IN 



MATHEMATICS. PRACTICAL GEOMETRY. 



1 niK ELLIPSB. 



CHAPTER X. 

 ON THE CONSTRUCTION OP THE CONIC SECTIONS. 



THK preceding propositions depend entirely on the pro- 

 perties of straight lines and the circle, and therefore 

 admit of construction by rule and compasses. Now, 

 beside the circle there are several curves which are used 

 more or less, by draughtsmen ; these are for the most 

 part drawn, when required, by determining accurately 

 several points in them, which are then neatly joined by 

 the hand. Of these curves the ellipse is the one oftenest 

 needed, and is used, in fact, almost as much as the circle, 

 in consequence of the perspective representation of a 

 circle being an ellipse. Besides the ellipse, the hyperbola 

 ami parabola are often needed in the delineation of 

 shadows. These three curves are generally called conic 

 sections, for this reason if a right cone ia cut by a plane, 

 the curve which bounds the section is one of these three, 

 except in the special cases when the elliptic section de- 

 generates into a circle, and the hyperbolic into two 

 straight lines. The object of the few following pages is 

 to prove such properties of these curves as shall enable 

 us to give rules for their construction. The complete 

 investigation of their properties, which forms a distinct 

 and very interesting branch of mathematics, is not here 

 intended. 



DEFINITION 1. Let dD (Fig. 

 60) be a given fixed straight line, 

 called the directrix ; S a given 

 fixed point, called the focus ; P 

 a movable point. Let Pd be 

 perpendicular to the given straight 

 line , then if P move in such a 



Fig. 60. 



manner that S P bears a constant 



ratio to P d, it traces out one of the curves called conic 



sections. 



DEFINITION 2. Suppose SP : P d : : e : 1, then if e 

 1, the curve is an ellipse; if e 1, the curve is a 

 pnrubiila: if e 7 1, the curve is a hyperbola. (e=the 

 eccentricity of the denoted curve). ( 



THE ELLIPSE. 



Let Dd, Sand 

 P (Fig. 61) be 

 the same as 

 in the fcr.-go- 

 ing definitions ; 

 through S draw 

 D A'i perpen- 

 dicular to D<1. 

 Then if A and 

 a are so taken 

 that S A : A D 

 : : e : 1, and S a 

 : a D : : e : 1, 

 then A and a 

 are points in the 

 ellipse ; also the line A a is the transverse diameter o r 

 miijnr axis ; bisect A a in C, then C is the centre of th e 

 ellipse ; through C draw B C b perpendicular to A a> 

 then B b is called to conjugate to the transverse diameter, 

 or the minor axis ; take H, so that a H = A S, or so 

 that C 8 C H. then H and S are called the foci (sin- 

 gular focus). 



(1.) To show that in the eUipie C S = e, A S. 



We have already seen that AS - e. AD, also that 

 8o-e.aD .'. aS-SA- e (aD-DA), and aH - AS. 

 .'. 8H-e. Ao, or 80-e. AC. Q. E. D. 



N.B t is called the eccentricity of the ellipse. 



(2.) To show that 8 B - A C. 



Manifestly 8 B - . D C - e A D + e A C = AS + SC 

 -AC. Q. E. D. 

 COR Hence, B <? - 8 B - S C 1 - A C - 8 C* - A C 1 



(3.) To show that 8 P- A C-t. C N. 

 Manifestly, 



SP=e D N=e (Dc-C N)=e (D A + A C)-e C N 

 -eDA + e AC-eCN = AS + SC-eCN 

 -AC-eCN. 



P N J C N 1 

 (4.) To show that c ' }> , + c A j - L 



For S P=SN 2 + P N (Eucl. L, 47). But S P=A C 

 -eCNandSN = SC-CN.-.(AC-eCN)=(SC-C\) 1 

 + PN 2 and S C=e AC. 



.'. AC'-2eAC. CN + e'CN> = e*AC'-2eACCN 

 + CN 2 +PN'. 



PN' 



-eV AC 2 (l-e*; 



NowAC 2 (l-e)=CB'. 



CN PN 



1. 



1 = 



PN CN 



Q. E. D. 



have 

 in a 



COR. With centre C and radius C A, describe a circle 

 Apa ; produce N P to meet the circumference in p ; join 

 Cp, which is plainly equal to A C. 



Now .pN 2 + C X* = C/> 2 = C A'. 



P N J C N 2 

 _ L. 4-_ =1 

 ' ' C A* T C A* 



- PN_/>N 



C B 2 C'A? 



Or, P N : pN : : C B : C A. 



This result, and the previous one, we shall 

 occasion to use in the article on Mensuration 

 future chapter. 



(5.) To show that H P = A C + e C N. 



For H P a = P N 2 + N H 3 =P N 2 + (C N + C S)'= 

 -KCN + eCA) 2 - CBJ 



Now from art. 4. PN'=C B 1 - g^, C N 2 = (1 - e)' 



CA 2 -(l-e 2 )CN I . 



HP 2 = CA-e !! CA s -CN l -|-e 8 CN 2 - r -CN 1 + 2 

 eCA, CN + e 2 CA*. 



, C N + e 2 C A 5 =(CA + e CN) 1 . 



CN. 



COR, Hence, SP + HP = CA-eCN + CA + 

 e C N = 2 C A = Aa. 



Or, S P + P H = the major axis of the Ellipse. 

 On this property of the F.llipse the first and third 

 practical methods of construction depeud. 



(6. ) If EFGH (Fig. 62) be a rectangular parallelogram, 



and D C and A B are the lines joining the bisections of the 



Pig g2. opposite sides, di- 



F p c tru/c G li into n, 



parts, and 



l 



O B into N, 

 ' equal parts. Let 

 Gp, <xm<ain p of 

 the parts info 

 which G P t* di- 

 vided, and let Op 3 

 contain p of the 

 equal parts into 

 which OB is di- 



nded, join Dp, and C/> 2 and produce it to meet D/>, in, P; 

 hen P u o point in the Ellipse, whose major and minor 



rided, 



the, 



axes are A B and O D. 



For, drawn PN parallel to DO and .'. perpendicular 

 to AB. 



