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 NCB Deposit  # 106

Gustavo Gordillo

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  Double Devil's Curve, Devil's Curve
Electric Motor Curve

with animation and historical figures

Electric Motor Curve
Electric Motor Curve

The NCB explores the historical Devil's Curve by selecting constants  a > b  followed simultaneously with  b > a . 
The two curves are then jointly animated using MATHEMATICA®.

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Double Devil's Curve

The Devil's Curve

The NCB is grateful to the Huntington Library for giving us an opportunity to compare the original publications from Gabriel Cramer (1704-1752) and Sylvestre Lacroix (1765-1843) with modern animations.  In investigating 18th century publications several observations are of interest to a contemporary reader.  First, Cramer wrote what surely was the definitive treatise on curves for his generation.  His Introduction à l'analyse des lignes courbes algébriques (1750) is an impressive collection of 250 figures or curves, each with a careful explanation of how the solutions and equations were calculated.  This must have required hours - possibly years - of meticulous work.   Moreover, his publication was surely appreciated by his contemporarties as we find multiple editions and translations also in the Huntington.   There was demand for his mathematics by his contemporaries!

Like Agnesi, who published her Instituzioni analitiche almost at the same time ( 1748 ), the exploration of curves, especially the algebraic foundations of curves, was considered a necessary background for the emerging calculus. 

Current readers of from a somewhat older generation will remember that most 20th century calculus texts were entitled Calculus and Analytic Geometry,  not simply Calculus.  Then as now, professors felt an emphasis on analytic geometry was good preparation for many areas of mathematics, especially applied math.  Thus, one can assume that Cramer, Agnesi, and Lacroix were directing their mathematical skills to neophites in algebra who wanted to understand the increasingly influential calculus.

Cramer's Rule

Readers today will recognize the name of Cramer by our continued use of  Cramer's Rule.  He is credited with finding a solution for square matrices of order n  by evaluating determinants.  Today the Rule is commonly taught in linear algebra and is no longer really considered a subtopic of analytic geometry. 

In his preface, Cramer speaks for himself on the importance of curves:
Cramer's Preface

Cramer's Example

Lacroix Example
Remarks on Notation:
Note both mathematicians used only the Cartesian form.  Neither had the polar or parametric forms at that time.  Moreover, true to the notation inherited from Descartes, Cramer used only xx and  yy  when we today use, of course, x^2 and y^2.  One-half century later Lacroix was using x^2 and a^2.  Both handled the fourth degree equation in the "ordinary method" for second degree equations.
Cramer's Illustration
Cramer figure
Lacroix's Illustration
Lacroix figure

An interesting question remains, "Who named this the Devil's Curve?"  Is it because the multiple branches and singularities make it so tedious to plot?  We were unable to find this name in the publications of Cramer or Lacroix.  Later authors must have considered the multiple roots at any abscissa and singularities to be diabolical.  Check this web site later as we plan to continue our search.

Implicit Differentiation

References and Comments
Anton, Howard and Rorres, Chris, Elementary Linear Algebra,  Wiley, 2010, pp. 112-113.
Brown, B. H., La Courve du diable, The American Mathematical Monthly, vol. 33 (5) May,1926,  pp. 273-274.
Cramer, Gabriel, Introduction à l'analyse des lignes courbes algébriques, Genève, Chez les Frères Cramer & Philibert, 1750.   
Fladt, Kuno, Analytische Geometrie spezieller ebener Kurven, Akademische Verlagsgesellschaft, 1962, p. 218.  Fladt uses "Die Teufelskcurve von Gabriel Cramer."  Teufelskcurve may be translated as either the Devil's Curve or the Devil's hairpen.
Gray, Alfred,  Modern Differential Geometry of Curves and Surfaces with MATHEMATICA®, CRC Press, 1998, p. 92.
Lacroix, Silvestre François, Traité du calcul différentiel et du calcul intégral, vol. 1, Libraire pour les Mathématique, quaai des Augustins, 1797.
Koestler, Arthur, The SLEEPWALKERS, Arkana: Penguin Books, 1989.
Rider, Paul R., The Devil's Curve and Abelian Integrals, The American Mathematical Monthly, vol. 34 (4) April, 1927, pp. 199-203.
Stewart, James,  Calculus: Early Transcendentals, 7th ed., Cengage: Brooks/Cole, 2012,  p. 215.
Venit, Stewart and Bishop, Wayne, Elementary Linear Algebra,  4th ed., ITP Publishing Co., 1996, pp. 183-184.  Proof: pp. 187-188.
ISBN:  053495190-2
Weisstein, Eric., Devil's Curve,  < >.
Weisstein, Eric, CRC Concise Encyclopedia of Mathematics, CRC Press, 1999, pp. 424 - 425.
Yates, Robert C., Curves and their Properties, NCTM, 1974, p. 203.  Yates writes, "This curve is found useful in presenting the theory of Riemann surfaces and Abelian integrals."  Yates' reference to Brown's publication is invaluable.
The NCB is grateful to the Huntington Library for permission to use the original illustrations first published by Cramer (1750) and later by Lacroix (1797) as well as many supporting publications.
For the Double Devil's Curve, let a=b+1 in order to produce the above animation.
Mathematics Devil and Motor curves may be created using Mathematica®.
and on the graphing calculator . . .
Graphing calculator

Gabriel Cramer  (1704 - 1752)
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