MATerials MATemàtics MAT 2 Volum 2023, treball no. 5, 27 pp. ISSN: 1887-1097 Publicació electrònica de divulgació del Departament de Matemàtiques de la Universitat Autònoma de Barcelona www.mat.uab.cat/web/matmat A problem on the approximation of n-roots based on the Viète’s work Pedro J. Herrero Piñeyro, Antonio Linero Bas, Ma. Rosa Massa-Esteve i Antonio Mellado Romero In this paper, after giving a brief approach about the historical question of how different authors and cultures handled approximations of the n-th root of N := An + R, we con- sider √ the particular case of the approximation of n N by A + Rd (here n, A, d are positive in- tegers, with n ≥ 2, d = (A + 1)n − An − 1, and 0 < R ≤ d), explicitly considered by Viète in his De Numerosa Potestatum. Viète estab- lished in one of the precepts of his work that R the n-th root is placed between A + d+1 and R A + d . Indeed, it is easily seen that, of course, R √ √ A + d+1 ≤ n N , but unfortunately n N can be greater than A + Rd when n ≥ 3. Then the question arises whether we are able or not of determining what values of R (fixing the inte- ger values n and A, so also d) √ give a true ap- n proximation by excess when N is estimated as A + Rd , that is, to find those values of R producing an excess in relation with the exact value of the root. We give a complete answer to the cases n ≤ 7, and we present without proof the main result for a general n. 1 Introduction There is not doubt that the History of Mathematics allows us to improve both our research and docent tasks. In the first case, the reading of ancient François Viète, image from [24], https://rcin.org.pl/dlibra/publication/16160/ edition/11661 2 A problem on the approximation of n-roots. . . texts or the compilation of works of some authors, as well as the rediscov- ering of some ideas hidden in those texts, provide us a valuable source for starting new investigations and for fitting them into a more general frame- work, thus avoiding the centrifugal tendency of specialization in the different fields of mathematics, or at least to make us see a certain internal unity in mathematics. Respect to our docent role, it is clear that the introduction of historical aspects in our different lectures can influence in the positive recep- tion by our students and, therefore, it contributes to trap their attention and to bring closer a subject considered a priori rather arid. Another possible benefit is the eventual elaboration of mathematical materials for breaking the monotony of classical classroom presentations. A lot of literature is de- veloped in this direction, let us to mention, for instance, [18], [21]. Bearing all this in mind, in this paper we are going to give a brief historical note about the approximation of n-th roots by French mathematician François Viète (1540–1603) in his De Numerosa Potestatum. This historical note will be the starting point for analyzing a simple mathematical question arising from the reading of the above mentioned Viète’s work, and dealing with the possibility of finding approximations by excess of the n-roots. The paper is organized as follows. After presenting some well-known historic episodes on the numerical computation of square and cubic roots, we present in Section 3, in a broad manner, the procedure employed by Viète in his De Numerosa Potestatum to compute general n-roots as well as a precept for locating irrational roots between two approximations by default and excess. This precept applies for square roots, but for greater roots the proposed approximation by excess can be in fact less than the searched root. This gives us occasion for proposing in Section 4 a mathematical question about the validity of the precept depending on the value N = An + R whose n-root we are looking for. The question is completely solved in Section 5 for 3 ≤ n ≤ 7 by using elementary mathematics and the Descartes’ rule of signs, whereas in Section 6 we advance without proof the result for a general value n. Our paper finishes with some comments about the Viète’s work, and the importance of a suitable implementation of the History of Mathematics in our mathematical task. 2 Historical approach of n-th roots Let us present some milestone about the topic of extraction of square, cubic, and general n-roots. For the reader interested in the topic of the numerical treatment of roots of equations along the history, we strongly recommend [7], a very complete treatise of arithmetical methods from the Antiquity to the end of 19-th century, as well as [23]. The computation of square and cubic roots just appears as a practical subject of undoubted value in the Babylonian and Egyptian cultures. For P. J. Herrero, A. Linero, Ma. R. Massa i A. Mellado 3 instance, in the Babylonian culture we can find, by a numerical procedure, the computation of the measure of the diagonal of a door in the clay tablet BM 96957 +VAT 6598, see [25]. To do it, they used the approximation a + 12 Ba for the square root of a2 + B (for a geometrical explanation of this approach, the reader is referred to [14], see also Figure 1). Number= Square of approx+Bit approx Bit+approx new approx new approx=approx+half of Bit×IGI approx + a = a + (1/2) B/a √ √ a2 B/a N = a2 + B ≈ a + 1 B 2 a a2 +B (1/2) B/a (1/4) B 2 /a2 √ √ Figure 1: Geometrical explanation for the approximation by excess of N = a2 + B by a + 21 B a , image composed from [14, p. 371] Later, we find the topic in the Greek civilization. √ A simple way to proceed to the approximation of an irrational square root N consists in approaching √ 2 N by a suitably nearly ratio m n2 , with m, n natural numbers, therefore N ≈ m n . This idea was applied by Aristarchus of Samos (c. 310 B.C.