DP (34. 1.) that is, to CA by conftruction; there-
fore the point F is in the circumference of a circle deſcribed about
AB; and becauſe the triangles PEG, FEC are fimilar, the fquare
of PE is to (the fquare of FE, or) the rectangle BEA, as the
fquare of (GP or) CM to the fquare of (CF or) CA; therefore
the point P is in the ellipfe (by Cor. XXXIV. Book I.) Q. E. D.
COR. Hence, if two right lines AB, MN bifect each other at
right angles in the point C, and DG be equal to the difference of
CA, CM, and GP (produced beyond the point G, which is in the
greater line AB) be equal to CM, and the line DG be moved
through four right angles fo that the point D be always in the line
MN and G in AB; the point P will defcribe an ellipfe, whoſe
tranfverfe axis will be AB, and conjugate axis MN, and from hence
an ellipſe may be deſcribed by an inftrument called the Elliptical
Compass, as will readily appear by confidering the conftruction of
that inftrument.
PROP. XXXIII.
PRO B L. III.
An indefinite right line being given in pofition, and a point
given in it, and a given right line bifected by it; to deſcribe a
parabola of which this indefinite right line fhall be a dia-
meter, and the given point its vertex, and to which the
given bifected line fhall be an ordinate.
the indefinite
IRST, let AB be the indefinite line given in pofition, bifect- FIG. 23.
ing the given line OS in C at right angles, and let A be the
given point in the line AB.
Find the line QR a third proportional to CA, CO, and from the
point A in the line AB and towards the point C take a fegment
AF equal to the fourth part of QR, and let AD in the line AB be
equal to AF, and draw through the point D the indefinite right
line DX perpendicular to AB: then place the ruler HEG (whofe
fides HE and EG are perpendicular to each other) on the plane
XDF, fo that the fide HE may be applied to DX, and let the
P 2
other.
بها
116
Book II.
Conic Sections.
other fide EG be on the fame fide of DX with the point F, and
let the extremity of a ftring GPF of the fame length with the fide
EG be annexed to the end G of the fide EG, and the other extre-
mity of it fixt in the point F: let part of the ſtring GP, by means
of the pin P, be applied to the fide of the ruler EG and ftretched
along it, then let the fide HE of the ruler be moved along DX,
and at the ſame time let the ſtring, ftretched by means of the pin,
be conftantly applied to EG the fide of the ruler; the line defcribed
by the motion of the pin P will be the parabola required.
For let a parabola paſs through the points O, A, S (by XXXIII.
Book I.) of which let AB be a diameter and A its vertex, and to
which OS be ordinately applied; AB will be the axis (by Def.
XXII. Book I.) and the right line AF or AD taken four times, its
parameter, by conſtruction, and Def. I. of this Book, and conſe-
quently the point F is the focus, and the line DX the directrix of
this parabola (by Def. III. and Cor. 2. to Def. VI. of this Book)
then becauſe the whole ftring GPF is equal to the fide of the ruler
EG; the part of. it PF, viz. the diſtance of the pin from the fo-
cus of the parabola, will be equal to the perpendicular PE drawn
from the pin to the directrix; therefore the pin P will always move
in the parabola OAS (by Cor. 2. XII. of this Book) and therefore
by its motion will defcribe it.
Secondly, let VL be the line given in pofition bifecting the given
line MN in Y, but not at right angles to it, and let LV be the
given point in the line VL.
:
Find QR a third proportional to YV, YM, and from the point
V in the line VL take the fegment VK equal to a fourth part of
QR, and on the fide of the point V oppofite to where the point Y
is and draw through V, VT parallel to MN, and the right line
VF equal to VK, making the angle TVF equal to the angle
TVK, and through K draw DX perpendicular to VL: then fix one
extremity of the ftring in the point F, and by means of the pin
and ruler let a parabola be deſcribed, as in the former cafe, whofe
directrix is the line DX and focus F, and this will be the parabola
required.
For
*
Book II.
117
Conic Sections.
For it will pass through the point V, becaufe VF, VK are equal
(by Cor. 2. XII. of this Book) and the line VL perpendicular to
the directrix will be a diameter (Cor. 1. Def. VI. of this Book) and
four times VK its parameter (Cor. 2. XXV. of this Book) and the
line VT touches it (by XV. of this Book) becauſe the angles TVF,
TVK are equal; and therefore the line MN parallel to this tangent
is ordinately applied to the diameter VL, and becauſe the fquare of
YM or YN, by conftruction, is equal to the rectangle under the
abfcifs YV and the parameter of the diameter VL, the points M
and N are in this parabola (by Cor. 1. Prop. I. of this Book.)
Q. E. F.
COR. 1. Hence, if the directrix DX be given in pofition, and
the vertex of the axis given, viz. the point A; the parabola may
be deſcribed by drawing AD perpendicular to the directrix, and
making AF equal to AD; for the point F will be the focus: and
likewife if the vertex A and focus F be given; join AF and pro-
duce it to D, ſo that AD be equal to AF, the line drawn through
the point D perpendicular to DA will be the directrix. But if the
axis AB be given in pofition, and its vertex A and parameter QR
given, let the focus be found by its Definition; likewife the axis.
being given in pofition, and the focus and parameter to the axis
given; the directrix may be found; therefore in thefe cafes a pa-
rabola may be deſcribed, as is demonftrated by this Prop.
COR. 2. Hence likewife, if a diameter LV of a parabola be FIG. 24.
given in pofition and its vertex V given, and its parameter be given
in magnitude, viz. QR, and a point P in the parabola; this para-
bola may be deſcribed. For in the line VL take the fegment VK
equal to a fourth part of QR, and through the point K draw the
right line KE perpendicular to VL, and through P, PE perpendicu-
lar to KE: then let two circles be deſcribed from the centers V, P,
and with the intervals VK and PE, meeting each other in the points
F, G; if a parabola be deſcribed, whoſe directrix is KE and focus
F or G, it will pass through the points V and P (Cor. 2. XII. of
this Book) and VL will be a diameter (Cor. 1. Def. VI.) and VK
taken four times, that is, QR will be the parameter of that dia-
meter
118
Book II.
Conic Sections.
:
FIG. 25.
meter (Cor. 2. XXV. of this Book). It is manifeft that in this cafe
two parabolas may be defcribed, which fhall folve the problem and
one only, if the circles defcribed meet in one point only; but if
theſe circles do not meet each other, the problem in that cafe will
be impoffible.
PROP. XXXIV. PROBL. IV.
Two right lines bifecting each other being given in pofition
and magnitude; to defcribe an ellipfe in which the given
lines fhall be conjugate diameters.
F¹
IRST, let AB, MN be the given lines bifecting each other in
the point C at right angles: and becauſe in this cafe, AB, MN
muſt be unequal (Cor. 2. XXXI. Book I.) defcribe a circle from
the center M, the extremity of the leffer line MN with an interval
equal to CA half of the greater line AB, and cutting AB in the
points F, O, and in theſe points let the end of a ſtring be fixed, of
the fame length with AB, and by means of the pin P let the ſtring
be ſtretched, and the pin carried round till it returns again to the
fame point it moved from; the line defcribed by this motion will
be an ellipfe, in which AB, MN fhall be conjugate diameters.
For let an ellipfe pafs through the points A, M, B, N, of which
AB, MN are conjugate diameters (by Prop. XXXIV. Book I.) and
becauſe theſe conjugate diameters are at right angles to each other,
the greater of them AB will be the tranfverfe axis and the leffer
MN the conjugate axis; and the points F, O the foci, by con-
ftruction, and Cor. 1. Def. V.
Then becauſe the fum of the lines which are drawn from the pin
P to the foci is always equal to AB, the pin P as it is carried round
will be in an ellipfe (by Cor. 2. XIV. of this Book) and confe-
quently by its motion defcribes an ellipfe.
Secondly, let the two given lines bifecting each other be not at
right angles. It is evident that two ellipfes cannot have the fame
center, nor the fame two conjugate diameters given in pofition and
magnitude; for if they could, every point in one ellipfe would be
alfo
Book II.
119
Conic Sections.
;
alfo in the other (by Cor. XXXIV. Book I.) that is, the ellipfes would
coincide; therefore two conjugate diameters being given in pofition
and magnitude the ellipfe will be given, and confequently the axes
will be given in pofition and magnitude, and thefe being found (by
Prop. VII. of this Book) the ellipfe may be defcribed, as in the
preceding cafe.
PROP. XXXV. PROBL. V.
Two right lines, which bifect each other, being given in
poſition and magnitude; to deſcribe oppoſite hyperbolas
of which the given lines fhall be conjugate diameters.
FH
IRST, let the two given lines AB, MN bifect each other in FIG. 26.
the point Cat right angles, join AM, and from the point C,
in the line AB produced both ways take the fegments CF, and CO
each equal to AM; at the point O let the end of the ruler OG be
fixed, fo that it may be freely carried round this point as a center:
and let the end of a ftring be affixed to the other extremity G of
the ruler, the length of which the ruler exceeds by a right line
equal to AB, and let the other end of the ftring be fixed in the point
F, and apply the ftring by means of the pin P to the fide of the
ruler OG, and let the ruler be moved about the point O, and at the
fame time the ſtring, by means of the pin, be conftantly applied
and kept cloſe to the ruler; the pin P by its motion will defcribe
one of the oppofite hyperbolas, of which the lines AB, MN are
conjugate diameters.
For let the oppofite hyperbolas be conceived to be defcribed (by
XXXV. Book I.) of which AB may be a tranfverfe diameter and
MN its conjugate; the point C will be the center, and becauſe
theſe conjugate diameters cut each other at right angles, AB will be
the tranfverfe axis and MN the conjugate axis, and confequently
the points F, O will be the foci of the hyperbolas by conſtruction,
and Cor. 2. Def. V. but becauſe the exceſs of the ruler GPO above
the ſtring GPF is equal to AB, the excefs of the line PO above PF
will
1
I 20
Book II.
Conic Sections.
書
​will be equal to the tranfverfe axis AB; therefore as the ruler is
moved the pin P will be in one of the hyperbolas (by Cor. 4. XIV.
of this Book). But if the extremity of the fame ruler which was
affixed to the point O, be now affixed to the point F, and the end
of the ſtring be fixed in the point O, and the fame as before ef-
fected an hyperbola oppofite to the hyperbola now deſcribed, will
be generated, as is evident.
Secondly, if the two given lines do not cut each other at right
angles: two conjugate diameters of an hyperbola being given in
pofition and magnitude, the hyperbolas will be given, and their axes
will be given in pofition and magnitude, as in the ſecond cafe of
the preceding Prop. therefore the axes being found (by Prop. VII.
of this Book) the hyperbolas may be defcribed, as in the preced--
ing cafe.
1
COR. To this and the preceding Propofition. Hence, if the tranf-
verſe axis AB of an ellipfe or hyperbola be given in pofition and
magnitude, and the foci F, O be given; the ellipfe or hyperbola
may be deſcribed. Or if any diameter AB be given in pofition and
magnitude, and the line PE, which from the given point P in an
ellipfe or hyperbola is ordinately applied to the diameter AB; an
ellipfe or hyperbola may be deſcribed: for bifect AB in C, and draw
through the point C a line parallel to PE, in which line take CM,
CN equal to each other, that the fquare of CA be to the fquare of
CM or CN, as the rectangle AEB to the ſquare of PE and by
the means of the preceding Propofitions deſcribe an ellipfe or hy--
perbola, in which AB, MN ſhall be conjugate diameters; it will paſs
through the point P, by Cor. XXXIV. and XXXV. Book I.
PROP.
1
Book II.
121
Conic Sections.
PROP. XXXVI. PROBL. VI.
The directrix and one of the afymptotes of an hyperbola
being given in pofition and the focus given, nearer to
this directrix; to defcribe the hyperbola.
L
ET DX be the directrix of an hyperbola, and F the focus FIG. 27.
nearer to that directrix, and let the line CY be parallel to one
of the aſymptotes; place a ruler on the fame plane with the hyper-
bola, ſo that the fide HE of it be applied to the line DX, and that
the other fide EG be on that part of DX where the point F is, and
let the fide of the ruler EG be inclined to the fide HE, that it
may be parallel to CY: then by means of the pin P, and of the
ftring applied to the fide EG, the extremity of which is fixed in
the point F, let all be effected as in the deſcription of the parabola,
Prop. XXXIII. of this Book; the line defcribed by the motion of
the pin P will be an hyperbola, its directrix DX, and afymptote pa-
rallel to CY, and focus F. For becauſe the ftring GPF is equal to
GPE, the right line PF drawn from the pin P to the focus is equal
to PE drawn from the fame pin to the directrix, and parallel to the
afymptote; and therefore the pin P moves in an hyperbola, of which
DX is the directrix and the point F the nearer focus to this direc-
trix, and whoſe aſymptote is parallel to CY (by Cor. Prop. XIII.
of this Book).
COR. If a plane cutting a conical furface and making the fection
of a parabola, be inclined in the leaft fo as to meet the oppofite fur-
face, this ſection will be changed into an hyperbola; and the fame
analogy may be obſerved between the defcriptions of thefe fections
upon a plane; for if the fide of the ruler to which the ftring is ap-
plied in the defcription of a parabola, be in the leaft inclined to the
other fide of the ruler, the pin will defcribe an hyperbola, and befide
when in the deſcription of thefe fections the fide of the ruler paffes
through the focus, the part PF of the ftring between the focus and
pin will be a fourth part of the parameter of the axis in both fec-
tions, as is evident from the Definition of the focus of a parabola,
and from (Cor. Prop. XXVI. of this Book).
е
PROP..
1
1
1
FIG. 28,
29,
30,
FIG. 28.
I 22
Conic Sections.
PROP. XXXVII.
Book II.
Let GVH be a right cone, and PAR a conic fection in its
furface, and LNO a circle which does not meet the fec-
tion let its diftance AL from the vertex of the fection
be equal to AF the diftance of the fame vertex from the
focus F neareſt to this circle; I ſay that the interſection of
the plane of this circle with the plane of the ſection will
be its directrix, and that PN a fide of the cone inter-
cepted between this circle and any point P in the ſection
will be equal to a right line drawn from the ſame point
to the focus F neareſt to this circle.
Lpendicular to the
ET the cone be cut through its axis VX by a plane GVH per-
pendicular to the plane PAR of the fection, and cutting it in
the right line AFB, this line will be the axis of the fection, which
in the ellipfe or hyperbola will be the tranfverfe, and confequently
the vertex A, the focus F of the fection, and the line AL will be
in the plane GVH: then becauſe the plane of the fection PAR,
and the plane of the circle LNO are both perpendicular to the plane
GVH, their common interfection DE will be perpendicular to the
plane GVH (19. 11.) and confequently to the axis AB of the
fection; and becauſe DE is in the plane of the circle LNO, it will
be parallel to the plane of the cone's bafe; let the line DL be the
interfection of the plane GVH with the plane of the circle LNO,
it will be parallel to the plane of the baſe, as is manifeft.
Cafe 1. First, let the fection PAR be a parabola, and draw from
a point P in it PQ an ordinate applied to the axis, it will be pa-
rallel to DE, and therefore to the plane of the baſe: let a plane paſs
through PQ parallel to the bafe, and interfecting the plane GVH
in the line GQH, which will be parallel to DL (16. 11.) then
becauſe the triangles GAQ, DAL are fimilar to each other and to
the triangle GVH, and the lines GV, HV are equal, becauſe the
cone
Book II.
123
Conic Sections.
cone is right, GA, QA, as likewife DA, LA will be equal, and
confequently QD will be equal to GL; therefore fince DA the dif
tance of the line DE from the vertex of the parabola is equal to
AL, that is to AF the distance of the focus from the fame vertex,
the line DE perpendicular to the axis AB will be the directrix of
the parabola; therefore the first part of this Prop. is evident when
the fection is a parabola. Wherefore if a right line PF be drawn
from the point P to the focus, it will be equal to the perpendicular
drawn from the fame point P to the directrix (by XII. of this
Book) that is, to QD or GL; but GL, PN are equal, being ſeg-
ments of the fides of a right cone intercepted between parallel cir-
cles, and therefore PN is equal to PF.
Cafe 2. Let the fection PAR be an ellipfe and C the center, and
draw the conjugate femiaxis CP, it will be parallel to the line DE,
and therefore to the plane of the bafe; let a plane pafs through the
line CP, parallel to the bafe, and interfecting the plane GVH in
the line MCK, which is parallel to DL (16. 11.) MK will be a
diameter of the circle MPK, and by a property of the circle, the
rectangle MCK is equal to the fquare of the conjugate femiaxis
CP; draw from the vertex V of the cone a line parallel to AB, and
meeting the diameter GH of the baſe, produced in S: then by
Prop. X. Book I. the rectangle ACB, or the fquare of CA, is to
the rectangle MCK, or the fquare of CP, as the fquare of VS to
the rectangle HSG; therefore, by converfion, the fquare of CA is
to its exceſs above the fquare of CP, that is, to the fquare of CF
(Cor. 1. Def. V. of this Book) as the fquare of VS, to its exceſs
above the rectangle HSG, that is, to the fquare of VG: but be-
caufe the triangles VSG, ACM are fimilar, the fquare of CA is to
the fquare of AM as the fquare of VS to the fquare of VG; there-
* The ſquare of VG.] The triangles VXS, VXG (Fig. 29, 30.) are right angled, and
therefore the difference between the ſquares of the hypotenuſes VS and VG is equal to
the difference between the ſquares of the fides XS and XG, that is, to the rectangle
HSG under GS, the fum of theſe fides, and HS their difference. So that in Fig. 29,
the fquare of VG is the excess of the fquares of VS above the rectangle HSG : and in
Fig. 30, the fquare of VG is equal to the fquare of VS and the rectangle HSG.
Q2:
fore
FIG. 29.
I24
Conic Sections.
1
FIG. 30.
Book II.
fore the lines CF and AM are equal; but becauſe MC, DL are pa-
fallel, MA or CF, the diſtance of the focus from the center, will
be to the femiaxis CA, as LA or FA to DA, the diſtance of the line
DE from the vertex A of the ellipfe; therefore the line DE, be-
ing perpendicular to the axis, will be the directrix (by Cor. 2. XI.
of this Book) and confequently the first part of the Prop. is evident
when the ſection is an ellipfe. Now draw from any point P in the
ellipfe the right line PC perpendicular to the axis AB, and let the
plane of a circle pafs through PC, and meet the fides VG, VH in
the points M, K, and interfect the plane GVH in the line MK pa-
rallel to DL; then becauſe CD is equal to the perpendicular drawn
from P to the directrix DE, if PF be drawn to the focus (by ·XI.
of this Book) PF will be to CD, as FA or LA to DA: but ML is
likewiſe to CD as LA to DA; therefore PF, ML are equal; but
ML, PN are equal, as in the cafe preceding; therefore PN is equal
to PF.
Cafe 3. Let the ſection PAR be an hyperbola, and let PR be the
interſection of its plane with the plane of the baſe HPG, it will be
parallel to DE (16. 11.) and therefore ordinately applied to the
tranſverſe axis AB, which produced let it meet in Q: let HQG be
the interfection of the plane HVG with the plane of the bafe, it
will be parallel to DL (16. 11.) draw from the center C of the
hyperbola a line parallel to HQG, DL, meeting the fides HV,
GV of the cone in the points K, M, and draw from the vertex of
the cone VS parallel to the axis AB of the hyperbola, meeting HQG
in S, and let CT be the conjugate femiaxis.
Becauſe by Prop. X. Book I. the rectangle ACB, that is, the
fquare of CA is to the rectangle KCM, as the fquare of VS to the
rectangle HSG, and in like manner the rectangle AQB (under the
abfciffes) is in the fame ratio to the rectangle HQG, or to the
fquare of the ordinate PQ, the rectangle KCM will be equal to the
fquare of the conjugate femiaxis CT, (by Def. XXIV. Book I.) and
therefore, by converfion, the fquare of CA is to the fum of the
fquares of CA and CT, that is, to the fquare of CF (by Cor. 2.
Def.
Book II.
125
Conic Sections.
Def. V. of this Book) as the fquare of VS to the ſum of the ſquare
of VS and the rectangle HSG, that is, to the fquare of VG :
but becauſe the triangles CAM, SVG are fimilar, the fquare of CA
is to the fquare of MA, as the fquare of VS to the fquare of VG;
therefore the lines CF, MA are equal: but becauſe MC, DL are
parallel, CA is to MA or CF, as DA to LA or AF, therefore the
line DE is the directrix of the fection (Cor. 2. XI. of this Book)
wherefore the firſt part of. this cafe is manifeft.
Now draw from any point in the hyperbola a right line PF to the
focus, and PQ perpendicular to the axis AB of the hyperbola, and
draw through PQ the plane of the circle GPHR, it may be fhewn
in the fame words as in the preceding cafe, that the line PF is equal
to GL and confequently to PN. Q; E. D.
COR. As there are two foci F, and ƒ in an ellipfe and hyper-
bola, there will be two circles LNO, Ino, fuch as are defcribed in
the propofition, whofe planes will interfect the plane of the ſection
in the lines DE, de, which will be directrices of the fection; and
becauſe the lines PF, Pf drawn to the foci are equal to PN, P#,
the fegment Nn of any fide of the cone intercepted between the
circles LNO, Ino, will be equal to the tranſverſe axis of the ſection,
by Prop. XIV. of this Book.
* See the note in page 123.
END OF THE SECOND BOOK.
U
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FIG. 28.
F
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Page·126.
FIG. 2 3.
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FIG.25
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FIG. 27.
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D
CONIC SECTIONS.
BOOK THE THIR D.
Of the Parabola.
PROPOSITION
SITION I
I.
If a right line FG touching a parabola in F meets two dia- FIG. 1.
meters VC, DH in the points G and R; the ſquares of
the ſegments FG, FR of the tangent between the point of
contact F and the diameters, will be to each other as the
fegments GV, RD of the diameters between their ver-
tices and the tangent. Or if a right line AB cutting a
parabola, meets two diameters FL, VC in the points K, E;
the rectangles AKB, AEB will be to each other as the
fegments FK, VE of the diameters between their vertices
and the fecant AB.
Part
1 28
Book II.
Conic Sections.
FIG. 2..
Part 1.
L
ET the right line P be the parameter of the diameter
which paffes through the point of contact F, the ſquare
of FG is equal to the rectangle under GV and the line P, and the
fquare of FR is equal to the rectangle under RD and the line P
(by Prop. II. Book II.) therefore the fquares of FG, FR are to
each other as the fegments GV, RD.
