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TREATISE
CONSTRUCTION, PROPERTIES, AND ANALOGIES
THREE CONIC SECTIONS
BY THE
Rev. B. BRIDGE, B. D. F. R. S.
FELLOW OF ST. PETER'S COLLEGE, CAMBRIDGE.
FROM THE SECOND LONDON EDITION,
WITH ADDITIONS AND ALTERATIONS BY THE AMERICAN EDITOR.
NEW HAVEN:
DURRIE AND PECK.
NEW-YORK : COLLINS, KEESE, AND CO,
1839.
t \
Entered according to the Act of Congress, in the year 1831,
by Hezekiaii Howe,
in the Clerk's Office of the District Court of Connecticut.
STEREOTYPED BY
FRANCIS F. RIPLEY,
New Yokk.
ADVERTISEMENT,
The present edition of Bridge's Conic Sections, is reprinted, with a few alterations, from the second London edition. Such changes only have been made, as seemed necessary to adapt the work to the purpose for which it is. intended, namely, that of furnishing a text- book for recitation in Colleges. To this end, the propositions have been enunciated without the use of letters and without reference to particular diagrams. As it is true, however, that a proposition is more readily comprehended, when it is asserted of the lines and an- gles of some particular figure, immediately before the eye, than when expressed in general terms ; it has been thought proper to introduce the general enunciations at the close of the demonstrations to which they belong, and to leave the author's statements at the commence- ment of those demonstrations unaltered. In this respect the conven- ience of students has been consulted, rather than the usual practice of writers.
The demonstrations, as they stand in the original work of Bridge, are in general so much distinguished for conciseness 'and simplicity, and leave so little to be supplied by the student, (a circumstance of great importance in a book designed for large classes,) that it has been thought best to vary from the author only in a very few instances. Alterations have, however, been made, when they seemed likely to be attended by any material advantage.
The number of the propositions has been somewhat increased, not for the purpose of completing the enumeration of particular properties of the Conic Sections, which, in a work like this, considering the fer- tility of the subject, would be equally impracticable and useless ; but in order to exhibit to the student how far many truths may be gener- alized, which he is apt to consider as limited by particular circum- stances. A few propositions, not of this kind, have been added, as
4 ADVERTISEMENT.
being among the more curious of those which Bridge has omitted to notice.
For the convenience of students, some references, particularly to- wards the close of the book, have been made to the mathematical treatises of President Day.
The original numbering of the Properties and of the Articles has been suffered to stand ; and whenever any thing has been inserted in the body of the work, the number of the preceding article has been repeated with a letter annexed. The additional Properties are dis- tinguished by the capitals A, B, C, &c. A few notes contain what- ever else is peculiar to this Edition.
F. A. P. BARNARD.
Yale College, June 20, 1831.
CONTENTS.
CHAPTER I.
Introduction.
Page.
Sect. I. On the nature of the Curves arising from the cutting of a
Cone obliquely to its base, ..... 9
II. On the mode of describing the Conic Sections upon a
plane, 12
CHAPTER II.
On the Parabola.
III. Definitions, . 18
IV. On the Properties of the Parabola, .... 19
CHAPTER III.
On the Ellipse.
V. Definitions, 34
VI. On the Properties, of the Ellipse, .... 35
CHAPTER IV.
On the Hyperbola. VII. Definitions, . 57
VHI. Properties of the Hyperbola analogous to those of the
Ellipse, 59
CONTENTS.
Page.
IX. On the Properties of the Hyperbola derived from its re- lation to the Asymptote, 73
X. On the Properties of the Equilateral Hyperbola, . 81
CHAPTER V.
On the Curvature of the Conic Sections.
XI. On Curvature, and the Variation of Curvature, • .85
XII. On the Curvature of the Parabola, • 92
XIII. On the Curvature of the Ellipse, 94
m
XIV. On the Curvature of the Hyperbola, ... 97
CHAPTER VI.
On the analogous Properties of the Three Conic Sections.
XV. On the changes which take place in the nature of the * Curve described upon the surface of a Cone, during the revolution of the plane of intersection, . . .99
XVI. On the mode of constructing the Three Conic Sections by means of a Directrix, and the Properties derived therefrom, . . 102
XVII. On the analogous properties of the Normal, Latus-rec- tum, Radius of Curvature, &c. &c. in all the Conic Sections, Ill
CHAPTER VII.
On the method of finding the dimensions of Conic Sections whose Later a-recta are given, and of describing such as shall pass, through certain given points.
XVIIL On the method of finding the dimensions of Conic Sec- tions, whose latera-recta are given, 117
CONTENTS. 7
Page.
XIX. On the method of describing Conic Sections which shall
pass through three given points, 121
CHAPTER VIII.
On the Quadrature of the Conic Sections.
XX. On the relation which obtains between the areas of Conic Sections of the same kind, having the same vertex and axis ; and on the Quadrature of the Parabola, Ellipse, and Hyperbola, 126
XXI. On the Quadrature of the Parabola, according to the
method of the Ancients, 133
CONIC SECTIONS.
CHAPTER I.
INTRODUCTION.
A cone is a solid figure formed by the revolution of a right an- gled triangle about one of its sides. (Euc. Def. 11. 3. Sup.) From the manner in which this solid is generated, it is evident that if it be cut by a plane parallel to its base, the intersection of the plane with the solid, will be a circle, since this section will coincide with the revolution of a perpendicular to the fixed side of the triangle ; and if it be cut by a plane passing through its vertex, the intersection will be a triangle, the sides of which will correspond to the hypothenuse of the generating triangle, in different positions, or at different periods of the revolution. If the plane by which the cone is cut be not par- allel to the base, or do not pass through the vertex, then the line tra- ced out upon its surface will be one of those curves more particular- ly distinguished by the name of Conic Sections, the properties of which are to be made the subject of the following Treatise.
(1.) Let BEFGp be a cone, and let it be cut by a plane EEnG perpendicular to its base and passing through its vertex ; then the section BEG will be a triangle. Next, let it be cut by a plane pAon at right angles to the plane BEwG, and parallel to a plane touching the side BE of the cone ; then the curve line pPAOo, which is form- ed by the intersection of this latter plane with the surface of the cone, is called a Parabola.
C.S 2
10
INTRODUCTION.
For the purpose of investigating the nature of this curve, let CPDON be a plane parallel to the base of the cone ; the intersection CPDO of this plane with the cone will be a circle. Since the plane BEnG divides the cone in- to two equal parts, CD (the com- mon intersection of the planes BEnG, CPDON) will be * di- ameter of that circle ; and for the same reason EG will be a diameter of the circle EpGoF. Let ANn be the common inter- section of the planes BEnG, p Aon, and PNO, pno, those of the plane pAon, with the planes CPDON, E^GoF respective- ly. Because the planes pAon, p 0 CPDON are perpendicular to the plane BEnG, PNO must be perpendicular to the plane BEnG, (Euc. 18. 2. Sup.) and conse- quently perpendicular to the two lines AN, ND drawn in that plane ; (Euc. Def. 1. 2. Sup.) for the same reason pno is perpendicular to the two lines An, nG. Hence by the property of the circle CNx
PN2
pit-
ND=PN2, or ND=TW f and EnxnG=pn2, or nG=%—
CN
En
Now since An is parallel to BE, and CD parallel to EG, the fig- ure CNnE is a parallelogram; .\CN=E?&. By similar triangles
AND, AnG ; AN : An : : ND : nG : : ^J ;?£l : : (since CN=En)
CN En v '
PN2:pn2.
(2.) Hence the nature of the curve APp is such, that if it begins to be generated from the given point A, and PN is drawn always at right angles to AN, AN will vary as PN2. And the same may be said with respect to the relation of AN and NO on the other side of ANn.
INTRODUCTION.
11
(3.) Next, let the plane MPAoM be drawn, as before, perpen- dicular to the plane BEG, but passing through the sides of the cone BE, BG ; then the curve MPAoM, formed by the intersection of this plane with the surface of the cone, is called an Ellipse.
In this case, draw two planes, CPDON, UpKon, parallel to the base of the cone ; then, for the same reason as before, PN will be perpendicular both to AN and ND, and pn will be perpendicular both to An and nK ; .-. NCxND=PN2, and rcHxrcK=^w2.
By sim. triangles AND, AriK ;
we have
AN NM
ANxNM
An rcM
AnxnM.
MNC, JVbiH; ND : nK, NC : nR ;
NCxND : nUxnK PN2 : pri*.
(4.) The nature of the curve APM therefore is such, that if A and M are given points, and PN be always drawn at right angles to AM between the points A, M, ANxNM will vary as PN2 ; and the same with respect to the relation between ANxNM and NO2.
12
INTRODUCTION.
(5.) Lastly, let the plane pAon be drawn, as before, perpendicu-
lar to the plane BEG, but cutting the side BG in A, and, when produced, meet- ing a plane drawn touching the other side EB produ- ced, in M ; then the curve pVAOo formed by the in- tersection of the plane pAon with the surface of the cone, is called an Hyperbola.
Let the plane CPDON be drawn parallel to the base ; then, by similar triangles,
AND, AnG ;
we have
MNC, MrcE ; F:(
|
c/^ /\ |
N>\ D |
|
-""rX |
|
|
/ JLL— |
|
|
ps \v |
n \ ^X |
AN
NM
An
wM
ND
NC
wG, wE;
ANxNM
AwxnM :
NDxNC : nGxnE : : PN2 : pn\
(6.) Hence the nature of the curve AVp is such, that if A and M are given points, and PN be always drawn at right angles to AN, the point A lying between M and N, then ANxNM will vary as PN2 ; and the same with respect to the relation of ANxNM and NO2.
II.
Having thus explained the nature of the curves arising from the intersection of a plane with the surface of a cone, we now proceed to show how these curves may be constructed geometrically.
INTRODUCTION.
13
(7.) Let ELF be a line given in position, and LZ another line drawn at right angles to it in the point L. In LZ take any point
S, and bisect SL in A. Let a point P move from A, in such a manner that it may always be at equal distances from S and the line ELF (or, in other words, let the line SP revolve round S as a cen- ter, and intersect another line PM moving parallel to LZ, in such a manner that SP may be always equal to PM ;) then the point P will trace out a curve OAP, having two similar branches, AP, AO, one on each side of the line AZ ; which curve will be a Parabola.
To show that this curve will be a parabola^ draw PNO at right angles to AZ ; then LNPM will be a parallelogram, and LN=PM =SP ; but LN=AN+AL=AN+AS (since AL=AS by construc- tion,) .-.SP=AN+AS.
Let AN=:r,
PN=y,
SA=a; thenSP=AN+AS
=x-\-a, and SN=AN— -AS, =x — a.
Now PN2=SP2— SN2, (Euc. 47. 1.) or y2=(#-fa)2— (x— a)2
* x*+2ax+a2— x*+2ax— a8
14
INTRODUCTION.
Since Aa is a constant quantity, x varies as y2, or AN ccPN2 ; the relation between AN and PN is therefore the same as in Art. 2. ; hence the curve AP is a Parabola.*
(8.) Next, take any line SH, and produce it both ways towards A and M. Let a point P begin to move from A, in such a manner that the sum of its distances from S and H may be always the same, (or, in other words, let two lines, SP, PH, intersecting each other in P, revolve round the fixed points S and H, in such a manner that SP+PH may be a constant quantity ;) then the curve APMO tra- ced out by the point P will be an Ellipse.
To prove this, it may be observed that when P is at A, then HA-f AS or HS-f-2AS, is equal to that constant quantity ; and when P is at M, SM-fMH or HS+2HM, is equal to the same
quantity. Hence HS+2AS=HS+2HM, from which it appears that 2AS = 2HM, or AS = HM. Now SP+PH = HA+AS= HA+HM=AM ; bisect therefore SH in C, and make CM equal to
* Geometrically demonstrated thus : Since SP=AN+AS, SP2=(AN+AS)2= (Euc. 8. 2.) SN2+4AS.AN. But (Euc. 47. 1.) SP2=SN2+PN2 ; .-. SN2+PN2=SN2-f4AS.AN, or PN2=4AS.AN.
INTRODUCTION. 15
CA, then M will be the point where the curve cuts the line SH produced ; and AM will be the constant quantity to which SP-f PH is equal.
Let Then
AC or CM=a, SC or CH=6,
and SP+PH=AM= 2AC=2a; if .-.SP=a— z,
HP will be equal to a+z.
AN=AC-CN=a— *,
NM=CM+CN=a+^
CN=-r, [-SN -SC— CN=6— x,
and PN=y; j NH =CH+ CN=6 -{-#,
Draw PNO at right angles to AM, then (Euclid, 47. 1.) we have,
HP2=PN2+NH*, or (a+z)* =y2 +(b+x)2 , (A)
and SP2=PN2-j-NS2, or (a— zy=y*+(b— x)\ (B)
bx Subtract (B) from (A), then 4.az=4.bx, or #=— ; substitute this value
for z in equation (A), and it becomes
(bx\ 2
which reduced is
a 4 +2a2^+62^2=a2y2 +a?b2+2a'ibx+a'ix*,
or a4— a262— a2^2+63^=a2y2,
i.e. (a2— 62)x(«2— #2)=a2y2. But since a and 6 are constant quantities, a2 — #2 varies as y2 ; now a*—a;*=(a—x) X (a+#) ; .-. (a—x)x(a+x) ocy2, or ANxNM ocPN2 ; hence, as N lies between A and M, the relation between ANxNM and PN2 is such, that the curve APM is an Ellipse.
(9.) Lastly, Take any line SH, and let the two lines SP, HP, inter- secting each other in P revolve round the fixed points, S, H, in such a manner that the difference of the lines HP and SP (viz. HP — SP) may be a constant quantity ; then the curve traced out by the point P will be an Hyperbola.
16
INTRODUCTION.
In this case, let A be the point where the curve cuts SH ; bisect SH in C, and take CM=CA. Since CH=CS, and CM=CA, HM will be equal to AS. Now when P comes to A, HA — AS= a con- stant quantity ; but HA — AS=HA — HM=AM ; v AM is that con- stant quantity. Hence AM=HP — SP.
Let Then
AC or CM=a, i AN =-CN-CA=^— a, SC or CH=6, I NM-ON+CM=:r+a, CN-ir, j NS =ON— CS-^— 6,
i
and PN=y; NH =CN+CH=^+6,
and HP— SP=AM= 2AO=2a; if ..BF=z-\-aJ
SP will be equal to z — a.
Draw PNO at right angles to AN, then we have
HP2=PN2+NH2, or (*+a)2=y2-f(:r-H>)23 (A)
SP2=PN2+NS2, or {z-af^y^x-bf. (B)
bx Subtract (B) from (A), then Aaz=ibx, and *— — ; substitute this
(bx \ 2 \-a 1 ^yi+^x+bf,
which reduced is
b*x*+2a?bx+a*=a2i/*+a9x*+2a*bx+a*b*, or b*x*—a2x*— a*b*+a*=a*y\ i. e. (6*-— a2) x (*•— a2)=a2y2.
INTRODUCTION. 17
Hence x2 — a? ooy2, or (ar— a) x (x+a) ccy2 ; i. e. ANxNM ocPN2 ; and since A lies between N and M, the relation between ANxNM, and PN2, is the same with that in the Hyperbola.*
Having thus established the identity of the curves generated by these two different methods, we now proceed to demonstrate their properties, beginning with the parabola.
* The same may be proved geometrically, as follows. The dem- onstration is applicable either to the Ellipse or Hyperbola.
Take AI=SP. Then IM=HP. .-. HP=CI+CA, and SP= CI co C A.
Now (Euc.47. 1.) (CI+CA)2(=HP2)=PN2-f(CN-fCS)2(=HN2); and, (CI^CA)2(=SP2)=PN2+(CN^CS)2(==SN2). That is, CI2+ 2CA.CI+CA2=PN2+CN2+2CN.CS+CS2, and CI2— 2CA.CI+CA2 =PN2+CN2— 2CN.CS+CS2. Subtract, and 4CA.CI=4CN.CS or CA.CI-CN.CS, .-. CA : CN : : CS : CI, and CA2 ; CN2 : : CS2 :CI2.
From A, draw AG at right angles to AC ; make AG a mean proportional between AS and SM, and join CG, meeting PN in D. Then AG2=AS.SM=CS2^ CA2, (Euc. 5. 2. cor.) and CS2=CA2± AG2.*
By sim. tri. CA2 : CN2 :: CA2±AG2(CS2) : CN2±ND2,* but (as above) CA2 : CN2 : : CS2 : CI2, .-. CI2=CN2±ND2.
In the first equation as expanded above, therefore, let CS2± AG2 1 be substituted for CA2, CS.CN for CA.CI, and CN2±ND2 for CI2, and we have CS2±AG2+2CS.CN+CN2±ND2=PN2+CS2+2CS.
CN+CN2, or ±AG2±ND2=PN2, that is, AG2^ND2=PN2. But (sim. tri.) AC2 : AG2 :: CN2 ^ CA2( AN.NM) ; AG2 * ND2(PN2). But the ratio AC2 : AG2 is constant. Hence AN.NMx PN2, which (N being between A and M) is the property of the Ellipse, and (A being between N and M) is the property of the Hyperbola.
* The sign — for the Ellipse, and + for the Hyperbola. t The sign -f for the Ellipse, and — for the Hyperbola. C. S. 3
18
ON THE PARABOLA.
CHAPTER II.
ON THE PARABOLA.
III.
DEFINITIONS.
(10.) Let pAP be a parabola generated by the lines SP, PM, moving according to the law prescribed in Art. 7. ; then the line ELF, which regulates the motion of the line PM, is called the Di- rectrix ; the point S, about which the line SP revolves, the Focus ; the line AZ, which passes through the middle of the curve, the Axis ; and the highest point A, the Vertex of the parabola.
ON THE PARABOLA. 19
(11.) Let fall the perpendicular PN upon the axis AZ, and through the focus S draw BC parallel to it, and meeting the curve in the points B and C. PN is then called the Ordinate to the axis, AN the Abscissa ; and the line BC is called the Principal Latus- rectum, or the Parameter to the Axis.
(12.) Produce MP in the direction PW, or, in other words, draw PW parallel to the axis AZ ; from any point Q, of the parabola draw QVq parallel to a tangent at P ; and through S draw be parallel to Q,V. PW is called the diameter to the point P ; Q,V the ordinate, PV the abscissa, and be the parameter, to the diameter PW.
(13.) Let PT touch the curve in P, and meet the axis produced in T, draw PO at right angles to PT, and let it cut the axis in O. PT is called the tangent, TN the subtangent, PO the normal, and NO the subnormal, to the point P.
IV.
On the Properties of the Parabola. Property 1. (14.) The Latus-rectum BC is equal to 4AS.
Draw BD (Fig. in page 13.) parallel to LZ, then SB=BD=SL. But since SA=AL, SL is equal to 2AS ; hence SB=2AS, and 2SBorBC=4AS.
This proposition may be thus enunciated.
The latus-rectum is equal to four times the distance from the fo- cus to the vertex.
Property 2. (15.) The tangent PT bisects the angle MPS.
Take Vp so small a part of the curve, that it may be considered as coinciding with the tangent, and consequently as a right line. Join Sjo, and draw pm parallel to AZ ; let fall po, pn, perpendiculars upon SP, PM.
20 ON THE PARABOLA.
The figure Mnpm is a parallelogram, .:nM==pm ; and since po is at right angles to SP, it may be considered as a small circular arc described with radius Sp, .-. So=Sp. Also SP=--PM, and Sp=pm.
Now Po-SP— So=SP— Sp, i and P«=PM— wM=SP- pm, V ..Vo=Vn. =SP— Sp, S
In the small right-angled triangles Vpo, Vpn, we have therefore "Pp common, and Po=P/i, .-. (47. 1.) po=pn ; having .:. their three sidss equal, the angle pVo must be equal to the <ipVn ; hence, since pT may be considered as the continuation of the line Pp, PT bisects the angle MPS ; which proposition may be thus expressed :
The tangent, at any point of the curve, bisects the angle formed at that point, by the perpendicular to the directrix, and the line drawn to the focus.*
* The reasoning in the text, though perfectly conclusive, is of a kind not always entirely satisfactory to the student, who is unaccus- tomed to its use. The same proposition may be demonstrated with- out the use of indefinitely small arcs, in the following manner.
It is first necessary to establish this position : — If a straight line? not parallel to the axis of the parabola, cut the curve in one point, it will, on being produced, if necessary, cut it again.
Let HP, not parallel to AZ, the axis, cut the curve in P. It will, on being produced towards P, intersect the curve in some other point.
Since HP and AZ are not parallel, they will meet, if produced. Let them meet in H. Draw the ordinate PN, and take AR a third proportional to AN and AH. Draw the ordi- nate RQ,. HP, produced, will meet the curve in a.
ON THE PARABOLA.
21
E
L I \ *> m
M F
|
A ^ |
||||
|
C,Z |
||||
|
s |
Eln |
|||
|
N |
^4p |
|||
|
0 |
(16.) Cor. Since the angle MPS continually increases as P moves towards A, and at A becomes equal to two right angles, the tangent at A must be perpendicular to the axis.
For, if not, let it take some other direction as P#, cutting RQ, in
M.*
By Hyp. AN : AH : : AH : AR (Euc. 12. 5.) AN : AH : : AN-f AH(NH) : AR+AH(RH) and AN2 : AH2 : : NH2 : RH2 : : (sim. tri.) PN2 : MR2 (Euc. Def. 11. 5.) AN2 : AH2 : : AN : AR : : (7.) PN2 : QR2, .-. PN2 : MR2 : : PN2 J QR2, or MR2=QR2, which is impossi- ble, unless HP produced pass through Q. Therefore, &c.
Cor. Hence if HQ, cuts the curve, AN : AH : : AH : AR.
The demonstration is not essentially dhferent from any other ar- rangement of the points A, P and Q,.
To prove that the tangent bisects the angle SPM, let the ordinate PN be drawn.
* M is taken between Q, and R, because, if taken on the other side of &, P# must cut the curve.
22
ON THE PARABOLA.
I
Property 3.
If PT meets the axis produced in T, then SP=ST, and TN=
2AN.
(17.) Since PM is parallel to TZ, the angle MPT=alternate an- gle STP ; but (by Prop. 2.) <MPT=<SPT, .-. <STP=<SPT, and consequently SP=ST. That is,
/ If a tangent to any point of the curve cut the axis produced, the points of contact and intersection will be equally distant from the focus.
(18.) Now (7.) SP=AN+AS ; and ST=TA+AS.
Hence, since SP=ST, we have AN+AS=TA+AS, .-. TA=AN, or TN=2AN.
Now if the tangent does not bisect SPM, some other line which cuts the curve, must do it. Let TP be that line, cutting the curve in P and again in p.* Draw the ordinate pm.
|
E |
L |
N M |
p |
|
|
A |
||||
|
' N n |
p |
|||
|
\ |
By Hyp. <SPT=<TPM=alternate <STP.
