Square root (history)

The history of the square Racine starts around. Its first known representation dates from. The value of $\ sqrt {2}$ was calculated in an approximate way in India at eighth century BC and China during second century BC. Between these two periods, the Greeks show his irrationality.

Mathematicians encountered these problems since the beginnings of the writing, in Mésopotamie (- 1700), they were fertile until XIXe century, it preserve a teaching interest .

The square root is a traditional question of Histoire of sciences. It makes it possible to trace the real operation of the discoveries, their lapse of memory or their transmission. Some Représentation S limit comprehension (epistemological obstacles), others support the speculation (Technique S, Social organization, Religion, Philosophie…) ; but beyond its advance, each company with its manner of making mathematics. To realize, nothing is worth a quotation of time, even translated, to discover it technicalities different, and often very elaborate. The square root is a very definite mathematical object, which allows a small voyage in space and time by the traces that it left.

Babylon

Old China

The intellectual culture of traditional China is generally arts person , so that a large sinologist dared to say: “The idea of quantity does not play as much to say any role in the philosophical speculations of the Chinese. The Numbers, however, passionately interest the Wise ones of old China”. CAD De Jing (v. -600) gives the tone: “Dào created one, of the one pushed both, two made three, and from the three were born the 10.000 beings” (42). The inspiration is continued by the Taoïsme (Wang Bi, 226~249), until the Néo-confucianisme (Zhou Dunyi, 1017~1073), in particular while speculating in Yi Jing. At best, one reads the dialectical ones on one and the multiple, but one would in vain seek there the equivalent of a Mystique Pythagoricien of the rationality. The Theorem of Pythagore and risks it irrationality were seen, when one notes the favor of the right-angled triangles at commensurable sides (3,4,5), but one also puts up with triplets approximate pythagoricians (4 ² +8 ² =9 ² - 1, 4 ² +7 ² =8 ² - 1). The number is especially emptied quantity to find of them a direction symbolic system, clean to label the diversity of reality. The imperial examinations to reach the functions mandarinales request a great knowledge of the Traditional Chinese, but do not comprise mathematical tests.

However, in 1983, one discovered the oldest mathematical text currently known (- 186) in the tomb of a well-read man imperial civil servant. The Suàn shù shū 算數書, “writings of account”, is a whole of 190 rods translated very recently (2004). They are examples of calculations, used for administrative operations (“rectangular Fields”, “Millet and rice”, “equitable Taxation”, “Surplus and deficit”…).

(- 186) Suàn shù shū 算數書, “writings of account”
√240: an acceptable approximation by the Method of the false position

Setting with the square of a field
Soit a field of one arpent: of how much step is it square? arpent” translates a unit of surface, '' mǔ '' 畝. “” not translated a unit of length. One knows that 240 步 bù square = 1 mǔ 畝. Is the problem thus √240?
Réponse: it is the square of 15 pas and 15/31 pas ; 15+15/31=15,483… .
Méthode: If it is a square of 15, there is a deficit of 15 ''; if it is a square of 16, there is an excess of 16 ''.
Réponse: To add excess '' and the deficit '' to make a divider ''. The numerator of the deficit multiplies the denominator of excess '', and the numerator of excess multiplies the denominator of the deficit ''; to add to make a dividend + 16x15) . To reverse to give the length + 16x15)/(15+16) =32x15/31=15+15/31 .

The approximation of √240 to which this succeeded text is acceptable. This first attested Chinese attempt at extraction of root is also the first formulation of the Méthode of the false position (known as Yíng bù zú “excess and deficit”) which one finds to Al-Khwarizmi (783~850). It is used elsewhere in the text with exactitude on linear equations. Here it functions like an approximation. Calculation can be difficult to follow, it proves in any case a great practice of verbal handling of the fractions. For us, of functional terms, that consists in carrying out a linear average between two framing values. That is to say F (X) =x ² - 240. One seeks X so that F (X) =0. By test and error one finds (1) F (15) =-15 and (2) F (16) =+16, X is between 15 and 16. If F were linear of ax+b type, it would be such as (1) a15+b=-15 and (2) a16+b=16. (2) - (1) => 16a+b-15a-b=a=16+15=31, from where a=31. Applied to (1) or (2), one finds b=-480, from where the linear approximation F (X) =31x-480=0 => x=480/31= ((31x15) +15) /31=15+15/31, solution initial of the problem. The algorithm is not applied in an iterative way.

