Search Results for root*

Biographies

1. Al-Khwarizmi biography
   
   • His equations are linear or quadratic and are composed of units, roots and squares.
     
   • For example, to al-Khwarizmi a unit was a number, a root was x, and a square was x2.
     
   • Squares equal to roots.
     
   • Roots equal to numbers.
     
   • Squares and roots equal to numbers; e.g.
     
   • Squares and numbers equal to roots; e.g.
     
   • Roots and numbers equal to squares; e.g.
     
   • For example, two applications of "al-muqabala" reduces 50 + 3 x + x2 = 29 + 10 x to 21 + x2 = 7 x (one application to deal with the numbers and a second to deal with the roots).
     
   • a square and 10 roots are equal to 39 units.
     
   • The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned.
     
   • Now the roots in the problem before us are 10.
     
   • Having taken then the square root of this which is 8, subtract from it half the roots, 5 leaving 3.
     
   • The number three therefore represents one root of this square, which itself, of course is 9.
     
   • Methods for arithmetical calculation are given, and a method to find square roots is known to have been in the Arabic original although it is missing from the Latin version.
     

2. La Roche biography
   
   • De la Roche published Larismetique nouellement composee in 1520 which was considered an excellent arithmetic book with good notation for powers and roots.
     
   • He used several notations for roots, so where we would write √5 he would write Recipe 2 5 where there is a line through the right hand leg of the R.
     
   • For the cube root he would write either Recipe 35 or Recipe squareshape 5.
     
   • As well as Recipe 45 he would use H Recipe 5 for the fourth root of 5, and H Recipe squareshape 5 for the sixth root.
     
   • Next find a number such that, multiplied by its root, the product is 10.
     
   • There results the cube root of 100, i.e.
     
   • But first express 3√100 as a sixth root, by multiplying 100 by itself, and you have 6√10000.
     
   • This multiplied by 6√100 gives 6√1000000, which is the square root of the cube root, or the cube root of the square root, or 6√1000000.
     
   • Extracting the square root gives 3√1000 which is 10, or reducing by the extraction of the cube root gives the square root of 100, which is 10, as before.
     

3. Rolle biography
   
   • In this treatise, he invented the notation n√x for the nth root of x and, as a consequence, it became the standard notation; it is used today.
     
   • Let us see how this idea worked: If P(x) = 0 is a given polynomial equation with real roots a and b then he constructs a polynomial P'(x), which he called the 'first cascade,' so that P'(b) = (b - a)Q(b) where Q(x) is a polynomial of lower degree.
     
   • Between any two consecutive roots of P(x) there is a root of P'(x), between any two consecutive roots of P'(x) there is a root of P''(x), etc.
     
   • His method is to start with a given polynomial, make a linear transformation to obtain a polynomial all of whose roots are positive (he never proves that his transformation always works but it does), then to continue to construct the cascade of polynomials until a linear polynomial is reached.
     
   • One can then move back up the cascade, finding approximately the roots of each polynomial.
     
   • It amplified the concepts of limits of roots of equations, provided the fundamentals from which Maclaurin derived his formula, began modern methods of series for determining roots, and discussed the relationship of imaginary roots in equations and their derivatives.
     

4. Jia Xian biography
   
   • The other contribution is an algorithm for root extraction but, as we shall see below, it uses the Pascal triangle method.
     
   • He generalised a method of finding square roots and cube roots to finding nth roots, for n > 3, and then extended the method to solving polynomial equations of arbitrary degree.
     
   • The algorithm is called the Zeng chang kaifang method by Jia Xian, which means the additive-multiplicative method for root extractions.
     
   • A fascinating historical account of methods of root extraction used by Chinese and Arabic scholars is given in [Arabic Sci.
     
   • Chemla defines precisely what constitutes the Ruffini-Horner method so that at each step of the algorithm precisely the same procedure, using multiplication and subtraction, is carried out until the root is obtained.
     
   • An examination of root extraction methods by Arabic authors leads to the conclusion that al-Samawal in the twelfth century was the first to use the Ruffini-Horner method.
     
   • 4 (2) (1994), 207-266.',4)">4] that both Jia Xian's method and al-Samawal's method end up with the same form for the approximation of nth roots.
     
   • If a is the integral portion of the nth root of A, then the approximation is given by .
     

5. Theodorus biography
   
   • [Theodorus] was proving to us a certain thing about square roots, I mean the side (i.e.
     
   • root) of a square of three square units and of five square units, that these roots are not commensurable in length with the unit length, and he went on in this way, taking all the separate cases up to the root of seventeen square units, at which point, for some reason, he stopped.
     
   • The first point is that Plato does not credit Theodorus with a proof that the square root of two was irrational.
     
   • The idea occurred to the two of us (Theaetetus and Socrates), seeing that these square roots appeared to be unlimited in multitude, to try to arrive at one collective term by which we could designate all these roots..
     

6. Wallis biography
   
   • In Treatise on Algebra Wallis accepts negative roots and complex roots.
     
   • He shows that a3 - 7a = 6 has exactly three roots and that they are all real.
     
   • He also criticises Descartes' Rule of Signs stating, quite correctly, that the rule which determines the number of positive and the number of negative roots by inspection, is only valid if all the roots of the equation are real.
     
   • One night he calculated the square root of a number with 53 digits in his head.
     
   • In the morning he dictated the 27 digit square root of the number, still entirely from memory.
     

7. Silva biography
   
   • In the following year he published Problems concerning rational functions of the roots of an algebraic equation (Portuguese) in the same journal.
     
   • The coefficients of a normalized factor belong to a field generated by the symmetric functions of p of the n roots.
     
   • In favorable cases, this field is generated by the sum or the product of these p roots.
     
   • Determine the equation whose roots are all sums of p roots; if there is a factor in the field, then this equation has a root in the field.
     
   • Assuming that it is possible to find all roots of an equation in a field, the preceding section furnishes a method of finding the coefficients of the factor, if it exists.
     

8. Hudde biography
   
   • He gave an ingenious method to find multiple roots of an equation which is essentially the modern method of finding the highest common factor of a polynomial and its derivative.
     
   • If in an equation two roots are equal and if it be multiplied by any arithmetical progression, i.e.
     
   • the first term by the first term of the progression, the second by the second term of the progression, and so on: I say that the equation found by the sum of these products shall have a root in common with the original equation.
     
   • Then Hudde is claiming that if a is a double root of f (x) = 0, then a is a root of f *(x) = 0.
     

9. Stampioen biography
   
   • This formula involved taking the cube root of expressions such as a + √b, both in the case where b is positive and negative.
     
   • Stampioen let this cube root be A + √B, where a, b, A and B are all natural numbers.
     
   • Having posed the problems as if by John Baptista of Antwerp, he then proceeded to give solutions to the problems under his own name, using his methods for finding the cube root of a + √b.
     
   • Descartes' method also starts off with an observation which we now easily see is true in Z[√b], namely a + √b has a cube root only if a2 - b is a cube.
     
   • 22 (2) (1969), 97-116.',2)">2] contains interesting observations on this argument and contains a French translation of Stampioen's method of finding the cube roots of a + √b as well as letters of Descartes to Mersenne on 30 September 1640 in which he criticises Stampioen's method.
     

10. Harriot biography
    
    • As an example of his abilities to solve equations, even when the roots are negative or imaginary, we reproduce his solution of an equation of degree 4.
      
    • Note how expert Harriot is in completing the square, knowing that whenever he takes a square root there are two answers to consider, and treating all answers equally whether positive or negative, real or imaginary.
      
    • As we have seen from the example above, Harriot did outstanding work on the solution of equations, recognising negative roots and complex roots in a way that makes his solutions look like a present day solution.
      
    • He made the observation that if a, b, c are the roots of a cubic then the cubic is (x - a)(x - b)(x - c) = 0.
      

11. Kushyar biography
    
    • Kushyar gives methods to construct exact square roots, as well as approximate methods to calculate the square roots of non-square numbers.
      
    • Similarly he gives methods to construct exact cube roots, and an approximate method to calculate the cube root of a non-square number.
      

12. Budan de Boislaurent biography
    
    • Budan is considered an amateur mathematician and he is best remembered for his discovery of a rule which gives necessary conditions for a polynomial equation to have n real roots between two given numbers.
      
    • In the early 19th century F D Budan and J B J Fourier presented two different (but equivalent) theorems which enable us to determine the maximum possible number of real roots that an equation has within a given interval.
      
    • If an equation in x has n roots between zero and some positive number p, the transformed equation in (x - p) must have at least n fewer variations in sign than the original.
      
    • Budan's goal was to solve Lagrange's problem - between which real numbers do real roots lie? - purely by methods of elementary arithmetic.
      
    • Let us note that Charles-Francois Sturm in his famous paper Memoire sur la resolution des equations numeriques published in 1829 completely solved the problem of determining the number of real roots of an equation on a given interval.
      

13. Fibonacci biography
    
    • In Flos Fibonacci gives an accurate approximation to a root of 10x + 2x2 + x3 = 20, one of the problems that he was challenged to solve by Johannes of Palermo.
      
    • Fibonacci proves that the root of the equation is neither an integer nor a fraction, nor the square root of a fraction.
      
    • For unity is a square and from it is produced the first square, namely 1; adding 3 to this makes the second square, namely 4, whose root is 2; if to this sum is added a third odd number, namely 5, the third square will be produced, namely 9, whose root is 3; and so the sequence and series of square numbers always rise through the regular addition of odd numbers.
      

14. Aitken biography
    
    • Aitken had an incredible memory (he knew π to 2000 places) and could instantly multiply, divide and take roots of large numbers.
      
    • For the latter Professor Aitken would ask for members of the class to give him numbers for which he would then write down the reciprocal, the square root, the cube root or other appropriate expression.
      

15. Al-Qalasadi biography
    
    • j from jadah meaning "root" .
      
    • In his arithmetic texts al-Qalasadi computed ∑n2, ∑n3 and used the method of successive approximation to determine square roots.
      
    • For example, the sequences ∑ n2 and ∑ n3 had been studied by al-Samawal and al-Baghdadi, and methods for computing square roots were known to the Babylonians.
      

16. Abel biography
    
    • The second of these explanations does seem the more likely, especially since Gauss had written in his thesis of 1801 that the algebraic solution of an equation was no better than devising a symbol for the root of the equation and then saying that the equation had a root equal to the symbol.
      
    • If every three roots of an irreducible equation of prime degree are related to one another in such a way that one of them may be expressed rationally in terms of the other two, then the equation is soluble in radicals.
      

17. Chebyshev biography
    
    • Chebyshev submitted a paper on The calculation of roots of equations in which he solved the equation y = f (x) by using a series expansion for the inverse function of f.
      
    • in the thesis I considered the case where the differential under the integral contains the square root of a rational function.
      
    • But it was interesting in several respects to extend those principles to a root of any degree.
      

18. Cotes biography
    
    • During the discussion they gave various approximations to the fourth root of 2 which is approximately 1.189207115.
      
    • In this Cotes explained gave a method of finding rational approximations as convergents of continued fractions, and the author of [6] suggests that this explains how he found the approximation 44/37to the fourth root of 2 which we mentioned above.
      
    • Cotes discovered an important theorem on the nth roots of unity, gave the continued fraction expansion of e, invented radian measure of angles, anticipated the method of least squares, published graphs of tangents and secants, and discovered a method of integrating rational fractions with binomial denominators.
      

19. Bennett biography
    
    • In the simple cases, when the modulus is a real number which is an odd prime, a power of an odd prime, or double the power of an odd prime, we know that there exist primitive roots of the modulus: that is, that there are numbers whose successive powers have for their rests the complete set of numbers less than, and prime to, the modulus.
      
    • A primitive root may be said to generate by its successive powers the complete set of rests.
      
    • It is also known that in general, when the modulus is any composite number, though primitive roots do not exist, there may be laid down a set of numbers, which will here be called generators, the products of powers of which give the complete set of rests prime to the modulus.
      

20. Narayana biography
    
    • Narayana also gave a rule to calculate approximate values of a square root.
      
    • He did this by using an indeterminate equation of the second order, Nx2 + 1 = y2, where N is the number whose square root is to be calculated.
      
    • If x and y are a pair of roots of this equation with x < y then √N is approximately equal to y/x.
      

21. Khayyam biography
    
    • This problem in turn led Khayyam to solve the cubic equation x3 + 200x = 20x2 + 2000 and he found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle.
      
    • In fact we can be fairly sure that Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients.
      
    • Muslim extraction of roots .
      

22. Sridhara biography
    
    • Following this algorithms are given for carrying out the elementary arithmetical operations, squaring, cubing, and square and cube root extraction, carried out with natural numbers.
      
    • Multiply both sides of the equation by a known quantity equal to four times the coefficient of the square of the unknown; add to both sides a known quantity equal to the square of the coefficient of the unknown; then take the square root.
      
    • and, taking the square root .
      
    • There is no suggestion that Sridhara took two values when he took the square root.
      

23. Napier biography
    
    • He is best known, however, for his invention of logarithms but his other mathematical contributions include a mnemonic for formulae used in solving spherical triangles, two formulae known as "Napier's analogies" used in solving spherical triangles and an invention called "Napier's bones" used for mechanically multiplying dividing and taking square roots and cube roots.
      
    • But amongst all, none more profitable than this which together with the hard and tedious multiplications, divisions, and extractions of roots, doth also cast away from the work itself even the very numbers themselves that are to be multiplied, divided and resolved into roots, and putteth other numbers in their place which perform as much as they can do, only by addition and subtraction, division by two or division by three.
      

24. Petersen biography
    
    • The interest he had shown in ruler and compass constructions when he was at school had continued to influence his research topic and his doctoral thesis was entitled On equations which can be solved by square roots, with application to the solution of problems by ruler and compass.
      
    • If the equation of degree 2n can be solved by square roots, one of the roots can be expressed by n such different square roots, where each can appear several times.
      

25. Heron biography
    
    • A method, known to the Babylonians 2000 years before, is also given for approximating the square root of a number.
      
    • Also in Book III, Heron gives a method to find the cube root of a number.
      
    • In particular Heron finds the cube root of 100 and the authors of [Elem.
      