SP=PM(7.)=LN.
From ST=LN take AS=(7.)AL, and AT=AN.
(Euc. 5. 1.) ST=
But by the
corollary above, AN : AT : : AT : An, or AN=Aw, which is ab- surd, if TP cuts the curve, .-. TP is the tangent.
Hence the tangent bisects <SPM.
It will be seen that while we are demonstrating Property 2d, we at the same time prove all that is laid down in Arts. 17 and 18. In- deed it would be better to demonstrate these latter propositions first, and infer Property 2d from them.
* It is not essential to the demonstration, on which side of P, p is taken.
ON THE PARABOLA. 23
The subtangent is bisected by the vertex ; or the subtangent is double the corresponding abscissa.
(18a.) Hence the tangent at C, the extremity of the latus-rectum meets the axis in L, the same point with the directrix. For (7.) SA=AL. Hence SL=2SA=CS, (14.) and the triangle CSL is isosceles.
Property 4.
(19.) The square of the ordinate (PN2)= latus-rectum x abscissa (BCxAN.)
By Art. 7, y2=4a#, or PN2=4ASxAN ; but by Prop. 1, BC= 4AS, .\PN2=BCxAN. Or, the square of any ordinate to the axis is equal to the rectangle of the corresponding abscissa and the latus- rectum.
PN2 PN2
(20.) Cor. Hence BC=-^ ; and * BC^-^
v ; AN ' 2AN
Property 5.
(21.) The subnormal NO=iBC.
Since TPO is a right angled triangle, (Euclid, 8. 6. cor.)
PN2
NO : PN : : PN : TN, .-.NO==^ ; but by Prop. 3, TN=2AN,
PN2 ,.NO=^— ^; hence (20.) NO=iBC. Or, the subnormal is equal
to half the latus-rectum.
Property 6.
(22.) The square of the ordinate (QV2)=4SPxPV. (See next figure.)
Produce Vd to H ; draw EQ, GV parallel to PN, and QD par- allel to AZ ; then the figures PTHY, PNGV will be parallelograms, and TH-=PV=NG ; .-.HN+NG=HN+TH, or HG=TN.
24
ON THE PARABOLA.
|
Let AN=a:, |
r HG=TN=*2AN=2# |
||
|
PV=NG=y, |
g |
HE=HG— GE =2x—z |
|
|
dD=EG=*, |
AG=AN+NG=2:-fy |
||
|
AS=a, |
AE=AG— EG =:r+y— z |
||
|
and.-.SP=AN+AS= |
=x-{-a. |
Now (19.) EQ,2=4ASxAE=4aX(3r+y— z) ; and by similar tri- angles, HEd, TNP, we have
HE2 : EQ,2 : : TN2 : PN2,
i.e. (2x — zf : 4ax(#-fy — z) :: 4#2 : Aaz,
&ax(x-\-y — z)x&x* Aax
... (2x— zj
=4#x(#-by — z), or 4#2 — kxz+z^^x^+kxy — 4xs. Hence z2=4#?/=Q,D2.
T
ON THE PARABOLA.
25
Again, by sim. As, HGY, Q,DY, we have HG2 : GV2 or PN2 : : QD2 : DY2,
i. e. 4#s
kax
. ■ kaxxbxy
: : Axy : DV2= — j-r-— =4ay.
4r
But av2=aD2+DV2
=4.ry+4ay=4(#-f-a)y=4SPxPV.
That is, the square of an ordinate to any diameter, is equal to four times the rectangle of the corresponding abscissa, and the distance from the vertex of that diameter to the focus.
* This proposition may be demonstrated geometrically as follows. In the first place QV2 ocPV.
From A draw AK, an ordinate to PV, and AG, the vertical tan- gent. Putting L for the latus-rectum, we have (19) L.AN (or AT)=PN2=EF2, and L.AF-FQ2, Again, by sim. tri. PTN, QRF
TN(=2AN) : FR : : PN(=EF) : FQ, or AN : FR : : EF : 2FGL And the rect's L.AN : L.FR : : EF2 : 2EF.FQ.
s. c.
26 ON THE PARABOLA.
The same demonstration, with a very slight alteration, is applica- ble to the case when P and Q, are on opposite sides of A.
Property 7.
(23.) If Q,V is produced to meet the curve in q, then qY=QV. (Fig. on p. 16.)
Draw qe at right angles to AZ, cutting PW in k ; then He=HG-f Ge, and Ae=AG-f-Ge ; if therefore Ge or Yk-=z. we have He=
But L.AN=EF2 .-.L.FR=2EF.FQ. To this add L.AT=EF2 L.AF=Fd2 .-. L(TA+AF-fFR)=L.TIWEF2-f2EF.Fa+Fa2=Ea2.
But (sim. tri.)AK2 : CIV8 : : AG2(L.AT) : EQ2(L.TR),
or AK2 : QV2 : : AT : TR : : PK : PV. But since AK2 and PK are constant Q,V2 ocPV.
Next, QY2=4SP.PV.
Upon the tangent PT, let fall the perpendicular SY, from the focus. Since STP is isosceles, PT is bisected by SY. AY also bisects PT, since AT=AN, (Euc. 2. 6.) Hence AY and SY inter- sect the tangent PT, in the same point.
By sim. tri. PN2 : PT2(AK2) :: SY2(=AS.ST Euc. 8. 6. cor.) : ST2 : : AS : ST(SP).
Hence, because AN=AT=PK and 4AN=4PK,
PN2 : AK2 : : 4AS.AN : 4SP.PK. But PN2=4AS.AN. .-. AK2=4SP.PK. But AK2 : GIV2 : : PK : PV : : 4SP.PK : 4SP.PV. •. Q,V2=4SP.PV.
These demonstrations are equally applicable to gY. The first will require one change of sign, ( — ef.fq) but in all other respects it may remain the same, only substituting the small for the large letters. And as ecf is proved, in this way, equal to L.TR=EQ,2, by sim. tri. Q,V=^V, and Prop. 7 requires no additional demonstration.
ON THE PARABOLA. 27
2x-{-z, and Ae=x+ y+z. In this case, the sign of z is changed throughout ; reasoning therefore as in the last Property, we should have z2 or Vk*=4&y.
By sim. As, HGV, Ykq, HG2 ; GV2 or PN2 : : W : kq\ i. e. 4#2 : 4a# : : kxy \ kq2.
Hence Ar^2= — 2 =4ay.
But Ytf^Vtf+kq*
=4ry+4ay=4(#+a)y=4SPxPV.
Since Q,V2 and V^2 are each equal to 4SPxPV, it follows that V<72=Q,V2, and consequently V<y«Q,V. Or, Every diameter bisects all lines in the parabola, drawn parallel to the tangent at its vertex, and terminated both ways by the curve ; or every diameter bisects its double ordinates.
Property 8.
(24.) The Parameter fa is equal to 4SP. (Fig. on p. 16.)
Let be and PW intersect each other in g", tfyen (23) cg=gb .% cg=-%bc, and cg*2=i&c2.
Since P^ST is a parallelogram, Pg-=ST=SP. Now (22) cg-2=4SPxPg-=4SPxSP=4SP2.
Hence £&c2=4SP2, and $bc =2SP, or 6c=4SP; that is, the parameter to any diameter is equal to four times the dis- tance from the vertex of that diameter to the focus.
(25.) Cor. Hence Q,V~=(4SPxPV=)^xPV ; and since be is constant with respect to the same diameter, P V ccQ,V2. That the square of the ordinate is equal to the parameter x abscissa, is there- fore a general property of the Parabola.
Property 9.
(26.) Draw Q,R parallel to PV, and meeting PT in R; then QRaPR2.
28
ON THE PARABOLA.
In this case (since Q, Vis parallel to RP) PVQJEfc is a parallelogram; .-.PR=QV, and QR=PV ; but (25) PV ocdV2, .-.QRocPR2.
Or, if diameters be produced to meet any tangent to the Parabola, without the curve, the parts of those diameters between the curve and the tangent will be as the squares of the intercepted parts of the tangent.
(27.) Cor. From this it fol- lows, that if PS, PW, be two lines meeting in a given angle, and a point Q begins to move from P in such a manner that its distance RQ, from the line PS (measured in a direction parallel to PW) shall vary as PR2, or, in other words, that RQ, TA, SM, &c. shall be to each other as PR2, PT2, PS2, &c. then the curve PQAM traced out by the motion of the point Q, will be a Para- bola.
Property 10.
(28.) If Q,Y be a tangent at Q,, and VP be produced to meet it in t, then V* is bisected in P.
Produce QY to q: and draw qY parallel to Yt ; then by sim. triangles, (since 0,^=20, V,) QY will be double of Q£, and qY double of V*.
By Art. 26, VtiqY :: Q*2 : QY2 : : 1 : 4 ; .-.P*=4?Y.
But Yt=$qY, .-. P*=4-Vif, or V* is bisected in P. That is, if a tangent and ordinate to any diameter be drawn from the same point, their intersections with the diameter and diameter produced will be equidistant from the vertex of that diameter.
ON THE PARABOLA.
29
The same may be proved of a tangent at q. Therefore the tan- gents drawn from the two extremities of any double ordinate inter- sect the diameter to which that double ordinate belongs in the same point.
(28 a.) This proposition may be thus generalized. Let PO be any tangent, and PK any line in the Parabola, drawn from the point of contact, and meeting the curve in K. Let AT be any di- ameter produced to meet the tan- gent in T, and cutting the line PK in I. Then,
AT : AI : : PI : IK.
O
/
K
For (26.) AT : KO : : TP2 : PO2 : : (sim. tri.) IT2 : KO2. Hence (Euc. Def. 11. 5.) IT is a mean proportional between AT
and KO ; or
AT : IT : : IT ; KO : : (sim. tri.) PI : PK.
Inverted division, AT : AI : : PI : IK.
That is, if from any point in the curve, there be drawn a tangent, and also a line to meet the curve in some other place ; and if any diameter, intercepted by this line, be produced to meet the tangent ; then will the curve divide the diameter in the same ratio in which the diameter divides the line.
The same demonstration, very slightly modified, will apply to di- ameters intersecting the line PK, produced either way, without the section, as at, and dt\ of which it may be proved that at : ai :: Vi : i K, and at' : ai : : Pi' : i'K,
Property 11.
(29.) Let fall SY perpendicular upon PT, and let AY be the ver- tical tangent. AY and SY intersect PT in the same point Y.
30
ON THE PARABOLA.
Since (17.) ST=SP, and SY is perpendicular to J>T, it will di- vide the triangle PST into two equal triangles ; consequently TY= YP ; but (18.) TA is also equal to AN ; .-. TY : YP : : TA : : AN ;
hence, (Euc. 6. 2.) AY is parallel to PN, and consequently perpen- dicular to the line AZ. That is, the vertical tangent intersects any other tangent, in the point where a perpendicular from the focus upon that tangent intersects it.
(30.) Cor. Since the normal PO is perpendicular to PT, it is parallel to SY, .-.TS : SO :: TY : YP; but TY=YP, .-.TS=SO.
Hence, (since SP=TS=SO,) if a circle be described with center S at the distance SP, it will pass through the points P, T, and O ; and the < OSP at the center will be double of the angle OTP at the circumference.
Property 12. (31.) PO is a mean proportional between BS and bg. Since PY-YT, OS=ST=SP, and TO=2SP=6^ (24.) Also (21.) ON=BS.
But (Euc. 8. 0. Cor.) ON : OP : : OP : OT .•.BS:OP::OP:&g-; that is, the normal is a mean proportional between the semiparame- ters of the axis and the diameter at the point of contact.
ON THE PARABOLA.
31
(32.) Cor. I. SA, SY and ST=SP, are severally halves of ON, OP and OT. .-. SA : SY : : SY : SP ; and SY2=SA.SP ; or SY= V(SA.SP), and as SA is constant, SY ocV(SP.)
(32.a.) Cor. 2. Since OP2=BS.^, and BS is constant ; OP2 oc bg cc2bg. And OPccV(2&g\) That is, the normal varies as the square root of the parameter to the diameter at the point of contact.
(32.6.) Cor. 3. Since SO=SP, <SPO=<SOP=<OPg- ; or, the normal bisects the angle made by the diameter at the point of con- tact, with the line drawn from that point to the focus.
(32.c.) Scholium. In optics, the angle made by a ray of light incident upon a reflecting surface, with a perpendicular to that sur- face, is called the angle of incidence ; and the angle made by a re- flected ray with the same perpendicular, is called the angle of reflec- tion. It is a general law that the angles of incidence and reflection are equal. Hence, if CAP represents a concave parabolic mirror, a ray of light falling upon it in the direction g-P, will be reflected to S. The same would be true of all rays parallel to gP. Hence the point S, in which all the rays would intersect each other, is called the focus.
Property A.
(32.d.) Let IH, OR be any two diameters intersected by the par- allels gG, ?Q, in H, R. Then, IH : OR : : GH.% : QR.R?, whether the points H and R, be within or without the section.
Let P represent the parameter to the diameter P W, of which Gg and Qq are double ordinates.
P I G H
32
ON THE PARABOLA.
Then (25.) P.PV=QV2. And P.PN=ON2=VR2.
Taking the diff.
P.NV(=P.OR)=QV2 kVR2=(Euc. 5. 2. Cor.)RQ.R?. In like manner P.IH=GH.Hg\
Hence, GH.Hg- : QR.R? : : P.IH J P.OR : : IH : OR.
That is, the parts of all diameters, intercepted by lines parallel to each other, whether within or without the Parabola, are as the rect- angles of the corresponding segments of the lines.
Property B.
(32.c.) Let the parallels CD, EF intersect the parallels GH, IK,
in the points N, P. Then
CN.ND : HN.NG : : EP.PF : KP.PI. For (32.d.) CN.ND : EP.PF And HN.NG : KP.PI
.-. CN.ND : EP.PF
LN:OP LN : OP. HN.NG : KP.PI ;
or (Euc. 16. 5.) CN.ND : HN.NG : : EP.PF : KP.PI ;
Or, the rectangles of the corresponding segments, into which par- allel lines in a Parabola divide each other, have to each other a con- stant ratio.
ON THE PARABOLA.
33
Property C.
(32/.) Let RX and PW be the diameters to which CD and HG are double ordinates. Let P represent the parameter of PW, and P' that of RX.
Then
CN.ND : HN.NG : : F ; P. p
By reasoning like that employed in Prop. A, it may be shown that
CN.ND=P'.IN and HN.NG=P.IN,
.-. CN.ND : HN.NG : : P'.IN : P.IN : : F : P.
This proposition may be thus enunciated :
If any two straight lines, which meet the curve in two points, in- tersect each other, the rectangles of their corresponding segments will be as the parameters of the diameters, to which those lines are double ordinates.
The last two propositions, like Property A, are applicable to lines both within and without the section, and the diagrams are lettered in such a manner that the demonstration may apply to either case. . C.S. 5
34
ON THE ELLIPSE.
CHAPTER III. ON THE ELLIPSE.
V.
DEFINITIONS.
(33.) Let APMO be an Ellipse generated by the revolution of the lines SP, HP, about the fixed points S, H, according to the law- prescribed in Art. 8. ; then B Q, the line AM, which passes through the two foci S and H, is called the Axis Major ; and if through the center C a line BCO be drawn at right angles to AM, it is called the Axis Minor of the Ellipse.
(34.) From any point P let fall the perpendicular PN upon the axis major AM, and through the focus S draw the straight line LST parallel to it. PN is then called the ordinate to the axis ; AN, NM, the Abscissas ; and the line LST is called the latus-rectum, or the Parameter to the Axis.
(35.) Draw any line PCG through the center, and another line DCK parallel to a tangent at P ; draw also Qv parallel to DCK. PCG is then called a Diameter, and DCK the Conjugate diameter to PCG ; dv is called an Ordinate to the diameter PCG, and Pv, vG, the Abscissas.
ON THE ELLIPSE.
35
YI.
On the Properties of the Ellipse. Property 1.
(36.) If SB, HB, are drawn from the foci to the extremity of the axis minor, then SB, HB, are each equal to AC.
Since SC=CH, and BC is common to the two right-angled tri- angles BCS, BCH, SB must be equal to BH ; .-. SB+BH-2SB or 2BH.
Again, by Sect. 2. Art. 8. SP+PH=AM=2AC ; and when P
B
|
- |
t |
~^\P |
|||
|
S |
c |
H 1 |
|||
|
T |
|||||
|
( |
> |
M
comes to B, SB+BH=2AC ; hence 2SB or 2BH=2AC ; .-. SB or BH=AC.
That is, the distance from either focus to the extremity of the axis minor is equal to the semi-axis major.
Property 2.
(37.) MSxSA=BC2
For BC2=SB2— SC2 (Euc. 47. 1.)
=AC2— SO2 (by Prop. 1.)
=(AC+SC)X(AC— SO)
=(CM+SC)x(AC— SC) (forCM=AC)
=MSxSA.
That is, the rectangle of the focal distances from the vertices is equal to the square of the semi-axis minor.
36 ON THE ELLIPSE.
(38.) Cor. In the same manner, it might be shown, that AH x HM=BC2.
Property 3.
(39.) The latus-rectum LST is a third proportional to the major and minor axes.
For SL+LH=2AC(by const:
.-.LH=2AC — SL, and LH2=4AC2— 4AC
xSL+SL2,
Again, LH2=SL2+SH2 (Euc. 47. 1.) =SL2+4SC2 (for SH=2SC) =SL2+4(SB2— BC2) =SL2+4(CA2— BC2).
Hence 4AC2— 4ACxSL+SL2=SL2+4AC2— 4BC2 ; .-.4ACxSL=4BC2.
And putting this equation into a ; 2AC : 2BC : : 2BC : 2SL,
proportion, we have \ or AM : BO : : BO : LT.
Therefore the latus-rectum is a third proportional to the major and minor axes.
(39a.) AS.SL=iLT.AM. For (39) B02=LT.AM.
.-. iB02(=BC2)=(37) AS.SM=^LT. AM.
Property 4.
(40.) Produce SP to p ; then if YZ bisects the angle HP/?, it will be a tangent to the Ellipse in P. (Fig. in page 29.)
For if YZ does not touch the ellipse, let it cut it in Q, ; take Pp= PH, and join pB, QS, QH, and Qp. Since P^=PH, PZ com- mon, and <CpPZ=HPZ, the side pZ will be equal to ZH ; and the <s PZjo, PZH, will be equal and consequently right <s. Again, since pZ=ZH, ZQ, common, and <s QZp, Q,ZH right <s, the side Qp is equal to the side Q,H.
Now (by Euc. 20. 1.) SQ,+Qp is greater than Sp or SP-fPp or SP+PH ; but QH=Qp ; therefore SGl+aH is greater than SP+PH ; but if Q, is a point in the curve, SQ,-J-Q,H must be equal
ON THE ELLIPSE.
37
to SP+PH ; Q, therefore is not a point in the curve. In the same manner it might be proved that YZ does not meet the curve in any other point on either side of P, it must therefore be a tangent at P.
Hence, if from the foci two straight lines be drawn to any point in the curve, the straight line bisecting the angle adjacent to that contained by these lines, is a tangent.
(41.) Cor. 1. It follows, from the above, that the <SPY=<HPZ ; for <SPY=vertical <pPZ ; but pPZ=HPZ ; .-. SPY=HPZ ; and this is a distinguishing property of the ellipse ; viz. That lines drawn from the foci to any point in the curve make equal angles with the tangent at that point.
(41.a.) Hence, also, (see Art. 32.c.) if rays of light proceed from one focus of a concave ellipsoidal mirror, they will be reflected by the mirror into the other focus.
(42.) Cor. 2. When P comes to A or M, the angle HPp becomes equal to two right angles ; at A or M, therefore, the tangent is per- pendicular to the axis AM.
Property 5.
(43.) If tangents be drawn at the extremities of any diameter of an Ellipse, they will be parallel to each other.
38 ON THE ELLIPSE.
Complete the parallelogram SPHG, of which SP, PH are two sides, and join PG ; then since the opposite sides of parallelograms are equal to each other, SG+GH is equal to SP+PH, and conse- quently G is a point in the Ellipse ; and since the diagonals of parallelograms bisect each other, SH is bisected in C ; therefore C is the center of the Ellipse, and PG a diameter (35).
Now let the tangents ef, gh be drawn at the extremities of the di- ameter PG; then by Art. 41. the <SPe=<HP/; but SPe+HP/ is the supplement of <SPH ; .-. SPe=4 supplement of SPH. For
the same reason, the <HGA=£ supplement of <SGH ; but the <s SGH, SPH are equal, being opposite <s of a parallelogram ; hence the <SPe=<HGA. Again, since SP is parallel to GH, the <SPG=<PGH ; therefore SPe+SPG=HGA+PGH, or GPe=PGA,
and consequently ef is parallel to gh. Therefore, if tangents, &c.
(44.) Hence, if tangents be drawn at the extremities of any two diameters of an Ellipse, they will form a parallelogram (eghf).
ON THE ELLIPSE.
Property 6.
(45.) If SP intersects the semi-conjugate diameter (CD) in E, then PE is equal to the semi-major axis (AC).
Draw HI parallel to CD or ef, then the <PIH=altemate <SPe, and <PHI=alternate<HP/; but <SPe=<HP/, .-. <PIH=<PHI, and consequently P1=PH. Again, since CE is parallel to HI, and SC=CH, SE must be equal to EI.
Hence FuBf* EI=SI.
,(PI+EI)orPE=Si±^(
SP+PH 2AC ______ AC.
Therefore, if from the extremity of any diameter, a line be drawn to the focus, meeting the conjugate diameter, the part intercepted by the conjugate will be equal to the semi-major axis.
Property 7. . 1
(46.) If the ordinate PN be drawn to the major axis, then ANx NM : PN2 : : AC2 : BC2 : : AM2 : BO2. (Fig. on p. 32.)
Let AC or CM=a, SC or CH=6,
CN=*, and PN=2/;
Then, by Art. 8, page 7, (a2— 62) X (a— x) X (a+ #)=a2y2 , or (AC2— SC2) x AN X NM=AC2 x PN2. But AC2— SC2=SB2— SC2=BC2 ;
.-. BC2xANxNM=AC2xPN2, and ANxNM : PN2 : : AC2 : BC2 : : AM2 ; BOa.
That is, as the square of the axis major is to the square of the axis minor, so are the rectangles of the abscissas of the former, to the squares of their ordinates.
40
(47.) Cor. Since ANxNM=(AC— GN)x(AC+CN)=AC2— CN2 ; .-. AC2— CN2 : PN2 : : AC2 : BC2.
Property A.
(47.a.) If from P, the line PR=AC, be drawn to BO, then PI= BC.
