# Who was V. Viskovatov?

I'd be interested to learn some biographical detail about Vasilii Viskovatov, whose name is associated with a method for converting (a ratio of) power series to a "corresponding" continued fraction, c.f. Handbook of Continued Fractions for Special Functions, p. 20 (Springer, 2008).

A few Internet hits (prominently Google books links, not wholly accessible) provide the following literature reference to his work, albeit with apparent mis-transliterated initial:

B. Viskovatov: 'De la méthode générale pour réduire toutes sortes des quantités en fractions continues', Memoires de L'Academie Impériale des Sciences de St. Petersburg, 1(1803-1806), pp. 226-247

[Trans.: 'A general method for reducing all types of quantities to continued fractions'.]

The algorithm itself can be described simply. Given the ratio of two power series $p(z)/q(z)$ with $q(0) = 1$, remove as a constant term $b_0 = p(0)$, and after factoring out the leading term $z$ in the numerator, invert the remaining factor from $p(z) - p(0)q(z)$:

$$\frac{p(z)}{q(z)} = b_0 + \frac{a_ 1 z \cdot r(z)}{q(z)} = b_0 + \frac{a_1 z}{q(z)/r(z)}$$

where for simplicity I've assumed a single power of $z$ in the numerator. Continue to apply the process to $q(z)/r(z)$ and a corresponding-type continued fraction emerges.

• Either that $\,1803-1806\;$ is a reference to the dates of the works, or else this guy was damn precocious... – DonAntonio May 13 '13 at 16:43
• I'm sure you are right; edited for clarity (I'd transcribed one citation from Google books). – hardmath May 13 '13 at 16:48
• There is an article about him in Russian Wikipedia. Here is an automatic English translation of the article: microsofttranslator.com/bv.aspx?from=ru&to=en&a=http://… – Yury May 13 '13 at 16:55
• There's also a nice discussion of Viscovatov's algorithm in Lorentzen/Waadeland's Continued Fractions with Applications and Cuyt/Wuytack's Nonlinear Methods in Numerical Analysis. Is this related to your quest to determine a CF expansion of the polylogarithm? – J. M. is a poor mathematician May 13 '13 at 17:30
• @J.M.: Right, related. I'm pretty sure I remember seeing this fairly simple transformation in Khintchin's little book, years ago. Seems like a guy who deserves an English Wikipedia article, so maybe I'll stub in a draft here as community wiki. I'd noticed some "c" rather than "k" in transliterations of his last name. – hardmath May 13 '13 at 17:52

## 2 Answers

It might be Vasilii Viskovatov:

http://encyclopedia2.thefreedictionary.com/Viskovatov,+Vasilii+Ivanovich

The time-frame and location is correct.

Don't forget that 'Vasilii' written in Cyrillic starts with a glyph that looks like a roman B:

• Nice find! This note about his son is really starting to fill in the details. I'd seen stuff about the son, whose Russian military illustrations have recently been "rediscovered", but of course I did not know what relation they might be. – hardmath May 13 '13 at 16:56
• FWIW, I came across that by searching permutations of common trasliterations of Slavic names: Viskovatoff, Biskovatov, Wiskovatov, and the combinations therein. – Emily May 13 '13 at 16:59
• I'm going to "correct" the initial in my Question's title, given your transliteration. – hardmath May 13 '13 at 17:43

Putting together some of the findings and references as community wiki.

From Russian Wikipedia article (tweaked Google trans.):

Vasily Viskovatov (December 26 1779 ( 6 January 1780 ), St. Petersburg - 8(20) October 1812, St. Petersburg ) - Russian mathematician. Well-known expert in the field of mathematical analysis and the calculus of variations, one of the most active followers of Simon Guriev in the promotion of new and innovative research ideas.

Released from the Artillery and Engineers Royal Cadet Corps in 1796 as bayonet-cadet in the corps officers.

In 1803 declared a major mathematician, he was elected an academician of the St. Petersburg Imperial Academy of Sciences.

In 1810 - Professor of Pure and Applied Mathematics at the Institute of Railway Engineers 1.

First used the Russian term "derivative". 2

The Free Dictionary entry mostly duplicates the above but cites The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved, so I won't give a quote. However it names a couple of the "major works" by Viskovatov, albeit in a transliterated form making automatic title translation fail.

From Mark Conrad's Russian Military History:

Major General Aleksandr Vasil’evich Viskovatov, Russian historian

This was V. Viskovatov's only son, "born in St. Petersburg on 22 April 1804." I presume this is a Julian calendar date, as Russia/Soviet Union did not adopt the Gregorian calendar until 1918 (which is the reason for two sets of dates in the previous Wikipedia article). It is mentioned that the family descended from a "branch of the Meshcherskii princes."

The son inherited his father's enthusiasm for mathematics, but sadly lost him at an early age (nine years old?). The date of the father's death is given here as 1813, which differs from that shown in the Wikipedia article, a detail I'll attempt to clarify.

The development of mathematics in Russia in XVIII-XIX centuries

After graduating from military school, he was left there for the teaching of mathematics. At the age of twenty (1799) he was elected corresponding member, and in 1804 an associate of the Academy of Sciences. Later, he was promoted to the extra-ordinary academician. With the establishment of the Institute of Railway Engineerss, he was appointed professor in 1812 but died 34 years old. Viskovatov published several memoirs in the publications of the Academy, and a guide to elementary algebra. He has translated and published "Principles of Mechanics" by Bossu and published a new edition of the Euler's "Algebra". Despite his premature death, Viskovatov already had many disciples.

I.V. Ignatushina. Formation of differential geometry as an academic subject in Russia in XVIII - the first half of the XIX century: disciples and followers of Leonhard Euler

V.I. Viskovatov, Guriev's student, taught mathematics and mechanics at the Artillery Cadet Corps, and then at the Institute of Railway Engineers and Mining Institute. In 1808 Viskovatov presented a work "Summary of the famous Lagrange method to interpret the differential calculus and the application thereof to the geometry of curves" (1810). The second part of this work, entitled "On the application of differential calculus to the geometry of curves", is one of the first presentation of the issues of differential geometry of plane curves in Russian. It discusses the concept of a tangent to the curve, convexity, curve inflection point, multiple points of cusps of the first and second kind, notions of evolute and involute of the curve and the concept of contiguous curves. He derived the equations of the tangent and normal line drawn at a given point of the curve, and the radius of curvature in the cases when the curve is set in a rectangular, oblique and polar coordinate systems.