Part 2. Let the right line P be the parameter of the diameter to
which the fecant AB is ordinately applied; the rectangle AKB is
equal to the rectangle under FK and the line P; and the rectangle
AEB is equal to the rectangle under VE, and the line P (Prop. III.
Book II.) and therefore the rectangle AKB, AEB are to each other
as the fegments FK, VE. Q, E. D.
Q.
- COR. Let a right line meet a parabola in the points A, B, and
let F be the vertex of the diameter meeting the right line AB in the
point K; if from any other point E in the line AB, EV be drawn
parallel to FK, ſo that it be to FK as the rectangle AEB to the
rectangle AKB; the point V will be in the parabola which paffes
through the points A, F, B, and whofe diameter is FK. When
the point E is between A and B, the line EV muſt be drawn on the
fame fide of AB with the point F, and when not, on the contrary
fide.
PROP. II.
If from the vertices of two diameters ordinates be drawn
applied to theſe diameters; the abfciffes between the ordi-
nates and vertices of the diameters will be equal..
FR
ROM: the vertices of the diameters FG, HM draw the ordi-
nates FL, HK applied to the diameters HM, FG; the ab-
fciffes HL, FK will be equal.
Draw through the vertex of the diameter HM a right line touch-
ing the parabola and meeting the diameter FG in A; then becauſe
HAFL is a parallelogram, the abfcifs HL is equal to AF, that is,
to the abfcifs FK, by Prop. XLVII. Book F. 2; E. D.
PROP.
Book III.
129
Of the Parabola.
PROP.
III.
J
If one fide of any triangle be parallel to the diameters of a
parabola; the ſquares of the other fides will be to each
other as the parameters of the diameters, whofe ordinates
are parallel to thoſe fides.
L
ET the fide MN of the triangle LMN be parallel to the dia- FIG. 3.
meters of the parabola BFH, and let the right lines P and Q
be the parameters of the diameters ED, FR, to which BC, BR
(parallel to LM, LN) are ordinately applied; the ſquare of LM
will be to the fquare of LN as P to Q
For through the vertices E and F of the diameters ED, FR,
draw the right lines EG, FG touching the parabola; they will be
parallel (by hypothefis, and Cor. 8. XXV. Book I.) to LM, LN;
let the tangent EG meet the diameter FR in A, and from the point
of contact E, draw EK an ordinate applied to the diameter FR, it
will be parallel to FG, and (by XLVII. Book I.) KF, FA will be
equal; therefore EG, GA are equal: then becauſe the fides of the
triangles LMN, GAF are refpectively parallel, thofe triangles will
be equiangular; therefore the fquare of LM is to the fquare of LN,
as the fquare of (GA or) GE to the fquare of GF, that is, as the
parameter P to the parameter Q, by Cor. 1. Prop. III. Book II.
Q. E. D.
COR. I. If from any point B in a parabola an ordinate BC be
drawn to a diameter ED, and any other right line BD to the fame
diameter; the fquare of BD will be equal to the rectangle under
the abſciſs EC, and the parameter of the diameter FR, to which
the line, parallel to BD, is ordinately applied. For let Q be the
parameter of the diameter FR, and P the parameter of the diameter
ED: by this Prop. the fquare of BC is to the fquare of BD as
(P to Qor as) the rectangle under P, EC to the rectangle under
Q, EC: but (by Prop. I. Book II.) the fquare of BC is equal to
the rectangle under P,.EC; therefore the fquare of BD is equal to
the rectangle under Q, EC.
R
COR.
130
Book III.
Conic Sections.
1
FIG. 4.
COR. 2. If a right line OE touches a parabola, and through a
point O taken in it a right line be drawn cutting the parabola in B
and H, and meeting in D the diameter drawn through the point of
contact E; the fquare of the fegment OD between the tangent and
the diameter will be equal to the rectangle BOH under the feg-
ments of the fecant between the tangent and parabola. For, by this
Prop. the fquare of OD is to the fquare of OE, as Q to P, and
(by Cor. 3. III. Book II.) the rectangle BOH is in the fame ratio to
the fquare of OE, therefore the fquare of OD is equal to the rect-
angle BOH.
COR. 3. The parameter of any diameter of a parabola is to the
parameter of its axis, as the fquare of radius to the ſquare of the
fine of the angle which that diameter contains with its ordinates, as
is evident from this Prop. and becauſe the fquares of the fides of
a triangle are as the fquares of the fines of the oppofite angles; and
therefore the parameters of the diameters of any parabola are reci-
procally, as the fquares of the fines of the angles which thoſe dia-
meters contain with their ordinates; and therefore the principal pa-
rameter, that is, the parameter of the axis is lefs than any other
parameter.
PROP. IV.
If the right line joining the extremities of two ordinates
meets the diameter to which they are applied: the part of
the diameter intercepted between its vertex and the right
line joining the extremities of the ordinates, will be a
mean proportional between the abfciffes.
L
1
ET EB, GD be ordinates applied to the diameter AD of the
parabola, and let a line joining the points E, G meet the dia-
meter AD in C; CA will be a mean proportional between the ab-
fciffes AB, AD.
Draw through the points E, G two tangents which may meet the
diameter AD in H and F; AH, AB, as likewife AF, AD will be
equal (XLVII. Book I.) but it is evident (from Prop. LIV. Book I.)
that
f
Book III.
131
Of the Parabola.
that AC between the vertex of the diameter and the line joining
the points of contact, is a mean proportional between AH and AF;
and therefore is a mean proportional between AB and AD. 2. E. D.
cant.
COR. 1. Hence, if two right lines be drawn from a point E in a
parabola, one of them a tangent, the other cutting the fection in
the points E, G, which fhall meet any diameter AL in H and C;
the ſegments EC, CG of the ſecant between the diameter and para-
bola will be to each other as the fegments AH, AC of the diameter
between the vertex and tangent, and the fame vertex and ſe-
For if EG be an ordinate applied to the diameter AL, EG
will be bifected in C and HC in A; but if EG be not an ordinate
to AL, draw from the points E, G the ordinates EB, GD to the
diameter AL and becauſe the triangles EBC, DGC are fimilar,
the fquare of EC is to the fquare of CG as the fquare of EB to the
fquare of GD (that is, as the abfcifs AB to the abfcifs AD, or be-
cauſe AB, AC, AD are proportional) as the fquare of AB to the
fquare of AC; therefore EC will be to CG, as (AB or) AH to AC.
PROP.
V.
Let the right line HK be an ordinate to the diameter FP: FIG. 5.
through its extremity H draw the diameter HC, and from
the vertex of the diameter FP draw a right line meeting
the parabola again in O, and the diameter HC in L, and
HK in N: FO, FL, FN will be proportional.
F
OR draw from the points O, L, two lines OP and LS, to the
diameter FP, and parallel to the ordinate HK; OP will be an
ordinate applied to the diameter FP, and LS will be equal to HK:
then becauſe OP, LS are parallel, the fquare of FO will be to the
fquare of FL as the fquare of OP to (the ſquare of LS or) HK,
that is, as the abfcifs FP to the abfcifs FK, or becauſe OP and HK
are parallel, as FO to FN; therefore FO, FL, FN are propor-
tional. 2. E. D.
--
R 2
SCHO-
132
Book III.
Conic Sections.
FIG. 5.
FIG. 6,
7.
FIG. 6.
FIC. 7.
JF.
SCHOLIU M.
1,
F two indefinite right lines DE, CM meet each other in a given angle
in the point H, and about any point F another indefinite right line re-
volves meeting DE, CM in the points L, N, and a point O be taken in
the revolving line on the fame fide of the point F where N is (the inter-
Jetion of the revolving line with the line DE) fo that FO, FL, FN be
continually proportional; the Locus of all the points O will be a para-
bola paffing through the points F and H, and to which CM will be a
diameter, and DE will be parallel to the ordinates applied to the dia-
meter which paffes through the point F: but if a mean proportional FO
be taken in this revolving line between FL, FN; the line defcribed by
the point O will be an hyperbola. See the Scholium to Prop. XLVI.
Book I.
PROP. VI.
If three right lines touching a parabola meet each other
they will be cut in the ſame ratio, viz. between their points
of concourfe and points of contact.
L
ET ABC be a parabola, which the lines AD, FE, DC touch
in the points A, B, C; AF will be to FD as FB to BE, and
as DE to EC. For let the point of contact B in the fegment of
the parabola fall between the points A, C: join AC and biſect it in
G, the diameter drawn through G will pafs through the point of
concourſe of the tangents AD, CD (Cor. 1. XXVI. Book I.)
Cafe 1. If the diameter DG paffes through the point B, the tan-
gent FBE will be parallel to AC, and therefore will be bifected in
the point B; and becauſe GB, BD are equal, and AC, FE parallel,
the tangents AD, CD will be bifected in the points F, E.
Cafe 2. But if the diameter DG does not pafs through the point
B; join AB, BC, and draw through the points F, B, E the dia-
meters FHL, BN and EKM; and (by Cor. 1. XXVI. Book I.)
AH, HB will be equal, and confequently AL, LN; and by the
fame
·
Book III.
133
Of the Parabola.
fame reaſoning NM, MC will be equal; therefore LM is equal to
(half of AC, that is) to GC, and therefore (taking away the com-
mon part GM) LG will be equal to MC or NM, and therefore
LN will be equal to GM: thefe being premiſed, AF will be to FD
as AL to LG (2. 6.) that is, as LN to NM, or GM to MC;
but as LN is to NM, fo is FB to BE, and as GM is to MC, fo is
DE to EC; therefore as AF is is to FD, fo is FB to BE, and DE
to EC. 2, E. D.
PROP. VII. PROBL.
I.
The directrix DX of a parabola being given in poſition, and FIG. 8.
the focus F given, and the right line MN, not parallel to
the diameters, given in poſition; to draw a tangent to the
parabola, which ſhall be parallel to this right line.
FR
ROM the focus F draw FO perpendicular to MN, meeting
the directrix in E, and FE being bifected in T, draw TP pa-
rallel to MN, which may meet in P the perpendicular to the di-
rectrix drawn through E; and PF being joined, the triangles PTE,
PTF will be equal (4. 1.) and therefore fince FP is equal to PE,
the point P will be in the parabola (Cor. 2. XII. Book II.) and
becauſe the angles EPT, FPT are equal, the line PT parallel to
MN touches the parabola. 2. E. F.
PROP. VIII.
VIII.
PROB L. II.
Through three given points to deſcribe a parabola, the dia-
meters of which fhall be parallel to a right line given in
pofition, provided it be not parallel to a right line join-
ing any two of the given points.
L
ET B, G, F be the given points but not in the fame line, and FIG. 9.
let QR be the line given in pofition: join GF, and draw
through the point B a line parallel to QR, meeting GF in E: if
the line GF be bifected in E, the propofition is the fame with
Prop.
Conic Sections.
134
Book III.
Prop. XXXIII, Book II. But if GF be not bifected in E, let it
be biſected in M, and let the line BE be to VM, drawn through M
parallel to BE, as the rectangle GEF to the rectangle GMF; then
by Prop. XXXIII. Book II. defcribe a parabola, of which the line
VM fhall be a diameter and V its vertex, and to which GF ſhall
be ordinately applied: this parabola will pafs through the point B
(by Cor. Prop. I. of this Book) and its diameters are parallel to QR
by conftruction. When the point E falls between G and F, the
line VM muſt be taken on the fame fide of GF, where the point B
is; but if the point E be not between G, F, then on the contrary
fide. Cor. Prop. I. of this Book. Q, E. F.
PROP. IX. PROBL. III.
Four points being given in a parabola; to deſcribe the pa-
FIG. 9.
Cafe 1.
L
rabola.
1
ET A, B, G, F be the given points, and let the right
lines joining theſe points form a trapezium, having none
of its fides parallel: let two of the oppofite fides produced meet in
D, and as the rectangle BDA is to the rectangle GDF, fo let the
fquare of DB be to the fquare of the fegment DE of the line
DGF; BE being joined, defcribe a parabola (by Prop. preceding)
paffing through the points G, B, F, whofe diameters are parallel to
BE; it will paſs through the point A.
For becauſe the rectangle BDA and the fquare of DB are un-
equal, the rectangle GDF and the fquare of DE will be unequal,
by conftruction; therefore the line DB does not touch the parabola
(as is evident from Cor. 2. III. of this Book) and therefore múft cut
it in another point, and if not in A, fuppofe in X; then (by Cor. 2.
Prop. III. Book II.) the rectangle BDX is to the rectangle GDF
as the ſquare of DB to the fquare of DE (by III. of this Book)
that is, by conftruction, as the rectangle BDA to the rectangle
GDF; therefore the rectangles BDX, BDA are equal, and there-
fore the lines DA, DX are equal, which is abfurd (for the point X
is on the fame fide of D as the points B and A, becaufe D is with-
out
Book III.
135
Of the Parabola.
out the parabola) therefore the parabola meets the line DB in the
point A. Becauſe the ſegment DE in the line DGF may be taken
on either fide of the point D, two parabolas may be deſcribed, which
will anſwer the problem.
Cafe 2. Let B, G, F, P be the four given points, and let the right
Hines joining the points G, F and B, P be parallel: the right line
MN bifecting theſe parallels will be a diameter of the parabola paff-
ing through theſe four points (by Cor. 6. XXV. Book I.) let then
a parabola be deſcribed paffing through the three points B, G, F,
whoſe diameters fhall be parallel to the line MN, and fince the line
BN is ordinately applied to a diameter of this parabola and PN is
equal to BN, this parabola will paſs through the point P; it is ma-
nifeft in this cafe that there can be but one pofition of the diameters
of the parabola which paffes through theſe points, and confequently
one parabola only can pafs through theſe points.
COR. From the conftruction a method of finding the pofition of
the diameters of a parabola from four points given in it, is ma-
nifeft.
PROP. X.
Two parabolas, having a common axis and the fame prin-
cipal parameter, produced indefinitely, continually ap-
proach each other, but never meet.
L'
ET AEF, DCB be two parabolas having the common axis EG, FIG. 10.
and let the right line PQ be the principal parameter of both :
let an ordinate be applied to the axis and meet it in G, the exterior
parabola in the points A, F, and the interior in D. Becauſe the
fquare of AG is equal to the rectangle under GE, PQ, the dif
ference of theſe fquares, that is, the rectangle ADF (5. 2.) will be
equal to the rectangle under CE, PQ; then becaufe the rectangle
CE, PQ is given, the rectangle ADF will be given in magnitude:
but the farther the point D recedes from the vertex C of the in-
terior
136
Book III.
Conic Sections.
terior parabola, the greater the line DF, will be, and therefore the
diſtance AD between the parabolas, will be lefs: but the points A, D
can never coincide, becauſe the rectangle ADF is always a given
magnitude. 2. E. D.
COR. Hence, the ſegment DH intercepted between the ver-
tices of two coinciding diameters is equal to the fegment EC be-
tween the vertices of the axes. For (by III. Book II.) the rectangle
ADF is equal to the rectangle under the lines PQ, DH, and by
this Prop. the fame rectangle ADF is equal to the rectangle under
PQ, EC; therefore the ſegments DH, EC are equal.
FIG. II.
The QUADRATURE of the PARABOLA.
PROP. XI.
If a right line BC be infcribed in a parabola, and tangents
be drawn through the points B, C, meeting each other in
the point A, and if the line BC be bifected in G and its
parts be again bifected, and in like manner the bifections.
of the parts be ftill continued; and if diameters be drawn
through all the bifecting points D, E, V, G, &c. meet-
ing the parabola in L, M, N, O, &c. and if all theſe
points B, L, M, N, O, &c. be joined by lines which may
form an inſcribed figure, and if tangents be drawn
through the fame points B, L, M, N, O, &c. which may
form a circumfcribed figure: the figure infcribed will be
always double the area contained within the triangle BAC
and without the circumfcribed figure.
FOR
1
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Book III.
137
Of the Parabola.
FOR
OR firft let the infcribed figure be the triangle BOC, the cir-
cumfcribed figure will be the trapezium BHZC, and the area
without it, will be the triangle HAZ: and becauſe GO is a diame-
ter, it will pass through the point A (Cor. 1. XXVI. Book I.) and
becauſe GA is double OA (XLVII. Book I.) BC will be double
of HZ; therefore the triangle BOC is double the triangle HAZ;
then if BG be bifected in E, the diameter EM will bifect BO, and
confequently will pafs through the point H; therefore, becauſe
FM, MH are equal, the triangle BMO will be double the triangle
KHI: but BMO is added to the firft infcribed figure, and KHI
is added to the area, which is without the circumfcribed figure and
in like manner, if the number of fides be increaſed in the infcribed
and circumfcribed figures, whatfoever is added to the area without
the circumfcribed figure, the double of it is always added to the in-
ſcribed figure; and therefore the infcribed figure will be always
double the area which is without the circumfcribed figure. 2. E. D.
COR. 1. Hence, it is evident, that no figure can be inſcribed (as
in the propofition) in the parabolic fegment BOQC which can be
double the area BACO, without the parabolic ſegment and within
the triangle BAC.
COR. 2. It is alfo evident that no figure can be circumfcribed
about the parabolic fegment BOQC (as in the propofition) fo that
the area without it and within the triangle fhall be half the parabo-
lic fegment BOQC.
PROP. XII.
The above conſtruction remaining, a figure may be infcribed F16. 11.
(as in the preceding Propofition) in the parabolic fegment
BOQC, terminated by the parabola and a right line, which
fhall not be leſs than the fegment by any given minute
fpace whatſoever: Or a figure may be circumfcribed about
this parabolic fegment (as in the preceding Propofition)
S
and
138
Conic Sections.
FIG. 11.
Book III.
and within the triangle BAC, which ſhall cut off from
the area BACO a ſpace leſs than any given ſpace whatſoever.
Part 1.
L
ET the triangle BOC be infcribed in the parabolic
fegment, as in the preceding Prop. Since it is half
of the whole triangle BAC, it will be greater than the half of the
parabolic fegment BOQC and in like manner if the triangles BMO
and CQO be inſcribed in the remaining fegments, they will be
greater than the halves of theſe fegments: and in the fame manner
it will be in the remaining fegments; wherefore a figure may be
inſcribed which fhall not be leſs than the parabolic fegment by any
given ſpace whatſoever (1. 10.)
Part 2. In the area BACO infcribe the triangle HAZ, whofe
baſe HZ fhall be parallel to the line BC, joining the points of con-
tact; becauſe GO and OA are equal, BH and HA, and likewife
CZ and ZA will be equal: therefore the triangle HAZ is half of
the triangles BOA and COA together, and therefore is greater than
the half of the area BACO; and in like manner, if in the remain-
ing fegments, BHOM and CZOQ, the triangles KHI, RZP be in-
fcribed, they will be greater than the halves of theſe fegments, and
if theſe inſcriptions be continued, the fame will always follow;
wherefore the circumfcribed figure at length fhall cut off from the
area BACO a ſpace leſs than any given ſpace whatſoever.
PROP. XIII.
Let BOQC be a ſpace terminated by a parabola and the
right line BC: draw a line parallel to BC, touching the
parabola in O, and meeting in X and T, the diameters
drawn through the points B and C; the parabolic feg-
ment BOQC will be to the circumfcribed parallelogram
BXTC as 2 to 3.
D'
RAW the tangents BA, CA, and through their point of con-
courfe A the diameter AG, which will pafs through the point.
of contact O, and will bifect BC in G; then becauſe AG is double
of
BookIII.
139
Of the Parabola.
of AO, the triangle BAC will be equal to the parallelogram BXTC;
but the parabolic fegment BOQC is double the area BACO; for if
it exceeded the double of that area, by any ſpace S, then by the
firſt part of the preceding Prop. a figure may be infcribed in the
fegment BOQC, which fhall differ leſs from this fegment by a ſpace
lefs than S, and therefore this figure will be greater than the double
of the area BACO, contrary to Cor. 1. Prop. XI. and therefore the
ſegment BOQC does not exceed the double of the area BACO;
nor is it lefs than the double of this area; for if the area BACO
could exceed the half of the fegment BOQC, by any ſpace what-
foever as S; then by the ſecond part of the preceding Prop. a figure
may be circumfcribed about this fegment which fhall cut off a
fpace leſs than the fpace S; and therefore the area without this
figure and within the triangle BAC would exceed the half of the
fegment BOQC, contrary to Cor. 2. Prop. XI. therefore fince the
fegment BOQC does not exceed the double of the area BACO,
nor is leſs than the double, it neceffarily must be double of this
area, and therefore is to the whole triangle BAC, that is, to the pa-
rallelogram BXTC as 2 to 3. 2. E. D..
COR. I. Two ſegments LDP, AKE of the fame parabola are to Fig. 12.
one another in the fefquiplicate ratio of the abfciffes DM, KT of the
diameters, which bifect the lines LP, AE. Draw from the points
L and A in the parabola the lines LR, AH perpendicular to the
diameters DM, KT, they will be the altitudes of the parallelograms
LNDM, ABKT; therefore LNDM is to ABKT in a ratio com-
pounded of the ratio of DM to KT and of LR to HA; but the
fquare of LR is to the fquare of HA, as DM to KT (by Cor. 1.
III. of this Book) wherefore LR is to HA in the fubduplicate ratio
of DM to KT, and confequently LNDM and ABKT are in the
fefquiplicate ratio of DM and KT, and therefore the whole parallelo-
grams circumfcribed about theſe fegments are in the fame ratio;
but the parabolic fegments LDP and AKE are to one another as
thofe parallelograms, by the Propofition, whence the Corollary is
evident.
$ 2
COR.
140
Book III.
Conic Sections.
COR. 2. Hence, if LDP, AKE be fegments of different para-
bolas, they will be to one another, in a ratio compounded of the
fefquiplicate ratio of the abfciffes DM, KT and the ſubduplicate ratio
of the principal parameters; for the perpendiculars LR, AH are in
this caſe in a ratio compounded: of the fubduplicate ratio of the ab-
fciffes DM, KT, and the fubduplicate ratio of the principal para-
meters, as is evident from Cor. 1. III. of this Book.
ུ་
DE 1
Another Demonftration of the QUADRATURE of the
PARABOLA *
Let BMOQC be a fpace terminated by a parabola and the
right line BC. Draw a line parallel to BC touching the
parabola in O, and let it meet in the points X and T, the
diameters drawn through B and C. The parabolic feg-
ment BMOQC will be to the circumfcribing parallelogram
BXTC, in the proportion of 2 to 3.