For (sim. tri.) Rrc2(PR2 — Pn2) : PN2 : : PR2 : PP. That is, AC2— CN2 ; PN2 : : AC2 : PI2. But (47) AC2— CN2 : PN2 : : AC2 : BC2. .-.PI2=BC2 and PI=BC.
Or, if from any point in the Ellipse, a line be drawn to the minor axis, equal to the semi-major, the part intercepted between that point and the major is equal to the semi-minor axis.
Property 8.
(48.) If the ordinate ¥?i be drawn to the minor axis, then Buy, nO : Prc2 : : BC2 : AC2.
In this case, P?i=CN, and Crc=PN ; .-. AN X NM=AC2-~ Prc2, (47) ; hence, by substitution in Art. 47, we have,
AC2— Pn2: Crc2:: AC2 : BC2,
... AC2 : AC2— Pn2 : : BC2 : C/i2 ; andPrc2: AC2:: BC2— Cw8 : BC2,*
* For AC2 : AC2 -(AC2— Pw2) or Prc2 : : BC8 ; BC2— Cn8, .-. (in- vertendo) Prc2 : AC2 : : BC2— Cn2 : BC2.
ON THE ELLIPSE.
41
Hence BnxnO : Prc2 :
(BC— Cro)x(BC+Cn) : BCS
BnxnO BC2 : AC2.
:BC2,
That is, as the square of the axis minor is to the square of the axis major, so are the rectangles of the abscissas of the former, to the squares of their ordinates.
(49.) Cor. Since B/ixnO=BC2— Crc2, we have BC2— Cn2 : Pn2 : : BC2 : AC2.
Property 9.
(50.) Describe the circle ARML upon the major axis AM, and draw an ordinate Q,PN cutting the ellipse in P ; then Q,N ; PN : : AC : BC.
By Art. 46. ANxNM : PN2 : : AC2 : BC2. But by Prop, of circle, AN x NM=aN2 ;
.-. QN2 : PN2 : : AC2 : BC3, and GIN : PN : : AC : BC.
In like manner, it may be shown that qn : Pn : : BC : AC. Hence, if a circle be described on either axis, then any ordinate in c. s. 6
42
ON THE ELLIPSE.
the circle, is to the corresponding ordinate in the Ellipse, as the axis of that ordinate is to the other axis.
(51.) Cor. 1. Since RC is equal to AC, QN : PN : : RC : BC. Hence it appears, that if upon AM as diameter, a circle be described, and if B be a given point in the line RC ; then if the ordinates of this circle are diminished in the given ratio of RC : BC, the curve APBM passing through the extremities of these lesser ordinates, will be an ellipse, whose axis major is to the axis minor in the same given ratio.
Also, if upon BO a circle be described, and if A be a point in Cr, produced ; then if the ordinates of this circle are increased in the given ratio of Cr : CA, the curve BPAO, passing through the extremities of these greater ordinates will be an ellipse, whose axis minor is to its axis major in the same given ratio.
(52.) Cor. 2. From hence also it may be shown, that the ortho- graphic projection of a circle upon a plane will be an ellipse. Sup- pose the circular plane ARML to be inclined to the plane of this
paper in such a manner, that the semicircle ARM may be above the paper, and the semicircle ALM below it, and let AM be the com- mon intersection of the two planes. Let the semicircle ARM be
ON THE ELLIPSE. 43
projected downwards upon the plane of the paper, by drawing perpen- diculars Q,P, RB, from each point of the circle, and let the semicir- cle ALM be projected upwards, by drawing the perpendiculars qp, LO, &c. ; then the curve ABMO, marked out by this projection, will be an ellipse. For draw Q,N, RC, at right angles to AM, and join PN, BC ; then the angles Q.NP, RCB, will measure the in- clination of the planes, and PN, BC will be perpendicular to their common intersection AM. Now Q,N : PN : : rad. ; cos. <Q,NP, and RC : BC::rad. : cos. RCB(=QNP) ; .-. QN : PN : : RC or AC : BC ; and consequently, the four lines QN, PN, RC, BC, bear the same relation to each other and to AM as they did in Cor. 1 ; hence P, B, &c. are points in an ellipse. In the same manner it may be proved, that the semicircle ALM is projected into a semi-ellipse AOM ; and thus the whole circle ARML is projected into an ellipse ABMO, whose axis major is AM.
This proposition is likewise manifestly true, when the plane of projection does not cut the circle, or cuts it unequally.
Property 10.
t
(53.) Let PCG be any diameter of an ellipse, and DCK its conju- gate diameter ; draw the ordinate Q,V, then
PG2 : DK2 : : PV.VG : QV2.
Let the circle AqMg be projected into the ellipse AQMG, accord- ing to the principles just now laid down, and let the diameter pCg of the circle be projected into the diameter PCG of the ellipse. Draw the diameter dCk} at right angles to pVg, and qv parallel to dCk, and let dCk, qv be projected into DCK, QV ; then since parallel lines are projected into parallel lines, QV will be parallel to DCK. Now it is evident that a tangent to the circle at p would be projected into a tangent to the ellipse at P ; dCk and qv therefore being par- allel to a tangent at p, (for they are both perpendicular to pCg) DCK and QV will both be parallel to a tangent at P ; hence DCK is the conjugate diameter, and QV the ordinate, to the diameter PCG. Again, since Qq is parallel to dD (for they are both at right <s to
44
the plane of the ellipse,) and QT parallel to DC, the plane ©f the triangle dDC must be parallel to the plane of the figure QqvY ; but qv is parallel to dO ; if therefore Q,V and qv are produced till they meet in L, they will form a triangle Qdh, similar to the triangle cZDC ; and since qL is in the plane of the circle, and Q,L in the plane of the ellipse, the point L must be in the common intersection (AM produced) of those planes. Now pP, vY being perpendicular to the plane of the ellipse, are parallel to each other, and to the lines Qq, dD ; hence it appears that the triangles Q.gL, YvL, dDC, And the triangles /?PC, vYC, are respectively similar ; we have then, by property of circle,
Cp2 : Cp2— Cv*(pv.vg Euc. 5. 2. Cor.) : : Cd2 : qv*.
CV2(PV.VG)
But, (sim. tri.) Cp* : C^2— Cv2 : : CP2 : CP And CcZ2 : qv* : : CD* : dV2.
.-. CP2 : PV.YG : : CD2 • aV2 ; Or, CP2 ; CD2 : : PG2 : DK2 : : PY.VG
avs
The same demonstration is, obviously, applicable to Ym, and CK. Consequently, as the square of any diameter is to the square of its conjugate, so are the rectangles of its abscissas to the squares of their ordinates.
ON THE ELLIPSE. 45
(54.) Cor. Since any diameter in the Ellipse is the projection of a corresponding diameter in the circle, and since all the diameters of the circle are bisected in the center ; it follows that all diameters of the Ellipse are bisected in the center. For similar reasons, every diameter in the ellipse bisects its double ordinates, or lines drawn in the Ellipse, parallel to the tangent at its vertex.
Property B.
(54.a.) Let PG, DK be any two conjugate diameters, and EF, aS any lines parallel to PG, DK, intersecting each other in M. Then PG2 : DK2 : : EM.MF : QM.MS.
Draw the ordinate EN.
Then (53.) PG2 : DK2 : : PN.NG(CP2— CN2) : EN2.
Also PG2 : DK2 : : CP2— CV2 : QV2.
.-. (Euc. 19. 5.) PG2 : DK2 : : CN2— CV2(=Ew2— Mn2)
: QV2— EN2(=MV2). That is, (Euc. 5, 2. Cor.) PG2 : DK2 : : EM.MF : QM.MS.
The same demonstration is applicable, (with a single change of sign) to E'F', which intersects QV in M' without the Ellipse. Wherefore, if straight lines in the Ellipse parallel to two conjugate diameters intersect each other, either within or without the Ellipse, the rectangles of their corresponding segments are to each other as the squares of the diameters to which they are parallel.
(54.6.) Cor. PG2 : DK2 or PC2 : DC2 : : En2~Mn2 : Q,V2— MV2.
46
ON THE ELLIPSE.
Property 11.
(55.) If Q,T, PT. are tangents to the circle and ellipse in the points Q, and P, they will meet in the axis produced at T ; and CA will be a mean proportional between CN and CT. And if Kt, let, are tan- gents to the points K and k, BC will be a mean proportional between Cn and Ct.
Let Q,T be a tangent to the circle in Q, and join TP. If TP does not touch the ellipse, let it cut it in P, p ; and through p draw the ordinate mpqr, meeting TQ, produced in r.
|
QLy^ |
t r R 0 |
B Y^\ |
|
|
i/Cdf* |
i |
P |
|
|
T *r \ |
^^y\ |
||
|
A I B |
* 1 |
n |
C j |
M
By sim. As, TNP, Tmp ; TNQ, Tmr ; we have TN : Tm i : PN : pro, and
TN:Tro::CiN:rro;
.-. PN :pm:: QN : rw=^pX^- ; but by Art. 50, the ordi-
nates of the circle and ellipse are to each other in a given ratio ;
therefore GIN : PN : : qm : pm, or qm=— -= — . Hence rm=qm,
which is impossible ; .-. TP does not cu* the ellipse, and consequently
ON THE ELLIPSE.
47
is a tangent to it in P ; and since TQ, touches the circle in Q,, CQ/T is a right-angled triangle, .-. (Euc. 6. 8.) CN : CQ, : : Ca : CT ; but CQ=CA, hence CN : CA : : CA : CT, and CNxCT=CA2.
(56.) Upon the axis minor BO describe the circle BkO ; draw the ordinate Kkn, and join Ck. By Art. 48, we have BnxnO : K?i2 : : BC2 : AC2 ; by the same process, therefore, as that in Art. 50, it may be proved that Kn is to kn in a given ratio. Draw Kt, kt, to the ellipse and the circle ; then proceeding in the same man- ner as in the former part of this demonstration, we might show that they will meet in the minor axis produced. Since kt touches the circle, Ckt is a right-angled triangle, .-. Cn : Ck :: Ck l Ct] but CA;=BC, .-. Cn : BC : : BC : Ct, or Cn xC*=BC2.
Hence, if a tangent and an ordinate to either of the axes be drawn to any point of the Ellipse, meeting that axis and axis produced, then the semi-axis is a mean proportional between the distances of the two intersections from the center.
(57.) Cor. In the right-angled triangle CQT, (Euc. 8. 6.) CN ; QN : : Q,N : NT, .-. CN X NT=Q,N2=CQ,2— CN2=AC2— CN2. For the same reason, Cn X nt=BC2 — Cn2.
Property C.
(57. a.) Let TLG be the focal tangent, or the tangent drawn at
F
48 OK THE ELLIPSE.
the extremity of SL, the ordinate from the focus. Let NPG be any ordinate produced to meet the tangent TLG. Then SP=NG.
If AI be taken equal to SP, IM=HP.
.-. SP-CA+CI, and HP=CA— CI.
(Euc. 47. 1.) (SP2)(CA+CI)2=PN2+(CS+CN)2(NS2).
And (HP2) (CA— CI)2=PN2+(CS— CN)8(HN2).
Expand and subtract 4CA.CL=4CS.CN, and CA.CI=CS.CN.
.-. ON : CI(SP— CA) : : CA : CS : : CT : CA (55.)
.-. CN+CT(TN) : SP : : CT : CA.
Again, (55.) CS.CT=AC2, and CS.ST-CA2— CS2=BCS (3G, and Euc. 47. 1.) .-. CS.CT : : CS.ST or CT : ST : : AC2 : BC2 : : AC : SL(39.)
.-. ST : SL : : CT : AC : : TN : SP.
But (sim. tri.) ST : SL : : TN : NO. .-. SP=NG ;
That is, the distance from the focus to any point of the curve is equal to the ordinate to that point, produced to meet the focal tan- gent.
Cor. 1. AS-AE, SM=MF, and C*=AC.
Cor. 2. Hence, also, since TN : NG is a constant ratio, TN : SP is a constant ratio. Therefore, if a line be drawn through T per- pendicular to AC produced, the distance of the point P from this line (=TN) is in a constant ratio to SP, the distance of the same point from the focus. This ratio, being (by demonstration above) =CT : CA, is a ratio of greater inequality. This perpendicular is the directrix of the Ellipse. (See Art. 138, et seq.)
Property 12.
(58.) If PCG, DCK, be conjugate diameters of the ellipse, and PF perpendicular to CK, then POxPF=BC2.
ON THE ELLIPSE.
49
Draw Cy parallel to PF. Then because PO is parallel to Cy,
and Ct parallel to PN, and the <s Cyt, PNO right <s, the triangles
PON, are similar ; .-. Ct : Cy : : PO : PN ; but Cy=PF, and
PN=Cw, (being opposite sides of parallelograms) ; ,%Ct : PF ::
PO : Cn, or PFxPO=Cwx07=-BC2 by Art. 56. That is, if from the extremity of any diameter, a perpendicular be drawn to its con- jugate ; then the rectangle of that perpendicular and the part of it intercepted by the axis major, will be equal to the square of the semi-axis minor.
Property 13.
(59.) Draw the ordinates DL, PN, to the major axis, then +CL2=AC2, and PN2+DL2=BC2. By Art. 47,
AC2— CL2 : DL2 : : AC2 : BC2, (A)
AC2— CL2 : DL2 : : CN X NT : PN2,
and AC2— CN2
.-. AC2— CL2 :
or AC2— CL2:CNxNT
DL2 CL2
: PN2,
: NT2 by sim. As ) DCL, PTN. $
c.s.
|
NT |
||
|
NT |
:CN, |
|
|
CT |
:CN, |
|
|
CTxCN |
: CN2, |
|
|
AC2 |
:CN2 |
(55.) |
50 ON THE ELLIPSE.
AP2 ri2 CNxNTxCL* CNxCL» Hence AC2— CL2= NT*-
From which we have,
CL2 : AC2— CL2 and compdo, AC2 : AC2— CL2
Hence AC2— CL2=CN2, or CN2+CL2-AC2.
(60.) Since AC2 — CL2=CN2, by substitution in proportion (A) we have CN2 : DL2 : : AC2 : BC2 ; but AC2— CN2 : PN2 : : AC2 : BC2, .-. CN2 : DL2 : : AC2— CN2 : PN2. And alternately,
CN2 : AC2— CN2 : : DL8 : PN2, Compdo, CA2 : AC2— CN2 : : DL2+PN2 : PN2 ;
.-. AC2 : DL2+PN2 : : AC2— CN2 : PN2 : : AC2 : BC2. Hence DL2+PN2=BC2.
Hence if ordinates to either axis be drawn from the extremities of any two conjugate diameters, the sum of their squares will be equal to the square of half the other axis.
Property 14.
(61.) PC2+CD2=AC2+BC2, PG and DK being conjugate diam- eters. (See last Fig.)
For by Prop. 13. CN2+CL2=AC2, and PN2+DL2=BC2 ; .-. CN2+PN2+CL2+DL2=AC2-f BC2, or CP2+CD2=AC2+BC2.
Therefore the sum of the squares of any two semi-conjugate di- ameters is equal to the sum of the squares of the semi-axes.
Property 15. (62.) CD x PF«AC x BC. (See last Fig.)
|
ON |
THE ELLIPSE. |
||
|
In case 2. of Prop. 13. it was proved |
that |
||
|
CN2 : DL2 : : |
AC2 : BC2 ; |
||
|
.-.CN :DL :: |
AC .: BC, |
||
|
andCN:AC :: |
DL : BC. |
||
|
By similar As, TCy, DCL, CT : Cy(PF) |
::CD |
||
|
Hence we have, CN : |
AC : |
:DL: |
BC, |
|
and CT: |
PF : |
:CD: |
DL, |
|
.-. CNxCT(AC2) : |
ACxPF : |
:CD: |
BC, |
|
or AC: |
PF : |
:CD: |
BC. |
|
.-. CDxPF==ACxBC. |
51
DL.
That is, if from the extremity of any diameter, a perpendicular be drawn to its conjugate, the rectangle of that perpendicular and the semi-conjugate, is equal to the rectangle of the semi-axes.
(63.) Cor. From this it appears, that all the parallelograms cir- cumscribing the ellipse, and having their sides drawn through the ex- tremities of any diameter and its conjugate, are equal to each other and to the parallelogram described about the major and minor axes. For the parallelogram eghf described about the conjugate diameters PCG, DCK, is equal to four times eDCP=4CDxPF=4ACxBC= right-angled parallelogram whose sides are 2AC and 2CB= paral- lelogram described about the major and minor axes.
Property 16.
(64.) If SY, HZ, be drawn from the foci perpendicular upon the tangent YZ, the points Y, Z, will be in a circle described upon the major axis AM. (Fig. in p. 52.)
Join YC, produce HP to W, making PW=PS, and join WY.
By Prop. 4. PY bisects the < SPW ; and since SP=PW, and PY is common, WY will be equal to YS, and < WYP=<SYP= a right angle ; hence WYS is a straight line. Now since WY=YS,
52
ON THE ELLIPSE.
and SC=CH, CY must be parallel to WH, and .-. SC : SH : : CY : HW, but SC=4SH ; .-. CY=£HW=4(HP+PS)=4AM=AC ; hence Y is a point in the circle whose center is C and radius CA. In the same manner it may be proved that Z is a point in the same circle.
Therefore, if perpendiculars be dropped from the foci upon any- tangent to the Ellipse, the intersections of those perpendiculars with the tangent will be in the circumference of a circle described upon the axis major.
Property 17.
(65.) SYxHZ=BC2
Since the <HZP is a right angle, it must be in a semicircle ; if .•. YC and ZH be produced, they will meet in the circumference of the circle at some point, and YK will be a diameter. Hence YC= CK; and as SC=CH, and <SCY=<KCH, the side HK must be equal to SY. But by the property of the circle, (Euc. 3. 35.) ZHxHK=AHxHM=BC2 by Art. 38. Hence (since HK=SY) SYxHZ=BC2.
That is, the rectangle of the perpendiculars from the foci upon any tangent, is equal to the square of the semi-axis minor.
ON THE ELLIPSE. 53
(66.) Cor. Since <s at Z and Y are right angles, and <SPY= <HPZ, the triangles SPY, HPZ, are similar ; hence,
SP : SY :: HP : HZ=^-5? ;
,SYxHZ=SYl^ ;
or BC2=
SP SY2xHP
PS
SP
From which it follows, that SY2=BC2 x^„ ;
% SY=BC X>/(|^) «\/(|f)> BC being constant'
Property D.
(66.a.) Let the vertical tangents AE, MF be drawn ; then EA.FM= BC2 and EF is the diameter of a circle, passing through S and H.
If P coincide with B, then E A and FM each =BC, and E A.FM= BC2. But if not, let the tangent PE intersect the axis in T.
Then (sim. tri.) EA : SY : : TA : TY, and HZ : FM : : TZ ; TM.
But (Euc. 36, 3. Cor.) TA.TM=TY.TZ, or TA:TY::TZ:TM; .-. EA : SY : : HZ : FM, and EA.FM=SY.HZ=(65.)BC*.
Again, (38.) AH.HM=BC2=EA.FM, .-. AH : EA : : FM : HM.
Hence (Euc. 6. 6.) the triangles EAH and HFM are similar, and <EHA=<HFM and <FHM=AEH. Whence <EHF is a right angle, and a circle described on EF will pass through H. The same may also be shown of S.
54
ON THE ELLIPSE.
Wherefore, if tangents be drawn from the vertices to meet any- other tangent, the rectangle of the vertical tangents will be equal to the square of the semi-minor axis ; and the intercepted part of the other tangent will be the diameter of a circle passing through the foci.
Property 18.
(67.) Draw the conjugate diameters PCG, DCK, then SPx HP=CD2.
By similar triangles, SPY, HPZ, PEF, we have
SP and HP
SY HZ
SPxHP : SYxHZ
PE :PF, PE :PF
PE2 : PF5
Now, by Art. 65, SYxHZ=BC2, Art. 45. . . PE2=AC2 ;
.-. SPxHP : BC2 : : AC2 ; PF2 or SPxHP-
BC2xAC2
PFa
But by Art. 62. CDxPF=ACxBC ;
k ACxBO ,_■_. AC2xBC2 ' CD~- PF~ and CD : PF2-"
Hence SPxHP=CD2.
ON THE ELLIPSE. 55
That is, the rectangle contained by the straight lines, drawn from the foci to the extremity of any diameter, is equal to the square of half the conjugate to that diameter.
Property E.
(67. a.) Let PG, RX be any two diameters, and let EF, parallel to PG, cut RX in L. Then PG2 • RX2 : : EL.LF : RL.LX. Draw DK conjugate to PG, and RN, Rrc, ordinates to PG, DK.
Then RN=Crc, HN=CI, and Rn=HI=CN.
By sim. tri. RraC, CLI, Crc2 : Cn2— CP : : RC2 : RC2— • CL2.
Also (54.6.) RN2(Cn2) : RN2— HN2(Crc2— CP) : : PC2— CN* : EI2— HI2(EP— Rrc2). But (sim. tri.) C^2 : Cn2— CI2 : : Rw2(CN2) : Rrc2— LP
: : RC2 : RC2— CL2.
Our proportions then, are
Cn2 : Cn2— CP : : PCs— CN* : El2— Rn2
Qrfi : Cn2— CP : : CN2 : Rn2— LI2 : : RC2 : RC2— CL2
Adding the terms of equal ) ratios, by Euc. 12. 5. |PC2 : El2~Ll2 : : RC2 : RC2-CL*
Alternation, and Euc. 5. 2, cor. PC2 : RC2 : : EL.LF : RL.LX, Or, PG2 : RX2 : : EL.LF : QJL.LX.
The same demonstration, (signs being changed whenever neces- sary,) is applicable to E'F', which intersects RX, produced, in L'.
Wherefore, the squares of any two diameters are to each other, as the rectangles of the segments of one of them, are to the rectangles of the corresponding segments of lines parallel to the other ; whether the point of intersection be within or without the ellipse.
56
ON THE ELLIPSE.
Property F.
(67.6.) Let PG, RX, be any diameters, and let EF, Q,S, parallel to PG, RX respectively, intersect each other in M.
Then PG2 : RXB : : EM.MF : Q,M.MS.
For, through M draw the diameter AB.
|
Then |
AB2 |
|
Or, |
AB2 |
|
In like manner, |
AB2 |
|
,« |
.PG2 |
PG3 : : AM.MB : EM.MF. AM.MB : : PG2 : EM.MF. AM.MB : : RX2 : QM.MS. RX2:: EM.MF ; QM.MS.
Which demonstration is equally applicable to lines intersecting within or without the ellipse.
Wherefore, if straight lines in the ellipse intersect each other, either within or without the curve, the rectangles of their corre- sponding segments are to each other as the squares of those diame- ters, to which they are parallel.