Remains information of context which we miss, which is the utility of this calculation? That seems to suppose that the mǔ 畝 is a unit of traditional surface which is not obtained by land surveying, perhaps by evaluation of the production of a piece, or as the character seems to indicate it, an agricultural working time (久 jiǔ “lasted”: 人 rén “a man” with a tool; 田 tián “field”). The civil servant who read this text took part in the imperial reform of Qin Shi Huang (- 260~-210). This one unified the weights and measures, applied a land reform, so that peasants become owners produce the grains necessary to military campaigns and great work. The many problems of conversions were to thus apply concretely in the calculation of the taxes, in confrontation with the traditional units of the local population.

This approach is very interesting for the history of mathematics and civilizations in general. Compared with the Greek tradition, mathematics is not the free theory of an idle class, they must be useful to a well-read man Confucéen who has a load towards the state and the people. It results a less clarity of expression from it from the problems, and the acceptance of pragmatic solutions, quite far away from the required religious perfection for example in the Indian Veda. However, that does not prevent an intention of generalization (more than of abstractions), guided by the paradigm Taoïste.

(- 200~200) Zhoubi Suanjing 周髀算經, “the shade of the cycles, delivers calculations”
Writes astronomical with mathematical results, published and commented on by Liu Hui

the Councils Taoist S of a mathematician to his pupil (~-100)
“you can include/understand this matter if you give him a sincere and continuous thought For the moment, you cannot generalize. Many things escape your knowledge. The Voie illuminates knowledge when simple words have a broad application. When you question only one problem and see a myriad of things there, there, you hear the Voie. ”
Zhoubi Suanjing , -100~100, in (Cullen 1996,175-178)

(0~263) JiuZhang SuanShu 九章算術 “Last nine Chapters on mathematical art”
a traditional mathematics published and commented on by Liu Today (263)

The following reference mark is a comment of Liu Hui (263), bringing back a text the Last nine Chapters on mathematical art going back about to our era (an inventory of the imperial library of -5 does not mention it whereas it is it then). In the chapter 4 少广 Shao guang “the least width”, questions 12 to 18 imply the extraction of square Racine according to the Méthode of the false position of the preceding book. Liu Hui is sometimes presented like the Euclide Chinese. That it is Chinese is out of doubt, the problems and the methods do not borrow anything from other cultures. On the other hand, the mode of generalization has nothing to do with an axiomatic development. To summarize, √2 was seen, but was not sought.

Ancient Greece

See also: Mathematical of ancient Greece, School pythagorician, Real number, Mode=approfondir

Greece knows and even shows the irrationality of √2, but this discovery is difficult to allot exactly.

Aristote (- 384~-322) holds the demonstration for acquired and uses it like an example of application of the reasoning by the absurdity and of the excluded thirds, he says just: “the diagonal of the square is incommensurable at its sides, or that would suppose that the odd numbers are even. ” (- 335~-323), its listeners are judicious to know the arithmetic demonstration.

Plato (- 427~-348) evokes a whole of the roots in the Théétète. The action is held after death of Socrate (- 399), very young Euclide (- 325~-265) brings back a dialog that Socrate would have had with the mathematician Théétète of Athens (- 415 ~ -395 or -369). The subject relates to the search for a principle to define true science. The definition of the irrational is used like an introductory example of the philosophical criterion to seek. Is also quoted the mathematician Theodore de Cyrène (- 470~-420), Master of Plato.

The incompatibility of the dates is obvious. However, it is in the style of Plato to call upon in his dialogs the figures representative of a problem. He wants clearly to mean that Théétète inherits Theodore. Remain that Euclide had not been born when Plato died. Several critics wanted to suppose another Euclide, with as many memory and talents. The probability is weak. Another assumption agrees completely with the composition of the text, the introduction of Euclide is posterior, after the death of the Master. It is known that before joining Alexandria, the mathematician was born in Athens, and the Académie remained a center of mathematical studies.