–c. 230 B.C.) in his work Aristachi de magnitudinibus et distantiis Solis et Lunæ Liber [3], when dealing with the approximation for the ratio between the distance of the Sun to the Earth and the distance between the Moon and the Earth ([3, Prop. VII]); to this effect, as a preliminary step in his study, he gives 75 as an estimate of the square root of 2 (notice that 2 ≈ 49/25). For further information, consult [3], a Spanish translation of Aristarchus’ work, with appropriate comments, by the third author of this paper. Figure 2: Front of Aristarchus’ work, and part of Proposition VII, image from https: //bibliotecafloridablanca.um.es/bibliotecafloridablanca/handle/11169/975 MATerials MATemàtics MAT 2 Volum 2006, treball no. 1, 14 pp. Publicació electrònica de divulgació del Departament de Ma de la Universitat Autònoma de Barcelona www.mat.uab.cat/matmat 4 A problem on the approximation of n-roots. . . We can mention Archimedes (287 B.C.–212 B.C.) who in his “Measure- ment of a circle” 1 shows us his ability for manipulating numerical computa- tions in order to give an approximation of the ratio between the circumfer- ence and the diameter of a circle; he needs to estimate the values of several square√roots in order to achieve his task and, for instance, he obtains that 265 153 < 3 < 1351 2 780 . Also, in the Greek culture, let us cite to Hero of Alexandria (about first century A.D.); he provides us in his Metric 3 the method of averages for approximating square roots: in [1, Livre I, VIII, p. 165] Hero illustrates his well known formula for the area of a triangle (namely, if p = a+b+c 2 denotes the semiperimeter of a triangle p of sides a, b, c, the corresponding area S is given by Hero’s formula S = p(p − a)(p − b)(p − c) ) √ taking sides of lengths 7, 8, 9 and, therefore, he needs to compute √ the value of 720 . This is done by taking 27 as thecloser integer to 720 , and then giving the first approximation 12 27 + 720 1 1 27 = 26 2 3 as a more accurately approach; even, Hero says us that if we do the same operations with the new approach we will obtain a better approximation. Notice that Hero presents √ an iterative method; in current  notation,  if xn is the approximation of N , the next term is xn+1 = 12 xn + xNn , the average between xn and its inverse; realize that this recurrence is nothing else but the Newton-Raphson method applied to the equation x2 − N = 0 4 . Moreover, in the problem XX of Book III of the Metric it is proposed the division of a pyramid by a plan parallel to the basis holding a certain proportion between the pyramids; in the development of the problem, it is necessary to obtain a numerical approximation, based on double false 9 position, of the cubic root of 100, and Hero gives us the value 4 + 14 , having, currently we will say, two exact decimals. In our current √ notation, if a3 < 3 3 3 3 N < (a + 1) , and d1 = N − a , d2 = (a + 1) − N , then N is approximated (a+1)d1 √ by a + (a+1)d 1 +ad2 , so for 3 100 we find —notice that a = 4, d1 = 36, d2 = 9 5 25— the value 4 + 14 . 1 The translation into English of the Archimedes’ works can be consulted in [2], in particular the referred ‘Measurement of a Circle’. 2 For an explanation of this approximation, based on geometrical considerations, we recommend the reading of [6]. 3 The reader can consult [1], a French translation, jointly with a critical edition, of the Hero’s Metric, one of his more important works. √ 4 As a delightful curiosity, and based on the idea that the recurrence to obtain N can be viewed as the mean between a term x and N , in [15] the author establishes the √ x following generalization for the approximation of k N : by considering the average between (k−1)x +N x1−k x, (k−1) . . times . , x and N x1−k , it is easily seen that the recurrence xn+1 = n n gives √ k k a sequence that converges quickly to N if the initial condition is close to the desired k-root, and surprisingly if we apply the Newton-Raphson method to f (x) = xk − N , we derive the same recurrence. 5 Hero says us ([1, p.351]): “Mais comment faut-il prendre un côté cubique des 100 unités, nous le dirons maintenant. Prends le cube le plus proche de 100, aussi bien celui P. J. Herrero, A. Linero, Ma. R. Massa i A. Mellado 5 Figure 3: Text of Hero’s formula as collected in [30, v. II] As another instance of the approximation of square roots, let us mention to Theon of Alexandria (c.335–c.405). In his Commentary on Ptolemy’s Syntaxis 6 , Theon explains how to make an approximation√ to the irrational side of a square, and he shows the approximation of 4500 ◦ to be (in sexa- √ gesimal numeration) 67◦ 4′ 55′′ , that is, 4500 = 67+ 60 4 + 55 60 . His procedure is based on a figure in which by a process of exhaustion he completes the square by using the well-known fourth proposition of Euclid’s Book II, to wit, if a straight line is cut at random, the square of the total is equal to the squares of the segments and twice the rectangle contained by the segments —see [16]. For a picture of this geometrical process, see Figure 47 . The fact of the universality of this mathematical problem about approxi- mating square and cubic roots is reflected, indeed, in its ubiquity in different cultures, apart from the Western one, as it occurs, for instance, in the Chi- nese and Hindu civilizations. In this way, from the Chinese culture, we can par excès que celui par défaut; or ce sont 125 et 64. Et ce par quoi l’un excède est 25 unités, ce par quoi l’autre est en défaut est 36 unités; et fais les 5 par les 36: il en résulte 180; plus les 100: il en résulte 280; et applique les 180 aux 280: il en résulte 9/14; ajoute- les au côté du plus petit cube, c’est-à-dire au 4: il en résulte 4 unités et 9/14. Autant que cela sera le côté cubique des 100 unités à très peu près.” /We are going to say how to take the cubic side for 100 units. Take the cube closer to 100, as well by excess as by default; they are 125 and 64. So, the excess is of 25 units, and the defect of 36 units; multiply 5 by 36: it outcomes 180; to add 100: it is 280; and apply (divide) 180 to 280: it outcomes 9/14; add them to the side corresponding to the less cube, that is, to 4: it outcomes 4 units and 9/14. Therefore, this will be the cubic side for the 100 units, more or less. [Our translation] 6 For example, consult A.Rome (ed.), Commentaires de Pappus et de Théon d’Alexan- drie sur l’Almageste. Texte établi et annoté. Tome II. Théon d’ Alexandrie, Commentaire sur les livres 1 et 2 de l’Almageste. (Studi e Testi 72). Città del Vaticano: Biblioteca Apostolica Vaticana, 1936. 