Cor. When a line becomes a tangent, its square corresponds to the rectangle in other cases. Therefore, the squares of tangents which intersect, are as the squares of the diameters to which they are parallel, and the tangents themselves are as the same diameters.
These are a few of the most useful properties of the Ellipse; a variety of others will be found in the Sixth Chapter, which treats of the analogous properties of the three Conic Sections. We now proceed to the Hyperbola.
ON THE HYPERBOLA. 57
CHAPTER IV. <m THE HYPERBOLA.
The Properties of the Hyperbola may be divided into two class- es : in the first class may be placed such properties as are analogous to those of the Ellipse ; in the second class, such as are derived from its relation to the Asymptote. We shall consider each of these classes separately, beginning with that which contains the properties analogous to those of the Ellipse.
VII.
DEFINITIONS.
(68.) Let PAQ, be an Hyperbola generated by the revolution of the lines SP, HP about the fixed points S, H, according to the law prescribed in Art. 9. Take HM=AS, and let the lines Sp, Up, re- volve round H, S, according to the same law ; then it is evident that the point p will trace out another curve pMq passing through M precisely similar to PAQ,. pMq is therefore called the opposite Hyperbola.
(69.) The point A is called the vertex ; and the part AM of the line HS which joins the two foci S and H, is called the Major axis of the Hyperbola.
(70.) If AM be bisected in C, C is called the center ; and if through C a line BCO be drawn at right angles to AM, and with center A and radius SC a circle be described cutting BCO in B and O, (in which case BC2=AB2— AC2=SC2— AC2), then BCO is called the Minor axis of the Hyperbola.
C.S. 8
58
ON THE HYPERBOLA.
(71.) From any point P let fall the perpendicular PN upon the axis major MA produced, and through the focus S draw LST par-
allel to it ; then PN is called the ordinate to the axis ; AN, NM, the Abscissas ; and the line LST is called the Latus-rectum, or the Parameter to the axis.
(72.) Produce BCO both ways; take O, Gh equal to CS or CH ; and with s, h as foci, BO major axis, and AM conjugate axis,
ON THE HYPERBOLA. 59
describe two other hyperbolas dBD, KOA: ; these are called conju- gate Hyperbolas. A figure thus arises consisting of four Hyperbolas, with their vertices A, B, M, O, turned towards each other, of which the opposite parts are similar and equal. If BCO=ACM, then these four Hyperbolas are exactly similar and equal ; and in this case the Hyperbolas are said to be Equilateral.
(73.) Any line PCG drawn through the center, and terminated by the opposite hyperbolas, is called a diameter ; the line DCK drawn parallel to a tangent at P, and terminated by the conjugate hyper- bolas, is called a conjugate diameter to PCG. From any point Q,, draw Q,V parallel to a tangent at P ; then Q,V is called the ordinate to the diameter PCG, and PV, VG the abscissas.
VIII.
Properties of the Hyperbola analogous to those of the Ellipse.
Property 1. (Prop. 2. of Ellipse.)
(74.) MSxSA=BC2 (See Fig. in p. 58.)
By Art. 70. BC2=SC2 — AC2.
=(SC+AC)x(SC— AC). =(SC+CM)x(SC— AC) (for CM=AC). =MSxSA.
That is, the rectangle of the focal distances from the vertices, is equal to the square of the semi-axis minor. Cor. For the same reason, AHxHM=BC2.
Property 2. (Prop. 3. of Ellipse.)
(75.) The latus-rectum LST is a third proportional to the major and minor axes. (Fig. in page 58.)
For HL-SL=2AC(by const.)
.-.HL=2AC-hSL, andHL2=4AC2+4ACx
SL+SL2.
Again, HL2=SL2+SH2 (Euc. 47. 1.)
=SL2+4SC2 (SH=2SC), =SL2+4AB2,
=SL2+4(AC2+BC2).
60 ON THE HYPERBOLA.
Hence 4AC2-HACxSL+SL2-SL2+4AC2-f-4BC2 , .-.4ACxSL=4BC2. and 2AC : 2BC : : 2BC : 2SL, or AM : BO : : BO : LT.
Or, the latus-rectum is a third proportional to the major and mi- nor axes.
(75.C.) AS.SM=*LT.AM. For (75.) B02-LT.AM. .-. *B02)=BC2)=(74.)AS.SM=*LT.AM.
Property 3. (Prop. 4. of Ellipse.)
(76.) If PyT bisects the angle HPS, it will be a tangent to the Hyperbola in P.
If PT be not a tangent, it must cut the curve in P. Let Q, be any point within the Hyperbola, in TP produced. Draw Sys at right angles to TP, meeting HP in s. Join HQ,, SQ, sQ. With H as center, and HQ, radius, describe the circular arc Q,^, cutting the curve in q. Join ^S, qH. Then, since qS, QS are the bases of the triangles ?HS, QHS and < ?HS> < QHS, ?S> QS. (Euc. 24. 1.)
In the right angled triangles SyP, syV} Vy is common, and <SPy=<sPy. .\Sy— sy, and Ps=PS. Also in the right an gled triangles QSy, Qsy, sy=Sy. and Qy is common ; .-. Qs=QS.
ON THE HYPERBOLA.
61
Since PS=-Ps, HP— Ps=HP— PS=AM ; i. e. Hs=AM. Also, since QS=Qs, Ha— as=HO— QS>H?— qs=AM ; .-.Hd— Gls> E.$. Or HQ,>Hs+Q,s ; which (Euc. 20. 1.) is impossible. Hence TP does not cut the curve, that is, it touches it.
Therefore, if from the foci two straight lines be drawn to any point in the curve, the straight line bisecting the angle contained by these lines, is a tangent.
(77.) Cor. When P comes to A, the <HPS= two right angles ; therefore a tangent at A is perpendicular to the axis AM.
Property 4. (Prop. 5. of Ellipse.)
(78.) If tangents be drawn at the extremities of any diameter of an Hyperbola, they will be parallel to each other.
62 ON THE HYPERBOLA.
Complete the parallelogram SPHG, of which HP, PS are two sides ; then since its opposite sides are equal, GS — HG will be equal to HP — PS, and by a process similar to that in the Ellipse, (Art. 43.), it may be proved that G is a point in the opposite hyperbola, C the center of the hyperbola, and PG a diameter.
Now in the parallelogram SPHG, the opposite <HGS=<HPS ; but by Art. 76, PT bisects the <HPS, and for the same reason G^- bisects the <HGS ; hence the <^GP is equal to <GPT, and there- fore ef is parallel to gh. Therefore, if tangents, &c.
(79.) Hence, (as in the Ellipse), if tangents be drawn at the ex- tremities of any two diameters PCG, DCK, they will form, by their intersection, a parallelogram eghf.
Property 5. (Prop. 6. of Ellipse.)
(80.) If SP and CD be produced till they intersect each other in E, then PE=AC.
Draw HI parallel to CDE or e/, and produce SE to meet it in I. Since HI is parallel to P/, the exterior <SP/=interior <PIH, and <HP/=alternate <PHI ; but <SP/=<HP/*, because PT bisects <HPS ; .-. <PHI=<PIH, and consequently P1=PH. Again, be-- cause CE is parallel to HI, and SC=CH, SE must be equal to EI.
n HP+PI Hence Pi= — ^ — ,
EI-SI-
.-.(PI-EI) or PE-SltpZ?!,
HP-PS 2AC
: - = — A.O.
Hence, if through the extremity of any diameter, a line be drawn from the focus, to meet the conjugate diameter produced, the part intercepted by the conjugate will be equal to the semi-axis major.
ON THE HYPERBOLA. 63
Property 6. (Prop. 7. of Ellipse.)
(81.) If the ordinate PN be drawn to the major axis, then ANx NM : PN2 : : AC2 : BC2. (Fig. in p. 64.)
Let AC or CM=a,
fijr^
Then, by Art. 9.
SC or CR=b, ) (b*— a?)x(x— a)x(x+a)=a?y\
CN=;r, PN=y;
or (SC2— AC2) x AN x NM=AC2 x PN2. But, by construction, SC2— AC2=BC2 ;
.-. BC2xANxNM=AC2xPN2, and ANxNM : PN2 : : AC2 : BC2.*
Therefore, as the square of the major axis is to the square of the minor, so are the rectangles of the abscissas of the former, to the squares of their ordinates.
Cor. 1. Si^ ANxNM=(CN— AC)x(CN+AC)=CN2— AC5 CN2-— AC2 : H2 : : AC2 : BC2.
(82.) Cor. 2. Produce NP to p, and draw any ordinate pm at t angles to Cm, then (since the conjugate hyperbola Bp is de- d with BC as major and AC minor axis) BmxmO : pm2 : : AC2, or (since BmxmO=(Cm— BC)x(Cm+BC)) Cm2— BC2 : pm2 : : BC2 : AC2.
[pa.
fright i
(83.) Cor. 3. Since 7?N=Cm, and pm=CN, we have (by Cor. 2.) pN2— BC2 : CN2 : : BC2 : AC2, or pN^-BC2 : BC2 : : CN2 : AC2, anddividendo,pN2-2BC2 : BC2 :: CN2— AC2 : AC2,
: : PN* : BC2.
* The general property of the Hyperbola analogous to the 10th Property of the Ellipse, viz. VvxvG : Gh?2 : : PC2 : CD2, will be found at the end of the Properties of the Hyperbola derived from its relation to the Asymptote.
64
ON THE HYPERBOLA.
Hence pN2— 2BC2=PN2, and />N2— PN2 = 2BC2 ; in the same manner, if nip be produced to q, it may be proved that qm2- pm*=2AC\
That is, the square of any ordinate to either axis is less than the square of the same ordinate produced to the conjugate Hyperbola, by twice the square of the semi-axis, to which it is parallel.
Property 7. (Prop. 8. of Ellipse.)
(84.) If the ordinate Vn be drawn to the minor axis, then BC2-}- Cn2 : Prc2 : : BC2 : AC2.
In this case, Prc=CN, and C/i = PN; therefore ANxNM
:CN,
.CTa
(CN2— AC2=)Prc2— ACT and PN2 = Cti2 ; hence, by substitution in Art. 81, Cor. 1, we have,
P^—AC2 : Cw2 :
.-. Prc2— AC2 : AC2 :
Compone?ido, Pra2 : AC2 :
or BC2+Cw2 : Pn2 :
AC2 :BC2;
Crc2 : BC2.
BC2+Cw2 :BC2,
BC2 : AC2.
ON THE HYPERBOLA. 65
That is, as the square of the minor axis is to the square of the major, so is the sum of the squares of the semi-minor, and of the distance from the center to any ordinate upon the minor, to the square of that ordinate.
Property 8. (Prop. 11. of Ellipse;) %
(85.) If the tangent PT cuts the major axis in T, and the minor axis in t, then CNxCT=AC2, and C*xCrc=BC2. (See last Fig.)
Since PT bisects the angle UPS, by Euc. 3, 6. we have
HT: TS :: HP :PS;
.-. HT— TS(2CT)* : HT+TS(SH) : : HP— PS(2AC) : HP+PS. (A) But by Euc. 12. 2, HP2=HS2+ PS2+2HSxSN ; .-. HP2— PS2=HS2+2HSxSN,
=(HS+SN)2— SN2, =HN2— SN2 ; .-. HN— SN(a^HP— PS(2AC) : : HP+PS ; HN+SN(2CN).| (B)
Hence we have, / § /
I : : 2AC : HP+PS, (A) tH^kY>^mj
•
SH : 2AC . 2CT : 2AC d#CT : AC
HP+PS : 2CN ; (B)
2AC : 2CN,
AC ; CN, or CNxCT=AC2.
C*t \ ( C**CTk A
AC : CT (and first : third
CN2 : CA2 ; CN2— AC2 : AC2 :: PN2 : BC2. <- N\ tj- But bv sim. As, PTfr, TC*, NT : CT : : PN : Ct. Hence, PN : Ct : : PN2 : BC2, or PNxC*=BC2, but PN=Crc, .-. C?*xC*=BC2.
(86.) Since CN : AC
first2 : second2) CN : CT dividendo, NT : CT
* For HT— TS=HC+CT— T%=SC+CT— (SC-CT)=2CT. t For HN+SN=HS+2SN=2CS+2SN=2(CS+SN)=2CN.
c.s.
66
ON THE IIYPERBOLA.
Therefore, if a tangent and ordinate be drawn from any point of the curve to either of the axes, half that axis will be a mean pro- portional between the distances of the two intersections from the center.
(87.) Coi^CN2— AC2=CN2— CN xCT=CNx(CN— CT)=CN x
NT.
Property A. (Prop. C. of Ellipse.)
(87. a.) Let TLG be the focal tangent, or the tangent drawn at the extremity of SL, the ordinate from the focus. Let NPG be any ordinate, produced to meet the tangent TLG. Then SP=NG.
*»
If AI be taken equal to SP, then IM=HP. z ''' 1 *SZr n *h ".• SP=Cf-CA, and HP=CI+CA. X PX* P H V * J^
- (Euc. 47. h) (SP2)(CI— CA)2=PN2+(CN— CS)2(NS2). H? * ^ And (HP2)(CI+CA)2=PN2+(C^+CS)2(NH2).
fexjmnd and subtract 4CI.CA=4CN.CS, and ^ ? y C\« \ % CI.CA=CN.CS. .-. CN : CI(SP+AC) : : CA : CS
: : CT : CA (85.)
... CN— CT(TN) : SP : : CT : CA. (Euc. 19. 5.)
Again, CS.CT=AC2, and (70.) CS2— AC2(=CS.ST)=BC2. .-. CS.CT : CS.ST, or CT : ST : : AC3 : BC2
: : AC ; SL (75.) .-. ST : SL : : CT : AC : : TN ; SP.
ptJ
i s
P+*£~
A C
r *4
/■
tC^c
m*
/AS
'
ON THE HYPERBOLA. 67
But (sim. tri.) ST : SL : : TN : NG. .-. SP=NG ;
That is, the distance from the focus to any point of the curve is equal to the ordinate to that point, produced until it meets the focal tangent.
Cor. 1. AS=AE and SM=MF. Also C£=AC. For (sim. tri.) CT : C* : : ST : SL : : CT : AC.
Cor. 2. Hence* also, since TN : NG is a constant ratio, TN : SP is a constant ratio. •Therefore, if a line be drawn through T per- pendicular to AC, the distance of the point P from that line (=TN) is in a constant ratio to SP, the distance of the same point 'from the focus. This ratio, being (by demonstration above) =CT : CA, is a ratio of less inequality. This perpendicular is the directrix of the Hyperbola. (See Art. 138, et seq.)
Property 9. (Prop. 12. of Ellipse.)
(88.) If PCG, DCK, be conjugate diameters of the Hyperbola, and OPF be drawn perpendicular to CD produced *if necessary, then POxPF=BC2. (See next Fig.)
all^| < POX, i
Draw Cy parallel to PF. Then because PO is parallel^ Cy, and Ct parallel to PN, the right-angled triangles tCy, PON7 are similar ; .-. Ct : Cy : : PO : PN. But Cy=PF, and PN=Crc, being opposite sides oQk parallelogram ;
.-. C* : PF : : PO ;Crc, or POxPF=C*xC?z=BC2 (86.)
Therefore, if from the extremity of any diameter, a perpendicular i , w drawn to its conjugate ; then the rectangle of that perpendicular and the part of it intercepted by the axis major, will be equal to the square of the semi-axis minor.
Property 10. (Prop. 13. of Ellipse.)
(89.) Draw the ordinates DL, PN, to the major axis, then CN2— CL2=AC2, and DL2— PN2=BC2. (See next Fig.)
68
ON THE HYPERBOLA.
• Draw the ordinate mD, and produce it to meet the hyperbola in q, and draw qk perpendicular to the major axis.
By Cor. 1. Art. 81. Or2-— AC2 : kq% :
and CN2— AC2 : PN2 :
.-.C&2— AC2: kf\
AC2 : BC2,
AC2 : BC2,
CN2— CA2 : PN2.
\/c=mqj mD=CL, and qk=T>Ij ; and by Art. 83. mq2- m* JI^AC2; .-.^2— AC2=(CA;2~AC2=)AC2-fmD2=AC2+CLs hence, by substitution, we have,
AC2+CL2 : DL2 : : CN2— AC2 : PN2 (A.)
By sim. As, DCL, PTN,
/ 1/4***] DL2:CL2
aequo, AC2-f CL2 ♦ CL2
Hence AC2-f CL2 : AC2
PN2 : NT2, CN2— AC2 :NT2, CNxNT :NT2, (87.) ffl CN : NT.
: CN— NT(CT) :CTxCN, : AC2 (85.)
ON THE HYPERBOLA.
69
.-. AC2+CL2=CN2, j. or CN2— CL2=AC.
(90.) Since AC2+CL2=CN2, by substitution in proportion (A), we have,
CN2 : DL2 :
or CN2 : CN2— AC2 :
and divd0, AC2 : CN2— AC2 :
.-. AC2 : DL2— PN2 :
CN2— AC2 : PN2,
DL2 : PN2,
DL2— PN2 : PN2,
CN2— AC2 : PN2,
AC2 : BC2 ;
-,&*
.-. DL2— PN2=BC2.
Hence, if ordinates to either axis be drawn from the extremities of any two conjugate diameters, the difference of their squares will be equal to the square of half the other axis.
Property 11. (Prop. 14. of Ellipse.)
(91.) PC2^CD2=ACVBC2, PG and DK being conjugate diameters.
For by Arts. 89, 90. CN2— CL2=AC2, and DL2— PN2=BC2,
.-. CN*-f PN2— (CL2+DL2)=AC2— BC2, or CP2— CD2=AC2— BC2.
Hence the din%§ence of the squares of any two semi-conjugate di- ameters is equal to the difference of the squares of the semi-axes.
Property 12. (Prop. 15. of Ellipse.) CDxPF-ACxBC.
Tn Art. 90. it was proved that CN2 : DL2 : .-.CN :DL : or CN :AC :
AC2 : BC2 ; AC :BC, DL : BC.
70 ON THE HYPERBOLA.
But by sim. *s, TCy, DCL, CT : Cy(PF) : : CD : DL.
P
Hence we have, CN : AC and CT: PF
.-. CNxCT(AC2) : ACxPF or AC: ' PF
fr-*1**** i .-. CDxPF=ACxBC.
DL : BC, CD : DL ; CD : BC, CD:BC;
That is, if from the extremity of any diameter, a perpendicular be drawn to its conjugate, the rectangle of that perpendicular and the semi-conjugate, is equal to the rectangle of the semi-axes.
(92.) Cor. Hence it appears, that all the parallelograms inscri- bed in the Hyperbolas, and having their sides drawn through the ex- tremities of any diameter and its conjugate, are equal to each other and to the parallelogram described about the major and minor axes ; for the parallelogram eghf (see Fig. in page 61.) described about the conjugate diameters PCG, DCK, is equal to four times eDCP= 4CD X PF=4AC x BC= right-angled parallelogram whose sides are 2AC and 2BC= parallelogram described about the major and minor axes.
Property 13. (Prop. 16. of Ellipse.)
(93.) If SY, HZ, be perpendiculars drawn from the foci to the tangent PYZ, then the points Y and Z are in the circumference of circle described upon the major axis AM.
Join YC, and produce SY to meet HP in W.
Since the tangent PYZ bisects the < HPS ; in the right-angled triangles WPY, SPY, we shall have PW=PS, and WY=YS. Now since WY=YS, and HC=CS, CY must be parallel to HW, and .-. SC : SH : : CY : HW ; but SC=|SH ; .-. CY=£HW=4(HP — PW)=^(HP— PS)=4AM=AC ; .-. Y is a point in the circle whose center is C, and radius C A. In the same manner it might be proved that Z is a point in the same circle.
ON THE HYPERBOLA.
71
Hence, if perpendiculars be dropped from the foci upon any tan- gent to the hyperbola, the intersections of those perpendiculars with
the tangent will be in the circumference of a circle described upon the axis major.
Property 14. (Prop. 17. of Ellipse.)
(94.) SYxHZ=BC2.
Since the <HZP is a right angle, it must be in a semicircle ; if .-. YC is produced to meet HZ in K, K will be in the circumference of the circle, and YK will be a diameter. Hence YC=CK ; and as SC=CH, and <SCY=<KCH, the side HK must be equal to SY. By the property of the circle (Euc. 36. 3.) HKxHZ=HMxHA=BC2, (74.) Hence (since HK=SY) SYxHZ=BC2.
That is, the rectangle of the perpendiculars from the foci upon any tangent is equal to the square of the semi-axis minor.
/
72 ON THE HYPERBOLA.
(95.) Cor. By sim. As, SPY, HPZ,
SYyHP SP : SY :: HP : HZ= * ;
a Y2 v ttp
.-.SYxHZ- gp— -B0».
op
' and SY2=BC2Xgp;
SP\ //SP
•••^-V(w)V(S>
Property B. (Prop. D. of Ellipse.)
(95.a.) Let the vertical tangents AE', MF' be drawn ; then E'A.F'M=BC2, and E'F' is the diameter of a circle passing through S and H.
By sim. tri. E'A : SY : : TA : TY, and HZ:FM::TZ:TM. But (Euc. 35. 3.) TA.TM=TY.TZ. Or, TA : TY : : TZ : TM. .-.E'A: SY::HZ : F'M, and E'A.F'M=SY.HZ=(94.)BC2.
: Again, (74.) AH.HM=BC2=E'A.F'M, .-. AH : E'A : : PI : HM.
Hence (Euc. 6. 6.) the triangles E'AH and HF'M are similar, and <E'HA=<HF'M and <F'HM=<AE'H. Whence <E'HF* is a right angle, and a circle described on E'F' will pass through H. The same may also be shown of S.
Wherefore, if tangents be drawn from the vertices to meet any
other tangent, the rectangle of the vertical tangents will be equal
to the square of the semi-axis minor ; and the intercepted part of
the other tangent will be the diameter of a circle passing through
f the foci.
* The points EH, HF' should be joined in order to form the tri- angles E AH, HMF'.
ON THE HYPERBOLA. 73
Property 15. (Prop. 18. of Ellipse.)
(96.) Draw the semi-conjugate diameter CD, then SPxHP= CD2.
By sim. As, SPY, HPZ, PEF,
SP and HP
.-. SPxHP
SY HZ
PE ; PF, PE :PF
SYxHZ(BC2)
.-. SPxHP
PE2(AC2) : PF2 ; AC2xBC2 PF2 * But by Property 12. CDxPF=ACxBC ;
Hence SPxHP=CD2.
Or, the rectangle contained by the straight lines drawn from the foci to the extremity of any diameter, is equal to the square of half the conjugate to that diameter.
IX.
On the Properties of the Hyperbola derived from its relation to
the Asymptote.