THE
HROUGH B and C draw tangents meeting each other in
the point A, and meeting the tangent XT in H and Z.
Through the point A draw the diameter of the fegment bifecting
the line BC in G, and paffing through the point of contact O (Cor.
1. Prop. XXVI. Book I.) In the parabolic fegment BMOQC let
the triangle BOC be infcribed, having the line BC for its bafe, and
its vertical angle at O the vertex of the diameter, and in the re-
maining fegments let there be infcribed, in like manner, the tri-
angles BMO, OQC; and in the correfponding external areas (ter-
minated by the parabola and its tangents) let the triangles KHT,
PZR be infcribed, by drawing the lines KT and PR parallel to
BO and OC, and touching the parabola in the points M and Q.
* This demonftration was drawn up by the author, and communicated by a friend
of his to the tranſlator.
Since
!
Book III.
139
Of the Parabola.
Since the fegment of the diameter AG is bifected in its vertex O
(Prop. XLVII. Book I.) and the lines HZ and BC are parallel,
the triangles BOC and HAZ have equal altitudes, and the baſe BC
is double the baſe HZ; therefore the triangle BOC will be double
the triangle HAZ. In the fame manner it may be proved, that the
triangles BMO and OQC infcribed in the leffer parabolic fegments.
are reſpectively double the triangles KHT and PZR infcribed in
the correfponding external areas; and fince the fame proportion
holds between the feveral triangles that may be infcribed in all the
remaining parabolic fegments, and their correfponding external areas,
and the number of triangles, infcribed at every operation, are equal
in each feries, it follows; that the contents of the whole feries of
triangles infcribed in the parabolic fegment BMOQC, after any
number of operations, will be double the contents of the like fe-
ries infcribed in the area BACO, which is terminated by the para-
bola and its tangents AB and AC. And fince the number of theſe
infcribed triangles may be fo far increaſed, that the contents of
each feries fhall differ from the fpace in which they are infcribed by
a quantity lefs than any given ſpace whatever, the parabolic feg-
ment BMOQC will be double the area BACO: for if it was
* The contents of each feries fhall differ from the space in which they are infcribed by a
quantity less than any given space whatever.] To prove this, it must be fhewn that a
triangle inſcribed either in a parabolic fegment, or in the area contained between the
parabola and two tangents, as in the above demonstration, is more than half of that
fpace in which it is infcribed. Thus the triangle BOC is more than half the fegment
BMOQC, becauſe it is half of the circumfcribing triangle BAC, as standing on the
fame bafe BC, and having its altitude OG one half of AG. So likewife the triangle
HAZ is more than half of the area BACO; for fince AG is bifected in O, and the
lines HZ and BC are parallel, the tangents BA and CA will be bifected in H and Z,
confequently the triangles BOH, AOH are equal, as are alfo the triangles COZ, AOZ;
and therefore the triangle HAZ is one half of the triangles BOA and COA, which con.
tain the area BACO, ſo that the triangle HAZ is more than half of that area. Now
the difference between the parabolic ſegment BMOQC, or the area BACO, and a ſeries
of triangles reſpectively inſcribed in each of theſe ſpaces, after any number of opera-
tions, will be the remaining fegments in which triangles have not yet been inſcribed,
and ſince every additional inſcription of triangles will diminish the remaining differ-
ence by more than one half, it follows that this difference may, at length, be fo far di-
miniſhed as to become less than any given space whatever: by Prop. 1. 10. Book of
Euclid.
greater
140
Book III.
Conic Sections.
I
T
greater than double that area by any given quantity, ſuppoſe the
ſpace S, then fince a ſeries of triangles might be inſcribed in the
fegment BMOQC, which would differ from it by a quantity leſs
than the ſpace S, this feries of triangles would be more than double
the area BACO, and confequently would be more than double the
ſeries of correfponding triangles infcribed in the area BACO, con-
trary to what was proved before. Or if the fegment BMOQC
was leſs than double the area BACO, then that area would be
greater than one half of the fegment BMOQC by a certain quan-
tity, ſuppoſe the ſpace S, and therefore fince a feries of triangles
might be infcribed in the area BACO, which would differ from it
by a quantity lefs than the ſpace S, this feries would be greater
than one half of the parabolic fegment BMOQC, and confequently
greater than half of the whole feries of correfponding triangles in-
ſcribed in that ſegment, contrary to what was proved before. There-
fore it appears that the parabolic fegment BMOQC is neither more
nor leſs than double the area BACO, and confequently the feg-
ment BMOQC is to the whole triangle BAC in the proportion of
2 to 3. But the triangle BAC is equal to the circumfcribing paral-
lelogram, as having double its altitude and being on the fame baſe
with it, and therefore the parabolic fegment BMOQC is to the
circumfcribing parallelogram BXTC in the proportion of 2 to 3.
2. E. D.
END OF THE THIRD BOOK.
1
CONIC SECTIONS.
BOOK THE FOURTH.
The Ellipfe and Hyperbola.
PROPOSITION I.
If through the vertices of two conjugate diameters, four right
lines be drawn touching an ellipfe or conjugate hyperbo-
las; the parallelogram formed by the tangents will be
equal to the rectangle under the axes.
ET C be the center of an ellipfe or hyperbola, AB, Mm the
Leare, and the FG Ray
axes, and ED, FG any conjugate diameters; the tangents
drawn through their vertices will form a parallelogram (by Cor.
XXVII. Book I.) let the tangents drawn through the points F and
D, meet each other in H; FCDH will be a fourth part of this pa-
rallelogram; draw from the center CK perpendicular to the tangent
FH
FIG. t
142
Book IV.
Conic Sections.
FIG. 3.
FH. Becauſe (by Cor. 1. XXXI. Book II.) the perpendicular CK
is to the tranfverſe ſemiaxis CA, as the conjugate femiaxis CM to
the femidiameter CD parallel to the tangent FH, the rectangle
under CK and CD, that is, the parallelogram FCDH will be equal
to the rectangle under the femiaxis CA and CM; whence the Pro-
poſition is evident.
COR. It is manifeft, that a parallelogram infcribed in an ellipfe
or conjugate hyperbolas, by joining the vertices of two conjugate
diameters is half of the parallelogram circumfcribed about theſe
conjugates, and therefore all fuch parallelograms are equal.
PROP. II.
If from the vertices of two conjugate diameters CF, CG of
an ellipfe, the ordinates FE, GH be drawn to a third dia-
meter AB; the fquare of the part, intercepted between ei-
ther of the ordinates and center, will be equal to the rect-
angle under the fegments between the other ordinate and
the vertices of the diameter, to which the ordinates are
applied.
FO
OR through the vertex of either of the diameters CF, let a
line be drawn touching the ellipfe, and meeting the diameter
AB in D, and its conjugate Mm in O: becauſe FD, CG, and FE,
GH are parallel, the triangles FED, GHC will be equiangular, and
becauſe CO, FE are parallel, the rectangles OFD, CED will be
fimilar; and therefore OFD is to CED, as the ſquare of FD to the
fquare of ED; or becauſe the triangles are fimilar, as the fquare of
CG to the fquare of CH: but the rectangle OFD is equal to the
fquare of CG (LI. Book I.) therefore the fquare of CH is equal
to the rectangle CED, that is, to the rectangle AEB (Cor. 1.
XLIX. Book I.) under the fegments between the other ordinate FE
and the vertices of the diameter AB; and thefe equals being taken
from
Book IV.
143
The Ellipfe and Hyperbola.
1
from the fquare of CB, the remainders, viz. (5. 2.) the rectangle
AHB and the ſquare of CE will be equal.
Q. E. D.`
Q;
COR. It is evident that the fquares of the fegments of the dia-
meter AB, to which the ordinates are applied, viz. between the or-
dinates and center, are together equal to the fquare of the femidia-
meter CB for fince the fquare of CH is equal to the rectangle
AEB, the fquare of CH, together with the fquare of CE, will be
equal to the fquare of CB (5. 2).
:
PRO P. III.
The fum of the fquares of any two conjugate diameters of
an ellipfe is equal to the fum of the ſquares of the axes.
L
ET CB, CM be the femiaxes of an ellipfe, and CF, CG two FIG. 3.
conjugate femidiameters, and let EF, GH be perpendicular to
CB, and FL and GN perpendicular to CM.
Becauſe, by the preceding Cor. the fquare of CB is equal to the
ſquares of CE, CH together, and the fquare of CM equal to the
fquares of CL, CN, that is, to the fquares of FE, GH; the
fquares of CB, CM together will be equal to the four fquares of
CE, CH, FE, GH, to which likewiſe the ſquares of CF, CG are
equal (47. 1.) therefore the fquares of CF, CG are equal to the
fquares of CB, CM together, and therefore the fquares of the dia-
meters are equal to the fquares of the axes. 2. E. D.
T
PROP.
4
144
Book IV.
Conic Sections.
FIG. 4.
PROP. IV.
If the angles contained by the aſymptotes of an hyperbola
are right angles; any two conjugate diameters will be
equal to each other but if the angles contained by the
afymptotes are not right angles, any two conjugate dia-
meters will be unequal, and the difference of their ſquares
will be equal to the difference of the ſquares of
two conjugate diameters.
Part I.
L
any other
ET AD, AE be the afymptotes of an hyperbola, and
its center A; draw from the center A any femidiame-
ter AB, and through the vertex B a tangent meeting the aſymptotes
in the points D, K; BD or BK will be equal to the femidiameter
conjugate to AB (XXXVIII. Book I.) therefore if DAK be a
right angle, a circle defcribed from the center B about the dia-
meter DK will pass through the point A (Converf. 31. 3.) and
therefore the femidiameter AB is equal to BD or BK, that is,
to the femidiameter conjugate to it.
In this cafe, the hyperbola is faid to be equilateral.
Part 2. Let the aſymptotes AD, AE of the hyperbola contain an
acute angle, and draw the femidiameter AB and tangent DBK, as
before becauſe DAE is an acute angle, it is without the femicircle
deſcribed upon the diameter DBK (Converf. 31. 3.) and there-
fore the tranfverfe femidiameter AB will be greater than BD or BK.
Draw any other femidiameter AC, and a tangent through its vertex
C meeting the afymptotes in the points F, E: it is required to fhew
that the difference of the fquares of AB, BD is equal to the dif-
ference of the fquares of AC, CF. From the points B, K, C, E,
draw BG, KL, CH and EM perpendicular to the afymptote DA:
becauſe DB, BK are equal, DG, GL will be equal, and becauſe
FC, CE are equal, FH, HM will be equal, and fince BG is a per-
pendicular, therefore the difference of the fquares of AB, BD will
be equal to the difference of the fquares of GA, GD or GL, that
is,
Book IV.
145
The Ellipfe and Hyperbola.
is, to the rectangle DAL (6. 2.) and in like manner the difference
of the fquares of AC, CF is equal to the difference of the fquares
of HA, HF or HM, that is, to the rectangle FAM: but becauſe
the triangles DAK, FAE are equal (Cor. 2. XLV. Book I.) the
fides about the common angle are reciprocal, that is, DA will be to
FA (as AE to AK, that is, becauſe of the parallels) as AM to
AL; therefore the rectangle DAL is equal to the rectangle FAM;
and therefore the difference of the ſquares of AB, BD is equal to
the difference of the fquares of AC, CF. It is evident that in an
hyperbola conjugate to BCN, èvery tranſverſe diameter is lefs than
its conjugate diameter. 2. E. D.
If the angle contained by the afymptotes be half a right angle,
the exceſs of the fquares of two conjugate diameters will be double
the parallelogram defcribed about them; and if the axes of an hy-
perbola and their difference be proportionals, that excefs will be
equal to that parallelogram.
PROP. V.
The two diameters of an ellipfe, which biſect the right lines
joining the vertices of the axes, are conjugate diameters
and equal to each other and on the contrary, if two con-
jugate diameters of an ellipfe be equal, they will biſect
the right lines joining the vertices of the axes.
Part I.
L
ET AB, Mm be the axes of an ellipfe, and join AM, FIG. 5.
BM, and draw the diameters ED, FG, which biſect
theſe lines, and let FG meet BM in I, and becauſe MI, IB, and
alfo AC, CB are equal, the diameter GF will be parallel to AM,
which is bifected by ED; therefore ED, FG are conjugate dia-
meters (XXVII. Book I.) and becauſe MCB is a right angle, and
IB, IM equal, the center of the circle deſcribed about the triangle
MCB will be in the point I; therefore IC will be equal to IB, and
T 2
confe-
146
Book IV.
Conic Sections.
confequently the angle ICB is equal to the angle IBC, that is, to
the alternate angie BCD; therefore the diameters ED, FG are equal,
by Cor. 3. Prop. VIII. Book II.
Part 2. If two conjugate diameters ED, FG of an ellipfe be
equal, and from the extremity B of one axis a line be drawn pa-
rallel to either of the conjugate diameters, fuppofe to ED, it will
pafs through the extremity of the other axis; for let this line meet
the ellipfe again in M, and the diameter FG in I, BM will be bi-
fected in I (Cor. XXVII. Book I.) join CM, and becauſe the dia-
meters ED, FG are equal, the angle ICB is equal to the angle BCD
(Cor. 3. VIII. Book II.) that is, to the alternate angle IBC, and
confequently IC is equal to IB or IM; therefore a circle may be
deſcribed from the center I paffing through the points B, C, M;
and therefore the angle MCB in a femicircle will be a right angle;
wherefore fince AB is an axis, MCm will be the other axis; and
therefore the equal conjugate diameters ED, FG bifect the lines
joining the vertices of the axes. 2. E. D.
COR. 1. Hence it is manifeft that there can be but two conjugate
diameters of an ellipfe equal to each other.
COR. 2. Hence likewife, if the lines MA, MB be drawn from
the vertex M of one of the axes to the vertices of the other axis,
the angle contained by theſe lines will be equal to the oppofite an-
gles ECF contained by the equal conjugate diameters.
J
I
PROP.
Book IV.
147
The Ellipfe and Hyperbola.
PROP. VI.
The ſquare of the fum of the equal conjugate diameters of
an ellipſe is equal to the fquare of the ſum of any other
two conjugate diameters, together with the fquare of their
difference.
ET the right lines GC, CF be equal conjugate diameters of an FIG. 6.
ellipſe, and AB, BD any unequal conjugates, and AE their
difference; the fquare of GF will be equal to the fquare of AD,
together with the fquare of AE.
Lellipfe, right AB, 16 2 CF the equal conjugate liau ters of air
For the fquares of GC, CF taken twice are equal to the fquares
of AB, BD taken twice (by III. of this Book) but the fquares of
GC, CF taken twice are equal to the ſquare of GF (4. 2.) and the
ſquares of AB, BD taken twice are equal to the fquare of AD, to-
gether with the fquare of AE (10. 2.) therefore the fquare of GF
is equal to the fquare of AD, together with the fquare of AE.
Q, E. D.
COR. Hence it appears that the fum of equal conjugate diameters
is greater, and the fum of the axes lefs than the fum of any other
conjugate diameters; for the difference of the axes is greater than
the difference of any other conjugate diameters, as is evident from
Prop. VIII. Book II.
PROP.
148
Book IV:
Conic Sections.
}
FIC. 7.
PROP. VII.
The obtufe angle contained by equal conjugate diameters of
an ellipfe is the greateſt, and the acute angle the leaſt of
all the angles which can be contained by any two conju-
gate diameters.
LE
ET AB be the tranfverfe axis, and Mm the conjugate axis of
an ellipfe: join AM, MB, the angle AMB is equal to the ob-
tufe angle contained by the equal conjugate diameters (Cor. 2. V.
of this Book) let CR, CT be any two unequal conjugate diame-
ters, and draw from the vertex A of the tranfverfe axis, AR an or-
dinate to CR, and meeting the ellipfe in D, it will be parallel to
CT, and join DB meeting CT in T; becauſe AC, CB and AR,
RD are equal, CR, TD will be parallel; therefore the angle ADB
is equal to the angle RCT; but it is evident that the angles ADB,
AMB are obtufe, being in a fegment less than the femicircle de-
ſcribed upon AB, as appears from (Prop. VIII. Book II.) if a cir-
cle be deſcribed through the points A, M, B, its fegment AMB will
be wholly within the ellipfe: for the line BK touching the circle in
B, and produced towards M, contains an acute angle KBA with
the axis AB becauſe the alternate fegment BEA is greater than a
femicircle) therefore this line, and confequently the arch BM fall
within the ellipfe; but if it could meet the ellipfe between the
points B and M, and a line be drawn from this point ordinately ap-
plied to the conjugate axis Mm, its other extremity would be in the
periphery of the circle AMB, and therefore the ellipfe and circle.
would meet in five points, contrary to (Cor. 3. LVI. Book I.) there-
fore the ſegment of the circle AMB is wholly within the ellipſe,
and therefore the angle AMB in this fegment is greater than the
angle ADB without the fame fegment: hence the Propofition is
evident.
(
PROP.
Book IV. The Ellipfe and Hyperbola.
149
PROP. VIII. PROBL. I.
Two conjugate diameters of an ellipfe or hyperbola being
given in poſition and magnitude, to find two other con-
jugate diameters which fhall contain given angles.
Cafe 1.
If the given angles be right angles, the diameters re-
quired will be the axes, and the problem is folved by
Prop. VII. Book II.
WHE
HEN the given diameters are the axes, let AB be FIG.
the greater, and Mm the leffer axis of the ellipſe or
hyperbola, and VZX, VZY the given angles; upon the greater
axis AB deſcribe the circle AFBG, ſo that the ſegments AFB, AGB
may contain angles equal to the given angles VZX, VZY, and let
FG be a diameter of this circle coinciding with the axis Mm, and
P be its center.
If the given obtufe angle VZX be equal to the angle contained
by the lines AM, BM in the ellipfe; the fegment AFB of the circle
meets the axis Mm in its vertex M, in which cafe the diameters
fought will be the equal conjugate diameters, by Cor. 2. Prop. V.
and if the given angle VZX be greater than the angle AMB, the
problem will be impoffible by the preceding Prop. but if the given
angle VZX be leſs than AMB, the fegment AFB of the circle meets
the axis Mm produced in F: then draw through the point F a line
parallel to MB, it will meet the circle fomewhere between the points
F and B, fuppofe in S; draw through the point S a line parallel to
BC, it meets CM in O, which will be between the point F and the
center C of the ellipfe; then becauſe, by a property of the circle,
GO, OS, OF are proportional, GO will be to OF as the fquare of
OS to the fquare of OF, that is (becauſe the triangles OSF, CBM
are fimilar) as the fquare of CB to the fquare of CM: in like man-
ner in the hyperbola take a point O fuch, in the diameter FG pro-
duced, that GO be to FO as the fquare of CB to the fquare of
CM, and in both fections from the center P, with the interval PO,
defcribe
178
150
Book IV.
Conic Sections.
1
defcribe a circle meeting the axis AB in two points, and let Q be
one of them; draw through the point Q a diameter of the circle
AFBG meeting it in the points D, N; the point D, which is nearer
to Q₂ will be in the ellipfe or hyperbola, whofe tranfverfe axis is
AB and Mm its conjugate. For draw from the point D a right line
perpendicular to AB, meeting it in H, and the circle again in L,
and join LN; becauſe the angle DLN is in a femicircle it is a
right angle, and LN, HQ will be parallel; but the rectangle HLD
is to the fquare of DH, as LH to HD, or as NQ to QD, that is,
by conſtruction, as GO to OF, or as the fquare of CB to the ſquare
CM, therefore the rectangle LHD or (by the circle) AHB is to
the fquare of DH, as the fquare of AB to the ſquare of Mm: there-
fore the point D is in the ellipfe or hyperbola, whofe tranſverſe axis
is AB and conjugate axis Mm (Cor. Prop. XXXIV. and XXXV.
Book I.)
Let therefore the right lines AD, BD be inflected to the point D,
and draw from the center of the fection CR, CT, bifecting theſe
lines, they will be the diameter fought. For becauſe AR, RD and
likewiſe AC, CB are equal, CR will be parallel to BD, and for the
fame reafon CT will be parallel to AD; therefore CR, CT are con-
jugate diameters (XXVII. Book I.) and contain an angle RCT
equal to ADB, and therefore contain angles (equal to thofe in the
fegments AFB, AGB of the circle, that is) equal to the given an-
gles VZX, VZY. Becauſe the circle deſcribed with the interval PO
meets the axis AB in another point befide Q, it is manifeft that in
the fame manner two other conjugate diameters may be found,
which ſhall contain angles equal to the given angles VZX, VZY.
2. E. I.
If the circle AFBG be defcribed upon the leffer axes, then, the
diameter DQN being found as before, the point N, more remote
from Q, will be in the ellipfe or hyperbola, as is evident.
When the axes of the hyperbola are equal, defcribe upon either
axis a circle, fo that its fegments fhall contain angles equal to the
given angles VZX, VZY, as before; and draw the right line DH
per-
Book IV.
151
The Ellipfe and Hyperbola.
perpendicular to the axis AB, meeting it in the point H, and touch-
ing the circle in D; becauſe the fquare of DH is equal to the rect-
angle AHB, the point D will be in the hyperbola, and the diame-
ters CR, CT, which biſect the right lines AD, BD, will be the dia-
meters fought.
Cafe 2. When the given diameters are not the axes: becauſe by
Prop. VII. Book II. the axes may be found from two conjugate
diameters given in pofition and magnitude, this cafe is reduced to
the preceding; and when the pofition of the diameters fought is
found, their vertices are found in the fame manner as the vertices
of the axes were found by Prop. VII. Book II.
Or this problem may be more generally folved, whatſoever con-
jugate diameters be given in pofition and magnitude, by defcribing
upon one of the given diameters a circle, fo that its fegments fhall
contain angles equal to the given angles, and by finding the inter-
fection of this circle with the ellipfe, or with the hyperbola, when
it is deſcribed about a tranſverſe diameter; which may be done by
a method fimilar to that, by which the interfection was found of the
ellipſe or hyperbola with the circle deſcribed upon its axis in the
firſt caſe of this Prop.
PROP. IX.
PROBL. II.
Two conjugate diameters of an ellipfe or hyperbola being
given in poſition and magnitude, and a right line given
in pofition; to find the points in which this right line
meets the ſection.
L
ET AB, Mm be the given diameters, and C the center of the FIG. 9,
ellipfe or hyperbola.
Cafe 1. Firſt, if the line given in pofition be parallel to one of
the conjugate diameters, and when the fection is an ellipfe, meets
the other diameter between its vertices, or when the fection is an
U
hyper-
10.
