|
|
CONTENTS
The surface under the hyperbole
|
 |
An example of the development of a concept in mathematics One should not see the history of mathematics like a triumphal parade in the medium of a boulevard without obstacles. Quite to the contrary, not only this history shows many jolts, but the borrowed paths are seldom similar to a straight line, and it even has some dead ends... There were also abrupt projections due to new concepts which answered questions sometimes very far away from the initial problems which had generated them. The logarithms are exemplary of this development chaotic and rich at the same time. Started from a simple idea, but whose implementation needed a hard work (the construction of the tables), they were initially the engine of a development of applied mathematics before proving to be the solution of a geometrical problem. Subjects of theoretical and thorough studies thereafter, they were also an essential tool for the modelisation of multiple physical phenomena. The traditional pedagogical presentation of the logarithms privileges the logarithm known as "Napierian". It is introduced as being the primitive function of the opposite function cancelling itself for value 1 of the variable. If this introduction is mathematically satisfactory, it is far from being obvious for the pupils and the students, and the essential property is hidden. Of course, the historical problem which led to the design of the logarithms is also absent, whereas its use to present this new concept has the advantage of simplicity: it is quite simply a question of building a table that makes it possible to perform quickly multiplications, divisions and exponentiations. Today the use of the logarithms for calculations is obsolete, but the concept remains fundamental in the basic mathematical culture and they are very present as well in physics as in chemistry. Their history undoubtedly remains a modest chapter, but its exemplarity, and its richness testify to what can present the development of Mathematics. The origin of the concept of the logarithms is to be found in a mathematical problem, undoubtedly, but in a problem of applied mathematics: it is a question of simplifying the heavy task of the calculating persons, excessively complicated as soon as it implies multiplications, divisions, even exponentiations or extractions of roots. To the XIV, XV and XVth centuries (and of course before) the fields concerned are less the economic questions than problems like land surveying, and especially astronomy, in particular in its applications to navigation. These operations ask for a certain . If progress of numeration could advance the things, like the use of the figures known as Arab, the algorithms of multiplication and division are unknown; the rational numbers, systematically written in whole part plus fraction of the unit, make even the additions very complicated. We are indebted to the Arab mathematician IBN JOUNIS for having proposed, in the XIth century a method, said prostapheresis, to replace the multiplication of two sinus by a sum of the same functions, and this method will remain a long time in force. The multiplication of the sines (and their division) is an essential operation, since any calculation in geometry, in particular the resolution of triangles, is an operation over lengths often nonmeasurable, therefore obtained from the values of the angles. It is ARCHIMEDES that had the fundamental idea which was to generate the logarithms: "When numbers are in continuous proportion starting from the unit, and that some of these numbers are multiplied between them, the product will be in the same progression, far away from largest from the multiplied numbers of as many numbers as the smallest one of the multiplied numbers is from the unit in the progression, and far away from the unit of the sum minus one of the numbers from which the multiplied numbers are far away from the unit " (Arinaire, trad. VERECKE) That is: The idea of ARCHIMEDES is found again in the work of CHUQUET and STIFEL, in the XVth century, but, neither one nor the other had sufficient influence to impose the comparison of a geometrical series and an arithmetic series as a method of calculation, or as a new field of investigation in mathematics. John NAPIER (also written NEPER) was born in 1550. Of Scottish minor nobility, he showed all his life a curious and dynamic spirit, in spite of an existence far away from the cultural centres of the time. The introduction of the logarithms is not his only claim to fame, since he also wrote a text on the equations and in addition he imagined a system of calculation by means of small graduated rules (Rabdologie). In 1614, he publishes " Mirifici logarithmorum canonis descriptio... " where, using a kinematic approach, he connects a geometrical series and an arithmetic series. The first is that of distances covered at a speed proportional to themselves, the second, that of distances covered at constant speed; these are then " the logarithms " of the first ones (the neologism is of NAPIER). The selected unit is 107, and the work includes a logarithmic table of the sinus, of which we mentioned the importance previously, the angles increasing minute by minute. In 1619 appears a second work, "Mirifici logarithmorum canonis constructio... " where the author explains how to calculate the logarithms. This work is posthumous, since NAPIER dies in 1617. Meanwhile, an eminent mathematician of London, Henry BRIGGS, had