The properties of the Hyperbola hitherto exhibited are perfectly analogous to those of the Ellipse ; we proceed now to explain some of the properties in which these two curves essentially differ. But we must first show what is meant by an Asymptote.
(97.) Since the two branches of the opposite hyperbolas are pre- cisely equal and similar on each side of the axis LK, if two ordinates PCI pa be drawn at equal distances AN, Mm, from the points A, M, then the tangents to the points P, Q, will meet in the same point T, and tangents to the points p, q in the same point t. Now by Art. 85, CNxCT=AC2, and since AC is a constant quantity, CT varies in- versely as CN ; when CN therefore becomes infinite CT will be
C. S. 10
74
ON THE HYPERBOLA.
equal to 0 ; i, e. if P, Q,, are points in the curve at an infinite dis- tance, the tangents PT, Q/F will meet in C ;* for the same reason
-K
if p, q are points, at an infinite distance in the opposite hyperbola, then the tangents pt, qt will also meet in O ; and since the <PTQ,= <iptq, these four lines will evidently then coalesce into two, viz. PT with tq and pt with TQ,. The tangents to the two opposite hy- perbolas at an infinite distance, may therefore be represented by two lines XCZ, UOY, intersecting each other in C, and making equal angles XCK, UCL, KCY, ZCL with the axis. These lines XCZ, UCY are called Asymptotes ; and we are now to determine their position with respect to the axes of the hyperbolas.
* To make this more intelligible, conceive PN to move parallel to itself in the direction NK, then since CNxCT= a constant quan- tity, whilst CN varies through ail degrees of magnitude, the point T will only pass from T to C so as to make CT=0 ; i. e. when P is a point in the curve at an infinite distance, the tangent PT will pass through C : and so of the rest.
ON THE HYPERBOLA. 75
(98.) Draw Aa at right angles to AM. When P is removed to an infinite distance, the triangle PNT becomes similar to the triangle a AC, and CN becomes the same as NT. Hence, in this case, PN : NT or NC : : Aa : AC (A); but by Cor. 1, Art. 81, CN2— AC2 : PN2 : : AC2 : BC2 ; and when CN is infinite, AC vanishes with respect to CN,* therefore this latter proportion becomes CN2 : PN2 : : AC2 : BC2, or CN : PN : : AC : BC (B) ; com- pare the two proportions (A) and (B), and we have Aa : AC : : BC : AC, or Aa=BC. Draw therefore Aa at right angles to AM, and make it equal to BC, join Ca, and this gives the position of the asymptote XCZ. In the same manner, by making A6=BC, and joining C6, we determine the position of the asymptote UCY ; in- deed, from what has been proved, it appears, that if a parallelogram acdb be described about the major and minor axes, the asymptotes will be merely the prolongation of the diagonals of such parallelo- gram.
(99.) These lines XCZ, UCY, will also be asymptotes to the con- jugate hyperbolas ; for by a similar process of reasoning it might be shown that the position of their asymptotes would be determined by drawing perpendiculars Ba, 06, at B and O, and making Ba and 06 each equal to AC. Thus these four hyperbolas are inclosed as it were between their asymptotes ; and by producing the ordinates to meet these asymptotes, new properties of the curves will arise, which we shall now proceed to investigate.
* To show that in this case CN2 — AC2 may be considered as equal to CN2, let CA=a, AN=:r, then CN=^+a, and CN2= z* _|_ 2ax + a2 ; hence CN2— AC2( = CN2— a2) = x2 + 2ax ; we have therefore CN2 : CN2— AC2 : : x% + 2a*+a2 : & +Hq* n *+
2a +— : x-\-2a\ but when x is infinite, becomes equal to 0; in
x x
this case, therefore, this latter ratio becomes a ratio of equality, from which it follows that CN2 may be substituted for CN2— AC2.
76
on the iiyperbloa. Property 16.
(100.) Let the ordinate Vp be produced to meet the asymptotes in the points L, I ; then PL.PZ=BC2 and />Z.pL=BC2 ; also PL.L/)= BC2 and pZ.ZP=BC2.
By Cor. 1. Art. 81,
CN2— CA2 : PN2 : : AC2 : BC2.
By sim. As, LNC, aAC,
CN2 : LN2 : : AC2 : Aa2 (BC2)
... ON2 : CN2— CA2 :
and dividendo, AC2 : CN2— CA2 :
or AC2 : LN2— PN2 :
LN2 : PN2, LN2— PN2 : PN2 ; ON2— CA2 : PN2,
AC2 : BC2 i
... LN2— PN2=BC2.
But LN2— PN2=(LN— PN) x (LN+PN)=PLxPZ .-. PLxPZ=BC2 or PL.Lp=BC2. For the same reason, p/xpL=BC2 or ^/./P=BC2.
ON THE HYPERBOLA. 77
Therefore, if an ordinate to the axis-major be produced to meet the asymptotes, then the rectangle of the segments intercepted be- tween the curve and either asymptote will be equal to the square of the semi-axis minor.
(101.) Cor. 1. Hence VLxP l=plxpL=YL.Lp=pl.lP.
Cor. 2. Draw any other ordinate Qq, and produce it to meet the asymptotes in X and Y, then will Q,XxQY=Aa2 ; hence we have QXxGlY=PLxPZ.
Property 17.
(102.) Draw any diameter PCG, and produce it tog) draw the ordinate Q,T to that diameter, and produce it to meet the asymptotes R, r ; then QRxQr=TrxTR. (See last Fig.)
Through the points P, Q,, draw hi, XY perpendicular to the axis of the hyperbola, and draw the tangent ef at P.
By sim. triangles Q,XR, PLe ; QrY, Vfl ; we have aX : aR :: PL : Pe,
andqY :__<^ : : PI t Vf)
.-. Q,XxQ,Y : QRxQr : : PLxPZ : PexP/.
But by Art. 101, aXxdY=PLxPZ ; hence QRxQr=PexP/.
In the same manner, by drawing an ordinate through T perpen- dicular to the axis, it might be shown that TrxTR=PexP/; hence QRxQr=TrxTR.
Therefore, if an ordinate to any diameter be produced to meet the asymptotes, the rectangle of the segments intercepted between the curve and one asymptote, will be equal to the rectangle of the segments intercepted between the curve and the other.
(103.) Cor. 1. Since Q,r=Q/T + Tr, and TR=Q,T-f-QR, we have
aRx(dT-fTr) =Trx(QT+QR), or QRxaT+QRxTr=TrxQ,T+TrxQR.
Subtract QRxTr from each side of this latter equation, and there results QRxQ,T=TrxQ,T, from which it appears that QR=Tr ;
78 ON THE HYPERBOLA.
in the same manner it may be proved that Q,X=</Y, and PL=//Z. If therefore Rr moves parallel to itself till it comes into the position of the tangent ef (in which case the points Q, and T coincide in P), we shall have Pe=P/", and consequently PexP/=Pe2.
(104.) Cor. 2. Since the triangles eCf RCr are similar, and since the diameter GCg bisects ef in P, it will bisect Rr in V ; hence VR^=Vr ; and as Q,R=Tr, we have YQ,=VT, i. e. the diame- ter GOg- bisects all its ordinates.
(105.) Cor. 3. Hence YR2— VQ2=Pe2. For VR2— VQ2=(YR— YQ) X (VR-t-VQ)=(VR— YQ) x (VR+YT) = QRxRT = QR xQr, (for RT=Q,R-hQT=Tr+QT==Qr). But QRxQr=PexP/=(103) Pe2; .-. VR2— YQ2=Pe2.
Property 18.
(106.) Join AB, and let it cut the asymptote XCZ in S ; draw PD parallel to the asymptote UCY, cutting the asymptote XCZ in R ; then CRxRP=AS2. (Fig. in next page.)
Since the diagonals BA, aC of the parallelogram aBCA are equal and bisect each other in the point S, the lines SA, SC, SB, Sa are equal j hence the < SAC = < SCA ; but < SCA = < ACY, .-. <SAC=<ACY, and consequently AB is parallel to UCY. If therefore Pr, Am are drawn parallel to the asymptote XCZ, then PRCr, ASCm will be parallelograms, and Pr will be equal to CR, and Am to SC.
By sim. triangles PrJ, Amb ; PRL, AS« ; we have
Pr(CR) : VI
and RP : PL
.-. CRxRP : PLxPZ
Am(SC) : A6, SA : Aa :
SC x SA : AaxAb.
But PLxPZ=AaxA6 ; hence CRxRP=SCxSA=SA2.
That is, if from any point of the curve a line be drawn to the nearer asymptote, parallel to the other asymptote, the rectangle of this line, and the distance of its intersection with the asymptote from the center, is a constant quantity ; and is equal to the square of half the diagonal of the rectangle of the semi-axes.
ON THE HYPERBOLA.
79
(107.) Cor. 1. Since XCZ is likewise an asymptote to the con- jugate hyperbola, by a similar process of reasoning it might be shown that CRxRD=SB2=SA2 ; hence CRxRD=CRxRP, and conse- quently RD=RP, i. e. PD is bisected by the asymptote.
(108.) Cor. 2. Since SA=4AB, SA2 is a constant quantity ; hence RP varies inversely as CR ; when CR therefore is infinite, HP will become equal to 0 ; which coincides with what has already been said as to the asymptote's touching the curve at an infinite distance.
Property 19.
Join CD, and produce it to K ; draw the diameter PCG ; then will DCK be the conjugate diameter to PCG. (Fig. in next page.)
(109.) Draw ef touching the curve in the point P, and meeting the asymptotes XCZ, UCY in the points e and/; then by Art. 103, P/ will be equal to Pe ; and since PR is parallel to C/; CR will be
80
ON THE HYPERBOLA.
also equal to Re. Hence, in the triangles CRD, PRe, we have CR=Re, RD=RP, and <CRD=<eRP, .-. (Euc. 4. 1.) CD is equal to Pe, and the <DCR equal to the <ReP; consequently DCK is parallel to the tangent ef, and is therefore the conjugate diameter to PCG (73.)
Therefore, if a parallel to either asymptote cut conjugate Hyper- bolas, the diameters passing through the points of intersection will be conjugate to each other.
(110.) Cor. Join Be, then eDCP will be a parallelogram,* whose diagonal is Ce ; and as Be is parallel to the diameter PCG, it touches the conjugate hyperbola in D. Complete the parallelogram eghf,
* For CD being equal and parallel to Pe, De must be equal and parallel to CP. (Euc. 33. 1.)
ON THE HYPERBOLA. 81
as in Art. 79, then in the same manner as it has been proved that Ce is the diagonal of the parallelogram eDCP, it might also be prov- ed that the point h would be found in the asymptote XCZ, and the points g, f in the asymptote UCY ; these asymptotes are therefore the prolongation not on! y of the diagonals of the parallelogram de- scribed about the major and minor axes, but also of the parallelo- gram described about any two conjugate diameters.
Property 20. (Prop. 10. of Ellipse.)
(111.) Draw the ordinate Q,v, then FvxvG : Q,r2 : : PC2 : CD2 : : PG2 : DK2.
5
Produce vQ, to X, then, by sim. As, OX, CPe, Cv2 : CP2 :: vX2 : Pe2;
... (V— CP2 : CP2 : : vX2— Pe2 : Pe2, and Cv2— CP2 : vX2— Pe2 : : CP2 : Pe2. (A)
But Ct?»— CP2=(0— CP) x (Cv+CV)=PvxvG. By Art. 105. vX2— vQ,2=Pe2, .-. vX2 — Pe2 =vQ,2.
Now (109.) Pe2=CD2. Hence, by substitution in Proportion (A), we have FvxvG : Qv2 : : CP2 : CD2 : : PG2 : DK2.
Therefore, the square of any diameter is to the square of ils con- jugate, as the rectangles of its abscissas are to the squares of their ordinates.
By reasoning similar to that employed in Arts. 54. a., 67.a., and 67.6., properties may be inferred, analogous to Props. B, E and F of the Ellipse.
On the Properties of the Equilateral Hyperbola.
In Art. 72, it was observed, that if the axes of the hyperbola be- come equal, it is then said to be equilateral ; in this case the figure possesses some peculiar properties, which it may be worth while to investigate. C. s. 11
82 ON THE EQUILATERAL HYPERBOLA.
(112.) Let the annexed figure represent an equilateral hyper- bola, with its opposite and conjugate hyperbolas : then, since the axes ACM, BCO are equal, it is evident that if a circle be
X
described upon the axis ACM, it will pass through the extremities of the axis BCO, and that the rectangular figure abdc which circum- scribes those axes will be a square. Draw the diagonals ad, cb, and produce them each way to X, V, U, Z ; then XCZ, UCV will be the asymptotes to the four hyperbolas ; and as the angles aCB, cCB, are each of them half a right angle, the angle aCc will be a right angle. Since the asymptote XCV cuts the asymptote UCV at right angles in the centre C, it will also cut all other lines BA, DP, pQ,, &c. (drawn parallel to UCV) at right angles. Now by Art. 106, CRxRP=sA2; and for the same reason CeXeQ*=$A* ; .-. CRx
ON THE EQUILATERAL HYPERBOLA. 83
RP=CeXeQ,, or CR : Ce : : eQ, : RP ; hence if any points R, e, &c. are taken in the asymptote, and from them ordi nates PR, eQ, &c. are drawn at right angles to it, then the abscissas CR, Ce, &c. will be to each other inversely as the ordinates RP, eQ, &c*
(113.) Since the Latus-rectum is a third proportional to the major and minor axes ; when those axes are equal, it must be equal to either of them ; LST is therefore equal to ACM or BCO. Now MSxSA=BC2=(since BC=AC) AC2 ; hence AC is a mean propor- tional between MS and SA ; and since SYxttz=BC2=AC2, AC is a mean proportional between the perpendiculars SY and Hz.
(114.) By Art. 91. PC2— CD2=AC2— BC2 ; but AC2— BC2=0, ... PC2— CD2=0, consequently PC=CD, and the diameter PCG= conjugate DCK. The sides eg, gh, hf, fe of the parallelogram eghf, drawn about those diameters, will therefore be equal ; and the parallelogram itself, a Rhombus whose area will be equal to the area of the square abdc described about the axes.
(115.) Draw PI at right angles to a tangent at P, and produce it to F ; then by Art. 88, PIxPF=BC2=AC2 ; but CDxPF=ACx BC=«AC2 ; .-. PIxPF=CDxPF, and PI=CD=PC, i. e. the normal PI is equal to the distance PC from the center.
* Let Cs or sA = a, CR = x, RP = y, then (since CR x CP =
sA2), xy=ai, and y = — , .-. y#= — '-, whose fluent is a2xlog. x-{-
Cor. ; suppose therefore the area AsRP to begin from 5, it would
vanish when a; = Cs or a ; hence a2 x log. a + Cor. = 0, and Cor.
= — a2xlog. a, the area A^RP is therefore equal to a2xlog. x —
x a2 X log. a = a2 x log. -. Suppose now that Cs=a=l, then a8
and a would each be equal to 1, and we should have area AsRP= log. x ; and thus if the abscissas CR, Ce, &c. are taken equal to the natural numbers in succession, the corresponding areas AsRP, AseQ,, &c. will be the Logarithms of those numbers. It is from this cir- cumstance that the system of logarithms whose modulus is unity are called Hyperbolic Logarithms.
84 ON THE EQUILATERAL HYPERBOLA.
(116.) Since ANxNM : PN2 :: AC2 : BC2, and AC2 = BC2, .-. ANxNM = PN2. Also, by Art. Ill, FvxvG : Q,v2 : : PC2 : CD2; but PC2=CD2, .-. Vv x vG = Qv2. Hence the rectangle of the abscissas is equal to the square of the ordinate, whether the ordinates be referred to the axis or to any diameter ; in this respect, therefore, the properties of the equilateral hyperbola are analogous to those of the circle.
We have just hinted at the analogy which obtains between the properties of the circle and those of the equilateral hyperbola when considered in a geometrical point of view ; but it appears more strik- ing when the nature of those curves is expressed algebraically.* To pursue the inquiry respecting this analogy, would lead to inves- tigations, which, though extremely curious and interesting in them- selves, are quite foreign to our present purposes. We therefore now proceed to consider the nature of the Curvature of the three Conic Sections.
* Let CA=a, CN=#, PN=y ; then, by Art. 81. Cor. 1. (since CA2=BC2 and .-.CN2— CA2=PN2) we have y2=#2— a\ or y=* V(#2 — a2) ; now let a be the radius of a circle, x the abscissa meas- ured from the center and y the ordinate, then, by the property of the circle, y = V(a2 — #2)=v( — l)XV(f — a2); the algebraic ex- pression therefore for the ordinate of the circle is the same with the expression for the ordinate of the equilateral hyperbola, except as to the imaginary factor V( — 1). This similarity in the algebraic expression for the ordinate, lays the foundation of some very curious analytical Theorems with respect to the analogy between these two curves.
ON CURVATURE IN GENERAL. 85
CHAPTER V. ON THE CURVATURE OP THE CONIC SECTIONS.
In order to become thoroughly acquainted with the geometry of curvilinear figures, it is necessary to acquire clear and distinct ideas of the nature of Curvature. Previous to the investigation, there- fore, of the theorems relating to the curvature of the Conic Sections, it will be very proper to consider the nature of Curvature in General.
XL
On Curvature, and the Variation of Curvature.
(117.) As a straight line (AB) is defined to be that which "lies
evenly between its extreme points,"* A - — -=s=- B
so a curved line (BC) may be said to be that which does not "lie even- ly between those points f and by C curvature is meant the continued deviation from that evenness of po-
* This is the original definition of Euclid, and it is retained by Simson, in his edition of that Geometer's works. If, however, we were left to conceive of a straight line solely from this definition, it is questionable whether our conceptions would be very clear. " The word evenly" as Playfair remarks, " stands as much in need of an explanation, as the word straight, which it is intended to define." The definition given by this latter mathematician is this. " If two lines are such, that they cannot coincide in any two points without coinciding altogether, each of them is called a straight line." A
86
ON CURVATURE IN GENERAL.
sition which takes place in the course of its description. The curv- ature, moreover, is said to be greater or less, according as that devia- tion is greater or less within a given distance of the point from which the curve begins to be described. We know not how to illustrate this definition better, than by referring the reader to the annexed fig- ure, where several circles AEM, AFL, AGK, &c. of different di- ameters AM, AL, AK, &c. begin to be described from the point A, all touching the straight line BC. At the given distance AD from the point A, draw the line DH at right angles to AD, and cutting the circles in the points E, F, G, &c., then the deviations of the cir-
cles AGK, AFL, AEM, &c. from the right line AB, are measured by the lines DG, DF, DE, &c. respectively; and since DG is greater than DF, DF than DE, &c. the curvature of the circle
straight line being thus defined, the best account that can be given of a curve is to say, that it is a line, which cannot have a common segment with a straight line ; or a line which continually deviates from a straight line.
ON CURVATURE IN GENERAL.
87
AGK is said to be greater than that of the circle AFL, of AFL greater than that of AEM, &c. &c.*
(118.) Suppose now XABCY to be any curve, to which tangents DA, EB, FC, &c. are drawn at the points A, B, C, &c. ; then, from what has been shown in Art. 117, it is evident that an unlimit- ed number of circles may be described at each of the points A, B,
C, &c. to which the lines DA, EB, FC, &c. shall be tangents as well as to the curve ; but that there can be only one circle, which shall have the same deviation from the tangent as the curve at each point. Let ALM, BNO, CPQ,, &c. be the circles which have the same deviation (i. e. which coincide) with the curve at the points A,
* We have here to observe, that although the lines DE, DF, DG, &c. are made use of to illustrate what is meant by greater or lesser curvature, yet the actual relation between the curvatures of these circles can only be ascertained by finding the relation of DE, DF, DG, &c just at the point of contact.
ON CURVATURE IN GENERAL.
B, C, &c. ; then these circles are called the Circles of Curvature to those points.*
(119.) The change which takes place in the curvature from the circumstance of its being measured at different points A, B, C, &c. by circles of different diameters, is called the variation of curvature of the curve ABC. ~)
Having thus denned what is meant by curvature and the variation of curvature, we are next to investigate the relation which takes place between the curve and the tangent just at the point of contact. This is a subject of considerable difficulty, inasmuch as it involves the consideration of quantities which will not admit of strict geo- metrical comparison, but require a species of minute analysis, the principles of which are exhibited in the following Theorems.
Theorem 1.
(120.) In the circle PQ,VL, take any arc QP ; from P, Q, draw any chords PV, Q,V, and the tangent PR to the point P ; from Q, draw Q,R parallel to PV, and Qv parallel to RP ; join QJP ; then, at the point of contact, the arc Q,P, the chord Q,P, the tangent RP, and the ordinate Qv, all become equal to each other.
* Since the curve and circle of curvature have the same devia tion from the tangent, at the point of contact, it is obvious that no other circle can be drawn between. This relation between the curve and circle of curvature is similar to that which exists between a circle and its tangent. Hence the circle of curvature is said to touch the curve. It will be observed, however, that the circle often cuts the curve, which it is said to touch in the point of contact. This must always be the case, except at points of maximum or min- imum curvature, when the circle falls wholly within or wholly without the curve.
ON CURVATURE IN GENERAL.
89
Since RP touches the circle, and PQ cuts it, the angle RPQ is equal to the angle QVP in the alternate segment ; and since QR is parallel to PV, the < RQP= alternate < QPV ; the triangles PQR, PQV therefore are similar ; hence we have PQ : PR : : PV : QV. Now suppose the chord PV to remain fixed whilst the chord QV revolves round the point V by the continual approach of the point Q towards P, then it is evident that the chords PV and QV continually approach towards a state of equality ; PQ, and PR therefore, which are to each other in the ratio of PV : QV, must also approach to a state of equality ; as must also the arc QP which lies between PQ, and PR, and the. ordinate Qv which is equal to PR. At the point of con- tact, QV becomes actually equal to PV ; hence the arc QP, chord QP, tangent PR, and ordinate Qv, (whose rela- tion is expressed by the equali- ty of the determinate lines PV actually equal.
QV) must at that point become
Theorem 2.
(121.) The chord PV is equal to ^^- ; assuming the rela- tion which QP and QR have to each other at the point of contact.
By similar triangles. QPR, PQV, QR : PQ : : PQ : PV ; but (by
Art. 120.) at the point of contact, chord PQ=arc PQ, .-. in this
Care QP? case QR : arc PQ : : arc PQ : PV ; hence PW aR~ •
In order to remove the objection which may arise from the cir- cumstance of representing the definite quantity PV by the quantity C. S. 12
90
ON CURVATURE IN GENERAL.
