Abstract
Minimum principles constitute one of the most beautiful and widespread paradigm in philosophy and the sciences.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
This is the law of parsimony often attributed to Ockham
Entities are not to be multiplied beyond necessity;
in science it is often stated as
What can be done with fewer assumptions is done in vain with more,
and was elevated to a virtue by Dante Alighieri, De Monarchia , Chapter XIV
All that is superfluous displeases God and nature. All that displeases God and nature is evil.
- 2.
Max Born wrote in his Physik im Wandel Meiner Zeit
It is science, not nature, to be economical.
- 3.
As stated in the Introduction of [112].
- 4.
Johann Bernoulli had posed his problem privately to him on 9 June 1696 and Leibniz’s answer is dated 16 June, see Section 2.3.
- 5.
The events connected with the brachistochrone problem were reported by Johann Bernoulli in a letter to Henri Basnage sieur de Beauval (1657-1710), editor in Rotterdam from 1687 to 1709 of the Histoire des Ouvrages des Savants a kind of follower of the Nouvelles de la République des Lettres de Pierre Bayle (1647-1706); see [22] and [118] pp. 283-290.
- 6.
Westfall [200] claims that the challenge was for Newton
Manifestly, both Bernoulli and Leibniz interpreted the silence from June to December as a demonstration that the problem had baffled Newton. They intended now to demonstrate their superiority publicly.
See Section 2.4 for more.
- 7.
Comunicatio suae pariter duarum alienarum ad esendum sibi primum a dn. Joh. Bernoullio, deinde a dn. Marchionne Hospitalio communicatarum solutionum problematis curbae celerrimi descensus a dn. Joh. Bernoullio geometris publice propositi, una cum solutione sua problematis alterius ab eodem postea propositi. A French translation is available in [167], pp. 351-358.
- 8.
He also noted that “l’Hôpital , Huygens were he alive, Hudde if he had not given up such pursuits, Newton if he would take the trouble” could also have solved the problem. In fact, Newton had published his answer anonymously in the January 1697 issue of the Philosophical Transactions; the paper was republished anonymously in the same issue of the Acta, see Section 2.3 and Section 2.4.
- 9.
Johann Bernoulli says in the Letter to Basnage [22] that he did not know about Galilei’s considerations when posing his problem and that he had only learned of Galilei later from Leibniz, a claim that sounds doubtful on account of the celebrity of Galilei and of his Dimostrazioni.
- 10.
The law was known also to Archimedes (287-212 BC) who had proved it by symmetry: If \(\theta _i\ne \theta _r\) for instance \(\theta _i>\theta _r\) then, by inverting the direction of the ray, we would get \(\theta _r>\theta _i.\)
- 11.
More precisely, the incident and reflected ray lie in the same plane through the source and the target and orthogonal to the mirror and \(\theta _i=\theta _r\), see next paragraph.
- 12.
Heron of Alexandria was an encyclopedic scholar who wrote mainly about geometry and mechanics mixing approximate and rigorous procedures. Not much is known about him — determining the period in which he lived has been one of the most debated question in the mathematical historiography —; with sufficient certitude we know that he lived between 100 BC and 100 AC.
- 13.
There seems to be no apparence that Descartes had heard about Snell’s result.
- 14.
We shall not enter such a topic and refer the reader, for instance, to [170] [110].
- 15.
In the De coelo Aristotle writes
The revolution of the heaven is the measure of all motions, because it alone is continuous and unvarying and eternal, the measure in every class of things is the smallest member, and the shortest motion is the quickest, therefore the motion of the heaven must clearly be the quickest of all motions. But the shortest path of those which return upon their starting-point is represented by the circumference of a circle and the quickest motion is that along the shortest path.
Quoted in [175] p. 46.
- 16.
Comparing figures with different forms but same perimeter seems to have been very attractive to Greek. Proclus (V century) says that the average ancient Greek would find paradoxical that triangles with the same basis and equal height had the same area. Polybius (II century BC) wrote that most people judge the size of cities simply from their circumference and Thucydides (V century BC) seemed to estimate the size of Sicily according to the time needed to circumnavigate it. [Actually, the area of a figure is estimated from above in terms of the perimeter, isoperimetric inequality, and one can even compute the area in terms of the boundary curve, but this is a little more complex.]