    • 51 (1) (1996), 28-34.',9)">9] give a general formula for the cube root of N which Heron seems to have used in his calculation: .
      

26. Rudolff biography
    
    • Rudolff calculated with polynomials with rational and irrational coefficients and was aware that ax2 + b = cx has 2 roots.
      
    • He used √ for square roots (the first to use this notation) and Rudolff3rt for cube roots and Rudolff4rt for 4 th roots.
      

27. Lehmer Derrick biography
    
    • He also made major contributions to studying the density of primes with a given primitive root and to the study of the partition function, in particular verifying certain conjectures by Ramanujan.
      
    • He was the first person to attack the Riemann Hypothesis by using a computer to check if the roots lie on the critical line.
      

28. Taylor biography
    
    • He gave an account of an experiment to discover the law of magnetic attraction (1715) and an improved method for approximating the roots of an equation by giving a new method for computing logarithms (1717).
      
    • It was, wrote Taylor, due to a comment that Machin made in Child's Coffeehouse when he had commented on using "Sir Isaac Newton's series" to solve Kepler's problem, and also using "Dr Halley's method of extracting roots" of polynomial equations.
      
    • Taylor initially derived the version which occurs as Proposition 11 as a generalisation of Halley's method of approximating roots of the Kepler equation, but soon discovered that it was a consequence of the Bernoulli series.
      

29. Fox Leslie biography
    
    • It contains chapters on: Matrix algebra; Elimination methods of Gauss, Jordan, and Aitken; Compact elimination methods of Doolittle, Crout, Banachiewicz and Cholesky; Orthogonalization methods; Condition, accuracy and precision; Comparison of methods, measure of work; Iterative and gradient methods; Iterative methods for latent roots and vectors; and Notes on error analysis for latent roots and vectors.
      
    • Throughout his life Fox remained true to his roots, a forthright socialist and above all a Yorkshireman.
      

30. Ferro biography
    
    • Firstly, in the middle of the 16th century in Europe, zero was not in use; secondly negative numbers were not in use; and thirdly there was no understanding of a quadratic having two roots.
      
    • He made an important contribution to rationalising fractions, extending methods to rationalise fractions which had square roots in the denominator (which were know to Euclid) to fractions whose denominators were the sum of three cube roots.
      

31. Artin biography
    
    • Second, there is Artin's conjecture on primitive roots.
      
    • Given any integer g not 1 or -1, and g not a power of some other integer, then Artin conjectured that there are infinitely many prime numbers p such that g is a primitive root modulo p in the sense of Gauss.
      

32. Cantor biography
    
    • the numbers which are roots of polynomial equations with integer coefficients, were countable.
      
    • A transcendental number is an irrational number that is not a root of any polynomial equation with integer coefficients.
      

33. Voronoy biography
    
    • He was awarded a Master's Degree in 1894 for a dissertation on the algebraic integers associated with the roots of an irreducible cubic equation.
      
    • In the essay I am now presenting, results from the general theory of algebraic integers are applied to the particular case of numbers depending on the root of an irreducible equation x3 = rx + s.
      

34. Springer biography
    
    • For the associatively-inclined this book expunges the dread word "nonassociative" from Jordan theory, since there is nothing nonassociative about inversion; most importantly, it makes Jordan structure theory accessible to the growing audience of persons familiar with root systems.
      
    • By placing the classification of Jordan algebras in the perspective of classification of certain root systems, the book demonstrates that the structure theories of associative, Lie, and Jordan algebras are not separate creations, but rather instances of the one all-encompassing miracle of root systems.
      

35. Vandermonde biography
    
    • The first of these four papers presented a formula for the sum of the mth powers of the roots of an equation.
      
    • It also presented a formula for the sum of the symmetric functions of the powers of such roots.
      
    • Vandermonde's real and unrecognised claim to fame was lodged in his first paper, in which he approached the general problem of the solubility of algebraic equations through a study of functions invariant under permutations of the roots of the equation.
      

36. Bortolotti biography
    
    • In the 1940 paper on Babylonian mathematics, Bortolotti gives a summary of problems published by Neugebauer but argues that the fact that large series of examples for quadratic equations are made up from the same roots demonstrates that this pair of roots has an 'arcane mystic property'.
      
    • It is also wrong to deny the existence of approximations to irrational square roots, to assume a geometrical basis of the quadratic equations or to deny the existence of texts of this type in the Hellenistic period.
      

37. Recorde biography
    
    • The book was the Second Part of Arithmetic, The Grounde of Artes being the first, covering the extraction of roots, the theory of equations and arithmetic with surds.
      
    • He makes good use of the sum and product of the roots stressing that for the equation .
      
    • the sum of the roots is p and their product is q.
      

38. Hurwitz biography
    
    • Further topics studied by Hurwitz include complex function theory, the roots of Bessel functions, and difference equations.
      
    • with positive leading coefficient a0 > 0 has only roots with negative real parts.
      
    • In 1893 the Swedish actuary and mathematics historian Gustaf Enestrom published a theorem on the complex roots of certain polynomials with real coefficients in a paper on pension insurance (in Swedish).
      

39. Al-Tusi Sharaf biography
    
    • Then Al-Tusi deduces that the equation has a positive root if .
      
    • Al-Tusi then went on to give what we would essentially call the Ruffini-Horner method for approximating the root of the cubic equation.
      
    • Although this method had been used by earlier Arabic mathematicians to find approximations for the nth root of an integer, al-Tusi is the first that we know who applied the method to solve general equations of this type.
      

40. Al-Karaji biography
    
    • He does not treat equations above the second degree except for ones which can easily be reduced to at most second degree equations followed by the extraction of roots.
      
    • He was able to give a rule for finding the square root of a composite quantity which is not completely general since it required the coefficients to be positive, but it is still a remarkable achievement.
      

41. Ajima biography
    
    • He produced log tables which were designed for taking 10th roots and powers of numbers.
      
    • For this purpose he set the log of the 10th root of 10 to 1.
      

42. Al-Nasawi biography
    
    • He also explains doubling, halving, taking square roots, and taking cube roots.
      
    • Each method for each of the four types is illustrated with worked examples and a checking procedure is explained which usually involves usually casting out nines The method al-Nasawi gives for taking cube roots is the same as the method described in the Chinese Mathematics in Nine Books, but quite how he learnt of this method is unknown.
      

43. Kostrikin biography
    
    • The meaning of an algebraic concept can be of a number-theoretic or geometric nature, and frequently its roots lie in computational aspects of mathematics and in the solution of equations.
      
    • The book deals with certain "concrete" aspects of the representation theory of finite (almost) simple groups, namely with the realization of certain classes of these groups as automorphism groups of integral lattices and of related algebraic and combinatorial objects (root systems, symplectic spreads).
      

44. Theaetetus biography
    
    • For Theaetetus had distinguished square roots commensurable in length from those which are incommensurable, and who divided the more generally known irrational lines according to the different means, assigning the medial line to geometry, the binomial to arithmetic and the apotome to harmony, as stated by Eudemus..
      
    • The idea occurred to the two of us [Theaetetus and the younger Socrates], seeing that these square roots appeared to be unlimited in multitude, to try to arrive at one collective term by which we could designate all these roots.
      

45. Raphson biography
    
    • His election to that Society was on the strength of his book Analysis aequationum universalis which was published in 1690 contained the Newton method for approximating the roots of an equation.
      
    • In Method of Fluxions Newton describes the same method and, as an example, finds the root of x3 - 2x - 5 = 0 lying between 2 and 3.
      

46. Brauer Alfred biography
    
    • He gave bounds for the least quadratic residues modulo a prime, and for the least primitive root for a prime.
      
    • He studied, for example, the location of characteristic roots using ovals of Cassini, publishing his first results on this in 1947.
      

47. Lagrange biography
    
    • The paper is the first to consider the roots of an equation as abstract quantities rather than having numerical values.
      
    • He studied permutations of the roots and, although he does not compose permutations in the paper, it can be considered as a first step in the development of group theory continued by Ruffini, Galois and Cauchy.
      

48. Schur biography
    
    • Schur was also interested in reducibility, location of roots and the construction of the Galois group of classes of polynomials such as Laguerre and Hermite polynomials.
      
    • Third, he handled algebraic equations, sometimes proceeding to the evaluation of roots, and sometimes treating the so-called equation without affect, that is, with symmetric Galois groups.
      

49. Bombelli biography
    
    • He then showed that, using his calculus of complex numbers, correct real solutions could be obtained from the Cardan-Tartaglia formula for the solution to a cubic even when the formula gave an expression involving the square roots of negative numbers.
      
    • Thus we have an engineer, Bombelli, making practical use of complex numbers perhaps because they gave him useful results, while Cardan found the square roots of negative numbers useless.
      

50. Al-Samawal biography
    
    • The treatise consists of four books: (1) On premises, multiplication, division and extraction of roots, (2) On extraction of unknown quantities, (3) On irrational magnitudes, and (4) On classification of problems.
      
    • He also gave methods for the extraction of the roots of polynomials.
      

51. Sun Zi biography
    
    • The first chapter describes systems of measuring with considerable detail, and gives instructions on using counting rods to multiply, divide, and compute square roots.
      
    • (1) (1987), 22-27.',11)">11], gives a detailed description of the algorithm used by Sun Zi for the extraction of roots and compares it with the method described in the Nine Chapters on the Mathematical Art.
      

52. Riemann biography
    
    • Except for a few trivial exceptions, the roots of ζ(s) all lie between 0 and 1.
      
    • In the paper he stated that the zeta function had infinitely many nontrivial roots and that it seemed probable that they all have real part 1/2.
      

53. Feigenbaum biography
    
    • I was allowed to use the new Friden calculating machine which, shortly before its transformation into a relic, could also extract square roots.
      
    • This was the first computer I ever used, and within an hour had programmed it to take square roots by Newton's method.
      

54. Wright Sewall biography
    
    • He was educated at home until he was eight years old and by the time he entered school in 1897 he had read his father's mathematics books, learning how to extract cube roots.
      
    • (How many mathematicians today know how to extract cube roots?) He was fascinated by mathematical models and calculating, learning arithmetical methods from his mother.
      

55. Walsh Joseph biography
    
    • Walsh was awarded his doctorate by Harvard in 1920 for his thesis On the location of the roots of a Jacobian of two binary forms, and of the derivative of a rational function.
      
    • He continued to publish a steady stream of papers with On the location of the roots of the derivative of a polynomial appearing in 1920 and then two papers A generalization of the Fourier cosine series and A theorem on cross-ratios in the geometry of inversion in 1921.
      

56. Zhang Heng biography
    
    • When roots are deep, they do not rot easily.
      
    • One interesting point to note in some of Zhang's mathematical work is that he leaves square roots as unevaluated.
      

57. Waring biography
    
    • In Meditationes Algebraicae Waring proves that all rational symmetric functions of the roots of an equation can be expressed as rational functions of the coefficients.
      
    • The most significant aspect of Waring's treatment of this example is the symmetric relation between the roots of the quartic equation and its resolvent cubic.
      

58. Archimedes biography
    
    • Archimedes also gave an accurate approximation to π and showed that he could approximate square roots accurately.
      
    • Kevin Brown (Some information about Archimedes' calculation of square roots) .
      

59. Al-Kashi biography
    
    • He found that al-Kashi had an algorithm for calculating nth roots which was a special case of the methods given many centuries later by Ruffini and Horner.
      
    • Muslim extraction of roots .
      

60. Dyson biography
    
    • The first, written in 1941 (published in 1944) is A proof that every equation has a root.
      
    • there are so many proofs of the theorem that every equation has a root that it seems almost criminal to produce another.
      

61. Sacrobosco biography
    
    • The work contains 11 chapters, one each on topics such as addition, subtraction, multiplication, division, square roots and cube roots.
      

62. Varignon biography
    
    • This states that the speed of a liquid flowing under the force of gravity out of an opening in a tank is proportional to the square root of the vertical distance between the liquid surface and the centre of the opening, and to the square root of twice the acceleration due to gravity.
      

63. Mersenne biography
    
    • In this work he was the first to publish the laws relating to the vibrating string: its frequency is proportional to the square root of the tension, and inversely proportional to the length, to the diameter and to the square root of the specific weight of the string, provided all other conditions remain the same when one of these quantities is altered.
      

64. Jordanus biography
    
    • Take the square root of g, call it h.
      
    • Now 4 is the root of this and also the difference of the two parts.
      

65. Cardan biography
    
    • One of the first problems that Cardan hit was that the formula sometimes involved square roots of negative numbers even though the answer was a 'proper' number.
      
    • Indeed Cardan gives precisely the conditions here for the formula to involve square roots of negative numbers.
      

66. Sripati biography
    
    • In verses 3, 4 and 5 of this chapter Sripati gave the rules of signs for addition, subtraction, multiplication, division, square, square root, cube and cube root of positive and negative quantities.
      

67. Ruffini biography
    
    • To solve a polynomial equation by radicals meant finding a formula for its roots in terms of the coefficients so that the formula only involves the operations of addition, subtraction, multiplication, division and taking roots.
      

68. Sturm biography
    
    • It considered the problem of determining the number of real roots of an equation on a given interval.
      
    • seeks to determine the mutual influence between A-L Cauchy's and Ch-F Sturm's research from 1829 to around 1840 on the roots of algebraic equations.
      

69. Praeger biography
    
    • In Enumeration of rooted trees with a height distribution (1985) written jointly with P Schultz and N C Wormald, the authors used generating functions to find a new solution to the problem of determining the number of rooted trees whose vertices have a given height distribution.
      

70. Ortega biography
    
    • In the second part of the book, devoted mostly to geometry, Ortega gives a method of extracting square roots very accurately using Pell's equation, which is surprising since a general solution to Pell's equation does not appear to have been found before Fermat over 100 years later.
      
    • Very little is known about Ortega and it appears that his importance rests entirely on the interesting method for taking square roots.
      

71. Goldbach biography
    
    • He also studied equations and worked out in his correspondence with Euler how to provide a quick test for whether an algebraic equation has a rational root.
      

