A Method of Defining the Ellipse, Hyperbola and Parabola as Conic SectionsAuthor(s): W. W. LandisSource: The American Mathematical Monthly, Vol. 5, No. 3 (Mar., 1898), pp. 72-73Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2968947 .

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72

The most probable value of z is that one which will make P a maximum. P is a maximum when (2 + 22 + e..e2) is a minimum; or, by equating to zero, the first derivative, when +2 +....... E,O. That is when z-wl +z-

W2 +... .Z-w-t O. Solving for z we find

(17) z=(w w2 + .wn)/n.

Hence it appears that in the long run the best way to solve the markings of judges of contests is simply to take the arithmetical means of the markings when the merits of the contestants are compared with ideal standards.

The methods for successive approximations have already been discussed in this journal.

The University of Chicago.

A METHOD OF DEFINING THE ELLIPSE, HYPERBOLA AND PARABOLA AS CONIC SECTIONS.

By W. W. LANDIS, A. M., Professor of Mathematics in Dickinson College, Carlisle, Pa.

ABC is a right circular cone, the angle at the vertex being 2a. DFG is a plane section making an angle 6a with the axis of the cone. Take D as the ori- gin, DH as the axis of x, and a perpendicular through D as the axis of y. We seek to find a relation between x and y, the parameters beingg(-AD) and a, and the variable parameter 6. In the circle BGC, y2= (a?rb)c=arc4bc.... .. (1), where a=BL, b=LH, and c=HC. In the isosceles triangle DLC, x2-=d2:Fbc

...... (2), where d=DC=DL. Adding (1) and (2), X2 +y2=ac+d2 ..... (3).

Now c-.=xsin6+dsina,

d=xcosf6seca=xcos88/cosa, and .4 a-=2gsi na.

Making this substitution we get

X2 +y 2 8c2 = 2g Fxsin a +x cos0sina1

COS 2 a * LCosa

or x2 [1cO lL+.2-2gxsina[sint0+cos6tana]-O ...... (4)

which we may write

x 2 [sin O.sina] +y 22gxsinta[sin9+cosStana]=0** (5).

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73

Now if 6=a, the plane thern being parallel to one and only one element, (5) reduces to y2-4gxsin12a=0, a parabola of latus rectum=4gsin2a.

If 19>a the section cuts all elements and the coefficients of x2 and y2 are both positive, and we have an ellipse whose center, axes, and eccentricity are readily found; and in particular if f=900, the section is parallel to the base, the coefficients of x2 and y2 are unity and we have a circle, whose center is (gsina, 0). If <a, the section cuts both iiappes, the coefficients of xl and y2 are of unlike sign and we have a hyperbola.

If g==O, (5) becomes y=?E COS 2-* .... (6). Now if 8=at, (6) be-

comes y=O, a straight line, the limit of the parabola. If 9<a, (6) represents two real straight lines, the limiting case of the hyperbola. And if #>a, (6) rep- resents two imaginiary lines itntersectinjg in the real point (0, 0), which is the lim- iting form of the ellipse.

The equations (5) and (6) showv the dependence of the nature of the conliC sections upon the angle which the cutting plane nmakes with the axis, and the de- pendence of their shape upon the angle of the cone and the distance from the ver- tex to the first elemenit cut.

REMARK 1. If the section be a parabola, the foot of the perpendicular from the middle point of the line throngh D parallel to BC, upon DH, is the foculs.

REMARK 2. The eccentricity of any conic section is e-=[cos/cosa].

NEW AND OLD PROOFS OF THE PYTHAGOREAN THEOREM.

By BENJ. F. YANNEY, A. M., Mount Union College, Alliance, Ohio, and JAMES A. CALDERHEAD, B. Sc., Curry University, Pittsburg, Pennsylvania.

[Continued from November Number.]

LXX. Fig. 31. AEMH=ACB of AEHO. MOL=A CB=ADK+ DKBC=BHI DKBC. LOI=BEK. ..ABBLMA BCDF4- AEHC.

LXXI. Fig. 31. AAlfPQ=c-= AMOC-c_ AEHC. BLPQ-c_ BLOC =cl .3CI)F. . . ABLM BCDF+AEHC.

LXXII. Fig. 31. MTE-ZBSF. .*. BS-MT. .A*B. A MI,B=2AAMTS. But AMTS- EAC+FBC; since MTETZBSF, and AEM -A CB.

2A MTS 22EA C 2FBC - AEHC + BCDF. A MLB?-o- BCDF A EHC.

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