OF QR
, in which QP and QR are confessedly too small for geometri-
cal comparison, it should be recollected that the measure of a ratio is entirely independent of the terms of a ratio, and consequently that the two ratios which compose the proportion QR : PQ : : PQ : PV are as much real ratios at that particular period when the arc PQ may be considered as equal to the chord PQ,, as at any other period of the progress of the point Q towards P. The conclusion therefore deduced from the reality of that proportion, viz. that PV is equal to
pa2
QR'
must be true in the case when the arc PQ= the chord PQ,
i. e. at the point of contact.
Theorem 3.
(122.) In different circles the curvature varies inversely as the radii of these circles.
Let AEL, AFK be two circles having a common tangent (BC) at A : in AB take any point D, and draw DF at right angles to AB ; draw the chords AE, EL ; AF, FK ; and let fall Ee, F/, perpen-
|
E |
<7^ |
|
|
fC> |
||
|
Sm |
/ \\ |
|
|
KX / |
||
|
L.^^/ |
dicular to the diameter AL J then will Ae be equal to DE, and A/ will be equal to DF. Now (by Euc. 8. 6.) in the right- angled triangles AEL, AFK, we have Ae : AE : : AE : AL ;
AE8 .-. Ae or DE=AL- ; also A/ : AF : : AF : AK ; .-. A/ or DF=
ON CURVATURE IN GENERAL. 91
AF2 AE2 AF2
-r^=r : hence DE : DF : : -r^- : -r-== . But the curvature of the
AK AL AK
circles AEL, AFK, (see note page 88,) is measured by the relation which obtains between DE and DF just at the point of contact ; and at that point, AE and AF both become equal to AD (by Art. 120.) and consequently equal to each other. At the point of contact, there- fore, (since AE2=AF2) we have DE : DF : : J* : -?- : : AK :
AL AK
AL ; i. e. curvature of circle AEL : curvature of circle AFK : : di- ameter of AFK : diameter of AEL : : radius of AFK : radius of AEL ; i. e. the curvature in different circles varies inversely as their radii.
Theorem 4.
(123.) Let now APQ, be any curve, PVO the circle op cur- vature to the point P ; take any arc PCI and through Q, draw RQ,q parallel to the chord P V passing through some given point S ; then (assuming the relation of the quantities PQ, and Q,R at the point of
PQ,2
contact) PV will be equal to :=-=-,
P<72 By Theorem 2, PV is equal to -*- ; but since the curve and
qix
circle of curvature coincide at the point of contact, at that point
92
ON THE CURVATURE OF THE PARABOLA.
P^ will become equal to PQ,, and qR equal to Q,R, and consequently
pa8
PV=
qr
(124.) Draw now VO at right angles to PV, and join PO ; then (PVO being a right angle, consequently in a semi-circle) PO will be the diameter of curvature to the point P. Bisect PO in r, then Vr will be the radius, and r the center of curvature to the point P.
XII.
On the Curvature of the Parabola.
Let AQ,P be a Parabola, whose axis is AZ, and focus S ; and let PVO be the circle of a curvature to any point P. Join SP, and produce it to meet the circle of curvature in V, then PV is the chord of curvature passing through the focus.
(125.) The Chord PV=4SP. Take any arc QP, so small that it may be considered as coinciding with the circle of curvature, and
ON THE CURVATURE OF THE PARABOLA. 93
draw Q,R parallel to SP ; draw also Qiv parallel to the tangent PT, cutting SP in x, and the diameter PW in v ; then Q,RP# will be a parallelogram, and P# will be equal to Q,R. Now since xv is par- allel to PT, and Vv parallel to TS, the &Vxv is similar to the APST ; but by Art. 17, SP is equal to ST ; .-. P#=Pv ; hence Vv is equal to Q,R. Let Qiv move up towards P parallel to itself, then, at the point of contact, Qiv will become equal to Q,P ;* since therefore Vv=
(Q P2 v Q v2
%s- by Art. 123. = )^~. But U,K / Pv
by Art. 22, 4SPxPv=Q,i;2 ; .-. ^=4SP ; hence PY=4SP= pa- rameter to the point P.
That is, the chord of curvature, passing through the focus, is equal to the parameter of the diameter at the point of contact.
(126.) SA.P02=16SP5. Draw YO at right angles to PV and join PO. Then (124.) PO is the diameter of curvature, and there- fore parallel to SY, which is perpendicular to the tangent PT. Hence the triangles PYO, SYP are similar
.-. PO : PY(=4SP) : : SP : SY, PO2 : 16SP2 : : SP2 : SY2(=SA.SP, by Art. 32.) PO2: 16SP2 ::SP : SA, .-.SA.P02=16SP3.
Therefore, a parallelopiped, whose base is the square of the diam- eter of curvature, and whose height is the distance from the focus to the vertex, is equal to 16 times the cube of the dist;mce from the focus to the point of contact.
4gpf Cor. 1. The diameter of curvature ■» ;>^>v.
V(SA)
# By Art. 120, Qx becomes equal to QP ; but at the point of con- tact P, the points x and v coincide ; therefore at that point the three lines QP, Q#, Qiv become equal to each other.
94 ON THE CURVATURE OF THE ELLIPSE.
PV3 pvi
Cor. 2. SA.P02=-.~ and PO=£r7^r-rV
4 2V(SA) ,
Cor. 3. The diameter PO (and of course the radius Pr) ocSP*
3.
or PV2, because SA is constant.
(127.) At the vertex A, where SP becomes perpendicular to the tangent, the chord and diameter of curvature will of course coincide ; and in this case each of them becomes equal to 4SA, i. e. to the la- tus-rectum.* The diameter (and consequently the radius) of cur- vature is therefore the least at A ; hence, by Art. 122, the curvature
itself will be greatest at A ; and since it varies as =-, i. e. , it
will keep continually decreasing as the point P recedes from A.
XIII.
Chi the Curvature of the Ellipse. (Fig. in next page.)
Let APM be an ellipse, PVLO the circle of curvature to the point P ; join PS, PC, and produce them to meet the circle of curvature in the points V, L ; draw YO, LO at right angles to PV, PL, and join PO ; then PV is the chord of curvature passing through the fo- cus ; PL the chord passing through the center ; and PO the diame- ter of curvature. Draw the conjugate diameter DCK ; then,
(128.) The chord of curvature (PL) passing through the center
2CD2 is equal to -«*• . Take any small arc Q,P as before ; draw Q,R
parallel to PC, and Q,v parallel to RP ; then will Pv be equal to RQ,. Suppose Qv to move up towards P, then, at the point of con-
* For at A, SP becomes equal to SA ; .-. PV=»4SP=~4SA ; and „> 4SP* 4SA* ...
ON THE CURVATURE OP THE ELLIPSE.
95
tact, Qv becomes equal to Q,P (120.), and vG becomes equal to PG, i. e. to 2PC. Now by Art. 53, FvxvG : Qv2 : : PC2 : CD2 ; substituting therefore for Pv, vG and Q,v, their values at the point of
contact, we have QRx2PC : GIP2 : : PC2 : CD2, or 2PC :^~ ::
2CD5
123.), .•.PL=-p^-.
(129.) The diameter of curvature (PO)=
2CDS
PF
The triangles
PCF, PLO have a common < at P, and right <s at L and F,
2CDS
-p-Q- J : : PC : PF,
2CD2xPC 2CD8 . L - /T> * .„
; the radius of curvature (Pr) will con-
•.PO=
PCxPF
PF
sequently be equal to „-.
96 ON THE CURVATURE OP THE ELLIPSE.
(129.a.) The chord of curvature (PY) passing through the focus
U ™—. The triangles PEF, PVO, have a common < at P and
AC
right <s at V and F, they are therefore similar. Hence PV : PO
(--)i:PP:PE(iC),,PV=™-^. A,B„H,
extremity of the minor axis) the semi-conjugate becomes equal to AC ; hence the chord of curvature passing from B through the
2 AC2
focus S=^=2AC.
(130.) At A (the extremity of the major axis) the diameter of
(2CD2,v 2RC2 ___._ j=___r= latus-rectum ;* at B (the extremity
2AC2
of the minor axis) it is equal to -^^ ; it is therefore least at A,
and greatest at B ; hence, by Art. 122, the curvature is greatest at the extremity of the major axis, and least at the extremity of the minor axis. At the intermediate points between the extremities of the axes, the curvature varies inversely as the cube of the normal.!
rectum, i. e. 2AC : 2BC : : 2BC : latus-rectum
* For by Art. 39, major axis : minor axis : : minor axis : latus-
4BC8=2BC2 2AC AC*
CD8
t The radius of curvature ==^^ . Now by Art. 58, (PI being the
Pr
normal) PIxPF=BC2= a constant quantity, .-.Plac^^. By Art. 62. CDxPF=ACxCB= a constant quantity, .-.CDoc^;
Pr 1 CD2/ 1 \
hence PI acCD. Again, since CD oc™ = li. e. CD2XppJ ocCD3 ocPI3 ; the radius of curvature therefore varies as PIS, con- sequently the curvature itself varies as p=-, or inversely as the cube.
ON THE CURVATURE OP THE HYPERBOLA.
97
XIY.
On the Curvature of the Hyperbola.
The process for rinding the chords and diameter of curvature in the Hyperbola is precisely the same as that for the Ellipse. Refer- ring the reader to the annexed Figure, we shall merely repeat the principal steps of the foregoing demonstration.
(131.) *By Art. 111., Vv x vG : Qv2 : : PC2 : CD2, and at the point' of contact, QJEtx2PC : QP2 : : PC2 : CD2, .-.S§- or PL-
of the normal. As PI ocCD, the curvature varies as p^pr-- gc^^—
OU3 Dn.3,
or inversely as the cube of the diameter conjugate to that at the
point of contact.
* The construction of the above figure is word for word the same as in the Ellipse. To avoid a confusion of lines, the circle of cur- vature is drawn entirely within the Hyperbola ; whereas, such part of the hyperbola as is of greater curvature than that at the point P, ought to have fallen within the circle of curvature, as in Figure, page 87.
c. s. 13
98 ON THE CURVATURE OF THE HYPERBOLA.
2CD2
chord of curvature passing through the center.
PC
(2CD2>v "per) ::
2CD2 CD2
PC : PF, .-. PO=— «^- ; and Pr the radius of curvature =^=.
-pp- ) :: PF
SCD
"AC
2CD2
: PE(AC), .-. PV= --^-= chord of curvature passing through the
focus.
r 2PT)2 2RC2
(134.) At the vertex A, the diameter of curvature — - «■ — r-^
= latus-rectum. Here the analogy between the Ellipse and the Hyperbola ends ; for with respect to the variation of curvature, since the normal PI keeps continually increasing from the point A,* the curvature will continually decrease as the point P recedes from A.
(135.) In the equilateral hyperbola (see Fig. in page 82) the latus- rectum is equal to the major axis ; the curvature therefore at the vertex A is the same with the curvature of the circle described upon the major axis. In this case PI=PC (Art. 115); .-.PI3ocPC3, and in the recess of P from the point A. the curvature varies in the
same
ratio, viz. ( - or ^pr3 ) with respect to the two sides of the
isosceles triangle CPI, one of which (PC) revolves round the fixed point C, and the other (PI) round the moveable point I, at right an- gles to the curve. Here then is an instance of great symmetry in the curvature of the equilateral hyperbola.
* That the radius of curvature varies as the cube of the normal, is proved in the same manner as in Note f, page 96.
ANALOGOUS PROPERTIES, &C. 99
CHAPTER VI.
ON THE ANALOGOUS PROPERTIES OF THE THREE CONIC
SECTIONS.
Hitherto we have noticed no other analogies than those which take place between the Ellipse and Hyperbola ; but as the three Conic Sections are derived from the same solid merely by changing the position of the plane which intersects its surface, it may naturally be expected that they will possess many properties common to them all. Previous to the investigation of these analogous properties, it may be worth while to consider the changes which take place in the nature of the section, during the revolution of the plane of intersec- tion from a position parallel to the base of the cone, till it becomes a tangent to one of its sides.
XV.
On the changes which take place in the nature of the curve describ- ed upon the surface of a cone, during the revolution of the plane of intersection.
(136.) Let the triangle BEZG represent the section of a cone perpendicular to its base, and passing through the vertex ; then if the cone be cut by a plane perpendicular to BEZG, and parallel to the base, the section AFD will be a circle. Draw the diameter AD of the circle AFD, and draw AZ parallel to the side BE of the cone. Conceive a plane (at right angles to the plane BEZG) to pass through AD, and afterwards to revolve through the angle DAG till it becomes a tangent to the side BG of the cone. From what was shown in Chapter I. it is evident that whilst this plane revolves
100
ANALOGOUS PROPERTIES OF
through the angle DAZ, its intersection APM with the surface of the cone will be an Ellipse, whose major axis is AM ; when it comes
into the position AZ, it will be a Parabola, whose axis is AZ ; and that whilst it revolves through the angle ZAG, it will be an Hyper- bola, whose major axis is AM', M' being the intersection of z A and EB produced.
It may further be observed, that in the revolution of the plane through the angle DAZ, so long as it cuts the side BE between D, and E, a whole ellipse will be formed upon the surface of the cone. When it comes into such a position as to cut the base, a part only of an ellipse will be formed ; and when it arrives at the position AZ, the point M moves off to an infinite distance, so that the Parabola thus formed may be considered as a part of an Ellipse, whose axis major is infinite. And as at the instant the plane leaves the position AZ in direction Zz, the curve of intersection becomes an Hyperbola,
THE THREE CONIC SECTIONS. 101
the Parabola may also be regarded as an Hyperbola, whose major axis is infinite. These three curves therefore approach to identity at the same time that the plane approaches to parallelism with the side BE of the cone.
(137.) The same conclusion may be drawn from the algebraic construction of these curves. Let the angle MAZ be equal to the angle ZAz, then the major axis (AM') of the Hyperbola will be equal to the major axis (AM) of the Ellipse.* In each case, find the center C or C, and let the abscissas AN or AN' =#, the ordin- ate PN or P'N'=y, semi-axis major (AC or AC')=a, semi-axis minor =&, AS or AS' (S or S' being focus) =c. Then in the Ellipse NM=AM— AN=2a— x, and MS=AM— AS-2a— c ; in the Hyperbola, N'M'=AM'+AN'==2a+^, and M'S'=AM'+AS'= 2a-fc. Now by Arts. 46, 81. (see Figs, in pp. 40, 64.) we have ANxNM or AN'xN'M' : PN2 or P'N'2 : : AC2 : BC2, or xx{2a±x) : y2 : : a% ; 62.
V- Hence y2=-j x(2ax±x*) is the general equation between the ab- a
scissa and ordinate of the ellipse and hyperbola.
But in the Ellipse MSxSA = BC2, and in the Hyperbola
M'S'xS'A = BC2, or (2a±c) Xc=l»2, hence by substitution y2=
2ac4-cz "L " . 2c*x 2cx2 , cV _ . *
^— x(2ax + x2)4.cx4- = 4- h — w • Conceive now the
a2 v ' a a a*
angles MAZ, ZA# to be continually diminished, then the axis major
both in the Ellipse and Hyperbola is continually increased, and just
at the instant of their approach to coincidence with the line AZ, each
of them becomes indefinitely great ; in which case, (supposing x and
c to be finite quantities) the three fractional terms of the last equa-
* Since AZ is parallel to M'BE, the angles MAZ, ZAz are re- spectively equal to the angles of the triangle MAM' ; which triangle is therefore isosceles.
102 ANALOGOUS PROPERTIES OF
tion become equal to nothing; .-. y2=4cr, or PN2=4ASxAN, which is the property of the Parabola. Hence it appears that a fi- nite part of an Ellipse or Hyperbola whose latus-rectum is finite, but whose axis major is infinite, may be considered as a Parabola ; and vice versa, that a finite part of a parabola may be considered as a part of an ellipse or hyperbola, whose axis major is infinite, and latus-rectum finite.
XVI.
On the mode of constructing the Three Conic Sections by means of a Directrix, and the Properties derived therefrom.
In Chap. I. we have already shown the method of constructing the Parabola by means of a directrix ; we now proceed to show that the Ellipse and Hyperbola may also be constructed by lines revolving in a similar manner.
(138.) Let MED be a line given in position ; and from the point E, draw CEC at right < s to MED ; in CEC take any point A, and set off AS : AE : : m : 1. Let the line SP begin to revolve from A round S, and PM move parallel to EC, in such manner that SP may be always to PM as AS to AE (i. e. in the given ratio of m : 1.) ; then the curve generated by the point of intersection P will be one of the Conic Sections.
Let fall PN at right < s to AC, and let AN=#, PN=y, AS=c ;
then, since AS (c) : AE : : m : 1, we have AE=— ; now PM =
m '
NE=AE + AN=— +ar, and SP : Pm(— +A : : m : 1 ; ... SP= m \m /
c+mx ; also SN=AN— AS=:r— c.
Hence we have, SP2 = (c+ra:r)2=c2-f2cra.r+raV2,
SN2 = (x—cf = c*—2cx+x* ;
.7. SP2— SN2(=PN2)-y2=(w-f-l)2c2:-l-(m2— l).r«r
ON THE THREE CONIC SECTIONS.
103
(139.) Let ra=l, or SP = PM, then m + l=2, and m2-— 1 = 0, ... y2=4c#, or PN2=4ASxAN; hence ALP is a parabola, whose vertex is A, focus S, and axis AC.
|
B |
M |
•'B |
||
|
C |
/ |
P |
||
|
E D |
a! s T |
N c |
(140.) Let m be less than 1, or SP less than PM. On the same side of A with PN, take AC : SC : : 1 : m, or AC : AC— SC(= SA=c) : : 1 : 1 — m, then c=(l — m).AC ; hence (m + \)2cx = (m -f 1) (l—m)x2AC.x=(l—m2).2AC.a;. From C draw BC at right angles to AC, and take BC2 : AC2 : : 1— m2 : 1, then 1— m% = BC2
B°2 A . 1
— 2, and m2— 1=
AC2'
Substitute these values for 1 — m2 and
BC5
m2 — 1, and we have (m + l)2c#=T7^x2AC.#, and (m2 — \)x2=
AC5
BC
""AC2*^5 nowlet AC==«> BC=6, then y2=((ra-hl)2c#-f(ra2-^
l)^2=)^x(2aa:— x*). Hence by (Art. 137) ALP is an ellipse, whose semi-axis major =*AC, semi-axis minor =BC, and focus S.*
* To prove that S is the focus, we have AC : SC : : 1 : m, .-. AC2 : SC2 : : 1 : m\ and AC2 : AC2— SC2 : : 1 : 1— m* ; but AC2 : BC2 :: 1 : 1— m\ .-. BC2 = AC2— SC2, and SC2 = AC2— BC2.
104 ANALOGOUS PROPERTIES OF
(141.) Let m be greater than 1, or SP greater than PM. Take
C on the other side of A in such a manner that AC : SC : : 1 : m,
or AC : SC— AC(=AS=c) : : 1 : m— 1, then c=(wfr— 1).AC, and
(m + l)2c=(m-f l)(m— l).2AC = (ra2— 1).2AC. From C draw
BC at right < s to AC, and take BC2 : AC2 : : m2— 1 : 1, then
RC2 m2 — I — xt^' ^et BC=6, AC=a, and substituting as before, we
b2 have y*——(2axJra;2)] hence ALP is an hyperbola, whose semi-
axis major is AC, semi-axis minor BC, and focus S.
Prom this mode of describing the three Conic Sections we deduce the following properties.
Property 1.
If a tangent be drawn to the extremity of the latus-rectum of any conic section, it will cut the axis, or the axis produced, in the same point with the directrix.
Draw the latus-rectum LST ; let LE be a tangent to the curve at L, and cut the axis in E ; then
(142.) In the Parabola, SP = PM, .-. AS = AE; hence SE = 2 AS. But by Art. 18, the sub-tangent SE= twice the abscissa AS, .-. E is the extremity of the sub-tangent, and also a point in the di- rectrix.
(143.) In the Ellipse, AS : AE : : m : 1, m being less than 1.
By construction (140) SC : AC : : m : 1.
.-. (Euc. 12. 5.) SC : AC : : SC-fAS(AC) ; AC+AE(EC).
Therefore EC is a third proportional to SC and AC ; which is also true (55.) if E be the point where the tangent cuts the axis pro- duced. Hence E is a point both in the directrix and tangent.
(144.) In the Hyperbola,
AS : AE : : m : 1, m being greater than 1.
THE THREE CONIC SECTIONS.
105
By construction (141.) SC : AC : : m : 1 ;
.-. (Euc. 19. 5.) SC : AC : : SC— AS(AC) : AC— AE(EC).
* Therefore, EC is a third proportional to SC and AC ; which is also true (85.) if E be the point where the tangent cuts the axis. Hence, E is a point both in the directrix and tangent.
(145.) This line LE, which is drawn touching the curve at the extremity of the latus-reclum, is called the focal tangent ; from what has just now been proved, it follows therefore that if a line be drawn at right angles to the axis from the point where it is intersected by the focal tangent, that line will be the directrix.*
Property A.
(145.a.) In the cone YYZ, let APQ be any conic section, and BHG an inscribed sphere, touching the cone in the circle BDG, and
the plane of the conic section APQ, in S. Then S is the focus of the conic section APQ,. Also, if the plane of the circle BDG be
* The substance of Arts. 138 to 145, inclusive, may very readily be inferred, without the aid of Algebra, from Arts. 18.a., 57.a., and
C. S 14
106 ANALOGOUS PROPERTIES OF
produced to intersect the plane of the conic section APQ in EF, then EP is the directrix of the conic section APQ,.
Let YYZ be a plane passing through the axis VC of the cone, cut- ting the plane of the conic section APQ, perpendicularly in AW, the axis of the conic section (1, 3 and 5,) and cutting the circle BDG in the line BG. Since YB and VG are tangents to the sphere from the same point Y, they are equal* and the axis VC of the cone, which bisects the angle BVG, cuts BG at right angles. For the same rea- son, the axis YC cuts all other lines passing through K in the plane of the circle BDG at right angles, and this plane is, therefore, per- pendicular to the axis YC, and consequently, to the plane YYZ, which passes through it. Since, therefore, the planes of the circle BDG, and the conic section APQ are both perpendicular to the plane YYZ, their common intersection EF is perpendicular to VYZ, and therefore to the lines BX, WX, which it meets in that plane.
Draw YL, in the plane YYZ, parallel to AW, intersecting GB, produced if necessary, in L. From any point P in the curve APQ, draw PM at right angles to EF. PM is parallel to WX, and con- sequently to VL. Join VP, intersecting the circumference of the circle BDG in D. Join LD, DM.
Since D is in the plane of the parallels PM, VL, the lines LD, DM are in that plane. But they are also in the plane of their circle BDG. Therefore they are in the common intersection of the two planes, and are in the same straight line. Now VD=VB, because both are tangents to the sphere from the same point Y. For the same reason PS=PD.