72. Stirling biography
    
    • In the minutes of a meeting of the Royal Society of London on 4 April 1717, when Brook Taylor lectured on extracting roots of equations and on logarithms, it is recorded:- .
      

73. Jordan biography
    
    • He applied his work on classical groups to determine the structure of the Galois group of equations whose roots were chosen to be associated with certain geometrical configurations.
      

74. Schmidt biography
    
    • He defined the norm ||z|| of the element z to be the square root of the inner product of z with its complex conjugate.
      

75. Wang Yuan biography
    
    • He also attacked other questions in number theory in papers such as On the least primitive root of a prime (1959) and On Diophantine approximations and numerical integrations.
      

76. Levi biography
    
    • It deals with arithmetical operations, including extraction of square roots and cube roots.
      

77. Kellogg biography
    
    • In 1908 he published three papers, namely Potential functions on the boundary of their regions of definition and Double distributions and the Dirichlet problem, both in the Transactions of the American Mathematical Society, and A necessary condition that all the roots of an algebraic equation be real in the Annals of Mathematics.
      

78. Stevin biography
    
    • He also made a strong plea that all numbers such as square roots, irrational numbers, surds, negative numbers etc should all be treated as numbers and not distinguished as being different in nature.
      

79. Bhaskara II biography
    
    • The six works are: Lilavati (The Beautiful) which is on mathematics; Bijaganita (Seed Counting or Root Extraction) which is on algebra; the Siddhantasiromani which is in two parts, the first on mathematical astronomy with the second part on the sphere; the Vasanabhasya of Mitaksara which is Bhaskaracharya's own commentary on the Siddhantasiromani ; the Karanakutuhala (Calculation of Astronomical Wonders) or Brahmatulya which is a simplified version of the Siddhantasiromani ; and the Vivarana which is a commentary on the Shishyadhividdhidatantra of Lalla.
      

80. Ahlfors biography
    
    • The post-war era was not a good time for a stranger to take root in Switzerland.
      

81. Tibbon biography
    
    • And here is geometry, the basis for all mathematical sciences, and this book is the basis, the root and the beginning for all later books on this science.
      

82. Gregory biography
    
    • However, we now summarise these and other contributions in the hope that, despite his reluctance to publish his methods, his remarkable contributions might indeed be more widely understood: Gregory anticipated Newton in discovering both the interpolation formula and the general binomial theorem as early as 1670; he discovered Taylor expansions more than 40 years before Taylor; he solved Kepler's famous problem of how to divide a semicircle by a straight line through a given point of the diameter in a given ratio (his method was to apply Taylor series to the general cycloid); he gives one of the earliest examples of a comparison test for convergence, essentially giving Cauchy's ratio test, together with an understanding of the remainder; he gave a definition of the integral which is essentially as general as that given by Riemann; his understanding of all solutions to a differential equation, including singular solutions, is impressive; he appears to be the first to attempt to prove that π and e are not the solution of algebraic equations; he knew how to express the sum of the nth powers of the roots of an algebraic equation in terms of the coefficients; and a remark in his last letter to Collins suggests that he had begun to realise that algebraic equations of degree greater than four could not be solved by radicals.
      

83. Barlow biography
    
    • These soon became known as Barlow's Tables and this work gives factors, squares, cubes, square roots, reciprocals and hyperbolic logarithms of all numbers from 1 to 10 000.
      

84. Brahmagupta biography
    
    • Another arithmetical result presented by Brahmagupta is his algorithm for computing square roots.
      

85. Mohr biography
    
    • Mohr corresponded with a number of mathematicians including Leibniz who had received a work written by Mohr on root extraction.
      

86. Saunderson biography
    
    • The introduction gives the reader the necessary arithmetical skills to begin the study of algebra, teaching the reader to carry out the standard arithmetical operations, take roots of numbers, calculate with fractions and become skilled in problems of proportion.
      

87. Pacioli biography
    
    • In [Sciences of the Renaissance (Paris, 1973), 93-106.',10)">10] the importance of Pacioli's work is discussed, in particular his computation of approximate values of a square root (using a special case of Newton's method), his incorrect analysis of certain games of chance (similar to those studied by Pascal which gave rise to the theory of probability), his problems involving number theory (similar problems appeared in Bachet's compilation), and his collection of many magic squares.
      

88. Bohr Niels biography
    
    • It has been suggested that the idea of complementarity came from outside physics, some arguing that the roots of the idea came from the discussions with his father, Christiansen and the philosopher Hoffding when he was still at school.
      

89. McCrea biography
    
    • He was born in Dublin and remained proud of his Irish roots throughout his life.
      

90. Frohlich biography
    
    • This paper, Artin root numbers and normal integral bases for quaternion fields, is described by the authors of [Biographical Memoirs of Fellows of the Royal Society of London 51 (2005), 149-168.',1)">1] as:- .
      

91. Smeal biography
    
    • Among Smeals' publications are (with Ernest Frederick John Love) The psychrometric formula (1911), (with S Brodetsky) On Graeffe's method for complex roots of algebraic equations (1924) and The equations of the gravitational field in orthogonal coordinates (1926).
      

92. Collatz biography
    
    • Among his early papers are Genaherte Berechnung von Eigenwerten (1939) in which he considers various methods of approximating characteristic values, Das Hornersche Schema bei komplexen Wurzeln algebraischer Gleichungen (1940) in which he presents a more efficient way of using Horner's method to approximate the complex roots of an algebraic equation, and Schrittweise Naherungen bei Integralgleichungen und Eigenwertschranken (1940) in which inequalities between the eigenvalues of certain integral equations are studied.
      

93. Torricelli biography
    
    • In De motu gravium which was published as part of Torricelli's 1644 Opera geometrica, Torricelli also proved that the flow of liquid through an opening is proportional to the square root of the height of the liquid, a result now known as Torricelli's theorem.
      

94. Dandelin biography
    
    • He gave a method of approximating the roots of an algebraic equation, now named the Dandelin-Graffe method.
      

95. Legendre biography
    
    • In "Elements" Legendre gave a simple proof that π is irrational, as well as the first proof that π2 is irrational, and conjectured that π is not the root of any algebraic equation of finite degree with rational coefficients.
      

96. Askey biography
    
    • Most notably, the authors demonstrate a superb familiarity with the historical roots of their subject.
      

97. Qin Jiushao biography
    
    • There is a remarkable formula given in this Chapter which expresses the area of a figure as the root of an equation of degree 4.
      

98. Zhang Qiujian biography
    
    • There are problems on extracting square and cube roots, problems on finding the solution to quadratic equations, problems on finding the sum of an arithmetic progression, and on solving systems of linear equations.
      

99. Runge biography
    
    • Runge then worked on a procedure for the numerical solution of algebraic equations in which the roots were expressed as infinite series of rational functions of the coefficients.
      

100. Yang Hui biography
     
     • The topics covered by Yang include multiplication, division, root-extraction, quadratic and simultaneous equations, series, computations of areas of a rectangle, a trapezium, a circle, and other figures.
       

101. Theon biography
     
     • This commentary is not calculated to give us a very high opinion of Theon's mathematical calibre, but it is valuable for several historical notices that it gives, and we are indebted to it for a useful account of the Greek method of operating with sexagesimal fractions, which is illustrated with examples of multiplication, division, and the extraction of the square root of a non-number by way of approximation.
       

102. Hammersley biography
     
     • This covered plenty of Euclidean geometry (including such topics as the nine-point circle) and algebra (Newton's identities for roots of polynomials) and trigonometry (identities governing angles of a triangle, circumcircle, incircle, etc), but no calculus.
       

103. Airey biography
     
     • John R Airey, The Roots of the Neumann and Bessel Functions, Proc.
       

104. Aryabhata II biography
     
     • Aryabhata II also gave a method to calculate the cube root of a number, but his method was not new, being based on that given many years earlier by Aryabhata I, see for example [Ganita Bharati 19 (1-4) (1997), 60-68.',5)">5].
       

105. Milne-Thomson biography
     
     • A year later he published Standard table of square roots and Jacobian Elliptic Function Tables.
       

106. Machin biography
     
     • Machin had explained to Taylor in Child's Coffeehouse how to use Newton's series to solve Kepler's problem and also how Halley's method finds roots of polynomial equations.
       

107. Guo Shoujing biography
     
     • The equation has two real roots, the smaller being the solution to the problem while the other, being numerically larger than the length of the arc, was rightly discarded by Guo.
       

108. Van der Pol biography
     
     • This paper presents, in graphical representation, a list of primes in the quadratic field defined by a primitive cube root of unity.
       

109. Lhuilier biography
     
     • Whereas the Poles found Lhuilier distinctly puritanical, his fellow citizens of Geneva reproached him for his lack of austerity and his whimsicality, although the latter quality never went beyond putting geometric theorems into verse and writing ballads on the number three and on the square root of minus one.
       

110. Eutocius biography
     
     • the main part concerns methods of sexagesimal computation: multiplication, division, square roots etc.
       

111. Roy biography
     
     • A sample of the papers Roy published during this period follows: The use and distribution of the Studentized D2-statistic when the variances and covariances are based on k samples (1940), On hierarchical sampling, hierarchical variances and their connexion with other aspects of statistical theory (1940), The distribution of the root-mean-square of the second type of the multiple correlation co-efficient (1940), Analysis of variance for multivariate normal populations: the sampling distribution of the requisite p-statistics on the null and non-null hypotheses (1942), Bernoulli's theorem and Tshebycheff's analogue (1945), On a certain class of multiple integrals (1945), Notes on testing of composite hypotheses (1947), and On the construction of an unbiassed and most powerful critical region out of any given statistic (1948).
       

112. Krylov Aleksei biography
     
     • numerical computation of its roots, does not present any difficulty.
       

113. Blichfeldt biography
     
     • Some of the many topics that he covered were diophantine approximations, orders of linear homogeneous groups, theory of geometry of numbers, approximate solutions of the integers of a set of linear equations, low-velocity fire angle, finite collineation groups, and characteristic roots.
       

114. Dehn biography
     
     • He realised that he could not lecture on advanced mathematics so he gave his lectures on Common roots of mathematics and ornamentics and Some moments in the development of mathematical ideas.
       

115. Andrews biography
     
     • Most notably, the authors demonstrate a superb familiarity with the historical roots of their subject.
       

116. Rado biography
     
     • His first paper On the roots of algebraic equations was published in 1921 and in the following year he published his first paper on conformal mappings.
       

117. Maclaurin biography
     
     • 28 (2) (1983), 149-164.',17)">17] the controversy between Maclaurin and George Campbell over complex roots is described.
       

118. Redei biography
     
     • The paper was Existence theorem for the primitive root of the congruence xφ(pa) - 1 equiv 0 (mod pα) (Hungarian).
       

119. Seki biography
     
     • He studied equations treating both positive and negative roots but had no concept of complex numbers.
       

120. Bethe biography
     
     • Recollections of both his own and of relatives include his ability to compute square roots at the age of four, a full understanding of fractions by the age of five, and the ability to find prime numbers by the age of seven.
       

121. Euler biography
     
     • We owe to Euler the notation f (x) for a function (1734), e for the base of natural logs (1727), i for the square root of -1 (1777), π for pi, ∑ for summation (1755), the notation for finite differences Δy and Δ2y and many others.
       

122. Hammer biography
     
     • Most of Peter Hammer's scientific production has its roots in the work of George Boole on propositional logic.
       

123. Cataldi biography
     
     • He used no clever tricks, merely checked that these numbers were prime by dividing each by all primes up to their square roots.
       
     • Of course, to do this he required a list of primes up to 724 (the approximate root of 219 - 1).
       
     • Cataldi found square roots of numbers by use of an infinite series leading to an early investigation into continued fractions.
       
     • In this work the square root of a number is found through the use of infinite series and unlimited continued fractions.
       
     • Cataldi calculates the square root of a number N by first taking the integer a such that a2 < N < (a+1)2.
       
     • We note that he doesn't use what are called today 'simple continued fractions' and his method will always give a continued fraction for a square root which has period 1.
       
     • Let us now proceed to the consideration of another method of finding roots continuing by adding row on row to the denominator of the fraction of the preceding rule.
       
     • But for greater convenience, I shall assume a number whose root may be easily taken and I shall assume that the first part of the root is an integer.
       
     • Then let 18 be the proposed number, and if I assume that the first root is 4.
       
     • The second root will be found by the above mentioned method to be 4.
       
     • He had not then the 'continued fraction', a mode of representation which he gave the next year in his work on the square root.
       

124. Gopel biography
     
     • Gopel's doctoral dissertation studied periodic continued fractions of the roots of integers and derived a representation of the numbers by quadratic forms.
       

125. Durell biography
     
     • Further topics encompass the special hyperbolic functions; projection and finite series; complex numbers; de Moivre's theorem and its applications; one- and many-valued functions of a complex variable; and roots of equations.
       

126. Lindemann biography
     
     • Ferdinand von Lindemann was the first to prove that π is transcendental, that is, π is not the root of any algebraic equation with rational coefficients.
       

127. Van Vleck biography
     
     • For example he published On the determination of a series of Sturm's functions by the calculation of a single determinant (1899), On linear criteria for the determination of the radius of convergence of a power series (1900), On the convergence of continued fractions with complex elements (1901), A determination of the number of real and imaginary roots of the hypergeometric series (1902), On an extension of the 1894 memoir of Stieltjes (1903), and On the extension of a theorem of Poincare for difference-equations (1912).
       

128. Klein biography
     
     • Plucker held a chair of mathematics and experimental physics at Bonn but, by the time Klein became his assistant, Plucker's interests had become very firmly rooted in geometry.
       

129. Hutton biography
     
     • The first volume looks at topics such as: arithmetic including discussion of square and cube roots, arithmetical and geometrical progressions, compound interest, double position and permutations and combinations; logarithms; algebra including the study of quadratic equations and the Cardan-Tartaglia method for cubic equations; geometry which follows the approach in Euclid's Elements; surveying; and conic sections.
       

130. Wantzel biography
     
     • By 1829, at the remarkably young age of 15, he edited a second edition of Reynaud's Treatise on arithmetic giving a proof of a method for finding square roots which was widely used but previously unproved.
       