152
Book IV.
Conic Sections.
hyperbola, if it meets the tranfverfe diameter produced; then be-
cauſe the line will be ordinately applied to this diameter, the points,
in which it meets the fection, may be found by Cor. XXXIV. and
XXXV. Book I. and if it meets the conjugate diameter of the
hyperbola, the points in which it meets the oppofite hyperbolas may
be found by Prop. XXXII. Book I.
Cafe 2. But if the line PQ given in pofition be not parallel to ei-
ther of the diameters AB, Mm: draw through the vertex of the
diameter AB, the line AT parallel to the other diameter Mm, it
will touch the ſection (Cor. XXVII. Book I.) draw through the
center a diameter parallel to PQ ánd meeting AT in H, and take
a ſegment AK in the line AT (towards the point H in the hyper-
bola, and the contrary way in the ellipfe) fo that the rectangle
KAH may be equal to the fquare of CM, and join CK; CH, CK
will be conjugate diameters (Cor. Prop. LI. Book I.) draw there-
fore AE to the diameter CK and parallel to CH, it will be an or-
dinate applied to CK. Let therefore CE, CF, CK be proportional,
and F will be the vertex of the diameter, which is in the line CK :
in the fame manner are found the vertices of the diameter, which
is in the line CH: therefore this cafe is reduced to the firft. But if
in either cafe, the line PQ paffes through the vertex of one of the
conjugate diameters, and is parallel to the other, it will touch the
fection, as is evident.
COR. Hence, two conjugate diameters being given in pofition
and magnitude, a right line may be drawn which fhall touch the
fection, and be parallel to a right line given in pofition.
1
:
PROP.
2
1
C
G
M
K
F
E
T
M
B
N
FIG.12.
B
H
FIG.I
I
R
A
L
m
II
H
TAB.II. Page 152.
M
FIG.2.
F
A
K
B
M
E
# WH
L
M
FIG LI·
A
FIG.3 ·
E
H
B
K
D H
B
FIG. 5.
N
G
FIG4
G
F
E
S
H
P
X
Z
T
M
K
m
ηι
G
F
M
D
ik
D
LFIG. 8 ·
R
S
P
H
B
FIG. 7.
P
D
R
M
H
2
F
m
B
DE V
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C
C
G-
F
FIG.6.
V
G
A-
B
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Z
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이
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OF
ICH
B
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Book IV. The Ellipfe and Hyperbola.
153
PROP. X. PROB L. III.
I.
To defcribe an ellipfe or oppofite hyperbolas about a given FIG. 11,
parallelogram EFGH, fo that the diameter of the ſection,
parallel to the right lines EH, FG, fhall be to its conju-
gate in a given ratio, viz. as TS to TQ.
D
RAW the right line AB bifecting EH, FG, and the right
line Mm bifecting EF, HG; AB will be parallel to the lines
EF, HG, and Mm to the lines EH, FG; it is manifeft that the in-
terſection C of thefe lines will be the center of the fection required,
and that the diameters lying in the lines AB, Mm will be conju-
gate; let the line AB meet EH in K, and the line Mm meet EF in
D and HG in V.
Cafe 1. First, let the fection to be deſcribed be an ellipfe, and as FIG. 11.
TS is to TQ, fo let EK be to KL, and let KL be perpendicular
to AB, from the center C, with the interval CL, defcribe a circle.
meeting AB in the points A, B; and as TS is to TQ take CM or
Cm to CA; the fection, whofe conjugate diameters are Mm, AB, is
the ellipfe to be deſcribed. For by conftruction, the ſquare of CM
is to the fquare of CA, as the fquare of TS to the fquare of TQ,
and the fquare of EK is in the fame ratio to the fquare of KL, or
(by a property of the circle) to the rectangle BKA, and the right
line EK is parallel to CM; therefore the point E will be in the el-
lipfe (Cor. XXXIV. Book I.) and becauſe EK, KH are equal,
the point H will be in the ellipfe; and in like manner, becauſe
ED, DF, and HV, VG are equal, the points F, G will be in the
ellipfe.
Cafe 2. Now let the oppofite hyperbolas be the fection to be de- FIG. 12.
fcribed, through the points E, F, G, H; in this cafe the lines TS,
TQ must not be to each other as EK to CK; therefore let EK be
to KC in a ratio lefs than the ratio of TS to TQ, and let the line
U 2
EK
154
Book IV.
Conic Sections.
EK be to KL, as TS to TQ; KL will be leſs than KC; deſcribe
a femicircle upon CK, and infcribe in it the line KL, join CL,
and take CA or CB equal to CL, and let CM or Cm be to CA, as
TS to TQ.
The oppofite hyperbolas, of which AB may be a tranſverſe dia-
meter, and Mm its conjugate, will be the hyperbolas to be deſcribed;
for by conſtruction, the ſquare of CM is to the ſquare of CA, as
the fquare of TS to the fquare of TQ or as the fquare of EK (to
the fquare of KL, viz. the difference of the fquares of CA, CK,
that is) to the rectangle BKA (6. 2.) but EK is parallel to CM;
therefore the point E will be in one of the oppofite hyperbolas
(Cor. XXXV. Book I.) and becaufe EK, KH are equal, the point
H will be in the fame hyperbola, and becauſe ED, DF, and HV,
VG are equal, and theſe lines are parallel to the diameter AB, and
meet its conjugate Mm in D and V, the points F and G will be in
the oppofite hyperbola.
But if EK be to KC in a greater ratio than TS to TQ, let the
ſquare of TS be to the ſquare of TQ as the ſquare of EK to the
fum of the ſquares of KC and of another line, fuppofe CA; then
defcribe from the center C, with the interval CA, a circle meeting
CK in the points A, B, and as TS to TQ, ſo let CM or Cm be to
CA; the oppofite hyperbolas, of which Mm may be a tranfverfe dia-
meter and AB its conjugate, will be the hyperbolas to be deſcribed:
for the ſquare of CM is to the fquare of CA as the ſquare of TS
to the fquare of TQ, or as the fquare of EK to the fum of the
fquares of CK and CA; therefore the hyperbolas will pass through
the points E, H, G, F, as is evident from Prop. XXXII. Book I.
2. E. F.
But if TS be to TQ as EK to KC, the problem in this cafe is
impoffible. For if the hyperbolas be defcribed, whofe conjugate
diameters fhall lye in the lines Mm, AB, and fhall be to each other as
EK to KC, the point E will be in one of their afymptotes.
COR. 1. As it appears from the conditions of this problem that
the diameters AB, Mm are given in pofition and magnitude, it is
evident
Book IV.
155
The Ellipfe and Hyperbola.
evident that one ellipfe only, or two oppofite hyperbolas only, can
be deſcribed to anfwer the problem.
COR. 2. Hence, an ellipfe or oppofite hyperbolas may be de-
ſcribed about the given parallelogram EFGH, fo that the ellipfe,
or one of the hyperbolas fhall paſs through a given point P. For
draw the right lines AB, Mm bifecting the oppofite fides of the pa-
rallelogram, the conjugate diameters of the fection to be defcribed
will be in theſe lines; draw through the point P a line parallel to
AB, and meeting Mm in R, and the fide EH in O, and let RN be
equal to PR : then deſcribe through the points E, F, G, H, an el-
lipfe, or the oppofite hyperbolas, fo that the fquare of the diameter
in the line AB fhall be to the fquare of the diameter in the line Mm
as the rectangle PON to the rectangle EOH; the point P will be
in this ellipfe, or in one of the hyperbolas, as is evident.
If the ſection to be deſcribed be an ellipfe, the point P muſt be
between the oppofite fides, produced, of the parallelogram; but if
it be an hyperbola, the point P must be within the parallelogram,
or between the adjacent fides produced.
COR. 3. Hence, an ellipfe or oppofite hyperbolas may be de-
ſcribed, ſo that the given point C fhall be their center, and the line
MCm given in pofition a diameter, and likewife the parallels EF,
PN bifected by MCm in the points D, R, fhall be ordinately ap-
plied to that diameter. For take in the line Mm the fegment CV
equal to CD, and through V draw GH parallel and equal to EF,
and bifected in V: join HE, GF, and defcribe by the preceding
Cor. the ellipfe or oppofite hyperbolas about the parallelogram
EFGH, ſo that one of them may pafs through the point P, this
will be the fection to be deſcribed: for the line Mm will be a dia-
meter, becauſe it biſects the parallels EF, GH terininated by the
ellipfe or oppofite hyperbolas, and the line PN parallel to EF, GH,
will be ordinately applied to it; and becauſe the equal ordinates
EF, GH are equally diſtant from the point C, that point will be the
center. If the given lines EF, PN be equal, and equally diſtant
from the point C, feveral ellipfes or hyperbolas may be defcribed,
which will anſwer the problem..
But
1
;
156
Book IV.
Conic Sections.
FIG. 13,
14.
But if thefe lines EF, PN be equally diftant from the point C,
when they are unequal; or be at different diftances from the fame
point when they are equal, the problem will be impoffible. If the
leffer line EF be at a greater diftance from the point C than the
greater line PN, the fection to be deſcribed will be an ellipfe; but
if the contrary happens, oppofite fections are to be deſcribed: and
fince in either cafe, it appears that the fections to be deſcribed muſt
neceffarily pafs through fix given points, from the conditions of the
problem, it is evident that one ellipfe only can be defcribed, or two
oppofite hyperbolas only, which fhall anſwer the problem, becauſe
two conic ſections or oppofite fections (Cor. 3. LVI. Book I.) can
not meet each other in five, much lefs, in fix points.
This Propofition and Corollaries, and likewife Prop. XIV. and
XV. of this Book, are taken from Marq. de l'Hofpital's Conic
Sections.
PROP. XI.
Let a parallelogram KLNO be infcribed in an ellipfe or op-
pofite fections given in pofition; if the diameter AB, bi-
fecting the oppofite fides in the points R and S, has al-
ways a given ratio to its fegment RS between thofe fides;
the parallelogram KLNO will be always of the fame
magnitude.
L
ET C be the center of the fection, CR, CS will be equal, as
is evident draw the diameter Mm parallel to the fides LN,
KO, it will be conjugate to AB; defcribe a parallelogram about
thefe conjugates, it will be equiangular to the parallelogram KLNO,
and therefore thefe parallelograms will be to each other in a ratio
compounded of the ratios of their fides; by hypothefis, the ratio
of AB to RS or KL is given; therefore the ratio of the fquare of
CB is given (to the fquare of CR, and therefore) to the rectangle
ARB (the difference of the fquares of CB and CR) and confe-
quently the ratio of the diameter Mm to LN ordinately applied to
AB
Book IV.
157
The Ellipfe and Hyperbola.
AB is given; therefore the ratio compounded of thefe given ratios.
of AB to KL, and of Mm to LN is given; and therefore the ratio
of the equiangular parallelograms is given: but the parallelogram
AMBm is given in magnitude (Prop. I. of this Book) and there-
fore the parallelogram KLNO is always of the fame magnitude.
Q, E. D.
PROP. XII.
340
If the fides of a parallelogram EFGH touch an ellipfe, or FIG. 13,
produced touch oppofite hyperbolas given in pofition, and
if the diameter AB of the ſection, which (if neceſſary
produced) paffes through the oppofite angles E and G,
has a given ratio to the fegment EG between thoſe angles;
the parallelogram EFGH will be always of the fame
magnitude.
L'
ET K, L, N, O be the points of contact, the right lines KN,.
LO joining the oppofite points of contact will be diameters
(Cor. 2. XXII. Book I.) and the four right lines joining theſe points
will form a parallelogram KLNO; draw the diameter AB bifect-
ing the parallels LN, KO in R and S, it will pass through the an-
gles E and G (Cor. 1. XXVI. Book I.) in like manner the dia-
meter Mm bifecting KL, ON will pafs through the other angles
F and H, and will be conjugate to AB. Then becauſe, by hypo-
thefis, the ratio of CB to CG is given, the ratio of CB tc CR is
alſo given; therefore, as in the preceding, the ratio of the femidia-
meter CM to the ordinate LR, or to CT equal to LR, is given;
therefore the ratio of CM to CF is given; and therefore if MP, FQ
be drawn perpendicular to the diameter AB, the ratio of MP to
FQ will be given; therefore the ratio compounded of theſe given.
ratios of AB to EG, and of MP to FQ is given, that is, the ratio
of the rectangle under AB, MP to the rectangle under EG, FQ ::
but the rectangle AB, MP is given in magnitude (Prop. I. of this-
Book)
1.
158
Book IV.
Conic Sections.
FIG. 15,
16.
Book) therefore the rectangle EG, FQ, that is, the parallelogram
EFGH is always of a given magnitude. 2. E. D.
PROP. XIII.
If the fides of a parallelogram EFGH touch an ellipfe, or
produced touch oppofite hyperbolas; this parallelogram
will be to a parallelogram defcribed about any conjugate
diameters of the fection, as one of its fides HG is to
that diameter of the fection which is parallel to that
fide.
ET the diameter AB of the fection be parallel to the fides
HG, EF, and draw the diameter Mm conjugate to AB, it will
pafs through the points in which the fides HG, EF touch the ellipfe
or oppofite hyperbolas (Cor. XXVII. Book I.) let the parallelogram
KLNO be deſcribed about theſe conjugate diameters, this and the
parallelogram EFGH will be between the fame parallels; there-
fore EFGH is to KLNO, as HG to (ON or) the diameter AB,
which is parallel to HG but KLNO is equal to the parallelogram
deſcribed about any other conjugate diameters, whence the Prop.
is evident.
COR. 1. It is manifeft from this Propofition, that the parallelo-
gram defcribed about any two conjugate diameters of an ellipfe is
the leaſt of all the parallelograms which can be deſcribed about the
fame ellipfe.
COR. 2. Let AM and BM be joined, it is evident that the tri-
angle AMB is the greateſt that can be infcribed in this femi-ellipfe,
and confequently the parallelogram infcribed in the ellipfe, by join-
ing the vertices of the conjugate diameters, is the greateſt that can
be infcribed in the fame ellipfe; therefore the greateſt parallelo-
gram inſcribed in an ellipfe is half the leaſt parallelogram circum-
ſcribed about the fame ellipfe. What has here been demonftrated
concerning parallelograms infcribed in, or circumfcribed about an
ellipfe
Book IV.
159
The Ellipfe and Hyperbola.
ellipfe or about the hyperbolas, holds good of parallelograms in-
ſcribed in, or circumfcribed about a circle.
PROP.
XIV.
If from the center of an hyperbola, the fegments CK, CL FIG. 17.
of an afymptote CN be taken, which fhall be to each
other as any other two parts CG, CH taken in the fame
afymptote CN and if the right lines GF, HD, KB, LE
be drawn parallel to the other afymptote CP, and meet-
ing the hyperbola in the points F, D, B, E: and the fe-
midiameters CF, CD, CB, CE be drawn; the hyperbolic
fectors CBE, CDF will be equal.
FOR
OR draw the right lines BD, EF meeting the afymptotes in
the points M, O and N, P, and becauſe KB, HD, CO are
parallel, MK will be to MB as HC to DO; therefore as MB, DO
are equal, MK, HC will be equal: and in like manner, becauſe
LE, GF, CP are parallel, and NE, FP equal, NL, GC will be
equal; therefore MK is to NL (as HC to GC, that is, by hypo-
thefis, as LC to KC, or (on account of the hyperbola) as KB to
LE: but the angles MKB, NLE are equal; therefore the triangles
MKB, NLE are fimilar, and conſequently the lines MB, NE, that
is, BD, EF are parallel. Draw then the femidiameter bifecting the
parallels BD, EF in the points A, Q; and becauſe it will bifect all
the lines terminated by the hyperbola, and parallel to BD, EF; it
will bifect the area EBDF; therefore fince the triangles CQF, CQE
are equal, if from them the equal areas QADF, QABE and the
equal triangles CDA, CAB be taken, the remainders, viz. the hy-
perbolic fectors CDF, CBE will be equal.
COR. If the parts CG, CH, CK of the afymptote CN be in con-
tinued proportion, and GF, HD, KB be drawn, as in the propo-
fition, and BF be joined; in the fame manner it may be proved
that
X
i
160
Book IV.
Conic Sections.
!
that the line BF is parallel to the tangent drawn through the point
D; and thence that the fectors CDF, CDB are equal; and there-
fore if any number of parts CG, CH, CK, CL, &c. be in continued
proportion, and the lines GF, HD, KB, LE, &c. be drawn parallel
to the afymptote CP; the ſectors CDF, CDB, CBE, &c. will be all
equal to each other.
FIG. 17.
PRO P. XV.
If the right lines DH, BK be drawn from the vertices of
the femidiameters CD, CB to an afymptote CN, and pa-
rallel to the other afymptote CP; the hyperbolic ſector
CBD will be equal to the hyperbolic trapezium DHKB.
FOF
OR the triangles CHD, CKB are equal, being the halves of
equal parallelograms (Cor. 2. XLIII. Book I.) therefore if
the common triangle CHR be taken from them, the triangle CDR
will be equal to the trapezium HKBR, and if DBR be added to
thefe equal ſpaces, the fector CBD will be equal to the hyperbolic
trapezium DHKB.
COR. Hence and from the preceding Corollary, if any number
of parts CG, CH, CK, CL of the afymptote CN, be taken in con-
tinued proportion, and the lines GF, HD, KB, LE, &c. be drawn
to the hyperbola, and parallel to the other afymptote CP, the hy-
perbolic trapeziums FGHD, DHKB, BKLE, will be all equal to
each other.
PROP. XVI.
Being the Vth. of Archimede, concerning the Conoid and
Spheroid.
The ſpace contained by an ellipfe is to the circle defcribed
upon the tranfverfe axis of the ellipfe, as the conjugate
axis to the tranſverſe axis.
Archi-
E
K
M
F
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FIG. 9.
E
M
B
A.
B
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D H
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Q
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FIGII
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TAB 12 Page 16
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FIG.IO.
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FIG.15.
IK
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FIG·12
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FIG.13.
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FIG.16.
A
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Book IV.
161
The Ellipfe and Hyperbola.
L
Archimede's Demonftration.
ET AC be the tranfverfe axis and BD the conjugate axis of FIG. 18.
an ellipfe, and let a circle be defcribed upon the diameter AC;
it is to be proved that the ſpace contained by the ellipfe will be to
this circle, as BD to CA or EF: let the circle Z have to the circle
AECF the fame ratio as BD to EF; I fay that the circle Z is equal
to the ellipfe. For if the circle Z be not equal to the ſpace ABCD
contained by the ellipfe; firft, if poffible, let it be greater: then a
polygon, having an even number of angles, may be infcribed in
the circle Z greater than the ſpace ABCD: let it be conceived to
be infcribed, and infcribe in the circle AECF a polygon fimilar to
that infcribed in the circle Z; and draw from its angles perpendi-
culars to the diameter AC, and let the points in which the perpen-
diculars cut the ellipfe be joined by right lines therefore there
will be fome polygon infcribed in the ellipfe, which will be to a po-
lygon infcribed in the circle AECF as BD to EF; for becauſe the
perpendiculars EH, KL are divided in the fame ratio in B and M
(Cor. 2. Prop. XXX. Book I.) it is evident, that the trapezium
LE has the fame ratio to LB, as HE to HB; therefore each of the
other trapeziums in the circle will have to each of the other trape-
ziums in the ellipfe the fame ratio as EH to BH; and the trian-
gles in the circle, at the points A, C, have the fame ratio to thoſe
in the ellipfe; therefore the whole polygon infcribed in the circle.
AECF will be to the whole polygon infcribed in the ellipfe as EF to
BD: but this fame polygon is to the polygon infcribed in the circle
Z in the fame ratio; becauſe the circles have this ratio; therefore
the polygon infcribed in the circle Z is equal to the polygon in-
ſcribed in the ellipfe, which is contrary to the hypothefis, for it was
fuppofed greater than the whole fpace contained by the ellipfe. Now
let, if poffible, the circle Z be lefs than the ſpace ABCD, a poly-
gon contained by an even number of fides, and greater than the
circle Z, may be infcribed in the ellipfe. Let it be infcribed, and
draw from the angles perpendiculars to AC, and produce them to
X 2
the
162
Book IV.
Conic Sections.
the circumference of the circle: then again a polygon will be in-
fcribed in the circle AECF, which will have the fame ratio to the
polygon inscribed in the ellipfe, as EF to BD, and if a polygon fimi-
lar to this be infcribed in the circle Z, it may be demonftrated, that
the infcribed polygon in the circle Z would be equal to the inſcribed
polygon in the ellipfe, which is impoffible; therefore the circle Z
is not less than the ſpace contained by the ellipfe; and it has been
proved that it is not greater, and therefore it is evident that the
faid ſpace has the fame ratio to the circle AECF as BD to EF.
Q. E. D.
COR. 1. Hence, it is manifeft that the ſpace contained by an el-
lipfe is equal to a circle whofe diameter is a mean proportional be-
tween the axes of the ellipfe.
COR. 2. Hence, an ellipfe is to the rectangle under its axes as a
circle to the fquare of its diameter. For an ellipfe is equal to a
circle whofe diameter is a mean proportional between the axes of
the ellipfe, and the rectangle under the axes is equal to the fquare
of the diameter of this circle; therefore the ellipfe is to that rect-
angle as that circle is to the fquare of its diameter.
END OF THE FOURTH BOOK.
CONIC
CONIC SECTIONS.
BOOK THE FIFTH.
Of fimilar Sections, and of thofe Properties of the
Conic Sections, which depend upon right lines
harmonically divided; of circles which have the
fame curvature with the Sections: and of de-
fcribing Sections, which fhall pass through given
points, and touch right lines given in pofition.
DEFI
:
164
Conic Sections.
Book V.
FIG. 1,
2.
I'
DEFINITION I.
F two conjugate diameters of an ellipfe or hyperbola be to each
other as two conjugate diameters of another ellipfe or hyperbola,
and contain the fame angles with them; theſe two ellipfes or hyper-
bolas are faid to be Similar.
PROPOSITION I.
Let two ellipfes or hyperbolas be fimilar to each other, and
fo placed that they may have the common center C, and
that two conjugate diameters AB, MN of one fection may
coincide with the two conjugate diameters ab, mn of the
other ſection, to which they are proportional; I ſay that
any other two conjugate diameters of the one fection will
coincide with two conjugate diameters of the other ſec-
tion, and that all the coinciding diameters will have the
fame ratio to each other.
Part 1.