7 See also [8, pp.204-205]. MATerials MATemàtics MAT 2 Volum 2006, treball no. 1, 14 pp. Publicació electrònica de divulgació del Departament de Ma de la Universitat Autònoma de Barcelona www.mat.uab.cat/matmat 6 A problem on the approximation of n-roots. . . Figure 4: The process for squaring 4500, image from [30, p.56, v.I] mention Liu Hui (c.220–c.280), who commented the treatise Computational Prescriptions in Nine Chapters (Jiuzhang suanshu) based upon the ancient Chinese tradition collected in more ancient manuscripts. In Chapter 4 (shao guang), we find methods for calculating square and cubic roots. For more information about the treatise and Liu Hui, consult [9, pp. 133-153] and [5, Chapter 7], and for a textbook on the Chinese Mathematics, see [20]. Concerning Hindu tradition we can cite the Sulvasutra (‘the rule of chords’), where we find for instance the construction of a square having dou- ble area than another given square of side L, under the rule L+ L3 + 12L L − 34·4·3 for the side of the new figure (for an interpretation of this formula, consult [10, p.189 and ss.]). From Figure 5 we see that 1+ 31 + 12 1 is an approximation √ 1 1 1 by excess to 2 ; to obtain 1+ 3 + 12 − 34·12 , a modern explanation may be to 577 √ realize that 34·12 is close to 2 (it is a convergent of the continued fraction √ associate to 2 ), thus making 1 + 31 + 12 1 577 − x = 34·12 1 yields to x = 34·12 . Let us notice that to Hindu tradition is credited to have contributed with the computation of square and cube roots by dividing the number in periods and to arrange the process into columns and lines (see [23, p. 8]). Moreover, according to [7, pp. 174-175], in the Lilavati of Bhaskara (1114–1185) we can appreciate the high level of perfection that the art of computation had reached among Hindus and, despite the Greek writers on root extraction, the importance of the transmission of Hindu methods to the Arabs and by them to the Europeans. P. J. Herrero, A. Linero, Ma. R. Massa i A. Mellado 7 (duply the area) = 1 + 1/3 + 1/12 Figure 5: Image elaborated from [10] In relation, thus, with the Arab tradition, the Arabic mathematicians also handled different numerical approximations for the resolution of square, cubic and affected equations. In fact, it is said that they extended the concept of square and cubic roots to general √ powers ([7, bp. 175]). For square roots, apart from the Hero’s rule a2 + b ≈ a + 2a , used among others √ by Al-Battani (c.858–929) ([23]), other rules were also applied, as b a2 + b ≈ a + 2a+1 in the case of Al-Karaji (c.953–c.1029) ([23]) or Al- √ Nasawi (c.1011–c.1075) (see [29]); concerning cubic roots 3 a3 + r√, the same 32 Al-Nasawi uses a + 3a2r+1 to give 154 + 71149 for approximating 3 3652296 , r even the formula a + 3a2 +3a+1 is considered according to [29]. Following with this quick tracing on the approximation of n-th roots, let us mention to Leonardo of Pisa (c.1170–c.1250), commonly known as Fibonacci, who in Chapter XIV of his Liber Abaci shows us how to proceed with the extraction of square and cubic roots (for an English translation of Liber Abaci, see [27]). We emphasize that this procedure is based on the developments, from a geometrical point of view, of the square and cube of a binomial a + b. Leonardo of Pisa also deals with square and cubic roots in Chapters II and V, respectively, of De Practica Geometria (for an English translation of this work, consult [17]). In Chapter √ II we find, √ for instance, 32 16 884 + 2·884 = 884 + 884 as the approach for 781488 = 8842 + 32 ; and r in Chapter V we find again a + 3a(a+1)+1 when, for instance, he approaches √ √ 3 900 = 9 + 171 by 9 + 271 or approximately 9 + 23 . We can say that 3 3 171 Fibonacci was aware of the Indo-Arabic numbering and of the calculation r methods associated with it, see [5]. Let us notice that the formula a+ 3a(a+1) was also employed for the cubic root, for instance we can mention the Spanish mathematician Juan de Ortega (1480–1568)8 . As an instance of a new technique, let us finish with a comment to the work of Rafael Bombelli (1526-1572), who in his Algebra proposes a 8 In his book “Tratado subtilissimo √ de Arismetica y de Geometria”, Juan Cromberger: Sevilla, 1537, we have, e.g., 3 18889 ≈ 26 + 1313 2106 . MATerials MATemàtics MAT 2 Volum 2006, treball no. 1, 14 pp. Publicació electrònica de divulgació del Departament de Ma de la Universitat Autònoma de Barcelona www.mat.uab.cat/matmat 8 A problem on the approximation of n-roots. . . √ recursive process to approximate 13 , which currently we identify with the generation √ √ of a continued fraction. Indeed, if in modern notation we 2 put 13 = 3 + 4 = 3 + x, then 13 = 9 + 6x + x or 2 4 √ x(x + 6) = 4, that is, x = 6+x . If we repeat the argument, we have 13 = 3 + x = 3 + 6+ 4 4 . The reader can consult a detailed explanation of Bombelli’s 4 6+ 6+··· √ √ method in [26]. For 2 , we obtain 2 = 1 + 2+ 1 1 , and its first conver- 1 2+ 2+··· 3 7 17 41 99 239 577 gents are {1; 2 ; 5 ; 12 ; 29 ; 70 ; 169 ; 408 }. 3 The Viète’s procedure During the 16-th century we know different texts devoted to Arithmetic in which we find the solution of square roots, both in exact and approximated manners, still following the guideline of Elements, and in general the ex- traction of roots of bigger powers by a numerical procedure based on that we actually we denominate Newton’s binomial. In this way, let us cite to Pérez de Moya (c.1512–1596), and his book Arithmetica practica y specula- tiva, 1562; when Pérez de Moya expounds how to compute the square root of 524176, that is, 724, after dividing 524176 in binary groups (52 · 41 · 76), he explains: “And you will say that the root of 52 is 7 and it remains 3. Con- tinue in order to extract the root of that 3 remaining and of those 4 placed between the two marked points, you will do that by doubling the number 7 obtained previously as the root. How it is shown in the fourth proposition of the Book II of the Elements, they are 14 and put these 14 under 34 as if 14 be a divisor, and do not worry about 7 of the first point,...” 