131. Borel Armand biography
     
     • It started with a draft of about 70 pages on root systems.
       

132. Abu Kamil biography
     
     • For example he uses the expression "square square root" for x5 (i.e.
       

133. Chow biography
     
     • Notably, "Chow motives" are naturally included in Voevodsky's triangulated category of motives, showing the deepest roots of this analogy.
       

134. Metzler biography
     
     • by Clark University, Worcester, Massachusetts in 1892 for his thesis On the Roots of Matrices written with William Story and Henry Taber as advisors.
       

135. Zu Chongzhi biography
     
     • To compute this accuracy for π, Zu must have used an inscribed regular 24,576-gon and undertaken the extremely lengthy calculations, involving hundereds of square roots, all to 9 decimal place accuracy.
       

136. Polya biography
     
     • It has applications such as the enumeration of chemical compounds and the enumeration of rooted trees in graph theory.
       

137. Skopin biography
     
     • Another of Skopin's results [p-extensions of a local field containing pM roots of 1 (Russian), Doklady Akad.
       

138. Naimark biography
     
     • Naimark's first work for his candidate's thesis was on the separation of roots of algebraic equations.
       

139. Wittich biography
     
     • Well born, and probably well off, Wittich was an enigmatic character whose roots remain mostly obscure.
       

140. Simpson biography
     
     • The method of approximating the roots did not use the differential calculus.
       

141. Killing biography
     
     • He also introduced the idea of a root system which appears throughout much of the algebra of today.
       

142. Ford biography
     
     • Ford read the paper On the Roots of a Derivative of a Rational Function to the meeting of the Society on Friday 14 May 1915, the paper On the Oscillation Functions derived from a Discontinuous Function to the meeting on 11 June 1915, and the paper A method of solving algebraic equations to the meeting on 12 January 1917.
       

143. Wilkins biography
     
     • to pull up any oak by the roots with a hair, lift it up with a straw, or blow it up with one's breath.
       

144. Stieltjes biography
     
     • Stieltjes examined the sequence of rational functions Pn(z)/Qn(z) and the connections between the roots of the polynomials Pn(z) and Qn(z).
       

145. Ramanathan biography
     
     • This paper is concerned with a sum which is, in fact, the sum of the nth powers of the primitive mth roots of unity.
       

146. Zhu Shijie biography
     
     • Now (2) has the roots -1, -8, 18, 25 but Zhu only gives the correct answer t = z + x = 18 bu.
       

147. Diophantus biography
     
     • Equations which would lead to solutions which are negative or irrational square roots, Diophantus considers as useless.
       

148. Stampacchia biography
     
     • These peculiar roots were particularly suited to his personality, rather nonconformist, and certainly contributed to his formation.
       

149. Ostrowski biography
     
     • His work on aglebraic equations involved a study of the fundamental theorem of algebra, Galois theory, and estimating the roots of algebraic equations.
       

150. Straus biography
     
     • Algebraic equations satisfied by roots of natural numbers.
       

151. Lax Anneli biography
     
     • This proved successful, however Lax was determined to follow this so-called 'interplay problem' to its very roots in high schools.
       

152. Al-Tusi Nasir biography
     
     • Another mathematical contribution was al-Tusi's manuscript, dated 1265, concerning the calculation of n-th roots of an integer; see [Mat.
       

153. Thomae biography
     
     • In the second of the papers Thomae also showed that the roots of a polynomial can be expressed in terms of hyperelliptic theta functions.
       

154. Gleason biography
     
     • He made a major contribution with his 1949 papers: Square roots in locally Euclidean groups; On the structure of locally compact groups; and A note on locally compact groups.
       

155. Ferrers biography
     
     • In 1853 Sylvester published On Mr Cayley's impromptu demonstration of the rule for determining at sight the degree of any symmetrical function of the roots of an equation expressed in terms of the coefficients in the Philosophical Magazine.
       

156. Aryabhata I biography
     
     • This supposition is based on the following two facts: first, the invention of his alphabetical counting system would have been impossible without zero or the place-value system; secondly, he carries out calculations on square and cubic roots which are impossible if the numbers in question are not written according to the place-value system and zero.
       

157. Viete biography
     
     • He knew the connection between the positive roots of equations and the coefficients of the different powers of the unknown quantity.
       

158. Bishop biography
     
     • B van Rootselaar, reviewing the text, wrote:- .
       

159. Lobachevsky biography
     
     • In 1834 Lobachevsky found a method for the approximation of the roots of algebraic equations.
       

160. Griffiths Brian biography
     
     • As well as illuminating classical mathematics, the book also provides a way forward into more recent topics, clearly demonstrating that subjects like topology and modern algebra have firm classical roots, unlike many expositions which give the impression of self-contained inventions which have superseded older mathematics.
       

161. Siegel biography
     
     • I see a pig broken into a beautiful garden and rooting up all flowers and trees.
       

162. Poinsot biography
     
     • In addition Poinsot worked on number theory and on this topic he studied Diophantine equations, how to express numbers as the difference of two squares and primitive roots.
       

163. Bondi biography
     
     • I think ethics must always be rooted in society and culture, and change as it changes, and I really hope that we become more tolerant in our attitudes ..
       

164. Al-Banna biography
     
     • For example it contains continued fractions and they are used to compute approximate square roots.
       

165. Chrysippus biography
     
     • Chrysippus was of Phoenician roots.
       

166. Mahavira biography
     
     • He also described a process for calculating the volume of a sphere and one for calculating the cube root of a number.
       

167. Seidel Jaap biography
     
     • Graphs related to exceptional root systems, Fifth Hungarian Colloq., Keszthely, 1976; .
       

168. Miller biography
     
     • He became a severe critic of historical methodology in mathematics and was zealous in rooting out error in conjecture or assumed fact.
       

169. Vinogradov biography
     
     • Vinogradov made many other contributions, for example to the theory of distribution of power residues, non-residues, indices and primitive roots.
       

170. Widman biography
     
     • The approach is unusual in that it contained symbols for addition, subtraction, and square roots.
       

History Topics

1. Quadratic etc equations
   
   • In about 300 BC Euclid developed a geometrical approach which, although later mathematicians used it to solve quadratic equations, amounted to finding a length which in our notation was the root of a quadratic equation.
     Go directly to this paragraph
     
   • He has six chapters each devoted to a different type of equation, the equations being made up of three types of quantities namely: roots, squares of roots and numbers i.e.
     Go directly to this paragraph
     
   • Squares equal to roots.
     
   • Roots equal to numbers.
     
   • Squares and roots equal to numbers, e.g.
     
   • Squares and numbers equal to roots, e.g.
     
   • Roots and numbers equal to squares, e.g.
     
   • The problem was to find the roots by adding, subtracting, multiplying, dividing and taking roots of expressions in the coefficients.
     Go directly to this paragraph
     
   • Now a is found by taking cube roots and b can be found in a similar way (or using b=m/3a).
     
   • Cardan knew that you could not take the square root of a negative number yet he also knew that x = 4 was a solution to the equation.
     Go directly to this paragraph
     
   • With this value of y the right hand side of (*) is a perfect square so, taking the square root of both sides, we obtain a quadratic in x.
     
   • The irreducible case of the cubic, namely the case where Cardan's formula leads to the square root of negative numbers, was studied in detail by Rafael Bombelli in 1572 in his work Algebra.
     Go directly to this paragraph
     
   • Now x = e - b2/e and both e and b2/e are cube roots of expressions given above.
     
   • This Leibniz did by reconstructing the cubic from its three roots (as given by the formula) as Harriot claimed in general.
     Go directly to this paragraph
     

2. Fund theorem of algebra
   
   • Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers.
     
   • Early studies of equations by al-Khwarizmi (c 800) only allowed positive real roots and the FTA was not relevant.
     Go directly to this paragraph
     
   • This discovery was made in the course of studying a formula which gave the roots of a cubic equation.
     Go directly to this paragraph
     
   • Descartes in 1637 says that one can 'imagine' for every equation of degree n, n roots but these imagined roots do not correspond to any real quantity.
     Go directly to this paragraph
     
   • Viete gave equations of degree n with n roots but the first claim that there are always n solutions was made by a Flemish mathematician Albert Girard in 1629 in L'invention en algebre .
     Go directly to this paragraph
     
   • They believed that a polynomial equation of degree n must have n roots, the problem was, they believed, to show that these roots were of the form a + bi, a, b real.
     Go directly to this paragraph
     
   • Now Harriot knew that a polynomial which vanishes at t has a root x - t but this did not become well known until stated by Descartes in 1637 in La geometrie, so Albert Girard did not have much of the background to understand the problem properly.
     Go directly to this paragraph
     
   • Euler was soon able to prove that every real polynomial of degree n, n ≤ 6 had exactly n complex roots.
     Go directly to this paragraph
     
   • Lagrange used his knowledge of permutations of roots to fill all the gaps in Euler's proof except that he was still assuming that the polynomial equation of degree n must have n roots of some kind so he could work with them and deduce properties, like eventually that they had the form a + bi, a, b real.
     Go directly to this paragraph
     
   • His proof was very elegant and its only 'problem' was that again the existence of roots was assumed.
     Go directly to this paragraph
     
   • He is undoubtedly the first to spot the fundamental flaw in the earlier proofs, to which we have referred many times above, namely the fact that they were assuming the existence of roots and then trying to deduce properties of them.
     Go directly to this paragraph
     
   • if one carries out operations with these impossible roots, as though they really existed, and says for example, the sum of all roots of the equation xm+axm-1 + bxm-2 + .
     
   • Gauss uses Euler's approach but instead of operating with roots which may not exist, Gauss operates with indeterminates.
     
   • In 1849 (on the 50th anniversary of his first proof!) Gauss produced the first proof that a polynomial equation of degree n with complex coefficients has n complex roots.
     
   • It is worth noting that despite Gauss's insistence that one could not assume the existence of roots which were then to be proved reals he did believe, as did everyone at that time, that there existed a whole hierarchy of imaginary quantities of which complex numbers were the simplest.
     
   • Euler gave the most algebraic of the proofs of the existence of the roots of an equation, the one which is based on the proposition that every real equation of odd degree has a real root.
     
   • The Argand proof is only an existence proof and it does not in any way allow the roots to be constructed.
     Go directly to this paragraph
     

3. Real numbers 2
   
   • He still only considers finite decimal expansions and realises that with these one can approximate numbers (which for him are constructed from positive integers by addition, subtraction, multiplication, division and taking nth roots) as closely as one wishes.
     
   • He considered approximations by continued fractions, and also approximations by taking successive square roots.
     
   • 2 (1964/1965), 168-180.',28)" onmouseover="window.status='Click to see reference';return true">28] van Rootselaar disagrees saying that "Bolzano's elaboration is quite incorrect".
     
   • Up to this time there was no proof that numbers existed that were not the roots of polynomial equations with rational coefficients.
     
   • Clearly √2 is the root of a polynomial equation with rational coefficients, namely x2 = 2, and it is easy to see that all roots of rational numbers arise as solutions of such equations.
     
   • A number is called transcendental if it is not the root of a polynomial equation with rational coefficients.
     
   • If x undergoes a continuous and constant increase from zero, then will pass successively through every state of positive ration b, and therefore that every determined positive ration b has one determined square root √b which will be commensurable or incommensurable according as b can or cannot be expressed as the square of a fraction.
     
   • When b cannot be so expressed, it is still possible to approximate in fractions to the incommensurable square root √b by choosing successively larger and larger positive denominators ..
     

4. Real numbers 1
   
   • Theodorus was writing out for us something about roots, such as the sides of squares whose area was 3 or 5 units, showing that the sides are incommensurable with the unit: he took the examples up to 17, but there for some reason he stopped.
     
   • The same idea of irrationality proof was expressed by Zeuthen and van der Waerden for the ratio of the diagonal and side of the square also, as well as for the square roots of 3, 5, ..
     
   • Knorr set out a new theory, trying especially to explain better why Theodoros stopped just at the square root of 17.
     
   • His theory is some kind of geometrical version on the irrationality proof of the square root of 2 known from school.
     
   • Hence mathematicians studied magnitudes which had lengths which, in modern terms, could be formed from positive integers by addition, subtraction, multiplication, division and taking square roots.
     
   • The Arabic mathematicians went further with constructible magnitudes for they used geometric methods to solve cubic equations which meant that they could construct magnitudes whose ratio to a unit length involved cube roots.
     
   • Fibonacci, using skills learnt from the Arabs, solved a cubic equation showing that its root was not formed from rationals and square roots of rationals as Euclid's magnitudes were.
     
   • Although no conceptual advances were taking place, by the end of the fifteenth century mathematicians were considering expressions build from positive integers by addition, subtraction, multiplication, division and taking nth roots.
     
   • By the sixteenth century rational numbers and roots of numbers were becoming accepted as numbers although there was still a sharp distinction between these different types of numbers.
     
   • He argued strongly in L'Arithmetique (1585) that all numbers such as square roots, irrational numbers, surds, negative numbers etc should all be treated as numbers and not distinguished as being different in nature.
     
   • Thesis 3:nnnThat any given root is a number.
     

5. Bolzano publications.html
   
   • Bernard Bolzano, Ein Lebensbild (German), Bernard Bolzano-Gesamtausgabe, Herausgegeben von Eduard Winter, Jan Berg, Friedrich Kambartel, Bob van Rootselaar.
     
   • Bernard Bolzano, Bolzano's corrections to his Functionenlehre, Edited by Bob van Rootselaar, Janus 56 (1969), 1-21.
     
   • Herausgegeben von Eduard Winter, Jan Berg, Friedrich Kambartel, Jaromir Louzil, Bob van Rootselaar (Friedrich Frommann Verlag (Gunther Holzboog), Stuttgart, 1972).
     
   • Zweite Auflage (mit Einleitung, Anmerkungen, Register und Bibliographie neu herausgegeben von Bob von Rootselaar) des Neudrucks von 1921 (durch Alois Hofler herausgegeben mit Anmerkungen von Hans Hahn) der Auflage von 1851.
     