DR
RAW the conjugate femidiameters CD, CF of the
exterior fection, viz. of which AB, MN are diame-
ters; then the femidiameters Cd, Cf of the interior fection coin-
ciding with CD, CF, will be conjugate.
For draw through the vertices M, m of the coinciding diameters
two right lines touching the fections, they will be parallel to each
other (by hypothefis, and Cor. XXVII. Book I.) let them meet
the femidiameters CD, CF, when neceffary produced, in the points
L, O, and H, K; then by a property of parallel lines, the rect-
angles OML, KmH will be fimilar; therefore will be to each
other as the fquares of OM, Km or of CM, Cm, that is, as the
ſquares of CA, Ca by hypothefis; but the rectangle OML is equal
to the fquare of the femidiameter CA (LI. Book I.) therefore the
rect-
Book V.
165
Conic Sections.
rectangle KmH is equal to the fquare of the femidiameter Ca of
the interior fection, and therefore Cd, Cf will be conjugate femi-
diameters of that fection (by Cor. LI. Book I.)
Part 2. Now draw any two femidiameters CM, CD of the ex-
terior ſection, and let Cm, Cd be the femidiameters of the interior
fection, coinciding with them; CM will be to Cm, as CD to
Cd.
Join MD and draw CR biſecting MD in P, and let Cr be the femi-
diameter of the interior fection, and coinciding with CR; becaufe,
by the first part, the diameters which are conjugate to CR, Cr, co-
incide, the ordinates applied to CR, Cr will be parallel: draw mp
an ordinate to Cr, and let it be produced to meet CD in any point
d; then becauſe MD, md are parallel, and MP, PD equal; mp, pd
will be equal: therefore becauſe mp is ordinately applied to Cr and
the point m is in the interior fection; the point d will be in the
fame fection; and therefore the point d is the vertex of the femi-
diameter Cd, and confequently becaufe MD, md are parallel, CM
will be to Cm as CD to Cd. Q. E. D.
COR. 1. If two ellipfes or hyperbolas are fimilar and fimilarly
placed (that is, placed as in the propofition) and the right line TX
terminated by the exterior fection touches the interior one; it will
be biſected in the point of contact: for let m be the point of con-
tact, and draw through the coinciding diameters CM, cm, and
becauſe the tangent TX is parallel to the diameter conjugate to
Cm, and the diameters conjugate to Cm, CM coincide, by the firſt
part of this Prop. the line TX will be an ordinate to the diameter
CM, and therefore will be bifected in the point m.
COR. 2. Or if the right line YZ terminated by the exterior fec-
tion meets the interior one in two points m, d; the fegments Ym,
dZ, intercepted both ways between the fections, will be equal:
for draw the diameter CR bifecting YZ in p; the diameter Cr co-
inciding with CR will bifect md in the fame point p; for becauſe
the diameters conjugate to CR, Cr, coincide, the ordinates applied
to
*
166
Bock V.
Conic Sections.
FIG. 2.
to CR, Cr are parallel; but the line YpZ is ordinately applied to
CR; therefore mpd is ordinately applied to Cr; therefore mp, pd
are equal, and confequently the fegments Ym, dZ are equal.
COR. 3. The fame things fuppofed, if the line TX terminated
by the exterior fection touches the interior fection in any point m,
and a line be drawn through another point d in the interior fection,
parallel to TX, and meeting the exterior fection in the points I, V;
the rectangle IdV will be equal to the rectangle TmX or to the
fquare of Tm: for join the points m, d, and let this line meet the
exterior fection in the points Y and Z; becauſe by the preceding
Cor. the fegments Ym, dZ are equal, the rectangles YdZ, YmZ
will be equal; and therefore the rectangles IdV, TmX under the
fegments of the parallels which the line YZ meets, will be equal
to each other (Cor. 1. Prop. XVIII. Book I.)
COR. 4. The above conftruction fuppofed, if two lines MQ, TX
terminated by the exterior fection touch the interior fection in the
points d and m, they will be to each other as the femidiameters
CF, CA, to which they are parallel: for through the point of
contact d draw parallel to TX a line meeting the exterior fection in
I and V; the rectangle MdQ (by XXXI. and XL. Book I.) is to
the rectangle IdV, that is, to the rectangle TmX (by preced. Cor.)
as the fquare of CF to the fquare of CA; but the rectangle MdQ_
and TmX are the ſquares of Md and Tm; therefore Md, Tm, or
the whole tangents MQ, TX are to each other as the femidiame-
ters CF, DA, or as Cf, Ca coinciding with them.
COR. 5. The afymptotes of fimilar hyperbolas contain the fame
angles for let two hyperbolas be fimilar and fimilarly placed, and
the point C their common center, and let M, m be the vertices of
the coinciding diameters, and A, a the vertices of their conjugate
diameters, and becauſe the ſections are fimilar and fimilarly placed,
the diameters CA, ca will coincide, and CM will be to Cm as CA
to Ca; therefore the right lines joining the points A, M and a, m
are parallel, and therefore if two right lines be drawn through the
center C, one of them parallel to AM, am, and the other bifecting
thefe
Book V.
167
Conic Sections.
thefe lines, they will be afymptotes to both the hyperbolas (by Cor.
2. XXXVIII. Book I.) whence the Corollary is evident.
PROP. II.
Let EAD, ead be two parabolas, whofe axes AB, aB coin- FIG. 3.
cide, and let them have the fame parameter; I ſay that
any other two diameters MN, mN coinciding will have
equal parameters, and will contain with their ordinates
equal angles.
Part I.
L
ET F, be the foci of the parabolas, and becauſe they
are equally diſtant from the vertices of their axes, by
hypotheſis and Def. of the focus, the diſtance Fƒ between the foci
is equal to the diſtance Aa between the vertices of the axes, that is,
to the distance Mm between the vertices of the diameters MN, mN
(by Cor. Prop. X. Book III.) therefore the lines MF, mf will be
parallel and equal to each other (33. 1.) and confequently the dia-
meters MN, mN coinciding, have equal parameters (Cor. I.
Prop. XXV. Book II.)
Part 2. Let the diameter MN be produced beyond the vertex to
K, becauſe MF, mf are parallel, the angles KMF, Kmf will be
equal; and therefore the right lines ML, mO bifecting thefe angles
will be parallel (29. 1.) but they touch the parabolas (by Prop.
XV. Book II.) therefore the coinciding diameters MN, mN contain
equal angles with the tangents ML, mO and confequently with
their ordinates. Q. E. D.
As circles differ only in the magnitude of their diameters, fo
parabolas differ only in the magnitude of their principal parameters,
and therefore all parabolas, as well as all circles, are faid to be
fimilar.
COR. 1. If a right line TO terminated by the exterior parabola
touches the interior in any point m, it will be bifected in that point;
Y
for
168
Book V.
Conic Sections.
:
for draw through the point of contact the diameter MN of the
exterior parabola, and it is evident, from the fecond part of this
Prop. that the line TO is ordinately applied to MN.
COR. 2. If a right line ED terminated by the exterior parabola
meets the interior in the points e, d; the fegments Ee, Dd will be
equal; for draw the diameter MN bifecting ED in P; and becauſe
by the ſecond part of this Prop. the line ed will be an ordinate ap-
plied to the diameter mN, which coincides with MN; eP, Pd will
be equal, and therefore the fegments Ee, Dd between the parabolas
are equal.
COR. 3. If a right line TO terminated by the exterior parabola
touches the interior in any point m, and a right line be drawn from
any point in the interior parabola parallel to TO, meeting the
exterior parabola in the points E, D; the rectangle EdD will be
equal to the rectangle TmO or the fquare of Tm; by joining the
points m, d, it is proved by the preceding Cor. in the fame manner
as Cor. 3. of the preceding Prop.
COR. 4. If two right lines TO, MQ terminated by the exterior
parabola touch the interior in the point m and d; their ſquares will
be to each other as the parameters of the diameters MN, OR,
which paſs through the points of contact; for thefe lines are ordi-
nately applied to the diameters MN, OR (by Cor. 1.) and (by
Cor. Prop. X. Book III.) the abfciffes Mm, Od are equal; there-
fore the fquares of the ordinates Tm, Md are to each other as the
parameters of the diameters to which they are applied; whence the
Corollary is evident.
Cor. 5. If two lines TO, MQ terminated by the exterior para-
bola touch the interior; they will cut off from the exterior para-
bola equal fegments TMO, MOQ; for the diameters MmN, OdR
being drawn through the points of contact; the abfciffes Mm, Od
will be equal; therefore the parabolic fegments TMO, MOQ will
be equal (by Cor. 1. XIII. Book III.)
DEFI
Book V.
169
Conic Sections.
II.
I
DEFINITIONS II, III.
F a right line AD be fo divided in the points C, B, that the FIG. 4-
whole line AD be to either of the extreme parts as the other
extreme to the middle part CB; the line AD is faid to be harmo-
nically divided.
COR. 1. It is evident that the middle part is lefs than either of
the extreme parts.
COR. 2. Two extreme points A, D of a right line to be harmo-
nically divided being given, and C one of the middle points; the
fourth point B may be found, fo that the given part AC fhall be
one of the extremes, viz. by dividing the fegment CD in the
point B, that the part CB, next to AC, be to BD as AC to AD.
COR. 3. Or the two middle points C, B being given, and A one
of the extreme points; the other extreme D may be found; for
draw the line AV from the point A, fo that the extreme part AC
may be to the middle part CB as AV to the fegment VQ taken to-
wards A; join BQ and draw VD parallel to QB, meeting ACB
produced in D, and AD will be to BD as AV to QV, that is, by
conſtruction, as AC to CB.
COR. 4. It appears from the fecond Corollary, that, in harmo-
nical divifion, two extreme points A, D, and the middle point C,
and one extreme part AC, being all given, no other point beſides B
can be found to be a fourth point of that divifion; and it is evi-
dent from the third Corollary, that the middle points C, B and A,
one of the two extreme points, and the middle part CB, all being
given, no other point befides D can be found, which fhall be a
fourth part of that divifion.
Def. III. If a right line be harmonically divided in the points.
a, c, b, d, and in any manner four right lines Va, Vc, Vb, Vd be
drawn through the points of divifion, either parallel to each other,
or meeting in the fame point V; thefe lines are called Harmonicals.
Y 2
LEMMA
170
Book V.
Conic Sections.
FIG. 4.
L'
LEMMA I.
ET right lines Va, Vc, Vb, Vd be harmonicals, meeting in
the point V; if any right line parallel to one of them Vd,
meets the other three right lines in the points E, C, F; it will be "
biſected in the intermediate point C. Or if any four harmonicals
meet in any manner a right line in the points A, C, B, D; this line
will be cut harmonically in thoſe points.
Part 1. Through the intermediate point C draw a parallel to the
line ad (from which line the harmonicals are formed) and meeting
them in the points A, C, B, D; it is evident that this line is divided
in the fame ratio as the line ad, that is, harmonically in the points.
A, C, B, D; therefore AD is to AC as BD to CB; but becauſe
VD, EC are parallel, VD is to EC (as AD to AC, that is, as BD
to CB, that is) as the fame line VD to CF (becauſe VD, CF are
parallel) therefore EC, CF are equal.
Part 2. Now let any right line meet four harmonicals in the
points A, C, B, D, and if thoſe harmonicals be parallel to each
other, the Prop. is evident (2. 6.) but if they meet each other
in a point V; draw through either of the middle points C the
line ECF parallel to VD, which paffes through the extreme point
D more remote from the point C, and meeting the other two lines.
in the points E, F; by the first part EC, CF will be equal, and
AD is to AC as VD to EC or CF, that is, as BD to CB; there-
fore AD is to AC as BD to CB. Q. E. D.
?
PROP.
Book V.
171
Conic Sections.
PROP.
III.
Let two right lines AB, AC, touching a conic fection or op-
pofite fections, meet each other, and draw BC joining
the points of contact: if a right line be drawn through
A the point of concourfe of the tangents, meeting the
fection or oppoſite ſections in the points E, K, and the
line joining the points of contact in O; it will be cut
harmonically in thoſe points A, E, O, K.
Cafe 1.
FI
IRST, let the line drawn through A not be a diameter,
and draw two lines through the points E, K parallel to
BC, joining the points of contact, meeting the tangents in D, G
and H, M, and the fection or fections in F and L: draw a dia-
meter through A meeting the lines DG, BC, HM in the points
N, P, Q, and becauſe it bifects the line BC in P (Cor. 1. XXV.
Book I.) it will bifect DG, HM in the points N and Q; and be-
cauſe EF, KL terminated by the ſection or fections are parallel to
BC, they will be bifected in N and Q; therefore the fegments DE,
FG and the ſegments HK, LM will be equal, and confequently the
rectangles DEG, HKM will be equal to the rectangles EDF,
KHL.
Becauſe DG, HM are parallel, HK will be to DE as KM to
EG; therefore the rectangles HKM, DEG are fimilar, and there-
fore theſe rectangles, or the rectangles KHL, EDF are to each
other as the fquares of HK, DE, or as the fquares of HA, DA;
but theſe rectangles KHL, EDF are to each other as the fquares
of HB, BD (by Cor. 4. XVIII. Book I.) therefore the fquares of
HA, DA are to each other as the fquares of HB, BD, and confe-
quently HA is to DA as HB to BD (22.6.) and from a property
of parallel lines, KA is to EA as KO to OE; therefore the line
AK is harmonically divided in the points A, E, O, K.
Cafe 2. When the lines touch the fame fection, and the line
drawn through A is a diameter; let it meet BC in P, and the fec-
tion or oppofite fections in R, T; the lines drawn through theſe
FIG. 5,
points
6.
FIG. 5.
172
Conic Sections.
Book V.
'
FIC. 7.
points parallel to BC will be tangents; let them meet the tangent
AB in S and V; by a property of parallel lines, VA is to SA as
VT to SR, that is, as VB to BS (by Cor. 5. XVIII. Book I.)
therefore, by a property of parallel lines, TA is to RA as TP to
PR; and therefore the diameter drawn through A is cut harmoni-
cally in the points A, R, P, T. 2, E. D.
COR. It appears from this demonftration that if the tangent VBS
meets two parallel tangents VT, SR in the points V, S, and the
line joining their points of contact in A; it will be harmonically
divided in the points V, B, S, A, viz. in the point of contact, and
in the points in which it meets the parallel tangents, and the right
line joining their contacts.
PRÖ P. IV.
If three right lines touch a conic fection or oppofite fections,
each of them will be harmonically divided, viz. in the
point of contact, and in the points in which it meets the
other two tangents and the right line joining their points
of contact.
I'
F two of the tangents be parallel, and the third meets the line
joining their points of contact, the propofition is evident from
the preceding Cor.
But if the three lines AB, AC, MD, touching the fection or op-
pofite fections in the points B, C, K, meet each other in A, D, M,
and the line BC joining two points of contact meets MD, if necef
fary produced, in H; HM will be to HD as MK to KD.
Through the point of interfection D of the tangents DK, DC,
draw a right line parallel to the other tangent AB, and cutting the
ſection, or either of the oppofite fections in the points E, G, and
meeting BC in F; the rectangle EDG will be equal to the fquare
of DF (LIII. Book I.) but, by a property of parallel lines, the
fquare of HM is to the fquare of HD as the ſquare of MB to (the
fquare of DF or) the rectangle EDG, that is (by Cor. 6. XVIII.
Book
Book V.
173
Conic Sections.
Book I.) as the fquare of MK to the fquare of KD; therefore HM
is to HD as MK to KD. 2. E. D.
COR. 1. Hence, if the right line BC joining the points of con-
tact of the tangents MB, DC meets the other tangent MKD in H,
and through its point of contact K a right line be drawn meeting
BC in L., and parallel to AB, AC, if theſe lines be parallel, but if
not, paffing through A their point of concourfe; the line BC will
be harmonically divided in the points H, B, L, C. For if MB, DC
meet in A; join HA, and becauſe by this Prop. the tangent MKD
is harmonically divided in the points H, M, K, D; the lines AH,
AM, AK, AD will be harmonicals; therefore, by Part 2. Lemma
preceding, the line HBC is harmonically divided in the points
H, B, L, C.
But if the tangents MB, DC be parallel, the tangent MKD is
harmonically divided in the points H, M, K, D, by Cor. to pre-
ceding Prop. therefore, becauſe MB, KL, DC are parallel, the line
HBC is harmonically divided in the points H, B, L, C.
COR. 2. Hence, if the two lines MB, DC be given in pofition,
touching the ſection in the given points B, C, and any other point
K in the fection be given; the pofition of the tangent paffing
through K may be found. For draw through the point K a right
line meeting BC in L, and parallel to MB, DC, if they be parallel,
but if not, paffing through their point of concourſe A; if LB, LC
be equal, the line AKL is a diameter, and therefore the tangent
paffing through K will be parallel to BC. But if LB be lefs than
LC, produce LB to H, fo that HB be to HC as BL to LC (Cor.
3. to Def. II.) the right line joining the points H, K will be a tan-
gent; for if not, draw a tangent through the point K, it neceffarily
meets fomewhere LB produced towards H, fuppofe in N. Then
by preceding Cor. the line HC, which by conftruction is harmoni-
cally divided in H, B, L, C, will alſo be harmonically divided in the
points N, B, L, C, contrary to Cor. 4. Def. II. therefore the line
HK is a tangent.
PROP.
174
Book V.
Conic Sections.
FIG. 8,
99
30,
11.
12.
PROP. V.
Let two points A, B be taken in a diameter DR of a conic
fection (on different fides of the center, if DR be a con-
jugate diameter of an hyperbola, otherwiſe on the fame
ſide) ſo that the femidiameter CD of the ellipſe or hyper-
bola be a mean proportional between CA, CB, the dif
tances of the points from the center C; but in the pa-
rabola fo that the diftances DA, DB of the points from
the vertex D be equal to each other; and draw through
theſe points A, B the right lines AT, BQ parallel to the
ordinates applied to the diameter DR: if through either
of the points A or B, fuppofe B, a right line be drawn
cutting the fection or oppoſite ſections in two points O
and P; FO, FP, which touch the ſection or oppofite fec-
tions in thoſe points, will be either parallel to the line
AT drawn through the other point A, or the point where
they meet each other will be in that line.
I
T is evident, if the fecant drawn through the point B coincides
with the diameter DR, that the tangents drawn through its ver-
tices will be parallel to the line AT, becauſe they are parallel to the
ordinates applied to the diameter DR.
But if the point B be within the fection, or between the hyper-
bolas, and if the fecant coincides with BQ, and meets the fection
or both fections in Q and S, it is evident that the tangents drawn
through Q and S will meet each other in the point A, becauſe in
the ellipfe or hyperbola CB, CD, CA are proportional, and in the
parabola DA, DB are equal.
A
Now let the fecant OP be drawn, which fhall not coincide with
the diameter DR nor with the line BQ; then the tangents drawn
through the points O and P will meet each other fomewhere as in
F: draw through the point F the diameter CEF bifecting OP in
G
1
>
K
F
B
า
M
1
}
FIG. 4.
VA
L
H
C
E
H<
F
FIG.18.
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FIG.I
ia
M
772
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2.
A
N
6
a
TAB.13
•
Page 174.
K
L
M
F
D
m
A
T
a
FIG. 5.
F
E
R
FIG. 3.
R
S
IN B
D
Ꮐ
E
N F
L
B
R
P
C
H
H
K
L
Z
\M
Q
B
T
B
F
M Q H
K
I
FIG. 2.
R
K
•
P: XH
M
K
D
N
FIG.6.
FIG. 7 B L
C
G
E
F
N
G
C
B
P
{
OF
CY
Q
Book V.
175
Conic Sections.
G (XXVI. Book I.) and from the vertex D of the diameter DR
between the points A, B, draw an ordinate applied to the diameter
CE, and DH touching the fection or conjugate fection, and meet-
ing the diameter CE in H; DH will be parallel to AT.
Becauſe, when the fection is an ellipfe or hyperbola (by XLVIII.
and XLIX. Book I.) each of the rectangles GCF, KCH is equal
to the fquare of the femidiameter CE, theſe rectangles will be
equal to each other, and therefore CF is to CH, as CK to CG (or
becauſe DK, OBG are parallel) as CD to CB, that is, by hypothe-
fis, as CA to CD; therefore if AF be joined, becauſe CF is to
CH as CA to CD, AF will be parallel to DH or AT; and there-
fore the point F is in the line AT.
But when the fection is a parabola, GE is equal to EF, and KE FIG. 12.
equal to EH (by XLVII. Book I.) therefore HF is equal to (GK
or BD, becauſe DK, OBG are parallel, that is, by hypothefis, to)
DA; therefore becauſe HF, DA are equal and parallel, if AF be
joined, it will be parallel to DH or AT; and therefore the point F
is in the line AT. 2. E. D.
COR. 1. The fame conftruction remaining as above, if a right
line be drawn through either of the points A or B, fuppoſe B,
meeting the ſection or fections in two points M, L, and alfo AT
drawn through the other point A, in F; it will be harmonically di-
vided in the points F, M, B, L.
II.
Firſt let the point B be within the ſection, and draw the diameter FIG. 8,
CE paffing through F, and an ordinate through B applied to CE,
meeting the fection or fections in the points O, P, the tangents
drawn through O, P, meet fomewhere in the diameter CEF (by
Cor. 2. XXVI. Book I.) and they alfo meet in the line AT, by this
Prop. and therefore meet in the point F, and confequently the line
drawn through B will be cut harmonically (by III. of this Book) in
the points F, M, B, L.
10.
But if the point B be without the fection, and the line AT meets FIG. 9,
the fection or fections in the points T, N; fince it appears from
(Prop. LII. Book I.) that BT, BN joining thefe points are tan-
Z
gents;
176
Book V.
Conic Sections.
gents; the Prop. is evident by the III. of this Book. Supply the
line LBM in Fig. 9, 10, 12.
COR. 2. If a conic fection be given in pofition, and any point B
given, neither the center of the ſection nor in an aſymptote of the
hyperbola; any two right lines touching the ſection or oppofite fec-
tions, and drawn through the extremities of the fecant paffing
through the point B, meet each other in a right line as in AT, the
pofition of which is determined by this Prop.