9 . To better understand the rule, firstly realize that from the division of the number into pairs of figures, we know that the integer part of the root has three digits, abc = 100a + 10b + c. Once we know that the number is placed between 700 and 800 (because 72 < 54 < 82 , so a = 7), for obtaining the second digit b in our current notation we would proceed as follows: (700 + 10b)2 = 524176 or 14000b + 100b2 = 524176 − 490000, that 100b is, 14000b(1 + 14000 ) = 34176; then, b ≤ 34176 14000 , and as an approximation we take b as the integer part of 34176 or, even, 34 the integer part of 14 , that 2 14000 √ is, b = 2; since 730 > 524176, at the moment we have 720 < 524176 < 730. To obtain the third digit, take into account the development 524176 = (700 + 20 + c)2 = 490000 + 28400 + 1440c + c2 , or 5776 = 34176 − 28400 = c 1440c(1 + 1440 ); now c ≈ 5776 577 1440 ≈ 144 , that is, we do the attempt c = 4, and we finally obtain the integer root 724. 9 “Y asi diras, que la rayz de 52 es 7 y sobran 3. Prosigue para sacar la rayz de los 3 que sobraron, y de los 4 que están entre los dos puntos, lo qual haras doblando los 7 que te han venido por rayz. Como muestra Euclides en la quarta del segundo, que son 14, pon estos 14 debaxo de los 34 como si fuessen los 14 algun partidor, y no cures del 7 que pusiste en el puncto primero,. . . ” (page 457). P. J. Herrero, A. Linero, Ma. R. Massa i A. Mellado 9 √ Figure 6: A fragment of 524176 in Arithmetica practica y speculativa by Pérez de Moya All these ideas and procedures will crystallize masterfully in the work De Numerosa Potestatum (1600/1646), from François Viète. The first edition of this work was in charge of Marino Ghetaldi (1568– 1626) in 1600 (see [31]). After, in 1646 a new version appeared, inside the posthumous book Opera Mathematica of Viète, edited by Frans van Schooten (1615–1660) (see [32]), also in Latin10 . The work is divided into three parts: De Numerosa Potestatum Purarum Resolutione ([31, ff. 2-6], [32, pp. 163- 172]), where Viète solved numerically the so-called pure equations, xn = a in our actual notation; De Numerosa Potestatum Adfectarum Resolutione ([31, ff. 7-34], [32, pp. 173-223]), devoted to the numerical resolution of (poly- nomial) equations having other terms apart from the biggest degree term and the independent coefficient (this kind of equations are called affected equations in the Viète’s terminology), and in that part Viète extended the method applied for the extraction of pure roots; finally, the last part, titled Consectarium Generale ad Analysim Potestatum Adfectarum, Et praecepto- rum quae ad eam pertinent, recollectio ([31, ff. 34-36], [32, pp. 224-228]), contains a summary of general consequences about the analysis developed in the previous part as well as an enumeration of precepts to be considered for the resolution of equations. For more information about this work, the reader is referred to the comments appearing in [33] and the study developed by J. Stedall in [28] and, more recently, by A. Mellado in [22, Cap. 7]. Concerning pure equations, Viète proposes the resolution of five equa- tions, Problems I-V, from the square root to the sixth root of a positive inte- ger number. In our current notation, they are x2 = 2916; x3 = 157464; x4 = 10 We have to mention that there exists an English translation, [33], although incomplete. MATerials MATemàtics MAT 2 Volum 2006, treball no. 1, 14 pp. Publicació electrònica de divulgació del Departament de Ma de la Universitat Autònoma de Barcelona www.mat.uab.cat/matmat 10 A problem on the approximation of n-roots. . . Figure 7: Extraction of the cubic root of 157464, [31] 331776; x5 = 7962624; x6 = 191102976, which correspond in the cosist nota- tion given by Viète to IQ æquari 29, 16; IC æquari 157, 464; IQQ æquari 331, 776; IQC æquari 7, 962, 624; ICC æquari 191, 102, 976, respectively. The resolution of these equations (all of them having an exact root) is carried out under very ingenious tables, original from Viète, on which he relies so that, supported in the development of the power of a binomial, he can obtain their solutions. (For instance, in Figure 7 we have the table to obtain the cubic root of 157464, [31, p.4]; see also [33, pp.317-320] for a detailed explanation). Respect to the numerical approach of n-th roots, in the precept 9 preced- ing the resolution of pure equations, Viète proposes to approach by above and below in the case of irrational roots, and explains: But if, although they are less, there is no dot for a power left over, it is clear that the root of the term to be resolved is irrational. To the collective root, therefore, add a fraction the numerator of which is the remainder of the term to be solved and [the de- nominator of which is] the same as what the divisors would be if another point were added to the term to be resolved. This fraction, added to the sum of the individual roots, makes a root for the power to be resolved larger than the true root. If, in the second power, the denominator is increased by 1, it makes a root smaller than the true one. The root lies implicitly between these divisors. It may be elicited as closely as you wish by, say, adding zeros to that which is to be solved and continuing the work. And P. J. Herrero, A. Linero, Ma. R. Massa i A. Mellado 11 this is necessarily within the limits of the tenth, otherwise the operation has not been carried out correctly11 . Translation from [33, pp. 313-314]. √ That is, in actual notation, the following approximation is proposed for n An + R , namely: Integer root + Rest/(Divisor+1)< Solution < Integer root + Rest/Divisor, or R R A+ < Solution < A + , d+1 d where the divisor d is 2A (square root), 3A2 +3A (cubic root), 4A3 +6A2 +4A (fourth root), etc. Let us say that the precept 9 is correct for the approxima- tion of square roots, but it fails to be true in some cases whenever the index of the searched root is greater √ than or equal to 3. To see it, consider for instance the computation of 4 20000 . This root appears at the end of Problema III in [31]. Viète itself provides us the corresponding estimations, and gives 5359 11+ 6095 and 11+ 5359 4 6094 , because in this example we have 20000 = 11 +5359 = 4 4 4 A + R, and the divisor is d = (A + 1) − A − 1 = 4A + 6A 3 2 + 4A = 6094. 