   • Wissenschaftliche Tagebucher, Band 2: Miscellanea Mathematica, 1 und 2 (German), Herausgegeben von Bob van Rootselaar und Anna van der Lugt (Friedrich Frommann Verlag (Gunther Holzboog), Stuttgart, 1977).
     
   • Band 2: Part II, Edited and with a forward by Bob van Rootselaar and Anna van der Lugt (Friedrich Frommann Verlag (Gunther Holzboog), Stuttgart, 1979).
     
   • Miscellanea mathematica 3, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1983).
     
   • 3 Part 2] Miscellanea mathematica 4, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt, With an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1989).
     
   • 4 Part 1] Miscellanea mathematica 5, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt.
     
   • With an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1990).
     
   • 4 Part 2] Miscellanea mathematica 6, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt.
     
   • With an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1991).
     
   • 5 Part 1] Miscellanea mathematica 7, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt.
     
   • With an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1993).
     
   • 5 Part 2] Miscellanea mathematica 8, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt.
     
   • With an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1993).
     
   • 6 Part 1] Miscellanea mathematica 9, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt.
     
   • With an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1994).
     
   • 6 Part 2] Miscellanea mathematica 10, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt.
     
   • With an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1995).
     
   • 7 Part 1] Miscellanea mathematica 11, With an introduction by Bob van Rootselaar.
     
   • Edited and with a foreword by van Rootselaar and Anna van der Lugt (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1996).
     
   • 7 Part 2] Miscellanea Mathematica 12, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1997).
     
   • 8 Part 1] Miscellanea Mathematica 13, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt.
     
   • With an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1998).
     
   • 8 Part 2] Miscellanea Mathematica 14, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt.
     
   • With an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1999).
     
   • 9 Part 1] Miscellanea Mathematica 15, With an introduction by Bob van Rootselaar.
     
   • Edited and with a foreword by van Rootselaar and Anna van der Lugt (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 2000).
     
   • [Theory of functions] Edited, with a foreword and an introduction by Bob van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 2000).
     
   • 9 Part 2] Miscellanea Mathematica 16, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt, and an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 2001).
     
   • 10 Part 1] Miscellanea Mathematica 17, With an introduction by Bob van Rootselaar.
     
   • Edited and with a foreword by van Rootselaar and Anna van der Lugt (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 2002).
     
   • 10 Part 2] Miscellanea Mathematica 18, With a foreword by Bob van Rootselaar and Anna van der Lugt (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 2004).
     
   • 11 Part 1] Miscellanea Mathematica 19, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt.
     
   • With an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog, Bad Cannstatt, 2005).
     

6. Fair book
   
   • (Note that the square root is done long hand without the use of logs.) .
     
   • Note that the square root is done long hand without the use of logs or without the obvious 12 × 6.
     
   • He doesn't use logs but uses long multiplication and square roots.
     
   • He then takes the square root (long-hand) to obtain 130.38.
     
   • The actual answer should be 614.4 and Walker's error comes from the fact that he calculates the square root of 38250 as 195.5 ending the calculation too soon since the correct answer should be 195.576.
     
   • Walker only computes the square root correct to two decimal places which is where the error arises.
     
   • Again error comes from stopping the square root computation after one decimal place without realising that the next figure would be 9.
     
   • Long-hand multiplication and square root to get 290 : 280 : : 609 : x .
     
   • He then multiplies 2304 by 1290 and takes the square root to get 1728.
     
   • (4000 × Tang 45°)/Sine 90°, takes the square root, then divides by 4 to obtain 15.80.
     
   • (3000 × Tang 40°)/Sine 90°, takes the square root, then divides by 4 to obtain 12.54.
     
   • Using six figure logs he multiplies 7.1111/9 by 18 and takes the cube root to obtain .
     
   • As in the previous problem, Walker computes 36 × 7.111 1/9 and takes the cube root to obtain .
     
   • As in the previous problem, Walker computes 42 × 7.111 1/9 and takes the cube root to obtain .
     
   • 6 × 57.3 then takes the cube root.
     
   • He divides the answer by .7854 (which is π/4) and takes the square root to obtain 50.
     
   • He then takes the square root, multiplies by 4 and obtains 942.
     
   • Finally he takes the cube root to obtain 22.25.
     
   • He then takes the square root of the product 78.54 × 314.16 to obtain 157.08 .
     

7. Tartaglia versus Cardan
   
   • Find me a number which when multiplied by its root plus three will make twenty-one.
     
   • Find me a number that when its cube root is added to it, the result is six, that is 6.
     
   • A man sells a sapphire for 500 ducats, making a profit of the cube root of his capital.
     
   • There is a tree, 12 braccia high, which was broken into parts at such a point that the height of the part which was left standing was the cube root of the length of the part that was cut away.
     
   • There are two bodies of twenty triangular faces the areas of which when added together make 700 braccia, and the area of the smaller is the cube root of the larger.
     
   • Of their cube roots subtracted .
     
   • You will take the cube roots added together, .
     
   • To which I reply, that, if by a mathematician you mean someone like you, that is, someone who spends the whole time on roots, fifth powers, cubes and other trifles, then you are quite right.
     
   • I promise you that if it were up to me to reward you, taking example from the custom of Alexander, I would load you up so much with roots and radishes that you would never eat anything else in your life.
     
   • Find me six quantities in continuous proportion starting with one, such that the double of the second and the triple of the third is equal to the root of the sixth.
     

8. Jaina mathematics
   
   • In Satkhandagama various sets are operated upon by logarithmic functions to base two, by squaring and extracting square roots, and by raising to finite or infinite powers.
     
   • The first square root multiplied by the second square root is the cube of the second square root.
     
   • The second square root was the fourth root of a number.
     
   • the second square root multiplied by the third square root is the cube of the third square root.
     
   • The third square root was the eighth root of a number.
     

9. Bakhshali manuscript
   
   • Another interesting piece of mathematics in the manuscript concerns calculating square roots.
     
   • In the case of a non-square number, subtract the nearest square number, divide the remainder by twice this nearest square; half the square of this is divided by the sum of the approximate root and the fraction.
     
   • this is subtracted and will give the corrected root.
     
   • The following examples also occur in the Bakhshali manuscript where the author applies the formula to obtain approximate square roots: .
     
   • 11 (2) (1976), 112-124.',6)" onmouseover="window.status='Click to see reference';return true">6] derives from the Bakhshali square root formula an iterative scheme for approximating square roots.
     

10. Babylonian Pythagoras references
    
    • A Ahmad, On Babylonian and Vedic square root of 2, Ganita Bharati 16 (1-4) 1994), 1-4.
      
    • E M Bruins, Square roots in Babylonian and Greek mathematics, Nederl.
      
    • D Fowler and E Robson, Square root approximations in old Babylonian mathematics: YBC 7289 in context, Historia Math.
      
    • S Ilic, M S Petkovic and D Herceg, A note on Babylonian square-root algorithm and related variants, Novi Sad J.
      
    • K Muroi, Extraction of cube roots in Babylonian mathematics, Centaurus 31 (3-4) (1988), 181-188.
      
    • K Muroi, Extraction of square roots in Babylonian mathematics, Historia Sci.
      

11. Babylonian Pythagoras references
    
    • A Ahmad, On Babylonian and Vedic square root of 2, Ganita Bharati 16 (1-4) 1994), 1-4.
      
    • E M Bruins, Square roots in Babylonian and Greek mathematics, Nederl.
      
    • D Fowler and E Robson, Square root approximations in old Babylonian mathematics: YBC 7289 in context, Historia Math.
      
    • S Ilic, M S Petkovic and D Herceg, A note on Babylonian square-root algorithm and related variants, Novi Sad J.
      
    • K Muroi, Extraction of cube roots in Babylonian mathematics, Centaurus 31 (3-4) (1988), 181-188.
      
    • K Muroi, Extraction of square roots in Babylonian mathematics, Historia Sci.
      

12. Babylonian mathematics
    
    • Notice that in each case this is the positive root from the two roots of the quadratic and the one which will make sense in solving "real" problems.
      
    • Take its square root (from a table of squares) to get 8; 30.
      
    • Kevin Brown (Information about Babylonian square roots) .
      

13. Group theory
    
    • In studying the cubic, for example, Lagrange assumes the roots of a given cubic equation are x', x'' and x'''.
      Go directly to this paragraph
      
    • Then, taking 1, w, w2 as the cube roots of unity, he examines the expression .
      Go directly to this paragraph
      
    • and notes that it takes just two different values under the six permutations of the roots x', x'', x'''.
      Go directly to this paragraph
      
    • His first paper on the subject was in 1815 but at this stage Cauchy is motivated by permutations of roots of equations.
      Go directly to this paragraph
      
    • Abel, in 1824, gave the first accepted proof of the insolubility of the quintic, and he used the existing ideas on permutations of roots but little new in the development of group theory.
      Go directly to this paragraph
      

14. Babylonian mathematics references
    
    • J Friberg, The third millenium roots of Babylonian mathematics.
      
    • S Ilic, M S Petkovic and D Herceg, A note on Babylonian square-root algorithm and related variants, Novi Sad J.
      
    • K Muroi, Extraction of cube roots in Babylonian mathematics, Centaurus 31 (3-4) (1988), 181-188.
      

15. Nine chapters
    
    • Problems 12 to 18 involve the extraction of square roots, and the remaining problems involve the extraction of cube roots.
      
    • Quadratic equations are considered for the first time in Chapter 9, are solved by an analogue of division using ideas from geometry, in fact from the Chinese square-root algorithm, rather than from algebra.
      

16. Chinese overview
    
    • He solved cubic equations by extending an algorithm for finding cube roots.
      
    • He improved methods for finding square and cube roots, and extended the method to the numerical solution of polynomial equations computing powers of sums using binomial coefficients constructed with Pascal's triangle.
      
    • He described multiplication, division, root-extraction, quadratic and simultaneous equations, series, computations of areas of a rectangle, a trapezium, a circle, and other figures.
      

17. Babylonian mathematics references
    
    • J Friberg, The third millenium roots of Babylonian mathematics.
      
    • S Ilic, M S Petkovic and D Herceg, A note on Babylonian square-root algorithm and related variants, Novi Sad J.
      
    • K Muroi, Extraction of cube roots in Babylonian mathematics, Centaurus 31 (3-4) (1988), 181-188.
      

18. Squaring the circle
    
    • His proof essentially attempted to prove that π was transcendental, that is not the root of a rational polynomial equation.
      
    • However, others such as Huygens, believed that π was algebraic, that is that it is the root of a rational polynomial equation.
      
    • The final solution to the problem of whether the circle could be squared using ruler and compass methods came in 1880 when Lindemann proved that π was transcendental, that is it is not the root of any polynomial equation with rational coefficients.
      

19. Mental arithmetic
    
    • [Wallis] occupied himself in finding (mentally) the integral part of the square root of 3 cross 1040; and several hours afterwards wrote down the result from memory.
      
    • This fact having attracted notice, two months later he was challenged to extract the square root of a number of 53 digits; this he performed mentally, and a month later he dictated the answer which he had not meantime committed to writing.
      
    • For example if asked for the decimal expansion of 1/851 he would think of 851 as 23 cross 37, if asked for the square root of 851 then he thought of it as 292 + 10, if asked for the decimal expansion of 17/851 then he would think of it as almost 0.02.
      

20. Chinese numerals
    
    • For example Sun Zi, in the first chapter of the Sunzi suanjing (Sun Zi's Mathematical Manual), gives instructions on using counting rods to multiply, divide, and compute square roots.
      
    • Arithmetical rules for the abacus were analogous to those of the counting board (even square roots and cube roots of numbers could be calculated) but it appears that the abacus was used almost exclusively by merchants who only used the operations of addition and subtraction.
      

21. Set theory
    
    • However Cantor examines the set of algebraic real numbers, that is the set of all real roots of equations of the form .
      Go directly to this paragraph
      
    • These give roots 0, 1, -1.
      
    • For each index there are only finitely many equations and so only finitely many roots.
      

22. Bolzano's manuscripts
    
    • Bob van Rootselaar published Bolzano's corrections to his Functionenlehre in 1969.
      
    • The first volume in the new series Bernard Bolzano-Gesamtausgabe published by Friedrich Frommann Verlag and edited by Eduard Winter, Jan Berg, Friedrich Kambartel, Jaromir Louzil, and Bob van Rootselaar, contains a biography of Bolzano together with details of the topics on which he worked: mathematics, logic, theology, philosophy and aesthetics.
      

23. Bolzano's manuscripts references
    
    • B van Rootselaar, Bolzano's corrections to his Functionenlehre, Janus 56 (1969), 1-21.
      
    • B van Rootselaar, Bolzano's theory of real numbers, Arch.
      

24. Babylonian Pythagoras
    
    • Take the square root to obtain x - y = 0;15.
      
    • Take the square root to obtain (x + y)/2 = 0;52,30.
      

25. Fair book insert
    
    • Construct another Plan, exactly similar to the given one, to any Scale you please; find its area:_ Then say as this Area is to the given Area of the Survey, so is the square of the Scale to which this Plan was laid down, to the square of the Scale required; the Square root of which will be the square of the Scale.
      
    • The method used claims to be to divide the area by .7854 (which is π/4), then take the square root.
      

26. Arabic mathematics
    
    • It allowed the extraction of roots by mathematicians such as Abu'l-Wafa and Omar Khayyam (born 1048).
      Go directly to this paragraph
      
    • Although not the first to do so, al-Kashi gave an algorithm for calculating nth roots which is a special case of the methods given many centuries later by Ruffini and Horner.
      Go directly to this paragraph
      

27. Bolzano's manuscripts references
    
    • B van Rootselaar, Bolzano's corrections to his Functionenlehre, Janus 56 (1969), 1-21.
      
    • B van Rootselaar, Bolzano's theory of real numbers, Arch.
      

28. Bakhshali manuscript references
    
    • M N Channabasappa, On the square root formula in the Bakhshali manuscript, Indian J.
      
    • M N Channabasappa, The Bakhshali square-root formula and high speed computation, Ganita Bharati 1 (3-4) (1979), 25-27.
      

29. Bakhshali manuscript references
    
    • M N Channabasappa, On the square root formula in the Bakhshali manuscript, Indian J.
      