This short passage shows that mathematics of then is always based on geometrical analogies, but that the speech continues without resorting to figures, in conclusive oral statements (see the logic of Aristote). Certain historians of sciences speak even about arithmetic reasoning precursory of an algebraic demonstration. Euclide thus collected a tradition already lengthily elaborate in its Book X of the elements. It draws some from other proposals, in particular a method of approximation of the roots known as of antiphérèse, or reciprocal subtraction. Definitions 1 to 3 point out those of the Théétète , in other terms.

Euclide thus received Théétète the definition of a class of numbers incommensurable, or irrational, the roots. Théétète learned from Theodore the proof of irrationality of the roots up to 17, probably by geometrical demonstration. But from where does Theodore hold the demonstration for the first, √2? Plato still, considers the demonstration for so elementary, that it can be explained to a slave in the Ménon (duplication of the square). A quite posterior Néoplatonisme allots it to the school pythagorician. Jamblique (250~325) estimates that it relates to the division of a segment in extreme and average reason, the Golden section $(1+ \ sqrt {5}) /2$ . Pappus (340) speaks about it for the diagonal about the square (√2), of the same Proclos (412~485). Does Jamblique add a very beautiful legend, criticized by Proclos, perhaps the first (and the last?) mathematical martyrdom.

Hippase de Métaponte, disciple of Pythagore (v. - 580~v. - 490), knew the incommensurability of √2 (perhaps discovered by its Master). The sect propagated an arithmetic mystic affirming the rationality universe with the mathematical direction, i.e. the sensitive one is made sizes being able all to be reduced to the unit. The existence of at least only one proven irrational size, √2, refutes the metaphysical principle. Hippase would have enfreint the rule of silence of the community, and to reveal the irrationality world. We do not know the theological stakes of the time, but Hippase was banished, and perhaps to have lost the friendship of his brothers, it would have been thrown to the sea.

The neo-pythogorician Théon of Smyrna (70~135), inspired of the method of antiphérèse of Euclide, used the principle of framing continuations to approach the value of √2 (a/b, (a+2b)/(a+b)…: 3/2, 7/5, 17/12…). Finally Diophante of Alexandria (200/214~284/298) will have need for a Théorie of the numbers unifying the entireties, the rational ones and the irrational ones, for the resolution of a equation diophantienne.

For ancient Greece, √2 is more than one number. It was initially a problem Métaphysique, which remained an introduction of choice to the Philosophie (because “no one does not enter there if it is not geometrician”). The question traverses one millenium of traditional culture, of Pythagore (- 580~-490) to Proclos (412~485).

Indian world

See also: Mathematical Indians, Mode=approfondir

The Age of the valley of Indus (- 2600~-1500), contemporary of the Mésopotamie and the did old Egypt know √2? Its writing was not deciphered yet, one can be at most tested with conjectures about material civilization. The presence of bricks standardized (1x2x4) on the same place that precise rules with the millimetre proves the practical knowledge of the reduction to the unit. Did they can make the lengths commensurable, the diagonal of the square their pose problem? It is in the state impossible to deduce speculative sciences from the know-how of the craftsmen. But the discovery of a system of weights and measures of a high degree of accuracy and of decimal nature, obliges to wonder what posterior mathematics owes with this culture.

The following time (- 1500~-400) is named vedic, because she is primarily known by the Veda, of the religious texts in a Indo-European Langue, the Sanskrit. In appendix, one can find Śulbasutras there. They are rules to build a furnace bridge with right proportions, with these same bricks of Indus, authorizing to wonder whether these texts do not compile quite former oral traditions. If he one wants that the gods like the sacrifice, he must be perfect, from where the motivation to seek the exactitude of the reports/ratios, like a Greek temple. Mathematics thus has a religious direction, without to be the subject of a worship of a kind pythagorician, even less demonstrations. Essence is that the rule functions.