And (sim. tri.) PD : PM : : YD : YL,
PS : PM : : VB : YL, a constant ratio.
* For the plane of the lines YB VG cuts the sphere in a circle, to which YB and YG are tangents. Hence it follows from Euc. 36. 3, that VB=YG.
THE THREE CONIC SECTIONS.
107
Hence, the distance SP, of any point of the curve P, from S, is in a constant ratio to the perpendicular PM, to the line EF ; which is the property of the focus and directrix of the conic section APQ,. Therefore S is the focus, and EF the directrix.
Property 2.
In any Conic Section, the distance SP= half latus-rectum
on •
1— ~,cos. <PSN
AC
(146.) Let radius = 1, then SP : SN : : 1 : cos. < PSN ; there- fore SN=SPxcos. PSN. Now, in the Ellipse and Hyperbola, SP : PM : : m : 1, and SC : AC : : m : 1 ; .-. SP : PM(=NE=
SE+SN) :: SC : AC; hence SPxAC = SExSC +SCxSN=
|
I) |
. |
B |
|||
|
c |
L. / |
P |
r, |
||
|
E D |
aI s T |
I |
¥ |
BC» + SC x SN = BO + SC x SP x cos. PSN ; or SP x AC— SPXSCXCOS.PSN-BO.; therefore SP-^^-^^-^
108 ANALOGOUS PROPERTIES OF
BC2 1 half latus-rectum*
x-
AC i_^.cos.PSN 1-^' cos. PSN AO AO
(147.) In the Parabola, SC may be considered as equal to AC ;t
_,_ half lat. rect. rnu . . , . , ,
.-. SP = ■: ==5, The same expression might also be dedu-
1 — cos. PSN
ced immediately from the properties of the Parabola, for since SE—
2AS, SP(=PM=NE==SE+SN)=2AS+SN=2AS + SPxcos. PSN,
*n m ticitvt oao j on ^A^ half lat. rect.
.-.SP— SPxcos. PSN=2AS, andSP=- sa^—i S5i5-
1 — cos. PSN 1 — cos. PSN
(148.) By means of this property we are enabled to find the va- riation of the distance SP in its angular motion round the focus S ; and in this respect it forms an important theorem in Physical Astron- omy. To put the expression just now deduced into the Algebraic form adopted by Mr. Vince (at page 26 of his ' Physical Astronomy') in tracing the radius vector (SP) round the elliptic orbit of the moon, let AC=1, BC=c, SC=w, < PSN=*; then
*When P is at L, <PSN=a right angle; .-.cos. PSN=0,
and SP=£ latus-rectum. When P is between L and A, cos. PSN
__ \ latus-rectum is negative ; .-. SP= «_■' '
1+AOXCOS,PSN
t For by Art. 137, the Parabola may be considered as an Ellipse, whose major axis is infinite ; in this case C goes off to an infinite distance, and the difference (AS) between AC and SC vanishes with respect to the quantities themselves, which may therefore be assum- ed as in a ratio of equality.
THE THREE CONIC SECTIONS.
109
SP
<
BC* *
AC— SCxcos. PSN 1
w
WXCOS. 2?
= CaX:
=(by actual division)
1 — wxcos. z
c2x(l + wXcos. ^+2^2x(cos. z)2+t03x(cos. z)3+, (fee.) For the trigonometrical transformation of this expression, and its practical application, we refer the reader to the work itself.
(149.) Before we leave this subject of the radius vector, it may not be improper to show its variation with respect to an angle de- scribed about the center of the Ellipse. Upon the major-axis AM describe the semi-circle AQ,M, produce NP to Q,, and join Q,C.
Q R
Draw PH to the other focus, then PN2=SP2— SN2=HP2— HN2 ; .-. HP2— SP2=HN2— SN2.
Hence we have,
(HP-fSP).(HP— SP)=(HN+SN).(HN— SN) ; -
HN— SN : HP— SP, 2CN :2AC— 2SP; CN : AC— SP, AC(orQ,C):CN 1 : cos. aCN ; .-.SCxcos. aCN = AC— SP,
.-. HP+SP : HN-f-SN i.e. 2AC:HSor2SC or AC : SC
.-.so: ac-sp
or SP= AC — SC xcos. Q,CN, which is an expression for the radius vector, with reference to the center.
Property 3.
(150.) If a conic section be cut through the focus (S) by a line (Vp) terminated at each extremity by the curve, then 4SPxSp= latus-rectum x P/>.
110
ANALOGOUS PROPERTIES OP
From the extremity of the latus-rectum LST draw LK at right < s to the directrix ; and from p draw pm parallel, and pn perp'en-
|
M |
||||
|
Ts |
L |
p |
||
|
K |
1 n |
ft |
||
|
aI |
s |
|||
|
in |
||||
|
P |
||||
|
T\ |
1 |
\^v |
dicular, to the axis. From the nature of the construction we have, SP : PM (or NE) : : SL : LK (or SE), or SP : SL : : NE : SE ; .-. SP— SL : SL : : NE— SE (SN) : SE, i. e. SP— SL : SN : : SL : SE ; for the same reason, ST— S/> (SL— S^) : Sn : : SL : SE ;
.-. SP— SL : SL— $p : : SN : Sn : : (by sim. As) SP : Sp.
Hence SPxS/?— SLxSp=SL xSP— SPxSp,
or 2SPxSp=SL x(SP+Sp)=SLxPp ; .-. 4SPxSp=2SLxPp=latus-rectumxPp.
Cor. 1. Hence p+^|
(151.) Cor. 1. Hence fp+o-=dT-* For since SLx(sp+sP)=
ocm a u SP+S^/ 1 , 1 \ 2
2SPxSP; we have g^ +_)-_.
THE THREE CONIC SECTIONS. . Ill
Cor. 2. Since SP— SL : SL— Sp : : SP : Sp, SP, SL and Sp are in harmonical proportion. Or, half the latus-rectum is an harmo- nical mean between the segments, into which the focus of a conic section divides any line which passes through it.
XVII.
On the analogous Properties of the Normal, Latus-rectum, Radius of Curvature, $*c. $*. in all the Conic Sections.
If S be the focus, A the vertex, and P any point in the Parabola, then (Arts. 125, 126.) 4SP= chord of curvature passing through
3 3
4SP^ 2SP2
the focus : —^-rr = diameter, and -t-&~t «= radius of curvature
V(SA) V(bA)
to the same point. In the Ellipse and Hyperbola, if C be center, S the focus, AC the semi-axis major, CD the semi-conjugate to the semi-diameter PC, and PF a perpendicular let fall from the point P
2CD2
to the conjugate diameter, then (Arts. 128, 129.) -p~- = chord of
2CD2 curvature passing through the center from the point P ; — r^r = chord
through focus ; -p^r = diameter, and p^~ = radius of curvature to the same point.
Property 1.
In every conic section, the cube of the normal divided by the ra- dius of curvature is equal to the square of half the latus-rectum.
(152. In the Parabola, (see Fig. in page 30.) since normal PO : SY::TP:TY, and TP = 2TY, .-.PO = 2SY; hence cube of
normal = 8SY3 = (by Art. 32.) 8SP* x SA^ ; radius of curvature
2SP^ cube of normal 8SP*xSA*xSA*
f,A rad. of curvature 2SP£
square of 2S A = square of half the latus-rectum.
= 4SA2
112 * ANALOGOUS PROPERTIES OP
(153.) In the Ellipse and Hyperbola (see Figures and Properties
RC2 in pp. 49, 67, 68.) POxPF=BC3; /NI»©-5=r, and cube of
, BC8 *S i CD2 . cube of normal
normal = 5==- ; radius of curvature =.™-; hence — = — ? —
PF 3 PF rad. of curvature
= CD^PF^=CD2^PF2==(by Pr°pertieS m PP- 5L 69')
BC« BC* ,BC2 _, .. . , ,
square of -r— r=square of half the latus-rectum.
AC3xBC2 AC2 u AC
(154.) Cor. Since half the latus-rectum is a constant quantity, the radius of curvature varies as the cube of the normal ; the curvature therefore varies inversely as the cube of the normal in all the Conic Sections ; which accords with what has already been demonstrated in Sections XIII and XIV.
Property 2.
In any conic section, if a perpendicular (OX) be let fall upon the line SP from the point O, where the normal intersects the axis, then the part PX cut off by this perpendicular is equal to half the latus- rectum.
(155.) In the Parabola. Draw the ordinate PN ; then, since (by Art. 30.) SP = SO, the angle SOP = SPO, and PO is common to the two right-angled triangles PXO, PON ; these two triangles are
therefore equal and similar ; T A S N o
hence PX = NO = (Art. 21.) half the latus-rectum.
(156.) In the Ellipse and Hyperbola. (Fig. in p. 113.) Draw the conjugate diameter DCK, then the right-angled trian- gles PEF, PXO are similar; .-.PE(AC) : PF :: PO : PX; hence
"PC2 AC x PX - PO x PF - BC2 ; .-. PX = ~ = half the latus-rectum.
THE THREE CONIC SECTIONS.
113
(157.) Cor. By means of- this Property, if SP be given in length and position, and the latus-rectnm and position of the tangent be also given, we can determine geometrically the position of the axis ; for we have only to make PX equal to half the latus-rectum, and draw XO at right angles to SP, and PO at right angles to the tangent at P, then O (the intersection of XO, PO) is a point in the axis, which being joined to S, gives SO the position of the axis.
Property 3.
In any conic section, take the arc PQ,, and from the point Q, draw GIT perpendicular and Q,R parallel to SP ; then (assuming the rela- tion of Q,T and Q,R just at the point of contact) the latus-rectum is
Q,Ta
equal to -^
(158.) In the Parabola.' Draw the perpendicular SY upon the tangent PY; then, since the arc Q,P coincides with the tangent at P, the triangle Q,PT continually approaches towards similarity with the triangle SPY as Q, moves up towards P ; and at the point of contact QP : QT : : SP : SY ; .-. QP2 : QT2 : : SP2 : SY2, and (dividing the first two terms by
C. S. 15
114 ANALOGOUS PROPERTIES OF
QR)^ :~:: SP2 : SY2 :: (by Art. 32.) SP2 : SPxSA : :
Q,P2
SP : SA. Now -^-=-(=chord of curvature passing through the fo-
cus) = 4SP; hence we have 4SP : ^=-::SP.-.SA, : ^— =
4SPxSA ACj. . ;
— — - — = 4S A = latus-rectum.
(159.) In the Ellipse and Hyperbola. Draw the conjugate diam- eter DCK, and the perpendiculars PF and SY upon it and the tan-
gent ; then the triangles QPT, PEF are similar, .-. QP2 : Q/T2 : :
PE2 (AC2) : PF2, and ^~ (-jfr^**^ l~i; AC2 : PF5 Q,K V AG / U,ri
QT2 2CD2xPF2 2AC2xBC2 2BC2
QR AC3 AC ^ AC
the latus-rectum.
The demonstration of this property of the Conic Sections forms the substance of the first three Propositions of the third Section (B. 1 .) of Sir Isaac Newton's Principia.
Property 4.
(160.) In every conic section,, the chord of curvature passing through the focus is to the latus-rectum in the duplicate ratio of
i
THE THREE CONIC SECTIONS. 115
SP : SY ; and the diameter of curvature is to the same in the trip- licate ratio of SP : S Y.
QP2
For the chord of curvature passing through the focus = ;=^5-; and
Q,T2
by Property 3, the latus-rectum = ?-=- ; hence the
, , * " &P2 T&2
chord of curvature : latus-rectum : : — — - : — — — : : GlP2 : Q.T2 : : SP2 : SY2 :
(dK. CIR
but diameter : chord of curvature (see Fig. in page 95.) : : SP : SY ;
.'. diameter of curvature : latus-rectum : : SP* : SY3 .
Property 5.
Let L = latus-rectum of any conic section ; then, in the Parabola, LxSP-=4SY2; in the Ellipse, LxSP is less than 4SY2 ; and in the Hyperbola, L xSP is greater than 4SY2.
(161.) In the Parabola. (32) SAxSP = SY2, .-.4SAxSP=4SY2, or LxSP=4SY2, for L=4SA.
RC2 v SP (162.) In the Ellipse. By Art. 66. SY2 = * , ... 4SY2
HP
4BC2 x SP / , 2BC2
HP
= ( for -xc ~h' and ' ' ' 4BC2 = 2AC X L )
^^— P 5 hence L x SP : 4SY2 : : HP : 2AC : : 2AC— SP* :
2 AC, and as 2 AC — SP is less than 2 AC, L x SP must be less than 4SY2.
(163.) In the Hyperbola, by a similar process we have LxSP : 4SY2 : : HP : 2AC : : 2AC+SPt : 2AC, and as 2AC+SP is greater than 2AC, LxSP must be greater than 4SY2.
* For SP+HP=2AC, .-. HP=2AC— SP. t For HP— SP=2AC, .. HP-2AC+ SP.
116 ANALOGOUS PROPERTIES, &C.
(164.) Before we conclude this Section, it will be proper to show the method of expressing the relation between SP and SY, in the form of an algebraic equation. In the Parabola, therefore, let SA— a,' SP=x, SY=y; then since SY2=SAxSP, we have y2=ax, or y=V(ax,) for the equation to the curve, in terms of the distance from the focus, and the perpendicular from the focus upon the tan- gent. In the Ellipse and Hyperbola, let AC=a, BC=^ SP=:r,
SY = y; then HP = 2AC±SP=2a±;r, .-.since SY2 = — -- — ,
b2 x b2x2 bx
we have y - ^— = g-^ and y- v(2g^ where the
negative or positive sign must be used according as the section is an Ellipse or an Hyperbola.*
(165.) To investigate the relation between CP and Cy (see Fig- ures in pages 49, 68,) let CP=x, Cy or PF=y ; then in the Ellipse, since AC2 + BC2 = CD2 + PC2, we have a2 -{- b2 = CD2 + x2, ... CD2 - a2 + b2—x2 or CD = V (a2 + b2— x2 .) Again, since
ACxBC=CDxPF, we have ab = CT)xy, .-. CD= — ; hence
— = V(a2-f62 — x*,) or y = — — — — — is the equation to the
y K ' V(a2 + Z>2— x2) *
curve in terms of the distance from the center, and 'perpendicular from the center upon the tangent.
In the Hyperbola, PC2 ^CD2 = AC2 ^CB2, or #2^CD2 = a2^b2 ;
,-. CD2 = *2-a2+&2, and y- "V^-t
V (#2 — a2-fo2)
* These expressions are the equations of the several conic sec- tions, considered as spirals, described by the revolution of the radius vector SP, about the focus.
t In these equations, the curves are considered as described by a radius vector CP, revolving about the center. This mode of con- sideration is, of course, inapplicable to the Parabola.
METHOD OF FINDING, &C.
117
CHAPTER VII.
ON THE METHOD OP FINDING THE DIMENSIONS OF CONIC SECTIONS WHOSE LATERA-RECTA ARE GIVEN, AND OF DESCRIBING SUCH AS SHALL PASS THROUGH CERTAIN GIVEN POINTS.
XVIII.
On the method of finding the dimensions of Conic Sections, whose later a-recta are given.
(166.) Let S be the focus of any conic section, P some point in the curve at a given distance from S ; join SP, and let it meet the tangent PT in the given angle SPT ; let the latus-rectum = L, and take PX=£L ; from X draw XO at right angles to SP, and from P draw PO at right angles to PT, then by Art. 157, O will be a point in the axis join SO, and it will give the po sition of the axis.
(167.) We are thus furnished with the means of determining geo- metrically the position of the axis of any conic section whose latus- rectum is given, and whose tangent at a given point meets a line drawn from the focus to that point, in a given angle. The position of the axis being found, its dimensions may be ascertained from the properties of each particular curve. In the Parabola, the latus-rec- tum is equal to four times the distance of the focus from the vertex ; if therefore in OS produced, we take SA equal to £L, A will be the vertex of the Parabola. In the Ellipse and Hyperbola, it will be necessary to find the center, as also the major and minor axis ; which is done in the following manner.
118
METHOD OF FINDING THE
(168.) In the Ellipse, the lines drawn from the foci to any point in the curve make equal angles with the tangent at that point ; if therefore the angle HPZ be made equal to the angle SPY, and SO be produced to meet PH in the point H, that point will be the other focus ; and this determines the length (SP+PH) of the major axis. Now by Art. 45, the conjugate diameter DCK cuts off from SP a part equal to the semi-axis major ; hence if PE be taken equal to J(SP-l-PH), and through E we draw DC parallel to the tangent at P, C will be the center of the ellipse. It only remains therefore to produce SH both ways, and make CA, CM each equal to PE, and we have AM the major axis of the curve. But (39) the latus-rec- tum is a third proportional to the major and minor-axis ; the minor
axis is therefore a mean pro- portional between the major axis and the latus-rectum ; from C then draw BC at right angles to AM, make BC a A mean proportional between AC and ^L, and B will be the extremity of the minor axis ; thus the dimensions of the el- lipse are determined.
BY
(169.) In the Hyperbola, the tangent bisects the angle SPH; in this case, therefore, the angle HPY must be made equal to the an- gle SPY on the opposite side of the* tangent ; then if OS is produced till it meets PH in Ihe point H, that point will be the other focus. Produce SP to E, and take PE equal to J(HP— SP) ; through E
DIMENSIONS OF CONIC SECTIONS.
119
draw EC parallel to the tangent at P, and C will be the center. Take OA, CM, each equal to PE, then AM will be the major axis. The minor axis is determined precisely in the same manner as in the Ellipse.
(170.) We have thus shown the method of solving this Problem, when the nature of the curve is given. Suppose now that the latus- rectum, the distance SP, and the position of the tangent be given as before, and it is required to find not only the dimensions, but the na- ture of the conic section. In this case we have recourse to Arts. 161, 162, 163 ; from which, when the latus-rectum and the relation between SP and SY are given, we can determine the particular na- ture of the curve. For it is there proved, that if LxSP be equal to 4SY2, the curve is a Parabola ; if LxSP be less than 4SY2, it is
H M
AS N
H M
an Ellipse ; and if LxSP be greater than 4SY2, it is an Hyperbola. In order to affect this general solution of the Problem, let the sine of the given < SPY=s, radius=l, then (by Trigonometry) SP : SY : : 1:5; .-. SY=s . SP, and SY2=s2 . SP2 ; consequently 4SY2=4s2 . SP2. Having therefore found the position of the axis, as in the former case ; then, to know whether the conic section, whose di- mensions are required, be a Parabola, Ellipse, or Hyperbola, we must compare LxSP with 4s2 . SP2. If LxSP be equal to 4s2 . SP2, i. e. if L be equal to 4s2 . SP, then the curve is a Parabola; take therefore SA=|L, and A is the vertex. If L be less than 4sf . SP, the curve is an Ellipse ; in which case, make the < HPZ (on the same side of the tangent with SP) equal to SPY, and proceed as in Art. 168. If L be greater than 4s2 . SP, the curve is an Hy- perbola ; make therefore HPY (on the other side of the tangent) equal to SPY, and proceed as in Art. 169.
120 METHOD OF FINDING THE
(171.) By Art. 160, the chord of curvature passing through the focus : the latus-rectum : : SP2 : SY2 : : 1 : s2 ; .-. the latus-rec- tum =s2x chord of curvature ; if therefore the chord of curvature and the relation of SP to SY be given, the latus-rectum will also be given. We are thus enabled to give the trigonometrical solution of the following
PROBLEM.
(172.) Given the chord of curvature passing from any point through the focus of a conic section, the distance of that point from the focus, and the position of the tangent ; it is required to find the nature and dimensions of the conic section.
e
Let the chord of curvature to the point P=40, SP=12, the angle SPY=30° ; then since the sine of 30°=half radius, s=£ ; .-. L«= (s2X chord of curvature =)£ x 40=10 ; also 4s2 xSP = 4 x^SP = SP=12 ; hence L is less than 4s2xSP, and consequently the conic section is an Ellipse.
Z
Since the < SPY -= 30°, the <XPO = 60; .-. <XOP = 30°, and PX = |PO, or PO = 2PX = (Art. 156.)L = 10. Hence, in the triangle SPO, we have SP = 12, PO = 10, < SPO = 60° from which we can determine the < PSO ; for POS + PSO = 120°,* .-. i(POS + PSO)=60°. Now SP f PO (22) : SP— PO (2) : : tan.
KPOS+PSOX60O) : tan. ^08^80) = ^^=^
.-.log. tan. } . (POS— PSO)=log. tan. 60°— log. ll=log.tan. 8° 57' ') Ijence < PSO = (|(POS +PSO)-i(POS— PSO) = )60°— 8° 57'= 51° 3'.
Since <HPZ= < SPY, the <OPH= < SPO, .-. < SPH = 120° ; in the triangle SPH we have therefore SP=12, < PSH=
* See Day's Trigonometry, Art. 153.
DIMENSIONS OF CONIC SECTIONS.
121
5° 57' : PH =
; but as sin. PHS 12 x sin. 51° 3' sin. 8°~5f '
8° 57'= log.
51° 3', <SPH = 120°, and .-. <PHS =
(8° 57')* : sin. PSH (51° 3') : : SP (12)
hence log. PH=log. 12-flog. sin. 51° 3' — log. sin 59.987; .-. PH= 59.987, and SB+PH = 12+59.987 = 71.987- major axis of the ellipse, and the minor axis= (mean proportional be- tween the major axis and latus-rectum=) V (10x71.987) = 26.83; from which the Ellipse may be constructed as in Art. 168.
XIX.
On the method of describing Conic Sections which shall pass through three given points.
(173.) Let SO, SP, SGI, be three lines given in length and posi- tion ; join PO, Q,P ; produce PO to p, making Op : Vp : : SO : SP ; and produce it both ways to m and D. Draw SE, On, PM, Qm, at right angles to mED ; then the conic section whose focus is S, di- rectrix MED, and determining ratio SO : : On, will pass through the points O, P, Q.
(174.) By sim. As Onp, PM/?, Op the construction Op : Vp : : SO : SP, .
; Pp : : On : PM ; but by SO : SP : : On • PM, and
as.
See Day's Trigonometry, Art. 150. 16
122 ON CONIC* SECTIONS
SO : On : : SP : PM. Again, by sim. A s FMq, QLmq, Vq:Q,q:: PM : am ; but Vq : Q? : : SP : SQ,-.SP : Sa : : PM : Qra, or SP : PM : : Sa : dm ; hence SO ; On : : SP : PM : : SQ, : am, i. e. the lines SO, SP, SO, diverging from S are in a given ratio to the lines On, PM, Q,ra drawn at right angles to the line MED. By Art. 138, therefore the curve OPQ, is a conic section whose focus is S and di- rectrix MED ; and it will be a parabola, ellipse, or hyperbola, ac- cording as the antecedent of that ratio is equal to, less or greater than, the consequent, or according as SO is equal to, less or greater than Ora.