    • M N Channabasappa, The Bakhshali square-root formula and high speed computation, Ganita Bharati 1 (3-4) (1979), 25-27.
      

30. Pell's equation
    
    • In the continued fraction of the square root of an integer the same denominators recur periodically.
      
    • The last number in the repeating sequence is double the integer part of the square root.
      

31. Perfect numbers
    
    • I think I am able to prove that there are no even numbers which are perfect apart from those of Euclid; and that there are no odd perfect numbers, unless they are composed of a single prime number, multiplied by a square whose root is composed of several other prime number.
      
    • For example, if 22021 were prime, in multiplying it by 9018009 which is a square whose root is composed of the prime numbers 3, 7, 11, 13, one would have 198585576189, which would be a perfect number.
      

32. Mathematics and Art references
    
    • K Andersen, Ancient roots of linear perspective, in From ancient omens to statistical mechanics (Copenhagen, 1987), 75-89.
      

33. Pell's equation references
    
    • K R Johnson, An iterative method for approximating square roots, Math.
      

34. Babylonian and Egyptian references
    
    • K Muroi, Extraction of cube roots in Babylonian mathematics, Centaurus 31 (3-4) (1988), 181-188.
      

35. Pi history references
    
    • E M Bruins, With roots towards Aryabhata's π-value, Ganita Bharati 5 (1-4) (1983), 1-7.
      

36. Babylonian mathematics references
    
    • K Muroi, Extraction of cube roots in Babylonian mathematics, Centaurus 31 (3-4) (1988), 181-188.
      

37. Real numbers 2 references
    
    • B van Rootselaar, Bolzano's theory of real numbers, Arch.
      

38. Quadratic etc equations references
    
    • J P Hogendijk, Sharaf al-Din al-Tusi on the number of positive roots of cubic equations, Historia Mathematica 16 (1) (1989), 69-85.
      

39. Mathematics and Art references
    
    • K Andersen, Ancient roots of linear perspective, in From ancient omens to statistical mechanics (Copenhagen, 1987), 75-89.
      

40. Pi history references
    
    • E M Bruins, With roots towards Aryabhata's π-value, Ganita Bharati 5 (1-4) (1983), 1-7.
      

41. Real numbers 2 references
    
    • B van Rootselaar, Bolzano's theory of real numbers, Arch.
      

42. Quadratic etc equations references
    
    • J P Hogendijk, Sharaf al-Din al-Tusi on the number of positive roots of cubic equations, Historia Mathematica 16 (1) (1989), 69-85.
      

43. Debating topics
    
    • Is the square root of 2 a number? .
      

44. Mathematics and Art
    
    • completely mathematical, concerning the roots in nature from which arise this graceful and noble art.
      

45. Wave versus matrix
    
    • The thought that the laws of the macrocosmos in the small reflect the terrestrial world obviously exercises a great magic on mankind's mind, indeed its form is rooted in the superstition (which is as old as the history of thought) that the destiny of men could be read from the stars.
      

46. Newton's bucket
    
    • However, he wrote in 1872 in History and Root of the Principle of the Conservation of Energy:- .
      

47. Ring Theory
    
    • In fact, numbers of the form a + b +c2 where a, b, c are integers and is a complex cube root of 1, also have unique factorisation, and this can be used to prove the n = 3 case of Fermat's last Theorem.
      

48. Indian mathematics
    
    • And plainly it is deeply rooted in Indian soil.
      

49. function concept
    
    • This is all very well but Euler gives no definition of "analytic expression" rather he assumes that the reader will understand it to mean expressions formed from the usual operations of addition, multiplication, powers, roots, etc.
      

50. Topology history
    
    • The problem arose from studying a polynomial equation f(w, z) = 0 and considering how the roots vary as w and z vary.
      Go directly to this paragraph
      

51. Mathematical classics
    
    • The Sunzi suanjing consists of three chapters, the first describing systems of measuring with considerable detail on using counting rods to multiply, divide, and compute square roots.
      

52. Mathematics and Architecture
    
    • Much has been written on the measurements of this pyramid and many coincidences have been found with , the golden number and its square root.
      

53. Maxwell's House
    
    • Other than the topics of Maxwell's described above by Tait, there are also manuscripts by Tait on Vanishing Fractions which is l'Hopital's rule, a manuscript on Maclaurin's Theorem and On the imaginary roots of negative quantities by the Rt Rev Terrot.
      Go directly to this paragraph
      

54. Calculus history
    
    • De Beaune extended his methods and applied it to tangents where double intersection translates into double roots.
      Go directly to this paragraph
      

55. Ten classics
    
    • The Sunzi suanjing consists of three chapters, the first describing systems of measuring with considerable detail on using counting rods to multiply, divide, and compute square roots.
      

56. Egyptian mathematics
    
    • In fact there is a numerical coincidence: the square root of the golden ratio times π is close to 4, in fact this product is 3.996168.
      

57. Elliptic functions
    
    • where r(x,y) is a rational function in two variables and p(x) is a polynomial of degree 3 or 4 with no repeated roots.
      

58. Pell's equation references
    
    • K R Johnson, An iterative method for approximating square roots, Math.
      

Famous Curves

1. Newtons
   
   • The five types depend on the roots of the cubic in x on the right hand side of the equation.
     
   • (i) All the roots are real and unequal : then the Figure is a diverging Parabola of the Form of a Bell, with an Oval at its vertex .
     
   • (ii) Two of the roots are equal : a Parabola will be formed, either Nodated by touching an Oval, or Punctate, by having the Oval infinitely small .
     
   • (iii) The three roots are equal : this is the Neilian Parabola, commonly called Semi-cubical .
     
   • (iv) Only one real root : If two of the roots are impossible, there will be a Pure Parabola of a Bell-like Form .
     

2. Lituus
   
   • Cotes discovered an important theorem on the nth roots of unity; anticipated the method of least squares and discovered a method of integrating rational fractions with binomial denominators.
     

Societies etc

1. BMC 1992
   
   • Brenner, S Modules, combinatorics and Weyl roots .
     

2. BMC 2002
   
   • Veselov, A Deformed root systems and quantum integrability .
     

References

1. References for Bolzano
   
   • B van Rootselaar, Biography in Dictionary of Scientific Biography (New York 1970-1990).
     
   • B van Rootselaar, Bolzano's corrections to his Functionenlehre, Janus 56 (1969), 1-21.
     
   • B van Rootselaar, Bolzano's theory of real numbers, Arch.
     

2. References for Jia Xian
   
   • K Chemla, Similarities between Chinese and Arabic mathematical writings I : Root extraction, Arabic Sci.
     
   • R Mei, Jia Xian's additive-multiplicative method for the extraction of roots (Chinese), Studies in the History of Natural Sciences 8 (1) (1989), 1 -8.
     

3. References for Skopin
   
   • A I Skopin, p-extensions of a local field containing pM roots of 1 (Russian), Doklady Akad.
     
   • A I Skopin, p-extensions of a local field containing roots of unity of degree pm (Russian), Izv.
     

4. References for Sun Zi
   
   • X T Xu, Sun Zi suan jing ( Master Sun's arithmetical manual) was the original source for the 'leap forward and regress method of place determination' in the extraction of roots (Chinese), J.
     

5. References for Quetelet
   
   • A Oberschall, The two empirical roots of social theory and the probability revolution, The probabilistic revolution 2 (MIT Press, Cambridge, MA, 1987), 103-131.
     

6. References for Fraenkel
   
   • B van Rootselaar, Biography in Dictionary of Scientific Biography (New York 1970-1990).
     

7. References for Maclaurin
   
   • S Mills, The controversy between Colin MacLaurin and George Campbell over complex roots, 1728-1729, Arch.
     

8. References for Aryabhata II
   
   • V N Jha, Aryabhata II's method for finding cube root of a number, Ganita Bharati 19 (1-4) (1997), 60-68.
     

9. References for Archimedes
   
   • D C Gazis and R Herman, Square roots geometry and Archimedes, Scripta Math.
     

10. References for Brouwer
    
    • B van Rootselaar, W P de Roever, Biography in Dictionary of Scientific Biography (New York 1970-1990).
      

11. References for Zermelo
    
    • B van Rootselaar, Biography in Dictionary of Scientific Biography (New York 1970-1990).
      

12. References for Lanczos
    
    • G Marx, The roots of Cornelius Lanczos, in Proceedings of the Cornelius Lanczos International Centenary Conference (Philadelphia, PA, 1994), liii-lvii.
      

13. References for Chuquet
    
    • H L'Huillier, Concerning the method employed by Nicolas Chuquet for the extraction of cube roots, in C Hay (ed.), Mathematics from manuscript to print 1300-1600 (Oxford, 1988), 89-95.
      

14. References for Al-Kashi
    
    • A-K Dakhel, Al-Kashi on root extraction, Sources and Studies in the History of the Exact Sciences 2.
      

15. References for Stolz
    
    • P Ehrlich, The rise of non-Archimedean mathematics and the roots of a misconception.
      

16. References for Sprague
    
    • T B Sprague, On the nature of the curves whose intersections give the imaginary roots of an algebraic equation, Transactions of Royal Society of Edinburgh 30 (1882), 467-480.
      

17. References for Brahmagupta
    
    • S C Kak, The Brahmagupta algorithm for square rooting, Ganita Bharati 11 (1-4) (1989), 27-29.
      

18. References for Tusi
    
    • S A Ahmedov, Extraction of a root of any order and the binomial formula in the work of Nasir ad-Din at-Tusi (Russian), Mat.
      

19. References for Frege
    
    • B van Rootselaar, Biography in Dictionary of Scientific Biography (New York 1970-1990).
      

20. References for Nilakantha
    
    • T Hayashi, A set of rules for the root-extraction prescribed by the sixteenth-century Indian mathematicians, Nilakantha Somastuvan and Sankara Variyar, Historia Sci.
      

21. References for Bombelli
    
    • M T Rivolo and A Simi, The computation of square and cube roots in Italy from Fibonacci to Bombelli (Italian), Arch.
      

22. References for Aryabhata
    
    • E M Bruins, With roots towards Aryabhata's pi -value, Ganita Bharati 5 (1-4) (1983), 1-7.
      

23. References for Al-Samawal
    
    • W C Waterhouse, Note on a method of extracting roots in as-Samaw'al, Arch.
      

24. References for Newton
    
    • M Bartolozzi and R Franci, A fragment of the history of algebra : Newton's rule on the number of imaginary roots in an algebraic equation (Italian), Rend.
      

25. References for Al-Tusi Sharaf
    
    • J P Hogendijk, Sharaf al-Din al-Tusi on the number of positive roots of cubic equations, Historia Math.
      

26. References for Horner
    
    • A T Fuller, Horner versus Holdred: an episode in the history of root computation, Historia Math.
      

27. References for Hertz Heinrich
    
    • L Corry, The empiricist roots of Hilbert's axiomatic approach, in Proof theory, Roskilde, 1997 (Kluwer Acad.
      

28. References for Heron
    
    • J G Smyly, Square roots in Heron of Alexandria, Hermathena 63 (1944), 18-26.
      

29. References for Couturat
    
    • I Grattan-Guinness, The Search for Mathematical Roots 1870-1940 (Princeton Uni.
      

30. References for Al-Tusi Nasir
    
    • S A Ahmedov, Extraction of a root of any order and the binomial formula in the work of Nasir ad-Din at-Tusi (Russian), Mat.
      

31. References for Aryabhata I
    
    • E M Bruins, With roots towards Aryabhata's p-value, Ganita Bharati 5 (1-4) (1983), 1-7.
      

Additional material

1. Muslim extraction of roots
   
   • Muslim extraction of roots .
     
   • The extraction of roots of numbers higher than cube roots, was, according to the writings of Omar Khayyam, an achievement of Muslim scholars.
     
   • From the Indians one has methods for obtaining square and cube roots, methods which are based on knowledge of individual cases, namely the knowledge of the squares of the nine digits 12, 22 , 32 (etc.) and their respective products, i.e.
     
   • In addition we have increased their types, namely in the form of the determination of the fourth, fifth, sixth roots up to any desired degree.
     
   • In fact al-Kashi had extracted the fifth root of 44 240 899 506 176.
     
   • For those who are not able to do this today, we inform the reader that the fifth root is 536 (which we admit we computed using Maple!).
     

2. Al-Khwarizmi and quadratic equations
   
   • Halve the number of the roots.
     
   • Extract its square root, 2, and subtract this from half the number of roots, 5.
     
   • This is the root you wanted, whose square is 9.
     
   • Alternatively, you may add the square root to half the number of roots and the sum is 7.
     
   • This is then the root you wanted and the square is 49.
     
   • In this case, both addition and subtraction can be used, which will not serve in any other of the three cases where the number of roots is to be halved.
     
   • Know also that when, in a problem leading to this case, you have multiplied half the number of roots by itself, if the product is less than the number of dirhams added to the square term, then the case is impossible.
     
   • On the other hand, if the product is equal to the dirhams themselves, then the root is half the number of roots.
     

3. Dickson: 'Theory of Equations
   
   • For instance, one may be sure that a given cubic equation has only the one real root seen in the graph, if the bend points lie on the same side of the x-axis.
     
   • It is surprising that the theorems of Descartes, Budan, and Sturm, on the real roots of an equation, are often stated inaccurately.
     
   • The treatment of roots of unity is concrete, in contrast to the usual abstract method.
     
   • This chapter should be read by everyone who thinks that complex numbers are "imaginary" and that we gain nothing by their use except to make certain equations have roots.
     
   • The theorem that an integral root of an equation with integral coefficients divides the constant term might well be supplemented by the similar theorem that if an equation with integral coefficients has a fractional root a/b, a must divide the constant term and b the coefficient of the highest power of x.
     
   • This gives in general a simpler way to find such roots than that given on page 62.
     
   • The usual theorems for the isolation of the roots are given in Chap.
     
   • The use of elementary calculus allows a clear treatment and a complete solution of the problem, "given an equation to locate its real roots," while the methods of Chap.
     
