Albert Girard () (11 October 1595 in

Saint-Mihiel Saint-Mihiel () is a commune in the Meuse department in Grand Est in north-eastern France. Geography Saint-Mihiel lies on the banks of the river Meuse. History A Benedictine abbey was established here in 708 or 709 by Count Wulfoalde and his wif ...

, France − 8 December 1632 in

Leiden Leiden (; in English and archaic Dutch also Leyden) is a city and municipality in the province of South Holland, Netherlands. The municipality of Leiden has a population of 119,713, but the city forms one densely connected agglomeration wi ...

, The Netherlands) was a French-born mathematician. He studied at the

University of Leiden Leiden University (abbreviated as ''LEI''; nl, Universiteit Leiden) is a public research university in Leiden, Netherlands. The university was founded as a Protestant university in 1575 by William, Prince of Orange, as a reward to the city of Le ...

. He "had early thoughts on the

fundamental theorem of algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...

" and gave the inductive definition for the

Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...

s. He was the first to use the abbreviations 'sin', 'cos' and 'tan' for the

trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...

in a treatise. Girard was the first to state, in 1625, that each prime of the form 1 mod 4 is the sum of two squares. (See

Fermat's theorem on sums of two squares In additive number theory, Fermat's theorem on sums of two squares states that an odd prime ''p'' can be expressed as: :p = x^2 + y^2, with ''x'' and ''y'' integers, if and only if :p \equiv 1 \pmod. The prime numbers for which this is true ar ...

.) It was said that he was quiet-natured and, unlike most mathematicians, did not keep a journal for his personal life. In the opinion of

Charles Hutton Charles Hutton FRS FRSE LLD (14 August 1737 – 27 January 1823) was a British mathematician and surveyor. He was professor of mathematics at the Royal Military Academy, Woolwich from 1773 to 1807. He is remembered for his calculation of the ...

, Girard was

...the first person who understood the general doctrine of the formation of the coefficients of the powers from the sum of the roots and their products. He was the first who discovered the rules for summing the powers of the roots of any equation.

This had previously been given by

François Viète François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...

for ''positive'' roots, and is today called Viète's formulas, but Viète did not give these for general roots. In his paper, Funkhouser locates the work of Girard in the history of the study of equations using

symmetric function In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f ...

s. In his work on the

theory of equations In algebra, the theory of equations is the study of algebraic equations (also called "polynomial equations"), which are equations defined by a polynomial. The main problem of the theory of equations was to know when an algebraic equation has an ...

, Lagrange cited Girard. Still later, in the nineteenth century, this work eventuated in the creation of

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...

, Galois and others. Girard also showed how the area of a spherical triangle depends on its interior angles. The result is called Girard's theorem. He also was a lutenist and mentioned having written a treatise on music, though this was never published.

References

{{DEFAULTSORT:Girard, Albert 1595 births 1632 deaths People from Saint-Mihiel Dutch people of French descent 17th-century French mathematicians 17th-century Dutch mathematicians Leiden University alumni French lutenists Dutch lutenists Emigrants from France to the Dutch Republic