5359 5359 √4 When checking the values, we find 11 + 6095 < 11 + 6094 < 20000 , that is, the two approximations are in fact by defect. If we look at [32], in this edition the precept 9 remains invariable with √ respect to the Ghetaldi’s edition, but now when estimating the value of 4 20000 , we appreciate a slight difference, because now we see that the text changes and, instead of the procedure of the former edition, now quaternary groups of zeros are added to the integer part in order to extract as many decimal digits as desired, in particular we 8917 8918 have 11 + 10000 and 11 + 10000 , see Figure 8 (notice that, although now the strategy is correct -we suppose that by truncating at some place the chains of quaternary √ groups-, however we also have two approximations by below 4 of 20000 ). At no time is noticed and justified the slight change in the new edition. Nor the question in [33] is mentioned, in fact, it is omitted (“Problems III and IV, here omitted, deal with the extraction of the roots of pure fourth and fifth powers”, [33, p.320]). Moreover, as far as we know, no mention to this fact has been signaled in the literature. With this note, we have tried to fill this lack. 11 “Quod si dum cedunt non supersit aliquod addictum Potestati punctum, argumentum est magnitudinis resolvendæ latus esse irrationale. Collecto itaque lateri adiungitur frag- mentum cuius numerator est numerus è magnitudine resolutâ reliquus. Divisores iidem, qui essent si aliquod punctum Potestati addictum superesset resolvendum, & tale frag- mentum singularium laterum summæ adiunctum facit latus Potestatis resolutæ maius vero. Et si denominatori addatur unitas, facit latus minus vero. In divisoribus enim inest implicitè latus, quod alioqui proximè esset eliciendum, ut pote productâ per numerales circulos eâ quæ resolvitur, Potestate, & continuato opere. At illud constat necesse est intra denarii metam, alioquin ritè non fuit operatum”. MATerials MATemàtics MAT 2 Volum 2006, treball no. 1, 14 pp. Publicació electrònica de divulgació del Departament de Ma de la Universitat Autònoma de Barcelona www.mat.uab.cat/matmat 12 A problem on the approximation of n-roots. . . Figure 8: Extraction of the cubic root of 20000, [31] (above), and [32] (below) 4 A question arises We look for an approximation of √ n An + R , where A ≥ 1 is an integer number, 0 < R ≤ (A + 1)n − An − 1 =: d (notice that we assume R > 0, that is,p the root is irrational, and that for n n n p R = (A + 1) − A , we in fact evaluate An + R e e = n (A + 1)n ; and recall that d is called the divisor of the approximation). √ Then we consider the two following approximations of n An + R , R AD(n, A, R) := A + (1) d+1 and R AE(n, A, R) := A + . (2) d It is clear that AD(n, A, R) < AE(n, A, R). Then, we propose the problem of determining whether these approximations √ are or not approximations by default or excess, respectively, of n An + R . For the case by default, it is a simple task to show that √ AD(n, A, R) < n An + R , (3)  n R or equivalently A + d+1 < An + R, merely use the Newton’s formula for the power of a binomial and notice that R < d + 1 = (A + 1)n − An . For the case of square roots, n = 2, indeed AD(2, A, R) and AE(2, √ A, R) 2 are true approximations by default and excess, respectively (for A + R < R 2 AE(2, A, R) consider that d = 2A and square to obtain A2 +R < A + 2A  ). Therefore R p R A+ < A2 + R < A + . 2A + 1 2A At the moment, we know that in general AD(n, A, R) always provides a default approximation for all n ≥ 2. Also, AE(n, A, R) gives an approxima- tion by excess when n = 2, that is, the precept 9 in [31] is appropriate when we look for the approximation of the square root of An + R. P. J. Herrero, A. Linero, Ma. R. Massa i A. Mellado 13 Next, fixing A and n ≥ 3, we are interested in knowing √ when the value AE(n, A, R) generates a true approximation by excess of n An + R . The next section is devoted to this task, for the cases 3 ≤ n ≤ 7 since they can be treated in a direct way (the case n ≥ 8 is more involved, see Theorem 4 below, and will be commented at the end jointly with the conclusions of this paper). 5 The solution for cases 3 ≤ n ≤ 7 According to the results presented in Section 4, we assume n ≥ 3. When we try to find the integer values √ R, with 0 < R ≤ d, such that A + Rd is an n n approximation by excess of A + R we have to analyze whether inequality R n   A+ > An + R (4) d holds or not. To stress the dependence of d on the values n and A, sometimes we will write d = d(n, A). We are interested in computing the amount of integer values R, 0 < R ≤ d, for which (4) is true. Then, it makes sense to define n √ o ν = ν(n, A) = Card R : 0 < R ≤ d(n, A), AE(n, A, R) > n An + R . (5) Putting s = d − R, the inequality (4) is converted to  s n 1+s+ A+1− > (A + 1)n , (6) d therefore ν(n, A) = Card {s ∈ Z : 0 ≤ s < d(n, A), (6) is true} . (7) Fix n ≥ 3, A ≥ 1, and hence d = d(n, A) = (A + 1)n − An − 1. We define the function fn,A (x) = f (x, n, A), fn,A : [0, d] → R as  x n fn,A (x) = 1 + x + A + 1 − − (A + 1)n . (8) d For simplicity of notation, we will write fn,A by f when no confusion can arise. It is straightforward to check that f (0) = 1 and f (d) = 0. Moreover, it is a simple exercise to check that fn,A (x) is a convex map, attaining its q  n−1 d minimum value at a point sm ∈ (0, d), in fact sm = d A + 1 − n . Figure 9 shows us the general aspect of the graph of fn,A : is strictly decreas- ing in the interval (0, sm ), having a zero ζ in such a zone, and it strictly increases in (sm , d). MATerials MATemàtics MAT 2 Volum 2006, treball no. 1, 14 pp. Publicació electrònica de divulgació del Departament de Ma de la Universitat Autònoma de Barcelona www.mat.uab.cat/matmat 14 A problem on the approximation of n-roots. . . Figure 9: One example for fn,A (x) R √ Figure 10: For n = 2, A + d approximates by excess the exact value of A2 + R Observe that if we extend f to the case n = 2, the corresponding map f2,A is non negative, which confirms the already known fact, seen in Section 4, stating that all the approximations A + Rd are by excess (see also Figure 10). In the sequel, it will be fruitful to apply the well-known Descartes’ rule of signs12 . 