Evangelista Torricelli

Evangelista Torricelli (born the October 15th 1608 with Faenza, in Emilie-Romagna - died the October 25th 1647) is a physicist and Italian mathematician of the 17th century.

Biography

Evangelista Torricelli began its studies in its birthplace, Faenza. Noticed for its gifts by the Jesuits, it was sent to Rome. As of 1626, he becomes the pupil of Benedetto Castelli, faithful friend of Galileo and author of a work of hydraulics, in 1628, very with the current of work of Galileo. Let us recall that in 1632, the Dialoguo of Galileo appears and causes a great agitation in Rome; it is worth with its author his famous lawsuit and its abjuration, the June 22nd 1633.

In 1641, Castelli, returning visit with already blind Galileo, in his villa of Arcetri, brings a treaty of Torricelli to him: the of Motu of 1641. At the request of Castelli, which deals then with the Lagoon of Venice, Torricelli leaves in Arcetri in October 1641 and will help Viviani during the last months of life of Galileo.

It then replaces it as mathematician of the Large-Duke Ferdinand II of Médicis, which releases it from any material concern. He is elected with the Academy of the Sound (of corn), academy whose objective is to purify the language as one releases the sound of the corn grinding. This election carries it to examine arts more than mathematics, which is worth remonstrances of his/her friends to him, Bonaventura Cavalieri and the pupils of Castelli, Raffaello Magiotti and Nardi.

Torricelli is with the acme of its career in 1644: publication of the Opera Geometrica , discovered vacuum “grosso” via the mercury barometer.

1647: at 39 years, in full scientific activity, the typhoid one carries it.

Many of its work were lost or published very tardily, which reduced its influence and its fame. One owes him however:

• the “barometric tube of Torricelli” into hydrostatic: $P\; =\; \backslash \; rho\; G\; H$

• the Formula of Torricelli in hydraulics: $V\; =\; \backslash \; sqrt\; \{2gH\}$.

Its mathematical work is however considerable.

Torricelli and the barometer (1644)

Torricelli is known to have highlighted, in 1644, the atmospheric pressure , by studying the water pump of Galileo, which enabled him to invent the Baromètre with tube of mercury which bears its name. A unit of Pressure, the Torr , is dedicated to him. It corresponds to the pressure of a millimetre of mercury. it is it [[Pascal (unit)|Pascal] who was retained like unit of the international Système in homage to Blaise Pascal, which continued and developed research in this field (1646-1648)].

Torricelli nothing published forever on this subject, nor even asserted the priority. And Blaise Pascal, in his work, does not quote Torricelli once, but, in 1651, states to have remade in 1646-48, an experiment made in Italy in 1644.

It is probable that Torricelli always wanted to move away from the contentions with the Enquiry. Michelangelo Ricci writes to him of Rome, on June 18th, 1644: “I believe that you are disgusted already enough by the ill-considered opinion of the theologists and their practice to interfere constantly and immediately the things God with the reasoning concerning nature. ” But the Jesuits, in particular Niccolò Zucchi, exclude the fact that there is vacuum in the barometric room: for Constantini ( Baliani E I Gesuiti , Firenze, 1969), it is to avoid a new conflict with the mathematicians. This could explain the silence of Torricelli on the subject.

Origin of the problem

This problem corresponds to a very practical consideration: for a long time the water provision of the cities convinced the fountain-makers whom the siphons dysfonctionnent with 18 pitch-stirrers (either 10,3 m). The plaster made it possible to raise the height of the water column, but without it being known why.

Galileo, in 1590, is opposed to the idea of the vacuum “grosso”, but under the influence of Jean-Baptiste Baliani conceded the vacuum between molecules in water, finally was solved with the vacuum “grosso” about 1635.

Baliani in 1630 with the vision right: Nature does not detest the vacuum, only the pressure of the air balances the pressure of the water column:

$P\; =\; \backslash \; rho\; G\; h$

where P is the pressure, $\backslash \; rho$ is the density of the liquid in $kg.m^\; \{-\; 3\}$, G the attraction of gravity in $m.s^\; \{-\; 2\}$ and H height in m in modern notation. And the pressure of the air is to be evaluated by the “weight of the air” to its different degree from tenuity .

In 1630, Galileo puts forth the assumption that the cohesion of the water cord is ensured by the resistance of the intramolecular vacuum: too much high, the cord breaks under its own weight. Argumentation distorts included in Discorso. Remarque: it is currently known that the “pressure interns water” is higher than 1000 bars!

Mersenne and Isaac Beeckman discusses it in 1628. Mersenne written with Galileo about 1640 to ask him the explanation of resistance to the spacing of 2 superimposed blades of glass.

In Italy, in Rome, Benedetto Castelli and Raffaello Magiotti decides to study the problem of the fountain-makers; Antonio Nardi, Gasparo Berti and Michelangelo Ricci also, with tiny Emmanuel Maignan (in favor of the vacuum grosso) and the Jesuit S Niccolò Zucchi and Kircher (opposites with the vacuum grosso, for theological reasons): Berti succeeds and shows it to the Romans (at the end of 1641): water goes up only to 10,3 Mr. But above, that is there?