(175.) In order to find the dimensions of the conic section ; divide SE at the point A, so that SA : AE : : SO : On, and A will be the vertex. If SO — Ow, then SA=AE, and the curve is a Parabola whose axis is EAS, vertex A, and latus-rectum 4SA. If SO be less than On, take AC : SC : : On : SO, then (by Sect. XVI.) C will be the center and AC the semi-axis major of the Ellipse ; the semi-axis minor (BC)=V(AC2 — SC2.) If SO be greater than On, take C on the other side of A, so that AC : SC : : On : 30, then C will be the center, and AC the semi-axis minor (BC) = V(SC2— AC2.)
This method of construction leads to the trigonometrical solution of the following
PROBLEM.
(176.) Three straight lines issuing from a point, being given in length and position ; it is required to find the nature and dimensions of the conic section which shall pass through the extremities of those three straight lines, and have its focus in the point of their inter- section.
Let SO =4, SP-7, SGI = 10, < OSP=60°, < PSQ = 20°; then SOP+SPO=120°, and SP+SO (11)* . SP— SO (3) : : tan.
* See Day's Trigonometry, Art. 153.
PASSING THROUGH THREE POINTS.
3 x tan. 60°
11
123
log.
* (SOP + SPO) (60°) : tan. i (SOP— SPO) tan. ) . (SOP— SPO)=log. 3+log. tan. 60°— log. ll=log. tan. 25°
Q
17'; hence < SOP=60° + 25° 17' = 85° 17', and < SPO = 60°— 25° 17'=34° 43'. Now, sin. SPO (34° 43')* : sin. OSP (60°) : : SO (4) ; PO =
4 x sin. 60°
.-.log. PO=log. 4+ log. sin 60°— log. sin. 34c43'=
sin. 34° 43''
log. 6.0825, .*. PO=6.0825. By a similar process it appears that the angle SPQ=125° 1', < SQP=34° 59', and PQ=4.1758. Thus all the sides and angles of the triangles SOP, SPO, are determined, and we now proceed to find the values of Pp and Qq.
By the construction Op : Vp : : SO : SP : : 4 : 7 ; .-. Op : Vp—Op (OP) : : 4 : 3. Hence 0^=f OP=f x 6.0825=8.11 ;
... Pp=Op+OP=6.0825+8.11=14.1925. Again, Vq : Qq : : SP : SQ : : 7 : 10 ;
... p^ : Qq—Vq (PQ) : : 7 : 3.
See Day's Trigonometry, Art. 150.
124 ON CONIC SECTIONS
Hence Vq = J x Pa = J X 4.1758 - 9.7435 ;
... Qq =Vq +PQ = 9.7435 +4.1758 - 13.9193.
Now the <SP?=180°— SPa=180°— 125° l' = 54° 59'; and the < pVq = S¥q— SPO = 54° 59'— 34° 43' = 20° 16'; hence, in the triangle qVp, we have Fq = 9.7435, Vp = 14.1925, and the in- cluded angle pVq = 20° 16'-; from which the angle qpF is found to be equal to 33° 45'. In the right-angled triangle Opn, we have therefore Op = 8.11, and the < Op?i = 33° 45', which gives* Ora = 4.5056.
Again, in the triangle PpM, we have Vp = 14.1925, and the angle P^M = 33° 45', from which PM is found to be equal to 7.8849. Finally, by similar triangles, P^M, Qqm, we have P^ (9.7435) :
Q? (13.9193) : : PM (7.8849) : am=^-|^== 11.264.
On reviewing the steps of this operation, we have, SO : On :: 4 : 4.5056 : : 1 : 1.1264, SP : PM:: 7 : 7.8849 :: 1 : 1.1264, SQ : am:: 10 : 11.264 :: 1 : 1.1264.t The given ratio therefore of SO : On is 1 : 1.1262; and as, in this case, SO is less than On, the curve is an Ellipse.
Having thus ascertained the nature of the curve, it now only re- mains to find its dimensions. For this purpose we must first find the length of SE, which (if ON be let fall perpendicular to it) is equal to SN +NE, i. e. to SN + On, for On is equal to NE, being the opposite side of a parallelogram.
Now < NOS = 180° — < SOP — < NOp (Opn) - 180°— 85° 17' — 33° 45' = 60° 58'.
* See Day's Trigonometry, Art. 134.
t It is not necessary to find all three of these ratios, since they are equal to one another. It would have been sufficient to have calcu- lated the length of On, merely ; for the ratio SO : On determines the nature of the curve.
PASSING THROUGH THREE POINTS. 125
In the triangle OSN we have therefore OS=4, and the angle SON=60° 58', from which we get SN=3.4973 ; and consequently SE(=SN+Orc)=3.4973-f-4.5056=8.0029.
Having found the value of SE, we must divide it in the ratio of SO : On ; i. e. in the given ratio of 1 ; 1.1264.
Thus SA : AE : : OS : On : : 1 : 1.1264 ;
.-. S A : SA -f AE(SE) : : 1 : 2.1264.
Hence SA=^— m== ' =3.7635 ; therefore, make the angle
OSA=90°— 60° 58'=29° 2', and take SA=3.7635, then A will be the extremity of the major axis.
To find the major axis itself, take
AC : SO : : On : OS : : 1.1264 : 1, or AC : AC— SC(AS) : : 1.1264 : .1264 ; .£ 1.1264 ACJ •'•AC~T1264XAS'
= ^^7 X 3.7635=33.4951. .1264
Hence SC=AC—AS=33.4951— 3.7635-29.7316.
and BC=V(AC2— SC2)=15.417.
Finally the dimensions of the Ellipse are as follow :
Major Axis=2AC=66.9902.
Minor Axis=2BC=30.834.
2BC2 Latus-rectum= ——^=14.185.
By means of this Problem the dimensions of the orbit of a Planet or Comet may be found from three observations made as to its dis- tance and angular position, at three different periods in the course of one revolution round the Sun.
126
QUADRATURE OF THE CONIC SECTIONS.
CHAPTER VIII. ON THE QUADRATURE OF THE CONIC SECTIONS.
XX.
On the relation which obtains between the areas of Conic Sections of the same kind, having the same vertex and axis ; and on the Quadrature of the Parabola, Ellipse, and Hyperbola.
(177.) Let AQq, APp, be any two curves having the same ver- tex A, and let them be referred to the same axis AM by ordinates
|
.B |
Q |
\m |
rB |
||
|
>b , |
V |
°\ |
•b |
M
N
If
GIN, PN, which are to each other in a given ratio ; then the areas AGIN, APN, generated by those ordinates, will be to each other in the same given ratio. For take the ordinate qpn indefinitely near to Q,PN, and draw dm, Vo parallel to the axis, then the flnxional or incremental areas QNn$ PNrcjo will approach to equality with the parallelograms QJSnm, PNno, as the ordinate qpn approaches to Q,PN ; but (by Euc. 1. 6.) these parallelograms are to each other in the ratio of GIN : PN ; the nascent increments therefore of the areas AGIN, APN are to each other in the ratio of GIN : PN ; and as these areas begin together from A, the areas themselves must also
QUADRATURE OF THE CONIC SECTION. 127
be to each other in that ratio, i. e. area AGIN : area APN : : GIN : PN.
(178.) Suppose now the curves AQq, APp to be two Conic Sec- tions of the same kind whose latera-recta are respectively L and I ; for instance, let them be two Parabolas ; then by the property of the parabola LxAN = QN2, and ZxAN = PN2, hence AQN : APN (::QN ; PN) : : V (LxAN) : V(ZxAN)::VL : W. If they be Ellipses or Hyperbolas which have the same major axis AM, and whose minor axes are respectively BC and bC, then
' ANxNM : PN2 : : AC2 : 6C2, and GIN2 : ANxNM : : BC2 : AC2 5
or>r<2 Q.hC*2
.-.QN2 : PN2 ::BC2 :&C2"
AC * AC : :L : I
Hence in this case also AGIN : APN( : : aN : PN) : : VL : Vl ; i. e. in Conic Sections of the same kind, having the same vertex and axis, the areas AGIN, APN are to each other in the given subdupli- cate ratio of their latera-recta.
(179.) Take any point S in the axis, and join SGI, SP ; then we have
area AQN : area APN : : QN : PN,
andASaN : A SPN ::QN:PN;
.•.AQN— SQN : APN— SPN ::GtN : PN, or area AGIS : area APS : : GIN : PN
::VL; Vl in all the Conic Sections.
(180.) But Ellipses and Hyperbolas having the same vertex and axis, will also have the same center.* Let C be that center, and in each case join GIC, PC ; then
* Although they have the same center, it should be recollected that (since the minor axes are not equal) they cannot have the same focus.
128
QUADRATURE OF THE PARABOLA.
In the Ellipse, AQN : APN : : QN
and A QCN : A PON : : QN
.-. AQN+QCN : APN+PCN : : QN
or sector ACQ, : sector ACP : : QN
::VL
PN, PN; PN, PN,
Vl.
In the Hyperbola, AQN : APN : : QN : PN,
and A QCN : A PCN : : QN : PN ;
.-. QCN— AQN : PCN— APN : : QN : PN,
or sector ACQ : sector ACP : : Q^N : PN,
:: VL : Vl.
On the Quadrature of the Parabola.
(181.) Let AP be a Parabola whose latus-rectum (BC)=a, ab- scissa (AN) =#, ordinate (PN) = y ; then, by the Property of the
Parabola, yz=ax, .:y=a2x2j and yX=a2x2'x now the fluent of
%xYa2x2=\xy\ hence the area APN=fANxPN=f
yx
= %a2x2=
the circumscribing parallelogram.
QUADRATURE OF THE PARABOLA. 129
(182.) Draw a tangent to the point P, and produce NA to meet it in T ; then since AN = |NT, the A PNT = (£TNxPN= )AN x PN; hence the area ANP=§ANxPN=f A PNT. Now sup- pose, in the Figure, at page 28, that a tangent be drawn to the point G, and that the line MS drawn parallel to the axis meets it in S, then the area AZG = |A TZG ; but A TZG : A SMG : : ZG2 : MG2 : : 1 : 4, .-.A SMG = 4 times A TZG= 6 times area AZG = 3 times area MAG ; hence area MAG =i A SMG.
(183.) But the area of a Parabola may be ascertained in terms of the square of its latus-rectum. For let AN : AS : : n : 1, then AN= n . AS ; but PN2 = 4AS x AN = An . AS2, .-. PN = 2AS. Vn ;
3
A?i2 hence area APN (| ANxPN)= f X n. AS X 2AS xVrc=-^-xAS2 =
3,
n2
(for AS=£BC, and .-. AS2=TV BC2V~ >cBC2; or if the whole
3
n2
Parabola is taken (as in Fig. p. 28,) then the area MAG =— x square of latus-rectum.
(184.) Not only the area ANP contained between the abscissa and ordinate, but also the area ASP described by the revolution of the line SP round the focus S, may be ascertained in the same man- ner. For since AN=n.AS, SN = (AN— AS=)(rc— 1).AS ; hence
A SPN = (iSN X PN = )n^- . AS x PN. Now area APS = area
APN— ASPN=fra.ASxPN ^- xASxPN= -y-.ASx
PN=(for PN = 2AS Vn) £+J^ xAS»-(-^±J^ xBC*.
(185.) Hence it appears, that if the latus-rectum be given, the par- abolic areas ANP, ASP may be found without any other irrationality C.S. 17
130
QUADRATURE OP THE ELLIPSE.
than that which arises from extracting the square root of numbers 5
for iff*-* 1, 4, 9, 16, &c. then
Area ANP = fe f, f, y, &c. of the square of the latus-rectum j and
Area ASP = fa fa f, if, &c. of the same ; but if n be not a square
number, then the expression for these areas will involve an irrational
quantity.
On the Quadrature of ^Ellipse.
(186.) Let ABMO be an Ellipse, and upon the major axis AM describe the circle ARML ; draw any ordinate Q,PN, then by Prop- erty 9, of the Ellipse, Q,N : PN in the given ratio of RC or AC : BC. But from what was proved in Sect. 20, area AQN : area APN : : QN : PN : : AC : BC ; and for the same reason, the semicircle ARM will be to the semi-ellipse ABM in the same ratio ; hence the whole Ellipse ABM : circle ARML described upon its major axis : : BC : AC : : minor axis : major axis.
(187.) As the area of the Ellipse bears this given ratio to the area of its circumscribing circle, the quadrature of the Ellipse must therefore depend upon the quadrature of the circle. Let
QUADRATURE OF THE HYPERBOLA. 131
p =3.1416 (=*areaof a circle whose radius is 1), then the area of the circle whose radius is AC -■ p X AC2 ; hence the area of the Ellipse : /;xAC2 :: BC : AC, .-.area of the Ellipse = p x AC X BC, i. e. the area of an Ellipse is found by multiplying the rectangle under its semi-axes by the same decimal number (p) as the square of the radius is multiplied by, to find the area of a circle. From this it also appears, that the area of an Ellipse is equal to the area of a circle whose radius is a mean proportional! between its semi- axes ; for the area of that circle is equal to (p x(rad.) 2=p xthe square of V ( AC x BC) - ) p x AC X BC.
(188.) The area of the parallelogram circumscribing the Ellipse is equal to 4 AC x BC, .-. area of Ellipse : area of that parallelo- gram : : p x AC xBC ; 4AC xBC : : 2h or 3.1416 : 4 : : .7854 : 1 ; i. e. the area of an Ellipse has the same ratio to the area of its cir- cumscribing parallelogram as the area of a circle has to its circum- scribing square.
On the Quadrature of the Hyperbola.
(189.) Let AP/> be an Hyperbola whose semi-axis major AC = a, semi-axis minor bC=b ; and let CN = .r, PN = ?/; then by Cor. 1. Prop. 6. of Hyperbola, CN2— CA2 : PN2 : : AC2 : BC2, or x%—
a1 iifiia*: 62, .-.y = - V(^2~«2); hence yx' = --. x V(^2— a2), a a
whose fluent found by a series and properly corrected would give the value of the area APN ; but this area may be ascertained by means of logarithms, when we have found the value of the hyperbo- lic sector ACP. (See Fig. in p. 126.)
(190.) Now the area of this sector is thus found. The area of A CPN = -£CN xPN = % .-. the fluxion of the A CPN = &p* .
* See Day's Mensuration, &c, Art. 30.
t Let x — mean proportional between AC and BC, then AC : x : : x : BC, .-. x> = AC x BC, or x= V (AC x BC.)
132 QUADRATURE OP THE HYPERBOLA.
but sector ACP = A PCN — area APN, .-. fluxion of sector ACP = fluxion of A PCN— fluxion of area APN= — ^— — Vx =
xy~vx ; we must therefore find the values of ~ and y~. Since y =
b , „ 0\ • bxx xy bx°- x u » -a. ion
-V(^-a2)5, = ^-2-_z^,.^==5^7^2-_z^; by Art. 189.
a^x*^a?f ' ~2 ... 2a V (x^d ■ ; hence xv — yx or ^
bx*x bx V (#2 — a2) aJa;'
^=^V(f— }; hence 2=£ or fluxion of sector ACP =
2 2a 2
the fluent or sec-
2aV(^2— a2) 2a 2v(x2— a2)'
tor ACP = u- X hyp. log. (x -f V (#2 — a2) ) + Cor. ; when x = a,
ACP = 0, ... sector ACP = $ x hyp. log. * + v(*'— g').
2 ° a
(191.) The triangle CPN=^=^^-2^^)J .-.area APN
/& /*a
(= A CPN- sector ACP)= tov(^-g)_g» x hyp. log.
j-L-v (#2 a2) *
— ' i. i. Suppose AQ,a to be an equilateral hyperbola, in
which a = 6 = l, then the area AQ,N=£rV(:r2 — 1) — £ hyp. log. (x-{-V(z2— 1) ). A portion of this hyperbola, whose abscissa is equal to its semi-axis major (in which case x = 2) will be numerical- ly expressed by the quantity V3— \ hyp. log. (2-f V3) = 1.7320 — .6584=1.0736 ; thus in Figure page 82, if the abscissa AN be taken equal to AC, then the area (APN) corresponding to this abscissa : square ACBa : : 1.0736 : 1, and area APN : quadrant ACB : : 1.0736 : .7854 : : 1.3669 : 1.
QUADRATURE OF THE PARABOLA, &C.
133
XXI.
On the Quadrature of the Parabola, according to the method of
the Ancients.
(192.) Let BQAPC be any portion of a Parabola cut off by the straight line BC ; bisect BC in the point D, and draw DA parallel to the axis ; then AD will be the diameter to the point A, and (by converse of Art. 23.) BC will be an ordinate to that diameter. Moreover, since a tangent to the point A is parallel to BC, A will be the highest point or vertex of the figure B&APC ; if therefore BA, AC, be joined, then this figure and the triangle ABC will have the same base and vertex.
(193.) Bisect BD in E, and draw EQ, parallel to DA ; through Q, draw GtNP parallel to BC, and from P draw PF parallel to AD ; then QNP will be an ordinate to the diameter AD in the point N, and Q,E, PF will be diameters to the points Q,, P respectively ; and since Q,EDN is a parallelogram, Q,N will be equal to ED, i. e. to £BD; hence* QN2 : BD2 : : 1 : 4 ; but by the property of the Parabola, AN : AD : : QN2 : BD2, .-. AN : AD : : 1 : 4, or AN = £AD ; hence ND or Q,E = f AD. Again, since EG is parallel to DA, and BE = ^BD, EG must be equal to £AD, .-. Q.G = { f AD, and EG: GQ,:: 2:1.
(194.) Join AE, Ad, GIB ; then since BD is bisected in E, the triangle ABE is equal to half the triangle ABD (by Euc. 1. 6. ;) and since GQ, is equal to £GE, the triangles A&G, BQ,G are res- pectively half of the triangles AGE, BGE ; hence the triangle AQ,B is half of the triangle ABE, and consequently ^th of the triangle ABD. In the same manner (if AP, PC, AF, be joined,) it may
134 QUADRATURE OF THE PARABOLA.
be proved that the triangle APC is {th the triangle ADC ; hence the two triangles AQB, APC, taken together, are equal to one fourth of the triangle ABC.
(195.) Now suppose BE, ED were bisected, and from the points of bisection lines were drawn parallel to DA (which will evidently bisect BG, G/.,») then the sum of the triangles formed within the parabolic spaces*' BQ, QA (by drawing lines from the points where those parallel lines cut the curve to the extremities of the chords BQ, QA) will be equal to ]th of the triangle AQBt ; and the sum of the triangles formed in a similar manner within the parabolic spaces AP, PC, will be equal to {th of the triangle APC ; .-. the sum of the tri- angles formed within the four parabolic spaces BQ,, QA, AP, PC is equal to ith of A AQB+Z^VPC, i. e. to Tyth of the triangle ABC. By bisecting the halves of BE, ED, &c. and drawing lines as be- fore, parallel to DA, and joining the points of their intersection with the curve to the extremities of the chords, a series of eight triangles would be formed in the remaining parabolic spaces, the sum of Which would be equal to {th of the sum of the triangles formed within the parabolic spaces BQ, QA, AP, PC, i. e. to T*Tth of the triangle ABC. We might thus go on bisecting the successive parts of the base BC, and forming triangles in a similar manner, till the whole parabolic figure BAC was exhausted, in which case it is evident that the area of that figure would be equal to the sum of the areas of all the triangles thus formed within it.
(196.) Let the triangle ABC = a, then to find the sum of the areas of all these triangles, we have merely to sum the series
* By parabolic spaces, we mean such portions of the Parabola as are contained between the arcs BQ, QA. AP, PC, and the straight lines BQ, QA, AP, PC respectively.
t For the same reason that the sum of the triangles AQB, APC is equal to ith the triangle ABC, this conclusion being evidently true for the triangles thus inscribed in any portion of a Parabola.
BY THE METHOD OF THE ANCIENTS. 135
a_|_a_j-iL_|-iL -f&c. continued ad infinitum, which is a geometric
series, whose first term is a, and common ratio J, Now the sum of
this series* = 1 7— =li =-tt 5 •'• tne area °f tne parabola
VI— r /l—\ a
BAC is equal to f x area of the A BAC. If a tangent was drawn to the point A, and from B, C, lines were drawn parallel to DA, then the triangle ABC would be the half of the parallelogram thus formed ; the parabolic area BAC is therefore fds of the circumscribing paral- lelogram ; which accords with what has already been proved respecting the quadrature of the Parabola in Section XX ; for it is evident the foregoing demonstration is true for the axis, since AD is any diameter.
(197.) From the given ratio which subsists between the parabolic area and its inscribed triangle, we may prove, that such portions of a Parabola as are cut oif by ordinates to equal diameters, are equal to one another. Let oAQ (Fig. in p. 136) be any Parabola, and draw the diameters P W, pw to the points P, p ; take PW=p«?, and through W, w, draw the ordinates OWQ, owq ; draw the axis AD ; take AD equal to PW or pw, and through D draw the ordinate BC ; and in the parabolic spaces BAC, OPQ, inscribe the triangles BAC, OPQ. Draw the tangent to the point P, and produce the axis to meet it in the point T ; let S be the focus, and join SP ; from.S let fall SY perpendicular upon the tangent, and draw QF perpendicular upon PW produced. Now 4SAxAD = CD2, and 4SPxPW=WQ,2; therefore WQ2 : CD2 :: 4SPxPW : 4SAxAD : : (since PW= AD) SP : SA. Again, since the ordinate WQ. is parallel to the tan- gent TP, and the diameter PW is parallel to the axis AD, the tri- angles WQF, STY are similar, .-. WQ2 : QF2 j : ST2 (or SP2) : SY2 : : (Euc. Def. 11. 5.) SP : SA ; hence WQ2 : CD2 : : WQ2 : QF2, .-. CD=QF. But the A PWQ=iPWxQF and the A ADC=JADxCD; since therefore PW, QF, are respectively equal to AD, CD, the A PWQ must be equal to the A ADC.
* See Day's Algebra, Art. 442.
136 QUADRATURE OF THE PARABOLA, &C.
T
Now these A s are the halves of the triangles OPQ and ABC ; hence the A OPQ, is equal to the triangle ABC, and consequently the Parabolic area OPQ, to the parabolic area BAC* In the same manner it might be proved that the parabolic area opq is equal to the area BAC ; .-. the area opq is equal to the area OPQ.
These observations upon the quadrature of the Parabola accord- ing to the method of the Ancients, contain the substance of the last seven propositions (viz. from 18 to 24 inclusive) of Archimedes De Quadratures Parabola, and of the fourth Proposition of his book De Conoidibus et Sphceroidibus.
* For by Art. 195, the parabolic areas OPQ, BAC are f ds of the triangles OPQ, BAC respectively.
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