- 17.
Minimum problems, as for instance minimum distance of a point from a curve, appear in the works of Archimedes and of Apollonius of Perga (II century BC).
- 18.
As it is typical of most ancient Greek mathematicians, including Euclid, not much is known of their life.
- 19.
An \(n-\)gon is a polygon with n sides; it is said to be regular if it has all its sides and all its angle equal.
- 20.
As we shall see, for a class of relevant problems, this is in fact not true, causing quite some problems, see Chapter 3.
- 21.
For a long period the existence of a minimizer was essentially taken for granted.
- 22.
Galilei of course writes only proportions of homogeneous quantities; we use modern notation that, on the other hand, were already in use at the time of the Bernoullis.
- 23.
The reader may consult one of the many available volumes on history of mathematics, for instance, [41] [40] [136] [119] and [111], and the references therein.
- 24.
Comments are mostly taken from [37] [111].
- 25.
An English translation can be found in [52].
- 26.
The sum of infinitely many infinitesimal lines produces a line.
- 27.
See [52] p. 147:
In any supposed (continuous) transition, ending in any terminus, it is permissible to institute a general reasoning, in which the final terminus may also be included.
In a way the principle had been called up already to produce the differentials and their properties .
- 28.
In the words of Bos [37] paraphrasing the letter of Leibniz to Varignon .
- 29.
However, we should say that mathematicians of the time were often able to handle difficulties and reach correct results, sometime by means of a sort of mysterious intuition.
- 30.
For Euler a generic function is essentially a power series.
- 31.
He also deals with functions of several variables and with non uniform distributions
$$x,\, x+ \omega ,\, x+\omega +\eta ,\, x+\omega +\eta +\mu ,\, \cdots ,$$ - 32.
Called also trochoid or, in French, roulette from the roulement des roues.
- 33.
For further information the reader is referred to any of the many history of mathematics, e.g. [44], and, in particular [201] and [110].
- 34.
This led to several unpleasant priority disputes. For whenever someone announced a result on the cycloid, Roberval would send out letters claiming he had found the same result earlier. A suggested motivation for his reluctance to publish is the following: Roberval held the chair of Ramus in the Collège de France, a position filled every three years by an open competition and the incumbent was responsible for writing the problems used in the competitions, so it was to his own advantage to keep his best methods secret.
- 35.
For an account of Cavalieri work and of its influence the reader may consult [1] [115] [2] [110].
- 36.
An anagram of Louis de Montalde, the name under which he published his Lettres provinciales.
- 37.
Tractatus duo De Cycloide, Oxoniae, 1659.
- 38.
This is the original in Italian:
Dei quesiti mandatigli di Francia non so che sia stato dimostrato alcuno. Gli ho con lei per difficili molto a essere sciolti. Questa linea arcuata sono più di 50 anni che mi venne in mente di descriverla [\(\ldots \)] per adattarla agli archi di un ponte [\(\ldots \)]. Parvemi da principio che lo spazio potesse esser triplo del cerchio che lo descrive, ma non fu così, benché la differenza non sia molta [\(\ldots \)]. Ebbi circa un anno fa una scrittura di un padre Mersenno dei Minimi di San Francesco da Paola mandatami da Parigi, ma scrittami in caratteri tali che tutta l’Accademia di Firenze non ne potesse intender tanto che se ne potesse trar costrutto alcuno [\(\ldots \)] io risposi all’amico che me la mandò che facesse intendere al detto padre che mi scrivesse in caratteri più intelligibili.
- 39.
It is worth noticing that apparently the equation of the cycloid appears to have been written for the first time by Leibniz in 1686 as
$$y=-\sqrt{2x-x^2}+\int \frac{dx}{\sqrt{2x-x^2}},$$compare [161], a translation in French is in [167], pp. 131-143.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Freguglia, P., Giaquinta, M. (2016). Some Introductory Material. In: The Early Period of the Calculus of Variations. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-38945-5_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-38945-5_1
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-38944-8
Online ISBN: 978-3-319-38945-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)