COR. 3. Or if a conic fection and any right line AT, not a dia-
meter of the ſection, be given in pofition, and from any point F
in AT two right lines be drawn, touching the fection or oppofite
fections; the right line joining the points of contact will pafs
through a given point. For becauſe the ſection and the line AT
are given in pofition, the diameter CD will be given in pofition,
whoſe ordinates are parallel to AT, and therefore its vertex D, and
the point A in which point it meets AT will be given, and therefore
in the diameter CD the point B will be given, placed with reſpect to
the points A, D, as in the propofition: if the tangents FO, FP be
drawn from any point F in AT, the line OP joining their points.
of contact will pafs through the point B. For if not, a line might
be drawn from the point B to one of the points of contact O, which
meets again the ſection or oppofite fection in another point; then the
tangent drawn through this point would meet the tangent FO in
the line AT, by this Prop. that is, it would meet FO in F; and
therefore three tangents would meet each other in the fame point F,
which is impoffible (Cor. 3. XXVI. Book I.) therefore the Corol-
lary is manifeft.
-
PROP.
Book V.
177
Conic Sections.
PROP. VI.
14.
The fame conſtruction fuppofed as before, if through either FIG. 13,
of the points A or B, fuppofe B, BOP be drawn parallel
to the afymptote CY, and meeting one of the oppofite
fections in O; a tangent drawn through the point O meets
the line AT (drawn through the other point A) in the
afymptote CY.
F
OR fince the tangent OF in Fig. 13. does not pafs through
the vertex D, viz. of the diameter DR between the points
A, B, it will not be parallel to AT, but meets it produced towards
the afymptote CY, and in Fig. 14. it is manifeft that the tangent
OF meets AT; then let OF meet AT in F; the point F will be in
the afymptote CY.
For if not, from the point F another right line may be drawn
(LII. Book I.) touching one of the oppofite fections in fome point.
as X; let OX be joined, and becauſe the point F, from which point
the tangents FO, FX are drawn, is in the line AT, and the point
B is placed with reſpect to the points A, D, as in preceding Prop.
the line OX will pafs through the point B (by Cor. preceding) that
is, will coincide with the line BOP; therefore BOP, which is pa-
rallel to the aſymptote, meets the fection or fections in two points
O and X, which is impoffible, and therefore the point F is in the
afymptote CY. Q. E. D.
COR. I. If an hyperbola be given in pofition, and a point F given
in its afymptote CY, and from the point F a right line be drawn
meeting the hyperbola or oppofite hyperbolas in the points N, T;
tangents drawn through theſe points meet each other in a right line
as OP, which will be given in pofition. From the point F draw a
right line touching the hyperbola in O; becauſe from the point F
one tangent only can be drawn, the point O will be given, and
therefore the line OP drawn parallel to CY will be given in pofi-
Z 2
tion;
178
Book V.
Conic Sections.
tion; let the tangents paffing through the points N, T, meet each
other in B, the point B will be in the line OP; for through B draw
the diameter CD, and let D be its vertex, and let it meet NT in
A; CB, CD, CA will be proportional, and AT an ordinate applied
to the diameter CD; therefore the points B, D, A, and the line
NAT are placed as in this Propofition: but if the point B be not
in the line OP, through B a right line may be drawn parallel to
CY, which ſhall meet the hyperbola in fome point different from
O, and the tangent drawn through that point will meet NAT in
the afymptote CY, by the Propofition; therefore it meets it in F,
and then two tangents meet each other in the fame afymptote, which
is impoffible; therefore the point B, the concourfe of the tangents
drawn through the points N, T, is in the line OP.
COR. 2. Or if an hyperbola and a right line OP parallel to its
afymptote CY be given in pofition, and if from any point B in
OP, two right lines be drawn touching the hyperbola or oppofite
hyperbolas in two points N, T; the right line joining thoſe points
will pass through a given point in the afymptote CY, to which OP
is parallel. Through B let the diameter CD be drawn meeting
NT in A, and through the point O, in which point OP meets the
hyperbola, draw a tangent meeting the afymptote CY in F, and
becauſe CB, CD, CA are proportional, and the line NAT an ordi-
nate applied to the diameter CD, and the tangent OF meets NAT
in the afymptote CY, by this Prop. therefore the line NT joining
the points of contact will pafs through the point F: but becauſe
OP is given in pofition, the point O will be given, and confequently
the tangent OF is given in pofition, but it meets CY given in
fition in F; therefore the point F is given.
po-
COR. 3. From any point F in the afymptote CY of an hyperbola,
draw a right line touching the hyperbola in O, and draw OP
through the point O parallel to CY: if through the point F a right
line be drawn cutting the hyperbola or hyperbolas in the points
N, T, and meeting OP in M, it will be divided harmonically in
the points N, M, T, F.
For
Book V.
179
Conic Sections.
For the tangents NB, TB drawn through the points N and T,
meet each other in the right line OP, by Cor. 1. let the tangent OF
meet NB, TB in E and G; becauſe, by Prop. IV. of this Book,
the tangent OF is divided harmonically in the points F, G, O, E;
BF, BG, BO, BE will be harmonicals: therefore the line NTF is
harmonically divided in the points N, M, T, F (by Part 2. Lem-
ma I.)
When the right lines NB, TB touch oppofite hyperbolas, one of FIG. 14.
them as NB may be fometimes parallel to the tangent OF; in this
cafe the right line joining the points O, N will be a diameter; and
becauſe the diameter BCA bifects TN in A, and OC, CN, and
likewife TA, AN are equal, the right line joining the points O, T,
will be parallel to BCA; therefore becauſe BOM is bifected in O
(Cor. 2. LIV. Book I.) MT will be equal to TA or AN: and in
like manner, becauſe OF, BN are parallel, MF will be equal to FN,
therefore TF, FA are equal; therefore TA or MT is double of
TF, and therefore MN is to FN, as MT to TF. Whence the
Corollary is evident.
PRO P. VII.
The fame conſtruction being fuppofed, if through either of FIG. 15.
the points A or B a right line parallel to an aſymptote of
an hyperbola be drawn to the right line paffing through
the other point; it will be bifected by the hyperbola.
F
IRST, through the point A without the hyperbola, draw a
line parallel to the afymptote CY, and meeting the line BQ in
P, and the hyperbola in O; becauſe CA, CD, CB are proportional,
and the line BQ meets the hyperbola in the points Q, S, and is or-
dinately applied to the diameter CD, the lines joining the points
A, Q and A, S will be tangents, and therefore the line AP is bi-
fected in the point O (by Cor. 2. LIV. Book I.)
In the fecond cafe, draw through the point B within the hyper-
bola a right line parallel to the afymptote CY, and meeting AT in
F
180
Book V.
Conic Sections.
་
FIG. 16.
F, and the hyperbola in M, BF will be bifected in M: for draw
AOP parallel to CY, as in the preceding cafe, and through the
point O a right line parallel to BQ, and meeting the diameter CDR
in G, and the line BF in L; by a property of parallel lines, BP is
equal to OL, and becauſe AP is double of AO, (BP, that is) OL
will be double of OG; therefore OL is bifected by the diameter
CDR in G but OL is an ordinate applied to that diameter, by
conſtruction, and its extremity O is in the hyperbola, therefore L
the other extremity of it is in the fame hyperbola; therefore the
line OGL meets BF in the hyperbola, that is, in the point M, and
therefore, becauſe BP, MO, FA are parallel, and PO, OA equal,
BM, MF will be equal.
PRO P. VIII.
If two right lines, not parallel to each other, be inſcribed in
a conic fection or oppoſite ſections, and their extremities
be joined by four right lines; their interfections, and the
two interſections of the tangents which are drawn through
the extremities of the two infcribed lines, will all four be
in a right line.
L
ET AC, BD be the two infcribed lines, which (produced if
neceffary) meet in E, and let AD, BC meet in F, and AB, DC
in G, and the tangents drawn through A and C in H, and the other
two tangents drawn through B, D in K; the four points F, G, H, K
are in a right line.
For draw through the point E, the diameter NO of the fection,
and (in the ellipfe and hyperbola) take NP a third proportional to
the diſtance of the point E from the center N, and the femidiame-
ter NO, placed in the order mentioned in Prop. V. of this Book;
but in the parabola let OP be equal to the diſtance of the point .P
from the vertex O, and in both cafes draw through the point P the
right line PQ parallel to the ordinates applied to the diameter NO;
the concourfe of the tangents AH, CH, and of the tangents BK,
DK
T
1 20
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TAB 14 Page 1
F
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B
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FIG. 8.
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FIG.9.
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FIG. 10.
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FIG.12.
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FIG. 13.
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FIG. 14.
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FIG. 14.
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TAB 14 Page
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180
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FIG. 10.
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TAB 14. Page 180.
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FIG. 12.
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FIG. 15.
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PICH:
Book V.
181
Conic Sections.
DK, will be in the line PQ (by Prop. V. of this Book) let PQ
meet AC, BD (if neceffary produced) in the points L, M; then
the line AC will be harmonically divided in the points A, L, C, E,
and likewiſe the line BD in the points B, M, D, E (by Cor. 1.
Prop. V. of this Book) let the line joining the points C, B meet
PQ in F, and if the line joining the points A, D does not pafs
through F, join DF meeting AC in X, and fince B, M, D, E are
points of harmonical divifion, if FE be joined, FE, FD, FM, FB
are harmonicals; therefore the line AC meeting thefe lines in the
points X, L, C, E will be harmonically divided in thoſe points (by
Part 2. Lemma I.) contrary to Cor. 4. Def. II. for it is divided
harmonically in the points A, L, C. E; therefore AD paſſes through
the point F: join AB meeting PQ in G, it may be fhewn in the
fame manner that the line joining the points C, D paffes through
G; therefore the four points H, K, F, G are in a right line, viz.
PQ. Q. E. D.
.
COR. I. But if the two lines AB, CD joining the extremities of
the two infcribed lines be parallel to each other, they will be parallel
to PQ, in which line the tangents drawn through the extremities.
of the infcribed lines meet; for if one of them AB meets PQ in
any point G then it may be fhewn as before. that the line DG
paffes through the point C, and therefore CD would meet AB in
G, contrary to the hypothefis; therefore the Corollary is manifeft.
COR. 2. If two right lines parallel to each other be inſcribed in a
fection or fections; tangents drawn through the extremities of theſe
parallel lines meet in the diameter bifecting them (Cor. 2. Prop.
XXVI. Book I.) and it is manifeft that the right lines joining the
extremities of the parallels meet in the fame diameter.
PROP.
182
Book V.
Conic Sections.
PROP.
IX.
1
FIG. 179
18.
FIG. 18.
If a right line cutting a ſection or oppofite fections in the
points P, Q, meets the directrix DX of the fection in B,
and three lines be drawn to the points P, Q, B, from the
focus F nearer to DX, the line FB drawn to the directrix
will bifect the angle QFP contained by the other two lines,
when the points P, Q are in the oppoſite ſections; but
it will biſect the angle KFP adjacent to the angle QFP,
when the points P, Q are in the ſame ſection.
FOF
OR draw from the points Q, P, the perpendiculars QM, PN
to the directrix, and QB will be to PB, as QM to PN, that
is, as QF to PF (by Cor. 3. Prop. XI. Book II.) therefore when
the points Q, P are in the oppofite fections, the line FB bifects the
angle QFP (3. 6.) but when the points are in the fame fection,
from the point P draw a right line parallel to QF, and meeting FB
in R, and QF will be to PR, as QB to PB, or QF to PF, as be-
fore; therefore PR, PF are equal, and therefore the angle BFP is
equal to the angle PRF, or to the alternate angle BFK; therefore
the line FB bifects the angle KFP adjacent to the angle QFP.
Q. E. D.
COR. 1. Hence, three points P, Q, V, being given in a conic
fection, and its focus F; the axis, paffing through the focus, may
be found both in pofition and magnitude; for join the points P, Q,
and draw FP, FQ, and the line FR bifecting the angle adjacent to
the angle PFQ, and meeting QB produced in B; the point B will
be in the directrix: 'and in like manner, two other points Q, V be-
ing joined, find another point in the directrix, and draw the direc-
trix DX: then draw from the focus and from the point P in the
fection the perpendiculars FD, PN to the directrix; and let FD be
divided in the point A, fo that FA be to AD, as PF to PN; the
point A will be the vertex of the axis (XI. Book II.) and if FA,
AD be equal, the fection will be a parabola; but if they be un-
equal,
Book V.
183
Conic Sections.
equal, take SF to SD, as FA to AD, and S will be the other vertex
of the axis; and if FA be greater than AD, the point S muft be
taken on the ſame fide of the focus where the point A lyes; but if
FA be leſs than AD, on the contrary fide; and in the firſt caſe the
fection will be an hyperbola, and in the other cafe an ellipfe.
COR. 2. Hence alfo, the directrix DX being given in pofition, FIG. 18.
and the focus F nearer to this directrix, and the point E in the
fection being given; a right line may be drawn to touch the fection
in the point E draw through the points E, F, the right line EFO,
and FB perpendicular to this line, meeting the directrix in B; the
line joining the points E, B will be a tangent; for if not, let EB
meet the fection again in T, and produce the right line joining the
points T, F to Y; by this Prop. the line FB bifects the angle EFY,
which is adjacent to the angle ETF; therefore the angle YFB is
equal to the angle EFB, that is, to the angle OFB, which is ab-
furd; in like manner, Fig. 17. it may be demonftrated, that the
line EB does not meet the ſection or oppofite fection again: there-
fore EB is a tangent. But if EFO be perpendicular to the direc-
trix, the right line drawn through the point E, and parallel to the
directrix, will be a tangent, as is evident.
COR. 3. Hence, if a right line EB touching the fection meets its
directrix DX in B, and EF be drawn from the point of contact E
to the focus nearer to DX; the right line joining the points F, B
will be perpendicular to EF; for if not, draw a perpendicular meet-
ing the directrix in N, the right line joining the points E, N, will
likewiſe be a tangent (by the preceding Cor.) which is abfurd.
COR. 4. Hence, if any right line EG, joining the points of con-
tact of two tangents BE, BG, paffes through the focus F; the
right line FB drawn from the focus to the point of concourſe of the
tangents, will be perpendicular to the line joining the points of con-
tact. For let DX be the directrix nearer to the focus F, and A
the vertex of the tranſverſe axis, between F and the line DX; then
becauſe the points F, A, D are placed, and DX drawn as in Prop. V.
and EG paffes through the focus F; the point B, the concourſe of
the tangents, will be in the directrix DX; therefore FB is perpen-
dicular to EG, by the preceding Cor.
A a
PROP.
184
Book V.
Conic Sections.
FIG. 17,
18.
PROP. X.
If two right lines HP, HQ touch a conic fection or oppofite
fections; the right line HF paffing through the point of
concourſe of the tangents and focus F, will bifect the
angle QFP contained by the right lines drawn from the
points of contact to the fame focus, when the lines HP,
HQ touch the fame fection; but when they touch the
oppoſite fections, the angle KFP, which is adjacent to the
angle QFP, will be biſected by the line HF.
L
ET the right line DX be the directrix of the fection nearer to
the focus F, which the right line QP joining the points of
contact meets in B, and which the axis meets in D; and let A be
the vertex of the axis between the points F, D; through H the
point of concourſe of the tangents, draw a diameter meeting QP
in L, and let V be its vertex, and let the line HF meet the ſection
in E; and becauſe in the ellipfe or hyperbola, of which C is the
center, CF, CA, CD, and likewiſe CH, CV, CL are proportional,
and becauſe FA, AD in the parabola, and likewife LV, VH are
equal, and the lines DX, LP are parallel to the ordinates applied
to the diameters DF, HL, and the line HF paffes through as well
the point F as the point H; according as this line may meet the
fection again in G, or only in E (when it is parallel to one of the
afymptotes) the tangent drawn through E meets either the tangent
drawn through G, or the afymptote parallel to HF, as well in the
line DX as in the line QP (by Prop. V. and VI. of this Book)
therefore the tangent drawn through E meets the directrix in B;
therefore the line FB will be perpendicular to EFG (by Cor. 3.
preceding) confequently the angles EFB, GFB are equal; but (QF
being produced to K) the angle PFB (Fig. 18.) will be equal to the
angle KFB (by the preceding Prop.) therefore the angle PFG is
equal to the angle EFK, and confequently to the angle QFG and
in like manner, in Fig. 17. becauſe the angles PFB, QFB are equal,
the angle PFG is equal to the angle QFE, and conſequently to the
angle KFG, and therefore the Propofition is evident. 2, E. D.
PROP.
Book V.
185
Conic Sections.
PROP. XI.
If two conic fections touch each other in one point (that is, FIG. 19.
if they have a common tangent in the fame point) they
cannot meet in three other points.
ET two fections meet each other in the point A, and let the
Le Gio's
right line GA touch both fections in this point; theſe ſections
cannot meet in three other points befides the point A.
For if poffible, let them meet each other in three other points B,
C, D, and join theſe points by three lines, and firſt let none of them
be parallel to GA; produce one of theſe lines BC that it may meet
GA in G, and draw from the common point D a right line parallel
to the common tangent GA, meeting GBC in N, and the fections
in the points O, P; then each of the rectangles DNO, DNP
will be to the ſquare of GA, as the rectangle CNB to the rectangle
BGC (by Cor. 3. XVIII. Book I.) therefore the rectangles DNO,
DNP are equal, and confequently the points P and O coincide;
therefore the ſections meet each other in five points, contrary to
Cor. 3.
LVI. Book I. Or if the line DN touches one of the ſec-
tions in D, and meets the other again in O, it may be demonftrated
in the fame manner that the fquare of DN is equal to the rectangle
DNO, which is abfurd. Or if the line DN touches both ſections
in D, becauſe it is parallel to the tangent GA, the right line join-
ing the points D, A will be a diameter of both ſections, and there-
fore if from the common point B an ordinate be applied to the com-
mon diameter DA, the other extremity of this ordinate will be
common to both fections, and confequently the fections meet each
other in five points, which is impoffible.
But if CD, joining two points common to both fections, be pa-
rallel to the common tangent GA, then a line drawn through the
point of contact A, bifecting CD in L, would be a diameter com-
mon to both fections; if therefore there be drawn from the other.
A a 2
com-
186
Book V.
Conic Sections.
FIG. 20.
common point B an ordinate applied to the diameter AL, the other
extremity of this ordinate will be in both ſections, and therefore
theſe ſections meet in five points, which is impoffible, and there-
fore the Prop. is evident.
PROP. XII.
If two conic fections meet each other in two points; they
will not meet in any other point.
L'
ET two conic fections touch each other in the points A, B,
and firſt let the common tangents drawn through A and B
meet each other in G, and draw through the point of concourſe of
the tangents, the right line GF bifecting the right line AB joining
the points of contact, it will be a diameter to both ſections (XXVI.
Book I.) then if theſe fections have another common point D, draw
DL an ordinate applied to the common diameter GF, the other
extremity L of this ordinate will be in both fections, and therefore
theſe ſections touching each other in the point A, would meet in
three other points B, D, L, contrary to what has been fhewn in the
preceding Prop. Or if the point C in the diameter GF be com-
mon to both ſections, then the right line drawn through C and
rallel to BA, would touch both fections in the point C (Cor. 8.
XXV. Book I.) let this tangent meet the tangent GB in E; then
the right line EH drawn through the point E, and bifecting the
line which joins the points B, C, will be a common diameter to
both fections, and therefore if there be drawn from the other com-
mon point A an ordinate applied to this diameter, the other extre-
mity of it will be in both fections, which is contrary to the preced-
ing Prop. as before therefore there is no other point common to
both fections, but the two points of contact A, B.
:
pa-
In the ſecond place, if the common tangents drawn through A
and B be parallel; the right line joining the points A, B will be a
diameter of both fections; if they could have any other common
point
Book V.
187
Conic Sections.
point befides the two points of contact; then, from this common
point, an ordinate being drawn to their common diameter, it may
be ſhewn as before, that the fections would have a fourth common
point, contrary to what has been proved in the preceding; and
therefore in all cafes, two fections which touch each other in two
points do not meet each other in any other point. 2. E. D.
PRO P. XIII.
If a circle touches a conic fection, or oppofite ſections in
two points; the right line joining the points of contact
will be an ordinate applied to the axis of the ſection.
And if the ordinate be applied to the tranſverſe axis of an
ellipſe or hyperbola; the circle falls wholly within the
fection; but if the ordinate be applied to the conjugate
axis of an ellipfe or hyperbola; the circle falls wholly
without the ellipfe, or without both the hyperbolas.
IF
F the common tangents be parallel, when the circle touches the
ellipfe or hyperbola in two points, the right line joining their
points of contact will be a diameter of the fection, and likewiſe of
the circle; and becaufe this diameter, by a property of the circle,
is perpendicular to the tangents, it will be an axis of the fection,
as is evident from (Cor. 2. VII. Book II.) and if it be the tranfverfe
axis, the whole circle falls without the ellipfe or both the hyper-
bolas; but if it be the conjugate axis of the ellipfe, the whole circle
falls within the ellipfe, by Prop. VIII. Book II.
22.
Part 1. But if the circle HRET touches the fection or oppofite FIG. 21,
fections in the points T, R, and the common tangents drawn
through theſe points meet each other in S, the right line TR join-
ing theſe points will be an ordinate applied to the axis of the fec-
tion. For draw through the point of concourfe S of the tangents
a right line bisecting TR in O; then becaufe (by the circle) the
tangents ST, SR are equal, the line SO will be perpendicular to
TR; but the line SO is a diameter of the fection (XXVI. Book I.)
and this diameter is the axis; for the line TR is ordinately applied
to
188
Book V.
Conic Sections.
to it at right angles; therefore the first part of the Propofition is
evident.
Part 2. If the circle HTER touches a conic fection in the points.
T, R, and the line TR joining theſe points be ordinately applied to
an axis AB, which is the tranfverfe in the ellipfe and hyperbola ;
the whole circle will be within the fection; and the center P of the
circle will be in the axis AB, as is evident; let B be the vertex of
the axis nearer to the point P, and let the circle meet the axis AB
in E on the fame fide of the center P, where B is; PE will be leſs
than PB.
For draw through the vertex B of the axis the tangent BL, and
through the point R a right line touching the circle and fection,
and meeting the tangent BL in L; and becaufe, in the ellipfe or
hyperbola, the conjugate axis, which is parallel to the tangent BL,
is leſs than the diameter parallel to RL, BL will be less than RL
(Cor. XXXI. and Cor. 1. XL. Book I.) and in the parabola BL is
lefs than RL (by Cor. 1. III. Book II. and Cor. 3. III. Book III.)
then join PL, BR, and becauſe RL touches the circle, PRL will
be a right angle; but PBL is a right angle; therefore, becauſe of
the common hypothenufe, the fquares of PB, BL together, are
equal to the fquares of PR, RL together: but the fquare of BL
is lefs than the fquare of RL, therefore the fquare of PB is greater
than the fquare of PR, and confequently the line PB is greater
than the line PR, that is, than PE; therefore the arch TER of
the circle meets the axis within the fection, and confequently that
whole arch is within the fection, otherwiſe the circle would meet
the fection in another point befides the two points of contact (con-
trary to what has been proved by the preceding) and as it is mani-
feft that the circle meets the axis AB again in H within the fection,
the whole arch THR, and confequently the whole circle will be
within the fection: and in the fame manner it may be demonftrated,
that the circle falls wholly without the ellipfe, when the right line.