In its Śulbasutra 52, Baudhayana (- 800) explains how to build a square of surface doubles of a given square, therefore side √2, in only one sentence: “that one increases the side of the square of a third and that of its decreased quarter of the thirty-fourth of itself”. The figure to be built is not commonplace, but the algebraic value totals 1 + 1/3 + 1/4 (1/3 - 1/34 X 1/3) = 1,414215…, precision with 5 decimals. The procedure is not shown, it is even difficult to guess how it was discovered. A indianist notes that the exact Indian sentence uses the word Sanskrit savi´e¸a , meaning “exceeds”. The writer knows that its result is higher than the sought number, and even from how much, the approximation would result from an iterative process with for stages: $\ sqrt2 \ approx 1 + \ frac {1} {3} \ approx 1 + \ frac {1} {3} + \ frac {1} {3}. \ frac {1} {4} \ approx 1 + \ frac {1} {3} + \ frac {1} {3}. \ frac {1} {4} - \ frac {1} {3}. \ frac {1} {4}. \ frac {1} {34} \ approx 1 + \ frac {1} {3} + \ frac {1} {3}. \ frac {1} {4} - \ frac {1} {3}. \ frac {1} {4}. \ frac {1} {34} - \ frac {1} {3}. \ frac {1} {4}. \ frac {1} {34}. \ frac {1} {1154}…$ (12 decimals of precision). Obtaining the signs (+) and the dividers (4, 34,1154…) rest on a rather complex procedure shown with geometry and the proportion of bricks; but with final very few calculations being carried out. The construction of furnace bridges east is enough to 5 decimals of precision. One should not however overestimate the posterity of these results. They seem to have been abandoned with the vedic religion. After the conquest of the Bactriane by Alexandre Large the (- 328), a Greek influence was continued until the beginning of our era (10) by the Royaume indo-Greek. He constitutes a gréco-Buddhist Syncrétisme there obvious in the statuary, but the preserved texts do not make it possible to determine if Euclide for example drew the attention. The Bouddhisme mahayana diffused since there in the remainder of Asia preserves astronomical speculations, but not notable mathematical contribition. The historians of sciences prefer to insist on the role of the Jaïnisme in the traditional revival of Indian mathematics, moreover more in the south.

In the Brahmasphuta-siddhanta “the revised system of Brahma” of Brahmagupta (598-668) one will find the zero, a method to calculate the square roots and some algorithms to solve quadratic equations. This book arrived at the arabes.

World arabo-Moslem

See also: Mathematical Arabic, Mode=approfondir

The mathematical in Arab language preserved and included/understood the Mathématiques of ancient Greece, it is already an enormous merit, they could have remained dead letters, such as for example Babylonian mathematics. The nestorien Hunayn ibn Ishaq (809~873) very early translated the Éléments of Euclide, read again by the sabéen Thabit ibn Qurra (826~901), which spoke Greek. The other decisive contribution is to have assimilated the Mathématiques Indians, in particular the positional decimal writing of the figures. However, it should be recalled that if Persian Al-Khwarizmi (783~850) speaks about the Indian figures in its Algèbre (الجبروالمقابلة - Al-jabr wa' l-muqâbalah), its demonstrations remain verbal (the square of the unknown factor is named “the square” or mâl , the unknown factor is “the thing” or shay or jidhr , the constant is the dirham or adǎd ). He proposes the first systematic model of algorithms of resolutions of the linear equations and quadratic. √2 is definitively domesticated, it does not have more mystery.

Occident

Rene Descartes, Regulae AD directionem ingenii “Rules for the direction of the spirit”, 1628 of Latin by Georges Roy
many relations which one seeks to express by several dimensions and several figures, and of which one names the first root, the second square, the third cube, the fourth biquadratic one, etc These terms myself misled me a long time, I acknowledge it; because after much of experiments I realized that, by this manner of designing the things, I had not discovered anything that I had not been able to know much more easily and distinctly without it; and that such denominations should be entirely rejected, for fear they do not disturb the thought, because, though one can call a cubic or biquadratic size, one should not never present it to imagination differently than like a line or a surface, according to the preceding rule. It thus should be noted before very that the root, the square, the cube, etc are only sizes in continuous proportion.

References

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