In mathematics, Hudde's rules are two

properties of polynomial roots Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property. Property may also refer to: Mathematics * Property (mathematics) Philosophy and science * Property (philosophy), in philosophy an ...

described by Johann Hudde. 1. If ''r'' is a

double root In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root. The notion of multipl ...

of the polynomial equation ::

a_0x^n + a_1x^ + \cdots + a_x + a_n = 0

:and if

b_0, b_1, \dots, b_, b_n

are numbers in

arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...

, then ''r'' is also a

root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...

of ::

a_0b_0x^n + a_1b_1x^ + \cdots + a_b_x + a_nb_n = 0.

:This definition is a form of the modern

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of ...

that if ''r'' is a double root of ''ƒ''(''x'') = 0, then ''r'' is a root of ''ƒ'' '(''x'') = 0. 2. If for ''x'' = ''a'' the polynomial ::

a_0x^n + a_1x^ + \cdots + a_x + a_n

:takes on a relative

maximum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...

minimum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...

value, then ''a'' is a root of the equation ::

na_0x^n + (n-1)a_1x^ + \cdots + 2a_x^2 + a_x = 0.

:This definition is a modification of Fermat's theorem in the form that if ''ƒ''(''a'') is a relative maximum or minimum value of a polynomial ''ƒ''(''x''), then ''ƒ'' '(''a'') = 0, where ''ƒ'' ' is the

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

of ''ƒ''. Hudde was working with Frans van Schooten on a

Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power ...

edition of

La Géométrie ''La Géométrie'' was published in 1637 as an appendix to ''Discours de la méthode'' (''Discourse on the Method''), written by René Descartes. In the ''Discourse'', he presents his method for obtaining clarity on any subject. ''La Géométrie ...

René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathe ...

. In the 1659 edition of the translation, Hudde contributed two letters: "Epistola prima de Redvctione Ǣqvationvm" (pages 406 to 506), and "Epistola secvnda de Maximus et Minimus" (pages 507 to 16). These letters may be read by the Internet Archive link below.

References

* Carl B. Boyer (1991) ''A History of Mathematics'', 2nd edition, page 373,

John Wiley & Sons John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, i ...

. * Robert Raymond Buss (1979) ''Newton's use of Hudde's Rule in his Development of the Calculus'', Ph.D. Thesis

Saint Louis University Saint Louis University (SLU) is a private Jesuit research university with campuses in St. Louis, Missouri, United States, and Madrid, Spain. Founded in 1818 by Louis William Valentine DuBourg, it is the oldest university west of the Mississip ...

ProQuest ProQuest LLC is an Ann Arbor, Michigan-based global information-content and technology company, founded in 1938 as University Microfilms by Eugene B. Power. ProQuest is known for its applications and information services for libraries, provid ...

#302919262 *

(1659
La Géométria, 2nd edition
via

Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music ...

. *

Kirsti Pedersen Kirsti Andersen (born December 9, 1941, Copenhagen), published under the name Kirsti Pedersen, is a Danish historian of mathematics. She is an Associate Professor of the History of Science at Aarhus University, where she had her Candidate exa ...

(1980) §5 "Descartes’s method of determining the normal, and Hudde’s rule", chapter 2: "Techniques of the calculus, 1630-1660", pages 16—19 in ''From the Calculus to Set Theory'' edited by

Ivor Grattan-Guinness Ivor Owen Grattan-Guinness (23 June 1941 – 12 December 2014) was a historian of mathematics and logic. Life Grattan-Guinness was born in Bakewell, England; his father was a mathematics teacher and educational administrator. He gained his ...

Duckworth Overlook Duckworth Books, originally Gerald Duckworth and Company, founded in 1898 by Gerald Duckworth, is a British publisher.Rules Theorems in algebra Polynomials Calculus