TR joining the points of contact is an ordinate applied to the con-
jugate axis.
Becauſe the.circle touching the oppofite hyperbolas is within the
angle
Book V.
189
Conic Sections.
!
1
angle contained by the common tangents, it is manifeft it falls
wholly without both the hyperbolas. 2. E. D.
COR. 1. If any right line TS touches a conic fection, and at the
point of contact a perpendicular be erected to the tangent, and
meeting the tranfverfe axis of the fection in P, PT will be the leaft
line that can be drawn from the point P to the fection: or if it
meets the conjugate axis of the hyperbola or ellipfe in the point. Q₂
the line QT in the hyperbola will be the leaft, and in the ellipſe the
greateſt of all the lines which can be drawn from the fame point
Qto the fection, viz. on the fame fide of the axis. For draw
from the point T a right line ordinately applied to the tranfverfe
axis, meeting the ſection again in R, and let the tangent TS meet
this axis in S; SR joining theſe points (Cor. 2. XXVI. Book I.)
will be a tangent, and equal to ST, and PR being drawn will be
equal to PT: therefore the triangles PTS, PRS are equiangular,
and confequently the angle PRS is a right angle; and therefore a
circle defcribed from the center P paffing through the points T
and R, touches the lines ST, SR, and confequently the fection in
T and R, and therefore that whole circle falls within the fection,
by this Prop. and therefore the radius PT of this circle is the leaft
line which can be drawn from the point P to the fection, viz. to
the fame part of the axis. It may be fhewn in the fame manner,
when a perpendicular to the tangent TS meets the conjugate axis of
the hyperbola or ellipfe in Q, that the circle defcribed with the ra-
dius QT falls wholly without the hyperbolas or ellipfe. Where-
fore the Corollary is manifeft.
COR. 2. If a right line TR terminated by an ellipfe or hyperbola
be an ordinate applied to its axis AB, and a circle paffing through
the points T and R, touches the ſection in one of the points T, it
will likewife touch it in the other point R; for P the center of the
circle will be in the axis AB of the ſection; draw through the point
T the common tangent TS meeting the axis AB in S, PTS (from a
property of the circle) will be a right angle, and the right line join-
ing the points S, R touches the fection; PR being joined, it may
be fhewn, as in the preceding Cor. that PRS is a right angle;
there-
199
Book V.
•
Conic Sections.
FIG. 21,
22.
therefore the circle touches the line RS, and confequently the fec-
tion in the point R.
PROP. XIV.
If two circles touch an ellipfe or hyperbola in the fame point
T, and touch again the fame ellipfe or oppofite hyperbo-
las in the points R, N; the rectangle under the diameters
of the circles will be equal to the ſquare of that diameter
of the fection which is conjugate to the diameter paffing
through the point of contact T, and if a circle touches a
parabola in two points, the ſquare of its diameter will be
equal to the rectangle under the parameter of the axis,
and the parameter of the diameter paffing through the
point of contact.
Part 1.
L'
ET AB, Mm be the axes of an ellipfe or hyperbola,
and C the center; the right lines TR, TN joining
the points of contact will be perpendicular to the axes, by Part 1.
of the preceding Prop. through the point T, in which both the
circles touch the fection, draw a common tangent meeting the axes.
in the points G, S, and at the point T to this tangent erect a per-
pendicular meeting the axes in P and Q; the points P and Q will
be the centers of the circles (1. and 19. 3.) and becauſe the dia-
meters of the ſection, which pafs through the points of contact,
are equal to each other, their conjugates will be equal to each other,
as is evident from (Prop. III. and IV. Book IV.) let FD be the dia-
meter of the fection conjugate to that which paffes through the
point of contact T. Becauſe the rectangled triangles TQG, TSP
are equiangular, QT will be to GT, as TS to TP; and therefore
the rectangle QTP is equal to the rectangle GTS, that is, to the
ſquare of CD, by Prop. LI. Book I. Whence the firſt part of the
Prop. is evident.
Part 2. Let the circle HE touch the parabola in the points T, R,
the right line TR joining theſe points will be perpendicular to the
axis,
Book V.
191
Conic Sections.
axis, as before; and therefore the parameters of the diameters,
which paſs through the points of contact, are equal (Cor. 3. XXV.
Book II.) draw through the point T the common tangent meeting
the axis in S, and erect at the point T a perpendicular to this tan-
gent, meeting the axis in P, the point P will be the center of the
circle HE as before; let TR meet the axis of the parabola in O,
and becauſe the rectangled triangles OPT, SPT are equiangular,
OP will be to PT, as PT to PS; therefore the ſquare of PT is
equal to the rectangle OPS, that is, to the rectangle under half the
parameter of the axis, and half the parameter of the diameter,
which paffes through the point of contact T (Prop. XXV. Book II.)
therefore the ſecond part of the Prop. is evident.
COR. 1. The fame things being premiſed, as in the firft cafe of
this Prop. the femidiameter PT of the circle touching the ellipfe
or hyperbola in two points, and falling within the fection, will be
to CD the femidiameter of the fection conjugate to that which
paffes through the point of contact, as the conjugate femiaxis CM
to the tranfverfe femiaxis CB; for draw from the center the line
CK perpendicular to the tangent GT: the rectangle under the lines
PT, CK will be equal to the ſquare of CM (by XXII. Book II.)
and the rectangle under the lines CD, CK is equal to the rectangle
MCB under the femiaxis, as is evident from (Prop. I. Book IV.)
therefore PT is to CD, as the ſquare of CM to the rectangle MCB,
that is, as CM to CB: therefore, by this Prop. CD will be to QT
(the femidiameter of the circle touching the ellipfe or oppofite hy-
perbola in two points, and falling without the ſections) in the fame
ratio, as CM to CB.
COR. 2. Hence it appears that the diameters of thefe circles will
be equal to each other, if they touch (as in the Propofition) equila-
teral hyperbolas, viz. whofe axes are equal to each other.
B b
PROP.
192
Book V.
Conic Sections.
FIG. 23,
24,
25,
PROP. XV.
If from the vertex of the tranſverſe axis of an ellipſe or
hyperbola, or from the vertex of the axis of a parabola,
a line be taken on the axis equal to its parameter, and a
circle be deſcribed about it as a diameter; it will fall
wholly within the ſection: but if from the vertex of the
conjugate axis of the ellipfe a line be taken equal to its
parameter; a circle defcribed about it as a diameter will
be wholly without the ſection.
L
ET AB be the tranfverfe axis of the ellipfe or hyperbola, or
the axis of the parabola, and take on the axis the line AC
26. equal to its parameter; the circle defcribed about the diameter AC
falls wholly within the fection: or if AB be the conjugate axis of
the ellipfe, the whole circle falls without the ſection.
For draw AD perpendicular to the axis and equal to AC, and
join CD, and in the ellipfe or hyperbola, let BLD be joined, but
draw DL in the parabola parallel to the axis; and through any
point E in the circle draw EF parallel to AD, meeting the fection
in G, and the lines AB, CD, LD in the points F, K, H; then be-
cauſe AC, AD are equal, FC, FK will be equal; and by a pro-
perty of the circle, the fquare of EF is equal to the rectangle AFC,
that is, to AFK; but the fquare of GF is equal to the rectangle
AFH (by Prop. I. IV. V. Book II.) but KF is less than FH, if
AB be not the conjugate axis of the ellipfe, in which cafe KF is
greater than FH; therefore in the firſt cafe, the rectangle AFK is
lefs than AFH, and therefore the fquare of EF will be lefs than
the fquare of GF, and the line EF lefs than FG, and confequently
the whole circle is within the fection in like manner it may be
fhewn in the other cafe that the whole circle is without the fec-
tion. 2. E. D.
:
COR.
Book V.
193
Conic Sections.
COR. 1. Hence, if from the vertex A of the tranfverfe axis of
the ellipfe or hyperbola, a line AP be taken on the axis within the
ſection, and not greater than half the parameter AC of this axis;
AP will be the leaft of all the right lines which can be drawn from
the point P to the fection: but if a line AP be taken from the
vertex of the conjugate axis of the ellipfe, not lefs than half the
parameter of the conjugate axis, PA will be the greateſt of all the
lines which can be drawn from the fame point P to the fection.
For, in the firſt caſe, becauſe the circle deſcribed with the radius PA
does not fall without the circle AEC, it will be wholly within the
fection and becauſe this circle in the ſecond cafe does not fall
within the circle AEC, it will be wholly without the ſection, where-
fore in both caſes the Corollary is manifeft.
:
23.
COR. 2. Hence, if a point P be given in the axis AB of a conic FIG. 21,
ſection (which in the ellipfe or hyperbola is the tranfverfe) and the
diftance of this point from the nearer vertex B of the axis is greater
than the half of its parameter; a right line may be drawn from the
point P, which ſhall be the leaſt of all the lines which can be drawn
from the fame point to the fection, viz. on the fame fide of the
axis.
When the fection is a parabola, take the line PO in the axis to-
wards the vertex B equal to half the parameter of the axis, and
when the fection is an ellipfe or hyperbola, take the line CO from
the center C in the axis towards the vertex B, fo that CO be to PO
as the axis AB to its parameter, and draw through the point O in
each fection a line perpendicular to the axis, and meeting the fec-
tion in T, the right line joining the points P, T fhall be the line
required. For draw through the point T a right line touching the
fection; the line PT will be perpendicular to this tangent, as is evi-
dent from (Part 1. Prop. XXV. and Prop. XXIV. Book II.) there-
fore PT is the leaft of all the lines which can be drawn from the
point P to the fection, viz. to the fame fide of the axis (by Cor. 1.
Prop. XIII. of this Book).
And if a point Q be given in the conjugate axis of the hyperbola
at any diſtance from the center, a line QT may be affigned in like
Bb 2
manner,
194
Book V.
Conic Sections.
1
manner, which fhall be the leaſt of all the lines drawn from the
fame point to the fame fection: or if a point Q be given in the
conjugate axis of the ellipfe, the diſtance of which point from the
remoter vertex M of the axis is less than half its parameter; in the
fame manner, QT may be affigned, which fhall be the greateſt of
all the lines which can be drawn from the point Q to the ſection,
viz. to the fame fide of the axis.
DEFINITION IV.
F a circle touches a conic fection in any point, ſo that no other
circle can be deſcribed between this circle and the fection, this
circle is faid to have the fame curvature with the fection in the point
of contact.
1
FIG. 27,
28.
PROP. XVI.
From the point C in a conic fection draw to its axis AH,
which in an ellipfe or hyperbola is the tranſverſe axis, an
ordinate meeting the fection again in B, and from the
point B, the diameter BK, and from the firſt point C an
ordinate applied to this diameter meeting the ſection again
in R: if a circle CRP be deſcribed touching the ſection
in C, and paffing through the point R, this circle will
have the fame curvature with the fection in the point of
contact C.
RAW through the point C a common tangent, meeting the
axis of the ſection in A, and through the point B a right line
touching the fection; this meets the other tangent in the point A
of the axis (Cor. 2. XXVI. Book I.) and AC, AB will be equal,
as is evident.
DRAW and
First,
រ
FIG.16.
H
E
H
P
K
t
M
Q
B
N
194.
G
TAB· 15 · Page.
· Page·1
M
N
B
X
K
G
R
E
TA
FIG. 18.
Y
L
FIG·17.
E
D
H
B
A
F
H
S
N
G
D
B
X
M
F
EA
FX
FIG. 22
G
M
K
а
M
K
m
G
N
F
K
S
A
D
H
E
B
E
R
P
HI
m
FIG · 21 ·
B
L
N
P
FIG 20-
G
FIG 19
Book V.
195
Conic Sections.
?
First, I ſay that the circle CRP does not meet the ſection but in
the points C and R; and that on one fide of the line CR it is with-
out, and on the other fide within the fection. For it cannot meet
it in the point B, becauſe it would touch the fection in that point
(by Cor. 2. XIII. of this Book) and then could not meet the fec-
tion in a third point R (by XII. of this Book) and by the fame
way of reaſoning it cannot meet the ellipfe in the other vertex of
the diameter BK, for CK, being drawn, is an ordinate applied to
the other axis. But if it be poffible, let it meet the fection in the
points C, R and O, and draw a line through O parallel to BA,
meeting the fection in N, and AC in L; the rectangle OLN
(XVIII. Book I.) will be to the fquare of LC as the fquare of BA
to the fquare of AC, that is, in a ratio of equality, and therefore,
becauſe the point O is in the circle which LC touches in C, N alfo
will be in the fame circle; and therefore the circle CRP meets the
fection in the three points R, O, N, befides the point of contact C,
contrary to Prop. XI. of this Book, wherefore the circle CRP meets
the fection only in the two points C, R; and becauſe it does not
touch the ſection in R (for then CR would be an ordinate applied
to the axis (by the first part of Prop. XIII. of this Book, contrary
to the hypothefis) the arch of that circle on one fide of the line
CR will be wholly without, and on the other fide wholly within
the fection.
Now if any other circle as COS be deſcribed lefs than CRP, FIG. 27.
touching the circle CRP and the fection in C, it falls within the
ſection on both fides of the point of contact C. For if that circle
paffes through B, it will touch the ſection in that point, and confe-
quently will be wholly within the fection (Part 2. Prop. XIII. of
this Book) then let it meet CB without the fection in S; and be-
cauſe the circle COS is wholly within the circle CRP, it neceffarily
muſt fall on one fide of the point of contact C within the ſection :
let the arch CM be within the fection, and fince the point S is with-
out the ſection, the arch MS meets the fection fomewhere, ſuppoſe
in the point O; draw a line through O parallel to BA, meeting.
the fection in N and AC in L; then it may be demonftrated as be-
fore,
196
Book V.
Conic Sections.
FIG. 28.
fore, that the rectangle OLN is equal to the fquare of the tangent
LC, and confequently the point N is in the circle COS; but that
circle does not touch the ſection in N (XII. of this Book) becauſe
it touches it in C, and meets it in the points N, O; therefore fince
the arch SN is on one fide of the point N without the fection, the
arch NC will be on the other fide of the point N within the fection,
and therefore that whole arch NC will be within the fection: for
otherwife the circle COS would meet the fection in another point,
befides in the points C, O, N, contrary to Prop. XI. of this Book ;
therefore a circle lefs than CRP falls on both fides of the point
of contact C within the ſection.
And if a circle CSOX greater than CRP be deſcribed, touching
the circle CRP and the fection in C, it falls on both fides of the
point of contact C without the fection. For becauſe it is wholly
without the circle CRP, it will neceffarily be on one fide of the
point C without the fection; therefore let the arch CXM be with-
out the fection, meeting the line CR produced in M; this circle
CSOX meets the parabola or hyperbola (whofe curves are indefinite)
ſomewhere between the points M and C, fuppofe in O and in the
ellipfe, becauſe CB is an ordinate applied to the tranfverfe axis, CK
will be an ordinate applied to the conjugate axis; therefore if the
circle CSOX does not meet CK within the fection, it falls wholly
without the ellipfe (by Part 2. Prop. XIII. of this Book) let it then
meet CK within the ellipfe in S; and fince the point M is without
the ellipfe, the arch MS meets the ellipfe fomewhere as in O;
draw therefore through the point O'in each fection a line parallel
to BA, meeting the fection in N and AC in L, it may be fhewn as
before, that the point N is in the circle CSOX, therefore becauſe
that circle touches the fection in C, it will not touch it again in the
point N or O (by Prop. XII. of this Book) wherefore becauſe the
point M is without the fection, the arch MO will be without the
fection, and confequently the arch OSN within the fame, and there-
fore the arch NC on the other fide of the point N will be without
the fection, and becauſe the circle CSOX does not meet the fection
but in the points C, N, O (XI. of this Book) the whole arch NC
will
Book V.
197
Conic Sections.
will be without the fection; therefore the circle CSOX falls on both
fides of the point C without the ſection.
Then becauſe the circle CRP, touching the fection in the point
C, falls on one fide of that point without, and on the other fide
within the ſection; and fince every other circle touching the ſection
in the point C falls on both fides of the fame point either within or
without the ſection, it is manifeft that no circle can pafs between
the ſection and the circle CRP; and therefore that circle has the
fame curvature with the fection in the point C, by the preceding
Definition.
L
LEMM A II.
ET DS touch a circle in D, and through a point C within the FIG. 29.
circle, draw the line CX parallel to the tangent DS; if two
right lines be drawn from the point D meeting the circle again in
M, K, and the line CX in C, V; the rectangles MDC, KDV will
be equal.
Let the line CX meet the circle in Y, and join DY, MY; in
the triangles DYM, DCY, the angle at D is common to both tri-
angles, and the angle CYD is equal to the alternate angle YDS,
that is, to the angle DMY in the alternate fegment (32. 3.) and
therefore the triangles DYM, DCY are equiangular; therefore MD
is to DY, as DY to DC (4. 6.) and therefore the rectangle MDC
is equal to the ſquare of DY: in the fame way it may be fhewn, KY
being joined, that the rectangle KDV is equal to the fquare of DY ;
therefore the Propofition is evident.
PROP.
198
Book V.
Gonic Sections.
PROP. XVII.
If a tangent be drawn through any point in a conic fection,
and a circle be defcribed touching this tangent in the fame
point, and cutting off from the diameter paffing through
this point a fegment equal to its parameter; that circle
will have the fame curvature with the ſection in the point
of contact.
FIG. 30. Cafe 1.
L
ET AB be the axis of the fection, and draw a tangent
AD through its vertex A, BAD will be a right angle,
and therefore if a circle ARC be defcribed, touching the line AD
and fection in the point A, its center will be in the axis AB, and if
that circle cuts off a fegment AC of the axis equal to its parame-
ter, it will be wholly within the fection (or without the fame, if
AB be the conjugate axis of the ellipfe) by Prop. XVI. of this
Book, and therefore (when AB is not the conjugate axis) a circle,
touching the ſection in A, and cutting off from the axis a fegment
leſs than AC, falls wholly within the ſection.
Then if a circle AHE cuts off from the axis of the conic fection
a fegment AE greater than AC; let P be its center, and becauſe the
fegment PA is greater than half the parameter of the axis, a line PT
or Pt on both fides of the axis AB may be drawn from the point
P, which fhall be lefs than PA (Cor. 2. Prop. XV. of this Book)
therefore the circle AHE defcribed about the diameter AE meets
thefe lines produced without the fection, fuppofe in the points M, L;
and becauſe the fame circle meets the axis in E within the fection,
the arch ME neceffarily meets the fection fomewhere between the
points M, E, fuppofe in H: and in like manner the arch LE meets
the ſection between the points L, E in b; and becauſe that circle
does not meet the fection but in the points A, H, b (XI. of this
Book) the arches AMH, ALb will be wholly without the fection,
that is, the circle AMEL falls on both fides of the point of con-
tact
1
Book V.
199
Conic Sections.
tact A, without the ſection: then becauſe every other circle touch- FIG. 29,
ing the fection in the point A falls on both fides of that point either
within the circle ARC, or without the fection, it appears that nỌ
other circle can pafs between the fection and the circle ARC; there-
fore that circle, viz. which cuts off from the axis AB a fegment
AC equal to its parameter, has the fame curvature with the fection
in the point A, by the preceding Definition.
The demonftration is the fame, when the point A is the vertex of
the conjugate axis of the ellipfe: for leffer reading greater, and for
within, without, and vice verſa.
Caſe 2. Let AB be the tranfverfe axis of an ellipfe or hyperbola, FIG. 29,
C the center, and through the point D, not the vertex of the axis,
draw the line DS touching the ſection, and let the circle DMK be
defcribed touching this line in the fame point D, and cutting off
from the diameter CD a fegment DM equal to its parameter; this
çircle will have the fame curvature with the fection in the point D.
For from the point D let the ordinate DE be applied to the tranf-
verſe axis, and meeting the fection again in F, and draw through
the point F the diameter FCG of the fection, to which let DH be
ordinately applied, meeting the fection again in L, and the circle
in K. Parallel to DS draw the diameter CQ meeting DL in V,
and draw the femidiameter CR conjugate to CF, meeting DS in
the point S, and let TD be an ordinate applied to this diameter;
becauſe the femidiameters CD, CF are equal, their conjugates CQ,
CR will be equal; let PD be half the parameter MD of the dia-
meter DCM, and the rectangle PDC will be equal to the fquare of
the femidiameter CQ, or to the fquare of CR, that is, to the rect-
angle SCT, by the XLVIII. and XLIX. Book I. or to the rect-
angle VDH becaufe CS, VD and CH, TD are parallel) then be-
cauſe the rectangles PDC, VDH are equal, their doubles, viz. the
rectangles MDC, LDV will be equal; but the rectangle KDV is
likewife equal to the rectangle MDC (by the preceding Lemma)
and therefore LD and KD are equal, confequently the points L, K
coincide; therefore becauſe the circle DKM touching the fection in
D, paffes through the other extremity L of the line drawn from the
point
Cs
200
Book V.
Conic Sections.
FIG. 31.
point D, and ordinately applied to the diameter FG, this circle
will have the fame curvature with the fection in the point D, by
the preceding Prop.
Now let the fection be a parabola, its axis AB, and draw through
the point D, not the vertex of the axis, a right line DS touching
the parabola, and draw the diameter DG; let a circle DMK be
defcribed touching the line DS in D, and cutting off from the dia-
meter DG the fegment DM equal to its parameter; that circle
will have the fame curvature with the parabola in the point D.