Contribution of Torricelli

Adopting the idea of Baliani, Torricelli has as a contribution essential that to propose the mercury of density 13,6, which should thus give a height of 10,3/13,6 = 0,76 Mr.

Experiment that Viviani realizes in the current of spring 1644:

• To build the tank and to fill it of quicksilver
• To take a balloon with long collar (of approximately 1m)
• to fill It at short-nap cloth-edge
• To stop with the inch and to turn over on the tank
• To remove the inch

Quicksilver runs out until a 76 cm height, whatever the type of balloon or its slope. In addition, so of water is versed on mercury, and that one withdraws the tube to water, this one “is aspired with horrible violence. ”

June 11th, 1644, Torricelli makes the critical analysis of this experiment with Ricci: Nature does not detest any the vacuum. The vacuum does not aspire: one can at will make slide the tube and reduce the barometric room.

There is no effect except if one heats . (It is thus, seems it, impossible to produce a barometer. But at that time, glass was undoubtedly not degassed, and it was to be established a light air pressure in the room. There remained perhaps also there steam of the former experiments.)

After 1644

Mersenne tried vainly (it is necessary to have a tube of glass which does not break). And they were the experiments of 1646-1648 of Small, Florin Périer and Pascal which magistralement solves the problem with the rise in Puy de Dôme, that Mersenne did not see because he dies in 1648.

The barometer was born.

Who the first made the calculation of the mass of the atmosphere? That is to say mass of a layer of mercury of 0,76 m on all the surface of the Earth:

$\backslash \; rho\_\; \{Hg\}\; \backslash \; cdot\; 76\; \backslash \; mbox\; \{cm\}\; \backslash \; cdot\; S$, with S = surface of the surface of the Earth = $4\; \backslash \; pi\; R^2$

In any case, it is not in papers of Torricelli, nor in those of Baliani.

Torricelli hydraulician, raises of Castelli

The De Motu Aquarum belongs to the treaty of 1644, Opera Geometrica , and it is stated there the law of Torricelli. Its translation had place in France, for Fermat, in 1664, preceded by the work of Castelli, on the running water , and of a speech on the junction of the Seas . (Recall: this speech, which must date from the years 1630, it will haunt Pierre-Paul Riquet (1609-1680) as of his youth.)

Is Torricelli the author of his law?

$v\; =\; \backslash \; sqrt\; \{2gh\}$

is the formula which makes known Torricelli in the world of hydraulics. It connects the speed v of flow of a liquid by the opening of one container to the height H of liquid contained in the container above the opening, G being the acceleration of gravity.

• Before, Mersenne wrote many letters with Peiresc in 1634. In 1639, it seems that Descartes congratulates it for its law which copies “by anticipation” that of Torricelli. These writings are preserved at Arts and Métiers in Paris.

• Descartes, Mersenne, Gassendi writes much until 1643 and afterwards. The difficulties are well encircled: constant water level, pressure loss if contracting, problem of the tail pipe, and obviously the water jet and the law of 1638 of Galileo (4th day). Hydraulica of Mersenne appears in 1644 and Mersenne met Torricelli in 1645. Clearly, if the communicating vases make go up water on the level of the lake, it is manifest that the water jet directed vertically does not go up there really, and each one sees well what the modification of the tube in a jet of watering produces. The demonstration obviously exceeds the physicists of the time.
• Subsequently, 1668, with the Academy of Science of Paris, Christiaan Huygens and Picardy Jean, then Edme Mariotte takes again the problem.

In 1695, therefore after Newton (1687), Pierre Varignon reasons as follows:

“the small mass of water $\backslash \; rho$.S.dx is ejected by the compressive force S.$\backslash \; rho$.g.h with a momentum $\backslash \; rho$.S.dx.v during time dt: that is to say v ² = g.h”

but it always misses factor 2 there.

In 1738, Daniel Bernoulli gives finally the solution in its Hydrodynamica . Leonhard Euler would come soon: all would become clear. This law thus made run… water much.

Torricelli, raises of Cavalieri

Torricelli, large admiror of Cavalieri, exceeded the Master in the use of the Méthode of indivisible the, because for the first time, it will consider homogeneous quantities: by summoning “small” surfaces, one obtains a surface, etc the width of the lines comes to solve the paradoxes of Cavalieri.

Torricelli and the indivisible ones

Cavalieri is undoubtedly the first to show, using the indivisible ones, that the surface under the curve of equation $y=\; x^\; \{n-1\}$ between the points of X-coordinates $x\_1$ and $x\_2$ is

$\backslash \; frac\; \{there\; there\; \_\; 2\; X\; \_\; 2\; \_1x\_1\}\; \{N\}$ for N whole higher than 1.

Fermat generalizes this relation for N rational positive higher than 1 and N whole lower strictly than -1 by using series.