12 In his original text, see [13, La Géométrie, Livre III, p.373], R. Descartes (1596–1650) states the rule as follows: “On connoist aussy de cecy combien il peut y avoir de vrayes racines, & combien de fausses en chasque Equation. A sçavoir il y en peut avoir autant de vrayes, que les signes + & − s’y trouvent de fois estre changés; autant de fausses qu’il s’y trouve de fois deux signes +, ou deux signes − qui s’entresuivent”. / We know how many true (positive) roots, and how many false (negative) roots, can exist in every equation. To wit, it can have so many true roots as changes of signs + and − occur; and it can have so many false roots as the times that two signs + or two signs − appear consecutively [our translation]. Descartes does not provide any proof of this fact, and he limits himself to give an example, the equation x4 − 4x3 − 19x2 + 106x − 120 = 0 whose positive roots (‘vrayes racines’) are x = 2, 3, 4, and x = −5 is the unique negative root (‘fausses racines’); let us notice that the above equation is constructed by multiplication of the equations x − 2 = 0, x − 3 = 0, x − 4 = 0, and x + 5 = 0. For a short proof, the reader is referred to [34]; for a historical account of this result, and its first completely rigorous proof, we suggest the reading of [4]. P. J. Herrero, A. Linero, Ma. R. Massa i A. Mellado 15 Theorem. (Descartes’ rule of signs) Let P (x) = a0 xb0 + a1 xb1 + . . . + an xbn be a polynomial with nonzero real coefficients aj , 0 ≤ j ≤ n, and where the bj are integers satisfying 0 ≤ b0 < b1 < . . . < bn . Then, the number of positive real zeros of P (x) (counted with multiplicities) is either equal to the number of variations in signs in the sequence of the coefficients a0 , a1 , . . . , an or less than that by an even whole number. Corollary 1. Under the hypothesis of the above Descartes’ rule of signs, if the number of variations in signs of a0 , a1 , . . . , an is exactly 1, then P (x) has a unique positive root. To obtain ν(n, A) we adopt the direct strategy of finding values x1 , x2 close enough such that fn,A (x1 ) > 0 and fn,A (x2 ) < 0. 5.1 Cubic, fourth and fifth roots In the case of cubic roots (n = 3), for A ≥ 1, d = 3A2 + 3A, 0 < R ≤ d, we only need to compute f3,A (A) and f3,A (A + 21 ) (remember (8)): (3A + 2)(3A + 4) f3,A (A) = > 0, 27(A + 1)3 1 (6A2 + 4A − 1)(6A3 + 14A2 + 8A − 1) f3,A (A + ) = − < 0. 2 216A3 (A + 1)3 We deduce that the root ζ of f3,A (x) = 0 belongs to the interval (A, A + 12 ). So, by the properties of f3,A presented at the beginning of this√Section, f3,A (s) > 0 for the integers s = 0, 1, . . . , A, that is, AE(3, A, R) > 3 A3 + R for d − A ≤ R ≤ d, and consequently, by (5), ν(3, A) = A + 1. √ √ √ Example 2. We approach 990 = 3 A3 + R = 3 93 + 261 by A + Rd = 9 + 3 261 29 29 3  270 = 9 + 30 . We obtain an approximation by excess since 9 + 30 > 990. 2 In this case, d − A = 3 · 9 + 3 · 9 − 9 = 261 ≤ R = 261 ≤ d = 270. √ On the other hand, A + Rd = 9 + 270 260 = 9 + 26 is less than 3 989 = e 27 p3 √ A3 + Re = 3 93 + 260 . Now, we obtain a default approximation, which is in accordance with 0 < R e = 260 < d − A = 261. □ For the case of fourth roots, now we obtain     2A 2A + 3 ν(4, A) = +1= . 3 3 Similarly to the cubic case, this is an immediate consequence of the values     2A 2A + 1 f4,A > 0, and f4,A <0 3 3 (easily checked by a personal computer) and of the properties of the convex function f4,A . MATerials MATemàtics MAT 2 Volum 2006, treball no. 1, 14 pp. Publicació electrònica de divulgació del Departament de Ma de la Universitat Autònoma de Barcelona www.mat.uab.cat/matmat 16 A problem on the approximation of n-roots. . . √ √ Example 3. Let us revisit the Viète’s example 4 20000 = 4 114 + 5359 . In this case, the approximation  2A+3  11 + 5359 6094 is by default as d = 6094, A = 11, R = 5359, and 3 = 8. □ For quintic roots, n = 5, we follow the same strategy of the previous cases and obtain that     A A+2 ν(5, A) = +1= . 2 2 To see it, with the help of a mathematical software,   A 1 f5,A = × (10A3 + 20A2 + 20A + 9) 2 100000(A + 1)5 (A2 + A + 1)5 × 5000A10 + 36000A9 + 127000A8 + 288500A7 + 464600A6 + 552400A5 + 490750A4 + 321860A3 + 149620A2 + 44770A + 6561 > 0,  and   A 1 1 f5,A + =− 2 2 100000(A + 1)5 (A2 + A + 1)5 × (10A3 + 10A2 + 10A − 1) × 10000A11 + 35000A10 + 79000A9 + 111000A8 + 115500A7 + 81400A6 + 41300A5 + 11250A4 + 1390A3 − 510A2 + 40A − 1 < 0.  Thus, the zero ζ of f5,A (x) = 0 lies in the open interval ( A2 , A2 + 12 ). Since  A is A a positive integer, we deduce that f5,A (s) > 0 only for s = 0, 1, . . . , 2 . This √ means that AE(n, A, R) > 5 A5 + R whenever R = d, d − 1, . . . , d − A2 . □   5.2 Sixth and seventh roots We compute now the value of ν(6, A), that √ is, the cardinal of values R ∈ {1, . . . , d(6, A)} such that AE(6, A, R) > 6 A6 + R . We will obtain     2A − 1 2A + 4 ν(6, A) = +1= . 5 5 To prove it, we do a discussion on the values of A. (i) Let A = 1. Then d = d(n, A) = (A + 1)n − An − 1 = 26 − 1 − 1 = 62. We have 1 6 123 6     58788614519 f6,1 (1) = 2 + 2 − − 26 = −62 + =− < 0. 62 62 56800235584 P. J. Herrero, A. Linero, Ma. R. Massa i A. Mellado 17 By the properties of fn,A , it follows that we have a unique approx- imation  2A+4  by 6excess  when s = 0, or equivalently R = d. Notice that 5 = 5 = 1 in this case. (ii) Let A = 2. Now, d = d(n, A) = 36 − 26 − 1 = 664. Again 1 6   −16543657942216431 f6,2 (1) = 2 + 3 − − 36 = < 0. 