   • Besides Horner's well-known method for the numerical computation of roots, Newton's is given and emphasized as one that is effective for non-algebraic as well as for algebraic equations; and Graffe's little known but very ingenious scheme of solution by forming equations whose roots are powers of the roots of the given equation, and Lagrange's solution by continued fractions are also explained.
     

4. Bolzano's publications
   
   • Bernard Bolzano, Ein Lebensbild (German), Bernard Bolzano-Gesamtausgabe, Herausgegeben von Eduard Winter, Jan Berg, Friedrich Kambartel, Bob van Rootselaar.
     
   • Bernard Bolzano, Bolzano's corrections to his Functionenlehre, Edited by Bob van Rootselaar, Janus 56 (1969), 1-21.
     
   • Herausgegeben von Eduard Winter, Jan Berg, Friedrich Kambartel, Jaromir Louzil, Bob van Rootselaar (Friedrich Frommann Verlag (Gunther Holzboog), Stuttgart, 1972).
     
   • Zweite Auflage (mit Einleitung, Anmerkungen, Register und Bibliographie neu herausgegeben von Bob von Rootselaar) des Neudrucks von 1921 (durch Alois Hofler herausgegeben mit Anmerkungen von Hans Hahn) der Auflage von 1851.
     
   • Wissenschaftliche Tagebucher, Band 2: Miscellanea Mathematica, 1 und 2 (German), Herausgegeben von Bob van Rootselaar und Anna van der Lugt (Friedrich Frommann Verlag (Gunther Holzboog), Stuttgart, 1977).
     
   • Band 2: Part II, Edited and with a forward by Bob van Rootselaar and Anna van der Lugt (Friedrich Frommann Verlag (Gunther Holzboog), Stuttgart, 1979).
     
   • Miscellanea mathematica 3, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1983).
     
   • 3 Part 2] Miscellanea mathematica 4, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt, With an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1989).
     
   • 4 Part 1] Miscellanea mathematica 5, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt.
     
   • With an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1990).
     
   • 4 Part 2] Miscellanea mathematica 6, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt.
     
   • With an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1991).
     
   • 5 Part 1] Miscellanea mathematica 7, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt.
     
   • With an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1993).
     
   • 5 Part 2] Miscellanea mathematica 8, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt.
     
   • With an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1993).
     
   • 6 Part 1] Miscellanea mathematica 9, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt.
     
   • With an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1994).
     
   • 6 Part 2] Miscellanea mathematica 10, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt.
     
   • With an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1995).
     
   • 7 Part 1] Miscellanea mathematica 11, With an introduction by Bob van Rootselaar.
     
   • Edited and with a foreword by van Rootselaar and Anna van der Lugt (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1996).
     
   • 7 Part 2] Miscellanea Mathematica 12, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1997).
     
   • 8 Part 1] Miscellanea Mathematica 13, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt.
     
   • With an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1998).
     
   • 8 Part 2] Miscellanea Mathematica 14, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt.
     
   • With an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 1999).
     
   • 9 Part 1] Miscellanea Mathematica 15, With an introduction by Bob van Rootselaar.
     
   • Edited and with a foreword by van Rootselaar and Anna van der Lugt (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 2000).
     
   • [Theory of functions] Edited, with a foreword and an introduction by Bob van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 2000).
     
   • 9 Part 2] Miscellanea Mathematica 16, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt, and an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 2001).
     
   • 10 Part 1] Miscellanea Mathematica 17, With an introduction by Bob van Rootselaar.
     
   • Edited and with a foreword by van Rootselaar and Anna van der Lugt (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 2002).
     
   • 10 Part 2] Miscellanea Mathematica 18, With a foreword by Bob van Rootselaar and Anna van der Lugt (Friedrich Frommann Verlag Gunther Holzboog GmbH & Co., Stuttgart, 2004).
     
   • 11 Part 1] Miscellanea Mathematica 19, Edited and with a foreword by Bob van Rootselaar and Anna van der Lugt.
     
   • With an introduction by van Rootselaar (Friedrich Frommann Verlag Gunther Holzboog, Bad Cannstatt, 2005).
     

5. St Andrews Mathematics Examinations
   
   • If α and β be the roots of the equation ax2 + bx + c = 0, prove α + β = -b/a and αβ = c/a.
     
   • The roots of 8x2 - mx + 9 = 0 are in the ratio of 2 : 1; find them.
     
   • Exemplify it to find the three cube roots of √-1, and also to find expressions for all the values of (a + b√-1)1/2.
     
   • Prove that in an equation with real coefficients, imaginary roots occur in pairs.
     
   • Prove that in any equation the number of positive roots cannot exceed the number of changes in the signs of the coefficients.
     
   • Prove that the equation x7 - 3x4 - 4x2 + x - 1 = 0 cannot have more than three real roots.
     
   • Find the positive root to 3 places of figures of x3 - 6x - 13 = 0.
     
   • and hence find the mass of a sphere whose density varies inversely as the square root of the distance from the centre.
     

6. Bombelli: 'Algebra
   
   • We quote from the text where Bombelli is using fractions to approximate to square roots.
     
   • Let us first assume that if we wish to find the approximate root of 13 that this will be 3 with 4 left over.
     
   • This is the first fraction to be added to the 3, making 3 2/3 which is the approximate root of 13.
     
   • Let us suppose we are required to find the root of 13.
     
   • The nearest square is 9, which has root 3.
     
   • I let the approximate root of 13 be 3 plus 1 unknown.
     
   • and the approximate value of the root is 3 2/3 since it has been set equal to 3 plus 1 unknown.
     
   • Of course we can also see that Bombelli's procedure leads to a continued fraction expansion of a square root.
     

7. Bronowski and retrodigitisation
   
   • In terms of primitive roots discussed in [22], this reflects the fact that 10 is a primitive root modulo 7.
     
   • Already Johann Carl Friedrich Gauss (1777--1855), in Disquisitiones Arithmeticae, raised the question of determining the primes p for which 10 is a primitive root modulo p.
     
   • Emil Artin (1898--1962) distilled the ensuing investigations of special cases in a general conjecture in 1927: any integer m, other than 0 or -1, and not divisible by a square, is the primitive root of infinitely many primes; and such primes have positive denisity in the set of primes independent of the choice of m.
     
   • Although much progress has been made on this conjecture, it has been of a conditional or non-constructive kind, and, as yet, no m is known which is a primitive root for infinitely many primes.
     
   • Robbins, Note 59.15: Calculating a primitive root (mod pe), Math.
     

8. Henry Baker addresses the British Association in 1913, Part 2
   
   • All equation of the fourth order can be solved by means of a cubic equation, because there exists a rational function of the four roots which takes only three values when the roots are exchanged in all possible ways.
     
   • Following out this suggestion for an equation of any order, we are led to consider, taking any particular rational function of its roots, what is the group of interchanges among them which leaves this function unaltered in value.
     
   • The attempt to extend the possibilities of integration to the case when the function to be integrated involves the square root of a polynomial of the fourth order, led first, after many efforts, among which Legendre's devotion of forty years was part, to the theory of doubly-periodic functions.
     
   • The rose is no less sweet because its sweetness is conditioned by the food we supply to its roots.
     

9. D'Arcy Thompson on Greek irrationals
   
   • Similar tables can be constructed, as the Greeks well knew, for other square roots; and the way to construct them is in each case easy to discover.
     
   • Of the two series which thus begin alike and then part company, the one leads to the square-root of 2 or the hypotenuse of an isosceles right-angled triangle, and the other leads to the Divine or Golden Section.
     
   • The cube root of 2 is another story.
     
   • I feel pretty sure that the peculiar importance and air of mystery attaching to the Delian Problem arose simply from the fact that for the extraction of a cube root you have no such arithmetical device as that by which a square root can be extracted so easily and so accurately.
     

10. Born Inaugural
    
    • The philosophy of the nineteenth century on which classical physics relied is deeply rooted in the ideas of David Hume.
      
    • Kant's work can be considered as a kind of enormous generalisation of this question; he attempted to formulate the postulates, which he called categories a Priori, necessary to build up experience in general, and, he discussed the roots of their validity.
      
    • Kant believed that the root of the validity of the first kind was "pure reason" itself, whereas the second kind came from a special ability of our brain, differing from reason, which he called "pure intuition" (reine Anschauung).
      

11. Ford - Mathematics for Field Artillery
    
    • Such terms as mil, grad, deflection, azimuth, dispersion diagram, probable error, centre of impact, nomogram, or alignment chart ' meant little or nothing to many college instructors in mathematics, while most college graduates could not approximate a square root by the algorithm, determine visibility on a map with the contour lines given, interpolate with second differences, and the like.
      
    • In the section on arithmetic the most interesting features emphasized are the conversion of metric into English units and vice versa, various units of angular measure, mil, grad, degree, the process of extracting the square root, and interpolation.
      
    • The tables are to four decimal places, and comprise (I) square roots of numbers, (II) logarithms of numbers, (III) natural trigonometric tables-mils, (IV) logarithmic trigonometric tables - mils, (V) natural and logarithmic tables - degrees.
      

12. ELOGIUM OF EULER
    
    • After having provided the steps to the roots of algebraic equations, and their general solvability, numerous new theories and some ingenious and insightful views, Mr.
      
    • Cotes had already provided the way in which to represent the roots of certain algebraic equations by sine and cosine.
      
    • This was done by searching for the sums or the expression of their general terms and to those of the roots or determinant equations, by which to obtain with a simple calculation the approximate value of the products or the indefinite sums of certain numbers.
      
    • Euler had the intention of exercising his grandson's memory to extract roots and to tabulate the first six powers of all numbers from 1 to 100 and to keep them firmly within the memory.
      

13. Euler Elogium.html.html
    
    • After having provided the steps to the roots of algebraic equations, and their general solvability, numerous new theories and some ingenious and insightful views, Mr.
      
    • Cotes had already provided the way in which to represent the roots of certain algebraic equations by sine and cosine.
      
    • This was done by searching for the sums or the expression of their general terms and to those of the roots or determinant equations, by which to obtain with a simple calculation the approximate value of the products or the indefinite sums of certain numbers.
      
    • Euler had the intention of exercising his grandson's memory to extract roots and to tabulate the first six powers of all numbers from 1 to 100 and to keep them firmly within the memory.
      

14. Ernest Hobson addresses the British Association in 1910
    
    • These ideals have their roots in irresistible impulses and deep-seated needs of the human mind, manifested in its efforts to introduce intelligibility into certain great domains of the world of thought.
      
    • The generalised dynamical scheme developed by Lagrange and Hamilton, with its power of dealing with systems, the detailed structure of which is partially unknown, has however proved a powerful weapon of attack, and affords a striking instance of the deep-rooted significance of mathematical form.
      

15. L'Hôpital: 'Analyse des infiniment petits' Preface
    
    • But since he was mainly concerned with the solution of equations he was interested in curves only as a way to finding roots.
      
    • Barrow's work did not stop there: he also invented a kind of calculus based on this method, but this calculus, like that of Descartes, could only be used once all fractions and roots had been removed.
      
    • This calculus is of immense scope: it can be used for the curves which occur in mechanics, transcendental curves such as the catenary, as well as for purely geometrical curves, squares or other roots do not cause any difficulty (and may even be an advantage), any number of variables may be considered, and it is equally easy to compare infinitely small quantities of any type.
      

16. Hans Hahn: 'The crisis in intuition
    
    • The task of completely formalizing mathematics, of reducing it entirely to logic, was arduous and difficult; it meant nothing less than a reform in root and branch ..
      
    • For it is not true, as Kant urged, that intuition is a pure a priori means of knowledge, but rather that it is force of habit rooted in psychological inertia.
      

17. G A Miller - A letter to the editor
    
    • For instance, recent discoveries relating to the finding of at least one root by the ancient Babylonians of certain numerical quadratic and cubic equations throws new light on the history of algebra and on the contributions made by the Greeks and the Arabians towards the solution of algebraic equations.
      
    • The given tables then enabled them to find a real definite root, at least approximately, when such a root exists.
      

18. EMS obituary
    
    • The attempt to extend the possibilities of integration to the case when the function to, be integrated involves the square root of a polynomial of the fourth order, led first, after many efforts, among which Legendre's devotion of forty years was part, to the theory of doubly-periodic functions.
      
    • House of the rooted hearts and long carouses, .
      

19. EMS obituary
    
    • The Turning Values of a Cubic Function, and the Nature of the Roots of a Cubic Equation.
      
    • The Turning Values of Cubic and Quartic Functions, and the Nature of the Roots of Cubic and Quartic Equations.
      

20. Phillip Griffiths Looks at 'Two Cultures' Today
    
    • confident at the roots ..
      
    • As modern as some of their views may seem to them, it has deep roots in the English Industrial Revolution of the late 18th century and the anti-industrial fervour of the Luddites.
      

21. Napier's rods
    
    • Let us note that Napier also designed "square root rods" and "cube root rods" but these did not become so popular.
      

22. Einar Hille: 'Analytic Function Theory
    
    • These general considerations have led to the following arrangement of the subject matter of Volume I: After a preliminary study of number systems, the geometry of the complex plane is developed, and simple functions such as linear fractions, powers, and roots are studied.
      

23. Mathematics at Aberdeen 1
    
    • This covered the basic operations of addition, subtraction, multiplication and division of integers; going on to fractions, proportion and the extraction of square and cube roots.
      

24. Poincaré on intuition in mathematics
    
    • M Meray wants to prove that a binomial equation always has a root, or, in ordinary words, that an angle may always be subdivided.
      

25. Three Sadleirian Professors
    
    • He also mentions the Hardy identity, the Hardy-Landau identity, and Hardy's theorem on the roots of the Zeta-functions.
      

26. Mandelbrot's Foreword to Dauben's Abraham Robinson
    
    • More specifically, in order to understand human creativity, I think we must investigate in detail how major creators have balanced the necessary but conflicting needs of deep-rootedness and personal boldness, how they have assessed the relative importance of their own private drumbeat, the drumbeat of the family and the professional community, and the drumbeat of society at large.
      

27. Mathematical Works of Colin Maclaurin
    
    • A Letter to Mr Folkes on Equations with Impossible Roots, May, 1726, No.
      