detected the importance of this work and moves to Scotland to meet its author. Taking up the fundamental idea, but by adopting a simple geometrical series, that of the powers of 10, he publishes in 1617 a first table, with 8 decimals. The logarithm of a number x is thus defined as the exponent ln of 10, such that x is equal to 10 power ln. Other tables will follow which will allow the diffusion of the method, in particular on the continent. In fact, the idea was in the air; a collaborator of KEPLER, the Swiss B\RGI, proposed at the same time, to simplify calculations which he was to carry out, to put in correspondence an arithmetic series (red numbers) and a geometrical series (black numbers); however his work was published only in 1620. It is initially in Germany that the logarithms will develop. At the beginning of 1617, KEPLER, fortuitously in Vienna, has the occasion to consult the first work of NEPER. After a quick glance, he misinterpretes it. He will comment this the following year in a letter to a friend: " A Scottish baron of which I did not retain the name, proposes a brilliant work in which he replaces the need for multiplication and division, by the simplicity of the addition and the subtraction, without employing the sinus: in exchange, he needs the rule of the tangents: and the variety, the length, the heaviness of the addition and the subtraction replace the difficulty of the multiplications and divisions " However KEPLER uses obviously the rule of the sinus, as well in plane as in spherical triangles; for him the work of NEPER does not show any interest. In the course of 1618, he however has in hand the work of Benjamin URSINUS: " Trigonometria Logarithmica John Neperi "; he recognizes his error then and is enthusiastic of this new calculation. In 1619, finally, the book " Mirifici Logarithmorum descriptio " arrives at Linz, to KEPLER, who rather quickly undertakes to modify the concept of it to adapt it to his needs. Its adhesion is such that he dedicates his ephemerides of 1620 (appeared end 1619) to the "famous and noble lord JOHN NEPER, baron of MERCHISTON ". The diffusion on the continent of this new concept is especially due to the tables published by Flemish Adrien ULACQ, in 1628, taking again the tables of BRIGGS. The goal was to provide a practical treaty of calculation, in particular for the use of the land-surveyors. The first tables were followed by others, increasingly precise, and mentioning their priority use for trigonometrical calculations. The method of construction of the tables passes initially, obviously, by the determination of the logarithms of the prime numbers; the others are then calculated by simple summation. It is a question in fact of taking either of " the proportional averages, or the square roots ". EULER will write in 1748: " thus taking proportional averages, one arrives to get Z=5,000000, that answers the logarithm sought 0,698970, by supposing the logarithmic basis=10. Consequently 10 69897/10000 = 5 with some approximationIt is in this manner that BRIGGS and ULACQ calculated the ordinary logarithmic table, though one imagined more expeditious methods since." THE SURFACE UNDER THE HYPERBOLE Of course, the essential stage of the mathematical development of the concept is in its approach to the hyperbola. It is due to the Jesuit GREGOIRE OF SAINT-VINCENT, born in Bruges in 1584. He had completed the drafting of a " Opus giomitricum... " in 1630, in which he claimed to have solved the problem of the hyperbola and quadratures of the circle. This work was published only in 1647, and if it is a failure as for the quadrature of the circle, it highlights that the surface under the hyperbola is connected with logarithms. The work of this author is not pointed specifically to the logarithms, but rather in an attempt at resolution of general problems of quadrature, very fashionable at the time, and in a completely traditional style; the innovative aspect lies in the use of a certain pass to infinity to justify the first part of his demonstration. We are however still before the era of LEIBNIZ and NEWTON... The approach of the calculation of the surface under the hyperbola with logarithms is thus not by GREGOIRE OF SAINT-VINCENT himself; his work, initially ignored, was the subject of criticisms, well founded, by the way, with regard to the quadrature of the circle. It is one of these defenders, the Jesuit SARASSA who will mention that " the hyperbolic surfaces can have to do with logarithms " The calculation by GREGOIRE OF SAINT-VINCENT rests on the fact that, when the abcisses are in geometrical series, surfaces are in arithmetical series. Let us take the simplest of the hyperbolas, equation x.y=1, in an orthonormal reference system. A, B, C... will be points of the axis of the abcisses (axis "x ") in geometrical series; D, E, G... will be then the points of the hyperbola with same abcisses. GREGOIRE OF SAINT-VINCENT shows first of all that surfaces between the curve and DE on the one hand, and EG on the other hand, are equal; trapeziums ADEB and BEGC having the same surfaces, then the surfaces under the hyperbola are equal. It will be found a few years later, in certain handbooks of geometry, like the one of PARDIES (1671), the annoucement of the result found by SAINT-VINCENT, but this is far from being the general case, and PARDIES was also a Jesuit! If the analytical aspect of the logarithm, in other words its status of function, was already considered by KEPLER, it due to TORRICELLI, and then to HUYGENS to study the logarithmic curve and to WALLIS, after a first work by MERCATOR, to propose a development in series (1667). This technique is then new, it is undoubtedly one of rare the interesting parts of the work of MERCATOR; indeed this author does not seem to have known how to develop the initial idea, namely the integration of the series:
in :
This new aspect then allows an easier calculation of the logarithms of the numbers and it will be found thereafter in the handbooks of XVIIIth century. With regard to the curve of the function logarithm, known as "logarithmic curve", TORRICELLI proposes its layout in 1646, in letters to his correspondents, but his death in 1647 delays its diffusion. It is rather due to HUYGHENS to expose their properties in his " Dissertation on the cause of weight ", appeared into 1690. HUYGENS had been interested since 1651 in the logarithms and their calculation, in particular within the framework of the quadrature of the hyperbola; he had taken again the problem much later (1666) when he was participating in the works of the very new Royal Academy of Sciences of Paris, and had used the concept in questions of probability and combinatorial. The logarithms at that time then form really part of the mathematical corpus; it is not anymore just a simple method of calculation, but a field with whole share. They can be found in many works, their theoretical status not being anymore questioned. The end of the XVIIth century sees the beginnings of the mathematical physics, and the tool proposed by the logarithms will be of an unquestionable help. It is found of course, as mentioned above, in the " Dissertation of the cause of weight " of HUYGENS, but they are also in various works on the atmospheric pressure, in particular those of MARIOTTE. It is necessary to see the use of the logarithms along four directions: - the first is the one that generated them, namely calculation of geometrical formulas, used in astronomy and applied in navigation, as well as, more simply, in land surveying. Many tables were published in a "pocket " format for its use in the land or on board of the vessels. These tables will be preceded by instructions for their use and will include of course a logarithmic table of the sinus. - the second, more simply still, is that of the application to multiplication calculations. It led to the construction of the "slide rules", to the use by any apprentice graduate of a table for any operation in physicochemical sciences and to the development of algorithms for the contemporary desk-top calculators. - the third consists in conjecturing models, based on experiments, were the logarithms will come into play in the comparison of values. To highlight a relationship between measurements in arithmetic series with another series in geometrical series will result in estimating the first phenomenon like a logarithm of the second. The logarithmic scales are current currency today... - the last is very theoretical; the introduction by LEIBNIZ and NEWTON of differential and integral calculus will allow a number of analytical reasoning, in relation with physical or chemical phenomena, being able to lead, by simple integration of the inverses, to logarithmic results. The logarithms used in the first three cases will be those of BRIGGS, that is decimal logarithms. On the other hand integration implements the "natural" logarithms, called "napierians " in honor of their father founder. In the field of pure mathematics, the logarithms introduce new transcendent sizes. They thus contribute to widen the field of comprehension of the numerical. However, one cannot speak about function, therefore of logarithmic function, in the modern sense, before the intervention of EULER in the second part of the XVIIIth century. This does not prevent LEIBNIZ and NEWTON to use the relations: (written in the modern way)
as attests a manuscript of the first author dated in 1675. It is again EULER who, in the "Institutions of integral calculus" published from 1768 to 1770 will treat in a masterly way the integration of logarithms. The use of integration by parts is systematic and it leads to a last operation either directly integrable or developable in whole series. In addition at the beginning of XVIIIth century, LEIBNIZ and JEAN BERNOULLI develop a controversy on the existence of logarithms of negative numbers, even imaginary. EULER, in 1749, will give a conclusion to the debate by giving up the univocal character of the logarithm; a number has infinite logarithms (complex) of which only one (except for a multiplicative constant) is real. At last, it is necessary to evoke the exponential, which is admitted to have been introduced by LEIBNIZ and JEAN BERNOULLI, within the framework of their work in analysis. This new concept will be developed by EULER, and will allow him to solve the problem of the Chain in its "Initiation to the infinitesimal analysis" of 1748. Since their introduction, the logarithms were found in the handbooks of arithmetic as well as in those of analysis. Object and method, they took part of the development of Mathematics, but also of the history of physicochemical sciences. The pH-metric for example could not have been conceived at the beginning of XXth century without the help of this mathematical concept. Started from an idea in fact very simple, they remain a tool maybe be modest, but despite everything essence of scientific knowledge.
|
with 
where 
up