664 85705457236443136 So, our unique approximation  excess corresponds to s = 0, or R = d.  2A+4by As in the previous case, 5 = 1. (iii) Let A = 3. We have d = 46 − 36 − 1 = 3366. By a direct computation (check it) we find that   6 f6,3 > 0, but f6,3 (2) < 0, 5 so the unique values giving us an approximation by excess are s = 0 and s = 1, that is, R ∈ {d, d − 1}. Notice that 2A+4 5 = 2. (iv) Let A ≥ 4. We claim that f6,A 2A  5 < 0. With the help of a computer, 2A N6 (A) we obtain f6,A 5 =: D6 (A) , where D6 (A) = 15625d6 and  N6 (A) = 30A4 + 75A3 + 100A2 + 75A + 28  × (−4050000A19 − 32400000A18 − 103005000A17 − 51975000A16 + 912481875A15 + 4658816250A14 + 13800777375A13 + 29961127500A12 51357591250 + A11 + 71924369500A10 + 83733302450A9 + 81679154000A8 + 66842104875A7 + 45673566250A6 + 25767455655A5 + 11766939100A4 + 4208223600A3 + 1113483200A2 + 195334080A + 17210368). Define Q6 (A) := 30A4 +75A3N 6 (A) +100A2 +75A+28 , a polynomial of degree 19. Then Q6 (3) > 0 and Q6 (4) < 0. From the Descartes’ rule of signs and its Corollary 1, we conclude that Q6 (A) has a unique positive real root, located at the open interval (3, 4) and, since Q6 (A) → −∞ when A tends to infinity, Q6 (A) <0 for all A ≥ 4. Thus, N6 (A) < 0 for all A ≥ 4. Therefore, f6,A 2A 5 < 0 for all A ≥ 4. This ends the claim. MATerials MATemàtics MAT 2 Volum 2006, treball no. 1, 14 pp. Publicació electrònica de divulgació del Departament de Ma de la Universitat Autònoma de Barcelona www.mat.uab.cat/matmat 18 A problem on the approximation of n-roots. . . On the other hand, we get f6,A ( 2A−1 5 ) > 0, because   2A − 1 1 f6,A = 5 15625A6 (6A4 + 15A3 + 20A2 + 15A + 6)6 × 30A5 + 75A4 + 100A3 + 75A2 + 28A + 1  × 8100000A24 + 121500000A23 + 875070000A22 + 4074840000A21 + · · · + 282630A3 + 8395A2  + 140A + 1 > 0. (all the coefficients of the last polynomial in A are positive). Now we are in a position to finish Case (iv), A ≥ 4: since f6,A ( 2A−1 5 )> 0 and f6,A ( 2A 5 ) < 0, we conclude that f 6,A (s) > 0 for the integer values 2A−1  s = 0, 1, . . . , 5 , realize that A is a positive integer. With our discussion of Cases (i)-(iv), we finish the case of sixth roots. □ Finally, for seventh roots we are going to show that     2A + n − 2 2A + 5 ν(7, A) = = . n−1 6 Similarly to sixth roots, we distinguish several cases. (i) A = 1. Then f7,1 (1) < 0so f7,1 (s) > 0 only for the integer s = 0. Thus, ν(7, 1) = 1 = 2A+5 6 = 7 6 = 1. (ii) A = 2. Again f7,2 (1) < 0 and ν(7, 2) = 1 = 96 = 2A+5     6 . N7 (A) (iii) A ≥ 3. Using a mathematical software, f7,A ( 2A 6 ) =: D7 (A) , with D7 (A) = 1801088541A35 + 37822859361A34 + 403443833184A33 + . . . + 403443833184A2 + 37822859361A + 1801088541 (all the coefficients of D7 (A) are positive) and N7 (A) = −600362847A34 − 10806531246A33 − 97258781214A32 − 577549058814A31 − 2514834199962A30 − 8420174695296A29 − 21944119719060A28 − 43061625563922A27 − 53191033284627A26 + 13234913195994A25 + 303648512134590A24 + 1063881793793574A23 + 2599443322376040A22 + 5157799213711410A21 + 8755187169969450A20 + 13031632337209290A19 + 17242848176742405A18 + 20448216949700610A17 + 21842457224825790A16 + 21075654616272900A15 + 18392229344961864A14 + 14515293924381222A13 + 10346477812600482A12 P. J. Herrero, A. Linero, Ma. R. Massa i A. Mellado 19 + 6644210201266050A11 + 3829261973222007A10 + 1970068132488204A9 + 898255956563028A8 + 359478031796604A7 + 124638816387666A6 + 36779824104114A5 + 9007142933166A4 + 1762931499924A3 + 259451944899A2 + 25624743060A + 1280000000. By the Descartes’ rule of signs, N7 (A) has a unique positive real root lying in the interval (2, 3) since N7 (2) > 0 and N7 (3) < 0 (see Fig- ure 11). Figure 11: The graph of N7 (A) near to its positive zero  N7 (A) Therefore, f7,A 2A 6 = D7 (A) < 0 for all A ≥ 3. On the other hand, also with the help of a mathematical software,   2A − 1 P7 (A) f7,A =: , 6 Q7 (A) with P7 (A) = 38423222208A41 + 1075850221824A40 + 14062899328128A39 + · · · + 118984124A4 + 2524592A3 + +34776A2 + 280A + 1 and Q7 (A) = 230539333248A42 + 4841325998208A41 + 51640810647552A40 + · · · + 51640810647552A9 + 4841325998208A8 + 230539333248A7 (all the non vanishing coefficients of P7 (A) and Q7 (A) are positive). Clearly, f7,A ( 2A−1 6 ) > 0 for all A ≥ 1. Since 2A−1  2A−1  6 ∈ / Z, we deduce  f (s)>  2A−1that 0 for 2A+5  the integer values s = 0, 1, . . . , 6 . So, ν(7, A) = 6 + 1 = 6 . □ MATerials MATemàtics MAT 2 Volum 2006, treball no. 1, 14 pp. Publicació electrònica de divulgació del Departament de Ma de la Universitat Autònoma de Barcelona www.mat.uab.cat/matmat 20 A problem on the approximation of n-roots. . . 6 The generalization It is worth mentioning that we were able to find a general result for the value ν(n, A). Because its proof is very involved to be presented in this simple note, we only trace j its maink steps. j We advance k jthat, in k general, the 2A+n−2 2A+n−2 2A−1 number ν(n, A) is either n−1 or n−1 − 1 = n−1 . In the first step we prove that for n ≥ 8 and A = 1, it holds ν(n, 1) = 1. This is due to the fact that, for n ≥ 8,  n 1 fn,1 (1) = 2 + 2 − n − 2n < 0. 2 −2 Next, the crucial step is to demonstrate that for n ≥ 8 and A ≥ 2 we have   2A fn,A < 0. (9) n−1 To reach it, we use that 1 − nx ≤ (1 − x)n ≤ 1 − nx + n(n−1) 2 x2 for x ∈ (0, 1) 2A and nx < 1, and then taking x = (A+1)(n−1)d we obtain the inequality   2A fn,A n−1  n 2A 2A = 1+ + A+1− − (A + 1)n n−1 (n − 1)d   n  2A n 2A = 1+ − (A + 1) 1 − 1 − n−1 (A + 1)(n − 1)d "  2 # 2A 2An n(n − 1) 2A ≤ 1+ − (A + 1)n − n−1 (A + 1)(n − 1)d 2 (A + 1)(n − 1)d 2A 2An(A + 1)n−1 2A2 n(A + 1)n−2 = 1+ − + n−1 (n − 1)d (n − 1)d2 (n − 1)d2 + 2Ad2 − 2An(A + 1)n−1 d + 2A2 n(A + 1)n−2 = . (n − 1)d2 So, the proof of (9) is complete if we show that gN (n, A) := (n − 1)d2 + 2Ad2 − 2An(A + 1)n−1 d + 2A2 n(A + 1)n−2 < 0. The proof of this last fact requires a laborious manipulation (and we omit it), as well as the final application of the Descarte’s rule of signs. In this way, if 2A ≤ n − 1 we can assure that ν(n, A) = 1 (because 2A fn,A (0) = 1 > 0 and fn,A (1) < 0 since n−1 ≤ 1, recall the properties of the convex function fn,A ); at the sametime, if 2A  > n − 1, we are able to prove that there exists k = k(n, A), k ∈ 0, 1, . . . , n−1 3 − 1 such that 2A − (k + 1) 2A − k ≤ s0 < , n−1 n−1 P. J. Herrero, A. Linero, Ma. R. Massa i A. Mellado 21 where d s0 = s0 (n, A) = n(A + 1)n − d is precisely the value of the intersection between the tangent line to fn,A at the point (0, 1) and the axis Ox (take into account that, by convexity, we have that fn,A (s0 ) > 0), see Figure 12. Figure 12: Function fn,A (x) and the tangent line passing by (0, 1) Joining all the above ingredients, we advance the following result: Theorem 4. Suppose that n and A are positive integers, with n ≥ 8, A ≥ 2. n n Let ν(n, A) be the number  (7); let d(n, A) = (A+1) −A −1  given by (5) or n x and fn,A (x) = 1 + x + A + 1 − d(n,A) − (A + 1)n . Then: (a) If 2A ≤ n − 1, ν(n, A) = 1. (b) Assume that 2A > n − 1.  