28. Von Neumann: 'The Mathematician' Part 2
    
    • are various important parts of modern mathematics in which the empirical origin is untraceable, or, if traceable, so remote that it is clear that the subject has undergone a complete metamorphosis since it was cut off from its empirical roots.
      

29. Kurosh's book The theory of groups 1st edition
    
    • A basic course of higher algebra is a prerequisite only for some initial examples of groups, such as matrices, permutations, roots of unity.
      

30. R L Wilder: 'Cultural Basis of Mathematics III
    
    • I refer of course to the drill type of teaching which may enable stupid John to get a required credit in mathematics but bores the creative minded William to the extent that he comes to loathe the subject! What essential difference is there between teaching a human animal to take the square root of 2 and teaching a pigeon to punch certain combinations of coloured buttons? Undoubtedly the symbolic reflex type of teaching is justified when the pupil is very young - closer to the so-called "animal" stage of his development, as we say.
      

31. Mathematicians and Music 2.2
    
    • The dominant and not the tonic is thus the root, of the whole scale.
      

32. George Temple's Inaugural Lecture I
    
    • The root of the difficulty is that the proper medium for the accurate expression of physical principles, methods, and conclusions is mathematics: and that physical mathematics, regarded as a language, has an almost untranslatable vocabulary.
      

33. Carl Runge: 'Graphical Methods
    
    • We may in a more generalized form state it thus: Find the integral numbers, which are the coefficients of an algebraic equation, of which is one of the roots.
      

34. The Works of Sir John Leslie
    
    • But his setting out of the Euclidean herbaceous border will be viewed with interest by the teacher, who may find incidentally many valuable rooted cuttings, suitable for transporting to the forcing frame of the examination paper.
      

35. Eddington on the Expanding Universe
    
    • R is the Einstein radius of the world, equal to the inverse square root of the cosmical constant; N is the number of electrons (or protons) in the universe.
      

36. Lucretius: 'On the Nature of Things
    
    • And all those which are driven together in more close-packed union and leap back but a little space apart, entangled by their own close-locking shapes, these make the strong roots of rock and the brute bulk of iron and all other things of their kind.
      

37. Kurosh: 'The theory of groups' 1st edition
    
    • A basic course of higher algebra is a prerequisite only for some initial examples of groups, such as matrices, permutations, roots of unity.
      

38. Eddington on the Expanding Universe
    

39. EMS obituary
    
    • A continuous process for root extraction by calculating machines (Proc.
      

40. EMS 1913 Colloquium
    
    • The Newtonian measure of momentum - mass multiplied by velocity - has in this system to be divided by the square root of the excess of unity over the squared ratio of the velocity of the particle to the velocity of light.
      

41. George Chrystal's First Promoter's Address
    
    • Such a competition is very rare, even in Germany, where the system of private teachers has been rooted for a long time.
      

42. E C Titchmarsh on Counting
    
    • The book covers Counting, Arithmetic, Algebra, The use of numbers in geometry, Irrational numbers, Indices and logarithms, Infinite series and e, The square root of minus one, Trigonometry, Functions, The differential calculus, The integral calculus and Aftermath (see this link).
      

43. Tietze: 'Famous Problems of Mathematics
    
    • It is not so much special preliminary knowledge that is required, as an emancipation from firmly rooted ideas, from ideas, moreover, which an outsider to mathematics usually has not pursued critically enough.
      

44. Sommerfeld: 'Atomic Structure
    
    • An almost greater enlightenment has resulted from the seven years of Rontgen spectroscopy, inasmuch as it has attacked the problem of the atom at its very root, and illuminates the interior.
      

45. A N Whitehead addresses the British Association in 1916
    
    • In the first place, science is rooted in what I have just called the whole apparatus of common-sense thought.
      

46. Cajori: 'A history of mathematics' Introduction
    
    • The easy credulity with which a young student supposes that of course every algebraic equation must have a root gives place finally to a delight in the slow conquest of the realm of imaginary numbers, and in the youthful genius of a Gauss who could demonstrate this once obscure fundamental proposition." The history of mathematics is important also as a valuable contribution to the history of civilisation.
      

47. Whittaker EMS Obituary.html
    
    • The advertisement intimating the opening of the laboratory in October 1913 specified the subjects to be taught (some of which were interpolation, method of least squares, solution of systems of linear equations, evaluation of determinants, determination of roots of transcendental equations, practical Fourier analysis, evaluation of definite integrals, numerical solution of differential equations, construction of tables of functions not previously tabulated such as parabolic cylinder functions) and also indicated that facilities were available for original research and that the University would grant recognition, under certain conditions, to research students who would be permitted to offer themselves for the degree of D.Sc.
      

48. Chrystal.html
    
    • When Chrystal came to Edinburgh he rooted up the humours of the classroom as a dentist draws teeth.
      

49. Edward Sang on his tables
    
    • In the usual way, by means of the extraction of the square root, the quadrant was divided into ten equal parts, and the sines of these computed to thirty-three, for thirty places.
      

50. EMS obituary
    
    • Muirhead had a rooted aversion to compulsion of any kind - an "anarchist" was what he used to call himself.
      

51. Fuss eulogy notes.html
    
    • He often proposes numbers for which the extractions of their roots were powers.
      

52. Heinrich Tietze on Numbers
    
    • [0ne would have to consider how the consequent change in manual skill would have influenced the whole history of tools and weapons and the division of peoples according to their linguistic roots; if not indeed a whole new world.] For convenience, we shall use the word 'year' for twelve and shall borrow the tenth and eleventh letters of the Greek alphabet, kappa (k) and lambda (l), for the patterns with six fingers of one hand and four of the second hand (6 + 4), and six fingers of one hand and five of the second hand (6 + 5), respectively.
      

53. Wave versus matrix mechanics
    
    • The thought that the laws of the macrocosmos in the small reflect the terrestrial world obviously exercises a great magic on mankind's mind, indeed it form is rooted in the superstition (which is as old as the history of thought) that the destiny of men could be read from the stars.
      

54. Halmos: creative art
    
    • At the same time, as the quantity of mathematics grows and the number of people who think about it keeps doubling over and over again, more new concepts need explication, more new logical interrelations cry out for study, and understanding, and simplification, and more and more the tree of mathematics bears elaborate and gaudy flowers that are, to many beholders, worth more than the roots from which it all comes and the causes that brought it all into existence.
      

55. E C Titchmarsh: 'Aftermath
    
    • The book covers Counting (see this link), Arithmetic, Algebra, The use of numbers in geometry, Irrational numbers, Indices and logarithms, Infinite series and e, The square root of minus one, Trigonometry, Functions, The differential calculus, The integral calculus and Aftermath.
      

56. Professor Chrystal
    
    • When Chrystal came to Edinburgh he rooted up the humours of the classroom as a dentist draws teeth.
      

57. Adam Ries: 'Coss
    
    • Adam Ries also called it radix, root or thing.
      

58. Mandelbrot: Foreword to 'Abraham Robinson' by Dauben
    
    • More specifically, in order to understand human creativity, I think we must investigate in detail how major creators have balanced the necessary but conflicting needs of deep-rootedness and personal boldness, how they have assessed the relative importance of their own private drumbeat, the drumbeat of the family and the professional community, and the drumbeat of society at large.
      

59. Inaugural Discourse by Julio Rey Pastor
    
    • They speak to us of "deeply rooted national traditions, which ought not to be destroyed nor erased" as if we were able to assert influence over the geographical factor in a discipline so essentially international as mathematics.
      

60. Eulogy to Euler by Fuss
    
    • However, the seed which he had planted in the soul of this young Geometer did not wait to grow deep roots.
      

61. Oswald Veblen Publications
    
    • 1906 (a) "The Square Root and the Relations of Order", Trans.
      

62. EMS obituary
    
    • Other titles were "On the uniqueness of the solution of the linear differential equation of the second order" (1903) and "The condition for the reality of the roots of an n-ic" (1906).
      

63. EMS 1913 Colloquium 4.html.html
    
    • The Newtonian measure of momentum - mass multiplied by velocity - has in this system to be divided by the square root of the excess of unity over the squared ratio of the velocity of the particle to the velocity of light.
      

64. EMS honours Maxwell and Tait
    
    • The question of this time scale is at the root of the major problems of cosmic physics.
      

65. Heath: Everyman's Library 'Euclid' Introduction
    
    • The propositions embody, in fact, the general method known as the "application of areas," which was of vital consequence to the Greek geometers, being the geometrical equivalent of the solution of the general quadratic equations ax ∓ bx2/c = S so far as they have real roots.
      

66. A de Lapparent: 'Wantzel
    
    • There he took classes at the coll6ge Charlemagne, and, in 1829, Reynaud appreciated him enough to not only allow him to correct proofs of the new edition of his Arithmetic but to also introduce in that volume a demonstration concerning the square root which was suggested by the young editor.
      

67. Otto Neugebauer - a biographical sketch
    
    • In 1932 appeared no less than six distinct contributions from his pen dealing with the history of ancient algebra, the sexagesimal system and Babylonian fractions, Apollonius, Babylonian series, square root approximations, and siege calculations.
      

68. John Maynard Keynes: 'Newton, the Man
    
    • I believe that his friends, above all Halifax, came to the conclusion that he must be rooted out of the life he was leading at Trinity which must soon lead to decay of mind and health.
      

69. Eulogy to Euler by Fuss
    
    • However, the seed which he had planted in the soul of this young Geometer did not wait to grow deep roots.
      

70. Footnote 13
    
    • He often proposes numbers for which the extractions of their roots were powers.
      

Quotations

1. Quotations by Aristotle
   
   • The roots of education are bitter, but the fruit is sweet.
     

2. A quotation by Tartaglia
   
   • Of their cube roots subtracted .
     

3. Quotations by Schrodinger
   
   • We are told such a number as the square root of 2 worried Pythagoras and his school almost to exhaustion.
     

4. Quotations by Knuth
   
   • Premature optimization is the root of all evil .
     

5. Quotations by Gauss
   
   • If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question.
     

6. Quotations by Galois
   
   • Go to the roots, of these calculations! Group the operations.
     

7. Quotations by Ezra
   
   • The One counts Himself, and on-one else counts him, and He is every number, He is root, and foundation and square and cube, and He is like the essence that carries all the cases, and every number is in His power, and He is in every number in deed, and He ispresent, and every number is present because of Him, and He is Ancient, and every [other] number is [re]newed, and He is the reason for every number, pair [even] and that is not pair, He is not a number, and will not multiply and will not divide.
     

8. Quotations by De Morgan
   
   • Imagine a person with a gift of ridicule [He might say] First that a negative quantity has no logarithm; secondly that a negative quantity has no square root; thirdly that the first non-existent is to the second as the circumference of a circle is to the diameter.
     

9. Quotations by Courant
   
   • Recent trends and fashions have, however, weakened the connection between mathematics and physics; mathematicians, turning away from their roots of mathematics in intuition, have concentrated on refinement and emphasized the postulated side of mathematics, and at other times have overlooked the unity of their science with physics and other fields.
     

10. Quotations by Cantor
    
    • How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all because I have followed its roots, so to speak, to the first infallible cause of all created things.
      

11. Quotations by Titchmarsh
    
    • I met a man once who told me that far from believing in the square root of minus one, he didn't believe in minus one.
      

Chronology

1. Mathematical Chronology
   
   • The Babylonians solve linear and quadratic algebraic equations, compile tables of square and cube roots.
     
   • Theodorus of Cyrene shows that certain square roots are irrational.
     
   • He uses zero and negative numbers, gives methods to solve quadratic equations, sum series, and compute square roots.
     
   • He explains the operations of arithmetic, particularly taking square and cube roots in each system.
     
   • Rudolff introduces a symbol resembling √ for square roots in his Die Coss the first German algebra book.
     
   • Cataldi uses continued fractions in finding square roots.
     
   • Cataldi publishes Trattato del modo brevissimo di trovar la radice quadra delli numeri in which he finds square roots using continued fractions.
     
   • It can multiply, divide and extract roots.
     
   • The paper is the first to consider the roots of a equation as abstract quantities rather than numbers.
     
   • He studies permutations of the roots and this work leads to group theory.
     
   • Euler introduces the symbol i to represent the square root of -1 in a manuscript which will not appear in print until 1794.
     
   • Barlow produces Barlow's Tables which give factors, squares, cubes, square roots, reciprocals and hyperbolic logs of all numbers from 1 to 10000.
     
   • Liouville finds the first transcendental numbers - numbers that cannot be expressed as the roots of an algebraic equation with rational coefficients.
     

2. Chronology for 1760 to 1780
   
   • The paper is the first to consider the roots of a equation as abstract quantities rather than numbers.
     
   • He studies permutations of the roots and this work leads to group theory.
     
   • Euler introduces the symbol i to represent the square root of -1 in a manuscript which will not appear in print until 1794.
     

3. Chronology for 1500 to 1600
   
   • Rudolff introduces a symbol resembling √ for square roots in his Die Coss the first German algebra book.
     
   • Cataldi uses continued fractions in finding square roots.
     

4. Chronology for 500 to 900
   
   • He uses zero and negative numbers, gives methods to solve quadratic equations, sum series, and compute square roots.
     

5. Chronology for 1810 to 1820
   
   • Barlow produces Barlow's Tables which give factors, squares, cubes, square roots, reciprocals and hyperbolic logs of all numbers from 1 to 10000.
     

6. Chronology for 1840 to 1850
   
   • Liouville finds the first transcendental numbers - numbers that cannot be expressed as the roots of an algebraic equation with rational coefficients.
     

7. Chronology for 500BC to 1AD
   
   • Theodorus of Cyrene shows that certain square roots are irrational.
     

8. Chronology for 1650 to 1675
   
   • It can multiply, divide and extract roots.
     

9. Chronology for 900 to 1100
   
   • He explains the operations of arithmetic, particularly taking square and cube roots in each system.
     

10. Chronology for 30000BC to 500BC
    
    • The Babylonians solve linear and quadratic algebraic equations, compile tables of square and cube roots.
      

11. Chronology for 1600 to 1625
    
    • Cataldi publishes Trattato del modo brevissimo di trovar la radice quadra delli numeri in which he finds square roots using continued fractions.