For draw from the point D an ordinate DE applied to the axis,
meeting again the parabola in F, and through the point F draw the
diameter FH, to which, from the point D, let DH be ordinately
applied, meeting again the parabola in L, and the circle in K: from
the vertex of the diameter FH, draw the ordinate FG, applied to
the diameter DG; this ordinate will be parallel to the tangent
DS, let it meet DH in C. Then becauſe the parameters of the dia-
meters DG, FH are equal (Cor. 3. XXV. Book II.) and the ab-
fciffes DG, FH are equal (II. Book III.) the ordinates DH, FG
will be equal; but becauſe the triangles DCG, FCH are fimilar,
and the fides DG, FH equal, DC will be equal to CH, and confe-
quently DC will be a fourth part of the whole line DL; therefore
the rectangle LDC will be equal to (the fquare of DH or FG,
that is, from a property of the parabola) to the rectangle MDG;
but by the preceding Lemma, the rectangle KDC is equal to the
rectangle MDG; and therefore as before, the circle DMK will pafs
through the point L, and confequently will have the fame curva-
ture with the parabola in the point D. 2. E. D.
1
COR. Hence, if a circle DBF touches a parabola in two points
D, F, and a circle DMO be deſcribed, paffing through the point of
contact D, and having the fame curvature with the parabola in the
point D; the parameter of the axis of the parabola, the diameter
of the circle DBF, the parameter of the diameter DG, and the
diameter of the circle of curvature, will be in continued propor-
tion the line joining the points D, F, will be an ordinate applied
to
1
Book V.
201
Conic Sections.
"
to the axis (XIII. of this Book) which let it meet in E, draw
through the point D a right line touching the parabola, and meet-
ing the axis in S; this line likewife touches both the circles; erect
at the point D a perpendicular to DS, meeting the axis in X, XD
will be the radius of the circle touching the parabola in the points
D and F, and the ſegment DM of the diameter DG cut off by the
circle of curvature, will be equal to the parameter of that diameter;
therefore if from the center of this circle the perpendicular NP be
drawn to DM, PD will be half of DM, and conſequently equal to
SX, and EX will be half the parameter of the axis (Prop. XXV,
Book II.) but EX is to XD as XD to XS, and as XD is to XS,
fo PD or XS is to ND, becauſe the triangles DXS, PDN are fimi-
lar; therefore the whole lines, viz. the parameter of the axis of
the parabola, the diameter of the circle DBF, the parameter of the
diameter DG, and the diameter DO of the circle of curvature, will
be in continued proportion.
PRO P. XVIII.
If a circle DMO touching an ellipfe or hyperbola in the FIG. 29.
point D, has the fame curvature with the ſection in that
point; its femidiameter ND will be to CQ the femidia-
meter of the ſection, conjugate to that which paffes
through the point D, as the fquare of CQ to the rect-
angle ACZ under the femiaxes,
I
F the circle touches the fection in the vertex of the axis, its dia-
meter will be the parameter of the fame axis, by Prop. XVI.
of this Book: therefore the Propofition is manifeft in this cafe.
:
But if the circle DMO touches the fection in a point D, not the
vertex of the axis; draw through the point D a common tangent
DS, and from the center of the fection the perpendicular Ca to
this tangent let DM be the fegment of the femidiameter CD cut
off by the circle DMO, and draw from the center of that circle
the perpendicular NP to the diameter CD; and PD will be half
the parameter of the diameter CD; therefore the rectangle PDC
Cc 2
is
}
202
Book V.
Conic Sections.
1
FIG. 32,
is equal to the fquare of the femidiameter CO conjugate to CD;
but becauſe the triangles NDP, DCa are fimilar, ND will be to
CD as PD to Ca; and therefore the rectangle under the lines ND,
Ca is equal to (the rectangle PDC, that is, to) the fquare of CQ;
but ND is to CQ_as (the rectangle under ND, Ca, that is, as) the
fquare of CQ to the rectangle under the lines CQ, Ca; but the
rectangle CQ, Ca is equal to the rectangle ACZ under the femi-
axes (I. Book IV.) and therefore the femidiameter ND of the circle
DMO, is to the femidiameter CQ as the fquare of CQ to the
rectangle ACZ under the femiaxes. 2. E. D.
}
COR. Becauſe it is fhewn in the demonftration of this Prop. that
the rectangle under the lines ND, Ca is equal to the fquare of CQ,
it is evident that the radius ND of the circle of curvature, the fe-
midiameter CQ of the fection parallel to the common tangent
DS, and the perpendicular Ca drawn from the center of the ſection:
to the tangent, are in continued proportion.
PROP. XIX.
If a right line cutting a conic fection in two points D, F,
meets two tangents AB, AC in the points E, G, and the
line BC joining their points of contact in H; the rect-
angle DEF will be to the rectangle FGD, as the fquare.
of EH to the fquare of GH.
D
RAW through the point E, where the fecant meets one of
the tangents AB, a line parallel to the other tangent AC, and
meeting the right line joining their points of contact in M, and
draw a line parallel to DF touching the fection in L, and meeting
the tangent AC in K; by Cor. Prop. LIII. Book I. the rectangle
DEF is to the fquare of EM as the fquare of KL to the fquare of
KC, that is, as the rectangle FGD to the fquare of GC (by Prop.
XVIII. Book I.) therefore, by permutation, the rectangle DEF
will be to the rectangle FGD as the fquare of EM to the fquare of
a parallel to the tangent Ac,
GC,
TAB· 16 · Page
e. 202
F
XG
P
K
H
Q
T
L
E
FIG 25.
•
B
F
D
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FIG. 29°
N
G
H
E
P
K
B
F
E
FIG.31.
Z
HN
H
P
G
L
FIG. 23.
M
K B
I
В
A
·D
G
E
K 11
P
FIG. 26.
F
G
H
_K
E
I
B
P
FIG. 24
C
A
D
M
K
O
X
A
B
C
H
FIG. 28.
P
N
A
D
K
R
MAT
C
HI
M
P
H
FIG. 30.
P
FIG 27.
M
R
い
​LINIV
CH
L
Book V.
203
Conic Sections.
GC, that is, becauſe the triangles HEM, HGC are fimilar, as the
fquare of EH to the fquare of GH. 2. E. D.
If the tangents, which the fecant meets, be parallel to each other,
the Prop. is manifeft, from Cor. 2. XVIII. Book I.
PROP. XX.
If two right lines DF, CA cutting a conic fection in the FIG. 33.
points D, F and C, A, meet each other in G, and meet in
the points E, H, a right line touching the fection in B;
the ſquare of EB is to the fquare of BH in a ratio com-
pounded of the ratio of the rectangle DEF to the rect-
angle CHA, and of the ratio of the rectangle AGC to
the rectangle FGD.
Cafe 1.
D
RAW through the point E, in which the fecant DF
meets the tangent, a line parallel to the other fecant
AC, which in the first place may meet the fection in the points
K, L; becauſe theſe fecants are parallel, the fquare of EB will be
to the ſquare of BH as the rectangle KEL to the rectangle CHA ;
but the ratio of the rectangle KEL to the rectangle CHA is com-
pounded of the ratio of the rectangle KEL to the rectangle AGC,
that is, of the ratio of the rectangle DEF to FGD, and of the ra-
tio of the fame AGC to CHA; therefore the ratio of the fquare of
EB to the fquare of BH is compounded of the ratio of the rect-
angle DEF to the rectangle FGD, and of the ratio of the rect-
angle AGC to CHA, that is, of the ratio of DEF to CHA, and
of the ratio of AGC to FGD.
Cafe 2. Now if any line be drawn parallel to DF, meeting the
fection in the points d, f, the line AC in g, and the tangent in e,
through which point a line drawn parallel to AC would not meet.
the fection; then from what has been demonftrated, the ratio of the
fquare of eB to the fquare of BH is compounded of the ratio of
the fquare of eB to the fquare of EB, and of the ratio of the fquare
of EB to the fquare of BH; therefore it is compounded of the
ratio
204
Book V.
Conic Sections.
FIG. 34.
ratio of the rectangle def to DEF (Cor. 4. XVIII. Book I.) and of
the ratio of the fame rectangle DEF to CHA, and of the ratio of
AGC to FGD, by the firſt caſe, that is, of the ratio of the rect-
angle def to CHA, and of the ratio of AgC to fgd, (becauſe the
rectangle DEF is both an antecedent and a confequent of the terms
which compound the ratio, and therefore may be omitted, and the
rectangles AgC, fgd may be ſubſtituted for the rectangles AGC,
FGD, which have the ſame ratio to each other (by Prop. XVIII.
Book I.) therefore in this cafe the Prop. is demonftrated.
The two preceding Propofitions are in part demonftrated by Sir
Ifaac Newton, in Prop. XXIII. and XXIV. Book I. Math. Prin.
Philof. Nat.
PROP.
XXI. PROB L. İ.
Five points being given in a conic ſection, to deſcribe the
fection.
L
ET the five points F, G, K, M, N be given in a conic ſection,
and join four of thefe points by two right lines FG, MN,
meeting each other in the point R, and draw through the fifth point
K two lines KD, KH parallel to FG, MN, and meeting the lines.
MN, FG in E and Q; and take the points D, H in the lines KD,
KH (if neceffary produced) fuch that the rectangle KED be to
MEN as GRF to MRN, and KQH to GQF as MRN to GRF;
(but the points K, D, or K, H must be on the fame or different
fides of the points E or Q, according as the points M, N, or G, F
are on the fame or different fides of the points E or Q) it appears
from Prop. XVIII. Book I. that the points D and H are in the
conic fection paffing through the five points F, G, K, M, N; then
draw the right line IL bifecting the parallels KD, FG terminated
by the ſection, it will be a diameter of the fection, and draw an-
other diameter AB bifecting the parallels KH, MN in O and P;
if theſe diameters are parallel to each other the fection will be a
parabola; and fince four points are given in it, viz. K, M, N, G,
the fection may be defcribed by the fecond cafe of Prop. IX. Book
III.
Book V.
205
Conic Sections.
III. But if the diameters meet each other, as in the point C, the
fection will be an ellipfe or hyperbola, which may be defcribed by
Cor. 3. Prop. X. Book IV. for its center C is given, and the dia-
meter AB is given in pofition, to which the parallels KH, MN are
ordinately applied.
In this and the following problems, when the ſection to be de-
fcribed is an ellipfe, it may fometimes be a circle, which is to be
confidered as a fpecies of the ellipfe.
PROP. XXII. PROB L. II.
To deſcribe a conic fection which fhall pass through three
given points, and touch two right lines given in poſition.
L
ET AV, AB be the lines given in pofition: and firſt let the FIG. 35.
fection to be deſcribed touch thefe lines in the given points
B, V, and paſs through the third given point within the angle BAV.
If the lines AB, AV be parallel, the fection to be deſcribed will
be an ellipfe, and the line BV joining the points of contact will be
a diameter, to which the line drawn through the third point given,
parallel to AB, AV, will be an ordinate, and therefore in this cafe
the ſection will be eaſily defcribed, by Cor. Prop. XXXV. Book II.
Cafe 1. Now let the lines AB, AV meet in the point A, and let
BV be bifected in P, and draw PA, it will be a diameter of the
fection to be defcribed (XXVI. Book I.) then let the third given
point O be in the line PA; if PO, OA are equal, deſcribe a para-
bola (by XXXIII. Book II.) of which PA fhall be a diameter,
whofe vertex is O, and to which BV fhall be ordinately applied;
the lines AB, AV touch this parabola in B and V, as is evident
from Prop. XLVII. Book I. But if the point O be between P and
A, and the lines PO, OA unequal, let the leffer line be produced
to C, fo that CP, CO, CA be proportional, and take CQ equal to
CO; then (by Cor. XXXV. Book II.) deſcribe an ellipfe or hy-
perbola, of which let OQ be a diameter, and the line VB be ordi-
nately applied to this diameter, the ſection touches the lines AB,
AV in the points B and V, as appears from Prop. XLVIII. XLIX.
Book
-
206
Book V.
Conic Sections.
FIG. 36.
Book I. if PO be lefs than OA, the fection will be in an ellipfe,
but if greater, an hyperbola. If the given point be Qin the line
PA, and not between P and A, take the point C between P and
Q, fo that CP, CQ, CA be proportional, and let CO be equal to
CQ, then deſcribe an ellipfe, as before, to the diameter OQ, it
touches the lines AB, AV in the points B and V, as is manifeſt.
If, the points B, V remaining the fame, the third given point K
be not in the diameter AP of the fection: draw KE parallel to
BV, meeting the tangent AB in L, and AP in F, and let FE be
equal to KF, and take from the point L towards K the line LG,
fo that the fquare of LG be equal to the rectangle KLE, join BG
meeting AP in O; then, as before, deſcribe the ſection touching
the lines AB, AV in B and V, and paffing through O, it will pafs
through the given point K; for if not, let the fection meet the
line KE in R, and take FT equal to RF, becaufe RF is parallel to
BV, it will be ordinately applied to the diameter AP, and therefore
the point T will be in the ſection, and KR, TE equal; draw through
the point O a right line touching the fection, and meeting the tan-
gent AB in N, it will be parallel to LRT; therefore the rectangle
RLT is equal to the fquare of LG (by Prop. LIII. Book I.) that
is, to the rectangle KLE, which is abfurd; therefore the fection
paffes through the given point K.
Cafe 2. Let the lines AB, AV be given in pofition, and the three
given points K, E, F between theſe lines; a ſection is to be de-
fcribed, which fhall touch theſe lines, and pafs through the three
given points.
Let the lines which join the points K, F and K, E, meet AB,
AV in the points D, G and L, N, and take in the line KF the
point H fuch that the fquare of HD be to the ſquare of HG as
the rectangle KDF to the rectangle FGK, and take in the line
KE the point P, fo that the fquare of PL be to the fquare of PN
as the rectangle KLE to the rectangle ENK, and let the right line
joining the points H, P be drawn; if this line meets in the points
B and V, the fides containing the angle BAV, within which the
three given points are taken, defcribe a conic fection, by the preced-
ing
!
Book V.
207
Conic Sections.
ing cafe, touching the lines AB, AV in B and V, and paffing through
the point K, it will likewiſe paſs through the points F and E; for
if this ſection does not paſs through F, let it meet the line KF in
X; then (by Prop. XIX. of this Book) the rectangle KDX will be
to the rectangle XGK as the fquare of HD to the ſquare of HG,
that is, by conftruction, as the rectangle KDF to the rectangle
FGK, which is abfurd; therefore the fection paffes through the
point F and in like manner it may be fhewn that it will pass through
the point E.
If the line KF, joining two of the three given points, be parallel
to AV, one of the lines given in pofition; then the point H may
be taken in this line on either fide of the point D, ſo that the ſquare
of DH be equal to the rectangle KDF, as is evident from Prop.
LIII. Book I.
When the fegment DK, FG, and confequently the rectangles
KDF, FGK be unequal, it is manifcft that the point H may be
taken either between the points K and F, or on the other fide of
the leffer ſegment DK, and in like manner the point P may be
taken on either fide of the fegment LK; and likewife if the line
drawn through the points E and F, meets the lines AB, AV, there
may be found in it two points in the fame way as the points H, P
were found, and fometimes twelve fuch lines as HP may be drawn,
and as many of theſe lines as meet the fides of the angle BAV, fo many
fections may be deſcribed, which fhall anſwer the problem in this cafe.
Cafe 3. If two lines AB, AV be given in pofition, and a fection.
be required to be defcribed, which fhall touch the line AV in the
given point V, and the line AB in another point, and pafs through
the given points K, F; join KF, and take in this line a point H as in
the preceding cafe, and draw VH meeting AB in B; then, as before,
let a ſection be deſcribed touching AB, AV in the points B and V,
and paffing through K, this fection will pafs through the point F, as
in the preceding cafe. If the point H can be taken between, or not
between the points K, F, two fections may be defcribed, which fhall
anfwer the problem.
In this Propofition all the cafes are folved of Prop. XXIII. Book I.
of Newton's Math. Prin.
D d
PROP.
208
Book V.
Conic Sections.
PROP. XXIII.
PRO BL. III.
FIG. 33.
FIG. 37.
Four points being given in a conic fection, and a right line
given in pofition touching the fection; to defcribe the
fection.
Cafe 1.
L'
ET A, F, D, C be the four points given in the fection,
and EH the line given in pofition touching the fection:
draw the lines AC, FD meeting the tangent in the points E and
H; if theſe lines be parallel to each other, take a point B in the
tangent between E and H, fo that the fquare of EB be to the
fquare of BH as the rectangle DEF to the rectangle CHA, B will
be the point of contact, by Cor. 4. XVIII. Book I. and if the ſeg-
ment EB be greater than the fegment BH, the point B may be
taken on the other fide of the point H; but if the lines AC, FD
meet each other in G, then the point of contact B may be found,
by Prop. XX. of this Book, which point, as before, may be taken
either between the points E and H, or not between; therefore fince
five points A, F, D, B, C are given in a conic fection, it may be
deſcribed by (Prop. XXI. of this Book).
Cafe 2. Let A, B, C, D be the given points in the fection, and
let GB touch it in B; join the four given points by right lines,
forming a trapezium; firſt, let none of the fides of this trapezium
be parallel, and let the diagonals meet in E,`and produce the fides
AD, BC to meet each other in F, fo that EF joining thefe two
points may meet GB in G, (which may be always done) then be-
cauſe two lines AB, DC, not parallel to each other, are infcribed in
a conic fection, and their extremities are joined by four lines meet-
ing each other in the points E and F, and becauſe the tangent drawn
through B meets EF in G, the tangent drawn through A meets EF
in the fame point G (by Prop. VIII. of this Book) therefore the
line joining the points A, G touches the fection required. Defcribe
then, by the firſt cafe of the preceding Prop. a ſection which ſhall
touch the lines GA, GB in the points A, B, and fhall paſs through
the point C, it will alfo pafs through the point D; for if not, let it
meet
Book V.
209
Conic Sections.
1
meet BD in X, then the line joining the points A, X, would meet
BC in F (by VIII. of this Book) which is abfurd; therefore the
fection will paſs through the point D.
Then, if the two fides AB, DC of the trapezium be parallel, the
right line EF bifecting theſe parallels will be a diameter, and if the
tangent GB meets EF, as in G, then the tangent drawn through A
will meet EF in the ſame point G, in which caſe the ſection must be
deſcribed as before: but if the tangent GB be parallel to the diame-
ter EF, the tangent drawn through A will be parallel to the ſame
diameter EF, and the ſection, to be deſcribed, an ellipfe, in which
the line AB will be the diameter conjugate to EF, and therefore if
a line be drawn from the point D or C to AB, parallel to EF, it
will be an ordinate applied to the diameter AB, and from hence (by
Cor. 1. XXXI. Book I.) the vertices of the diameter EF may be
found, and the ellipfe defcribed by Prop. XXXIV. Book II.
But if the right lines joining the four given points form a paral-
lelogram the fection will be an ellipfe, and the lines which bifect
the oppofite fides of this parallelogram will be conjugate diameters,
and their magnitudes are found, by Prop. XLVIII. Book I.
2. E. F.
PROP. XXIV. PROBL. IV.
Five right lines, which touch a conic fection, being given in
pofition; to find their points of contact, the fection not
being given in pofition.
L
ET the lines AB, BC, CD, DE, EA touch a conic fection; FIG. 38.
their points of contact are to be found.
Let ABCDE be the figure contained by the five tangents, and
call AB the first fide, BC the fecond fide, and in like manner the
other fides: and let FBCD be a quadrilateral contained by the firſt
four fides, and draw the diagonals BD, FC meeting each other in
M: now omitting the firft fide AB, let ICDE be a quadrilateral
contained by the other four fides, and draw the diagonals ID, CE
meeting each other in N; and draw MN; this line will pafs
Dd 2
through
210
Book V.
Conic Sections.
FIG. 39.
through the points in which the fecond fide BC and the fourth fide
DE touch the fection.
For let G, H, K, L, O be the points of contact of the lines
AB, BC, CD, DE, EA, and join GH, LK and GL, HK: then
becauſe the lines GH, LK are inſcribed in a fection, and B is the
point of concourfe of the lines which touch the fection at the ex-
tremities of the line GH, and D the point of concourfe of the lines
which touch the ſection at the extremities of LK; the points B, D,
and the point of interfection of GK, LH will be in a right line (by
Prop. VIII. of this Book). Again, becauſe GL, HK are infcribed
in a ſection, and the lines which touch the ſection in G, L meet
each other in F, and the lines touching the fection in H, K meet in
the point C; the points F, C, and the point of interfection of GK,
LH are in a right line; but the fame point of interſection has been
fhewn to be in the line BD; therefore it muſt be in the point of in-
terfection of BD, FC, that is, in M; which point therefore will
be in the line LH. Join LO, and becauſe the lines OH, LK are
inſcribed in a ſection, and the tangents which touch the fection in
O, H meet in the point I, and the tangents which touch the ſec-
tion in L, K meet in the point D, the points I, D, and the point
of interfection of OK, LH will be in a right line; and becauſe
OL, HK are inſcribed in a ſection, it may be fhewn in the fame
manner that the points E, C, and the point of interſection of OK,
LH are in a right line; wherefore it will be in the point of interſec-
tion of the lines ID, EC, that is, in the point N: therefore N is in
the line LH, and it has been fhewn that M is in the fame line;
therefore MN paffes through the points L, H, viz. of contact of
BC, DE and in like manner the points of contact of the other
tangents may be found.
:
COR. And likewiſe, if in a conic fection five points A, B, C, D, E
be given; the lines which touch the ſection in thoſe points are eaſily
found, the fection not being given in pofition.
Let A be called the first point, B the fecond, and fo on; join the
four firſt points A, B, C, D by the right lines A B, B C, CD, DA,
meeting each other in the points F, G: likewife join AC, BD; then
becauſe
Book V.
211
Conic Sections.
becauſe the lines AC, BD are inſcribed in a ſection, (by Prop. VIII.
of this Book) the points F, G, and the point of concourfe of the
lines which touch the fection in B, D will be a right line, that is,
FG will paſs through the point of concourſe of theſe tangents.
Omitting the point A, let BCDE be a quadrilateral contained by
the other four fides; and let the points H, K be the interfections of
the fides, and join CE; then becauſe EC, BD are inſcribed in a
conic fection, the points H, K of interfection of the fides, and the
point of concourſe of the tangents, which touch the fection in B, D,
will be in a right line, that is, HK will pafs through this point of
concourſe of the tangents; but this fame point has been fhewn to
be in the line FG; therefore it must be in L the point of interfec-
tion of the lines FG, HK; and therefore LB, LD touch the fec-
tion in the points B, D: and in the fame manner, the tangents may
be found, which fhall touch the fection in the other points.
This Propofition and
of Dr. Robert Simfon.
of the first edition.
Corollary are taken from the Conic Sections
See Cor. 2. and 3. Prop. XLVII. Book V.
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Lately published by the fame Author,
I.
PHILOSOPHICAL ESSAYS
On the Following Subjects.
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