n−1  (b1) If n − 1 ∤ 2A − k for all k = 1, 2, . . . , 3 − 1,   2A + n − 2 ν(n, A) = . n−1 (b2) If n − 1 | 2A − k for some k = 1, 2, . . . , n−1   3 − 1, and we put d(n,A) s0 = n(A+1)n−1 −d(n,A) , then in turn: j k j k (b2I) If s0 ≥ 2A−k n−1 , ν(n, A) = 2A+n−2 n−1 . j k j k (b2II) If s0 < 2A−k n−1 and f ( 2A−k n,A n−1 ) > 0, ν(n, A) = 2A+n−2 n−1 . j k j k (b2III) If s0 < 2A−k n−1 and fn,A ( 2A−k n−1 ) < 0, ν(n, A) = 2A+n−2 n−1 −1. MATerials MATemàtics MAT 2 Volum 2006, treball no. 1, 14 pp. Publicació electrònica de divulgació del Departament de Ma de la Universitat Autònoma de Barcelona www.mat.uab.cat/matmat 22 A problem on the approximation of n-roots. . . 7 Conclusions In this note, we have presented a little piece of history relative to the numer- ical approach of n-th roots, in particular we have seen in the Viète’s work De Numerosa Potestatum a numerical procedure (based upon the develop- ment of a binomial) strongly related with the extraction of square roots in the Babylonian or Greek tradition. According to De Morgan, [12], the Viète method was improved by T. Harriot (1560–1621), and used by W. Oughtred (1574–1660), J. Wallis (1616-1703) as well as the generality of at least En- glish writers (sic), which demonstrates its spread in England; concerning the European continent, also several authors considered the method either in its original version, as Dechales (1621-1678) in his Cursus seu Mundus Mathematicus, first published in 1674, or trying to adapt it to an incipient algebraic language, as it occurs in the Cursus Mathematicus of P. Hérigone (1580–1643), published at Paris in six volumes during 1634 and 1642. Even, we know that famous mathematicians of the 17-th century referred to it in cases where was necessary its application, as G.W. Leibniz (1646-1716) who indicates its suitable use 13 . Anyway, following again [12], we can say that the method continued to be used up to the time at which the Newton approximation appeared at the last quarter of 17-th century. In this direc- tion, let us also mention to T. Fantet de Lagny (1660–1734), who in [11, pp. 9-10] writes: “Il y a prés d’un siecle que le sçavant Mr. Viete a donné une methode generale pour l’extraction des racines de toutes les puissances pures, dans son livre de numerosa potestatum resolutione... Tous ceux qui ont écrit depuis sur l’extraction des racines, n’ont fait que copier Viete, & se copier les uns les autres, & on ne doit pas en estre surpris. Cette methode paroit si naturelle & si conforme à ce grand principe des Analystes, que la resolution doit se faire par le voye opposée à la composition, qu’il ne tombe pas mesme dans l’esprit qu’on en puisse inventer une meilleure” 14 . From the detailed reading of De Numerosa Potestatum, we have detected a lack in the different editions of this work, in relation with its precept 9 about approximation by excess of the n-roots, n ≥ 3. In this direction, imagine the difficulty in Viète’s time for computing by hand all the problems appearing in De Numerosa Potestatum, and compare it with the current easy treatment by 13 Leibiniz writes “... per extractiones radicum appropinquatrices methodo Vietaea” -by Viéte’s method of approximately root extraction- in [19, p. 780] for the root extraction operation, that is, he referred to the De Numerosa Potestatum as a reference for the root extraction. 14 “It is almost a century since the learned Mr. Viete gave a general method for extracting the roots of all pure powers, in his book de numerosa potestatum resolutione. Anyone who has written about root extraction since then has only copied Viete, and copied each other, and we should not be surprised. This method seems so natural and so according to this great principle of the Analysts, that the solution must be done by the way opposite to the composition, that even our spirit does not think of inventing a better one” (our translation). P. J. Herrero, A. Linero, Ma. R. Massa i A. Mellado 23 a personal computer, realize how actually the task is enormously simplified. In turn, this study gave rise to the pure mathematical question of count- ing√the values of remainders R for which A+ Rd is an approximation by excess of An + R , 0 < R ≤ d, d = (A + 1)n − An − 1. We have done a complete n (and direct, basic) study of the value ν(n, A) in the cases 3 ≤ n ≤ 7, and we have advanced the general result in Theorem 4, whose proof was only slightly outlined. Although, of course, the significance of the found main result in Theo- rem 4 is not transcendental, at least allows us to illustrate the importance and benefits of the study of History of Mathematics, given rise to questions and reflections through, in particular, the reading of primary sources. They can help us to elaborate materials for classroom, from a more human and didactic approach to the learning of mathematics. Realize that in this note we have employed a series of basic mathematics, and we think that materials for classroom could be implemented from it, dealing with different aspects, among others, for instance, the importance of the algebraization of Mathe- matics during 17-th century (compare the cosist notation in Viète with our modern notation), the handling of inequalities and combinatorial numbers, properties of convex functions and their drawing, calculus of extremal points, location of roots, even the student could develop an algorithmic procedure for testing the validity of Theorem 4. Acknowledgements This work has been supported by the grant MTM2017-84079-P (AEI/FEDER, UE) from Ministerio de Ciencia, Innovación y Universidades (Spain). References [1] F. Acerbi, B. Vitrac, Metrica. Héron d’Alexandrie. Introduction, texte critique, traduction française et notes de commentaire. Pisa-Roma: Fab- rizio Serra Editore, 2014. [2] Archimedes, The works of Archimedes. 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Licenciado Francisco Cascales, Murcia antonio.mellado2@um.es Publicat el 27 de novembre de 2023 MATerials MATemàtics MAT 2 Volum 2006, treball no. 1, 14 pp. Publicació electrònica de divulgació del Departament de Ma de la Universitat Autònoma de Barcelona www.mat.uab.cat/matmat