• M. I. Zelikin
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 86)


The Riccati equations were named in honor of the famous Italian mathematician1 Count Jacopo Francesco Riccati (1676–1754), who published the paper “Animadversationes in aequationes differentiales secundi gradus” in the journal Acta Eruditorum in 1724 (see [90]). This paper was devoted to methods for separating variables and for reducing the order of differential equations. It should be noted that the content of the paper had become known to the mathematical community somewhat earlier [12]. When studying the problem of reconstructing a plane curve using its curvature properties, Riccati had obtained the equation
$$b\frac{{dx}}{{dt}} = a{{t}^{a}} + {{x}^{2}}.$$
In the same volume of Acta Eruditorum where Riccati’s paper was published, a paper by 22-year-old Daniel Bernoulli followed it, in which he wrote that both the Nicholas Bernoullis (elder and younger), Johann Bernoulli, and Daniel Bernoulli himself had studied this equation and they all, independently of each other, had found conditions for the parameter α under which this equation admits a separation of variables and is therefore integrable in quadratures. Using a popular manner of that time, D. Bernoulli wrote the answer in the form of an anagram. Riccati answered this note in the next number of the journal. Without regard to the integrability conditions, which were familiar to him,2 Riccati in fact refused to dispute the priority, saying that he had no intention to challenge the Bernoulli family, which he deeply respected.


Riccati Equation Grassmann Manifold Curvature Property Homogeneity Domain Double Ratio 
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  1. 1.
    It suffices to note here that a number of universities asked Riccati to be a professor, Peter the Great proposed the presidency of the St. Petersburg Academy of Sciences to him, and the Court of Vienna invited him to be the Imperial Chancellor. Riccati refused all these offers, preferring to live with his family and study mathematics.Google Scholar
  2. 2.
    These conditions had been discussed in an exchange of letters with the younger Nicholas Bernoulli during five years before the paper was published. The paper itself was sent to Johann Bernoulli, editor of Acta Eruditorum,through the mediation of Nicholas Bernoulli, and its content therefore became known to Daniel Bernoulli [40].Google Scholar
  3. 3.
    Liouville, J. (1841): Remarke nouvelles sur l’équation de Riccati. J. Math. Pures et Appl. 6, 1–13.Google Scholar

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© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • M. I. Zelikin
    • 1
  1. 1.Department of MathematicsMGU, Vorob’evy GoryMoscowRussia

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