THE STORY OF LANDEN, THE HYPERBOLA AND THE ELLIPSE

Transcription

1 THE STORY OF LANDEN, THE HYPERBOLA AND THE ELLIPSE VICTOR H. MOLL, JUDITH L. NOWALSKY, AND LEONARDO SOLANILLA Absrac. We esablish a relaion among he arc lenghs of a hyperbola, a circle and an ellipse.. Inroducion The problem of recificaion of conics was a cenral quesion of analysis in he 8 h cenury. The goal of his noe is o describe Landen s work on recifying he arc of a hyperbola in erms of an ellipse and a circle. Naurally, Landen s language is ha of his ime, in erms of fluens and fluxions, and his argumens are no rigorous in he modern sense. The main resul presened here is a special relaion beween he lengh of an ellipse, he lengh of a hyperbolic segmen, and he lengh of circle. The proof is based on a generaliaion of Euler s formula for he lemniscaic curve as described in [4]. (.). The hyperbola The arc lengh of he equilaeral hyperbola saring a = is given by (.) h() =, L h (x) = x as a funcion of he erminal poin = x. The angen line o he hyperbola a = x is (.3) T h () = x x + ( x), x whose inersecion wih he -axis is = /x (, ). The line x (.4) N h () = x is he perpendicular o L h passing hrough he origin. The lines T h and L h inersec a he poin ( ) x P h = x, x (.5) x. Dae: April 7,. 99 Mahemaics Subjec Classificaion. Primary 33. Key words and phrases. Landen ransformaions.

2 VICTOR H. MOLL, JUDITH L. NOWALSKY, AND LEONARDO SOLANILLA The disance from (x, h(x)) o he common poin P h is (.6) x g h (x) = x x. I was observed by Maclaurin, D Alamber, and Landen ha x x (.7) f h (x) := g h (x) L h (x) = x x is easier o analye han he arc lengh L h (x). Proposiion.. Le (.8) Then (.9) where (.) F h () = F h () = f h (x), =. x. Proof. Make he change of variable (.) in (.7). Then f h (x) becomes F h () = + (.) ds s 3/ s. in erms of he new variable = /(x ). Since d s s = ds s s 3/ s, inegraing from o reduces (.) o (.8). (3.) 3. The ellipse The equaion of he ellipse can be wrien as e() = ( ),. In his case he angen line a = r is T e () = r ( r ) ( r), r and he line r N e () = r is he perpendicular o T e hrough he origin. These wo lines inersec a he poin ( ) r r( P e = + r, r ) (3.) + r, and he disance from (r, e(r)) o he common poin P e is r (3.3) g e (r) = r + r.

3 3 We express he funcion g e in erms of he new variable = r as ( ) (3.4) g e () = The connecion We now evaluae he funcion F h () in (.8) a wo poins y, (, ) relaed via he bilinear ransformaion = ( y)/( + y). We have F h () + F h (y) = s s ds + s s ds. y The change of variable σ = ( s)/( + s) in he second inegral yields F h () + F h (y) = s s ds + y σ ( + σ) 3/ σ dσ. Now recall he funcion g e () in (3.4) and is differenial dg e = d ( + ) 3/. Therefore y F h () + F h (y) = g e () g e () +. Now observe ha g e () = and inroduce he absolue consan L := (4.) so ha (4.) F h () + F h (y) = g e () + L. Thus we have esablished he following inegral relaion. Theorem 4.. Le y (, ) and = ( y)/( + y). Then s s ds + s ( ) (4.3) s ds = + L + y wih he absolue consan L in (4.). Proof. Le so ha (4.4) Inegraing (4.4) gives (4.5) G h () = F h () + F h (y) = s s ds + dg h () d ( )/(+) = ( + ) 3/. G h () = ( ) + L + s s ds,

4 4 VICTOR H. MOLL, JUDITH L. NOWALSKY, AND LEONARDO SOLANILLA By leing =, he consan L is easily evaluaed as L := (4.6) using Wallis formula. = = π/ π π Γ (/4) sin θ dθ We now follow Landen o esablish he value of L in erms of ellipic arcs. The equaion (4.) simplifies if we evaluae i a he fixed poin = of he ransformaion = ( y)/( + y). In erms of he x variable, he fixed poin is (4.7) Indeed x = + = cos(π/8). (4.8) F h ( ) = ( + L). Now inroduce he complemenary inegral (4.9) and observe ha M := L + M = L e () = ( ), where L e () is a quarer of he lengh of he ellipse. Theorem 4.. The inegrals L and M saisfy L + M = L e () L M = π 4. + () Therefore L = ( L e () ) L e () π M = ( L e () + ) L e () π. Proof. Observe ha for q Q we have (4.) d( q ) and inegraing from o we obain (4.) q = q + q = qq (q + ) q+ q+.

5 5 For example, wih q = 3/ i yields L = / = 5 3 Ieraion of his recurrence yields, afer m seps, (4.) L = m+ j= 5/ m+/ (j ) ( )j+. Similarly, saring wih q = / we ge afer m seps (4.3) and M = m j= (j ) ( )j m /. Ieraion of (4.) wih iniial values q = and q = yields for he expressions and and A = B = A := B := m j= m j= j ( )j = π =, m (j + ) ( )j m+ m L A = (j ) ( )j+ j ( )j+ m+/ ( ) / (4m + ). m ( ) / M B = j= m j= (j ) ( )j j ( )j m / ( ) / m+ ( ) / m +. As m, he quoien of he inegrals converges o and we obain (4.4) L M = π lim 4m + m m + = π. We now wrie π/ = L c () as a quarer of he lengh of he circle in analogy o L e (). Theorem 4.3. The lengh of he hyperbolic segmen is given by ( ) + (4.5) L h = (Le () 4 4L c ()) L e ().

6 6 VICTOR H. MOLL, JUDITH L. NOWALSKY, AND LEONARDO SOLANILLA References [] CAJORI, F.: A hisory of he concepions of limis and fluxions in Grea Briain from Newon o Woodhouse. Open Cour, Chicago, 99. [] EULER, L.: De miris proprieaibus curvae elasicae sub aequaione y = xx/ x 4 dx conenae, Commen 65 Enesroemianus Index. Aca academia scieniarum Perop. 78: II (786), Reprined in Opera Omnia, ser.,, 9-8. [3] LANDEN, J.: A disquisiion concerning cerain fluens, which are assignable by he arcs of he conic secions; wherein are invesigaed some new and useful heorems for compuing such fluens. Philos. Trans. Roy. Soc. London 6 (77), [4] MOLL, V. - NEILL, P. - NOWALSKY, J. - SOLANILLA, L.: A propery of Euler s elasic curve. Elemene der Mahemaik 55,, [5] NOWALSKY, J.L.: Properies of he Generalied Euler s Elasic Curve y = x n / x n dx. Maser Thesis, Tulane Universiy, 998. Deparmen of Mahemaics, Tulane Universiy, New Orleans, Louisiana 748 address: vhm@mah.ulane.edu Deparmen of Mahemaics, Tulane Universiy, New Orleans, Louisiana 748 address: judihn@mah.ulane.edu Deparmen of Mahemaics, Tulane Universiy, New Orleans, LA 78 address: solanill@mah.ulane.edu