Torricelli will generalize the work of indivisible of Cavalieri so that current thermodynamics calls a process polytropic:

That is to say $PV^n$=cste, $\backslash \; int\_\; \{V\_1\}\; ^\; \{V\_2\}\; P\; FD\; =\; \backslash \; frac\; \{P\_1V\_1-P\_2V\_2\}\; \{n-1\}$

It is the case known as “hyperbolic” (with N different from 1; the case N = 1 will be able to be obtained only by using the Logarithme S)

Let us call $A\_1$ and $A\_2$ the two points on the polytropic curve, $B\_1$ and $B\_2$ their X-coordinate; $C\_1$ and $C\_2$ their ordinate. The feat of ingenuity of Torricelli is to compare the surfaces of the curvilinear trapezoids $S\_1\; =\; surface\; (A\_1B\_1B\_2A\_2)$ and $S\_2\; =\; surface\; (A\_1C\_1C\_2A\_2)$. It shows that $S\_2\; =\; nS\_1$. Then it is enough for him to make the difference of the two surfaces

$S\_1\; -\; S\_2\; =\; -\; (n-1)\; S\_1$

$S\_1\; -\; S\_2\; =\; surface\; (OB\_2A\_2C\_2)\; -\; surface\; (OA\_1C\_1)\; =\; P\_2V\_2-P\_1V\_1$

from where

$S\_1\; =\; \backslash \; frac\; \{P\_1V\_1-P\_2V\_2\}\; \{n-1\}$

To show the equality: $S\_2\; =\; nS\_1$, Torricelli compares the surfaces of gnomons y.dx and x.dy. Nowadays, calculation is done easily by differentiating $PV^n$

$V^ndP\; +\; nPV^\; \{n-1\}\; FD\; =\; 0$ from where $VdP\; =\; -\; nPdV$

But to say it in 1646 remains a feat of ingenuity. What has just discovered Torricelli, it is that, in this infinitesimal figure, the surfaces of gnomons y.dx and x.dy is to be considered and these are these surfaces that one summons: it is thus necessary to consider the thicknesses “lines of Cavalieri. ”

For volumes of revolution, it also understood that the discs of salami have a volume, and that thus should be summoned infinitesimal volumes.

It redémontre, thanks to the indivisible , celebrates it relation of Archimedes, registered on its tomb, concerning the volume of the bowl (S) hemispherical, that of the right cylinder (D) of the same ray R and height R and that of the circular cone of ray R and height R :

$\backslash \; frac\; \{V\; (D)\}\; \{3\}\; =\; \backslash \; frac\; \{V\; (S)\}\{2\}\; =\; \backslash \; frac\; \{V\; (C)\}\; \{1\}$

He is also the inventor of the trumpet of Gabriel , a geometrical figure which presents the astonishing characteristic to have an infinite surface but a finished volume. It is the result of the rotation of part of the hyperbole équilatère y=1/x, for x>1, around the axis (OX) .

He shows also the formula of the volume of the barrel or formula of the three levels (Roberval 1643 letter):

Is a solid of revolution of meridian conical. Let us cut it in 2 levels $z\_1$ and $z\_2$. That is to say $z\_2-z\_1=h$. That is to say $S\_1$ the surface of the disc of dimension $z\_1$, $S\_2$ that of the disc of dimension $z\_2$, $S\_m$ the surface of the disc of average dimension $z\_1+z\_2\; \backslash \; over2$ then

$V\_\; \{barrel\}\; =\; H\; \backslash \; frac\; \{S\_1+S\_2+4S\_m\}\; \{6\}$

It from of deduced volume from the napkin ring:

Is a sphere of ray R. Ôter by drilling the cylinder, centered, vertical, basic circular, which leaves only one annular volume, vulgarly called “napkin ring”, height 2:00.

$V\; =\; \backslash \; frac\; \{4\}\; \{3\}\; \backslash \; pi\; h^3$

This volume is calculated by removing with the volume of the barrel height 2:00 the cylinder of ray $\backslash \; sqrt\; \{R^2-h^2\}$. One can notice that this result is independent of the ray R of the starting sphere and depends only on the height H of the napkin ring (for R > H )

Calculation of the barycentres

Torricelli also is interested in the Barycentre studied solids. Thus, in the same letter addressed to Roberval, it defines the position of the center of gravity of the barrel:

Is a solid of revolution of meridian conical. Let us cut it in two levels $z\_1$ and $z\_2$ and note $h\; =\; z\_2-z\_1$. Either $S\_1$ the surface of the disc of dimension $z\_1$, $S\_2$ that of the disc of dimension $z\_2$, $S\_m$ the surface of the disc of average dimension $z\_1+z\_2\; \backslash \; over2$ then the center of gravity G has as a dimension $z\_G$ such as:

$z\_2\; -\; z\_G\; =\; H\; \backslash \; frac\; \{S\_1+2S\_m\}\; \{S\_1+S\_2+4S\_m\}$

Lastly, generalizing a stone cut in bevel which Cavalieri had proposed to him, it publishes on April 7th, 1646, the formula for the center of gravity (rewritten in modern notation):

Is X = F (Z) the equation of meridian of a surface of revolution, limited by the plans Z = has and Z = B . Then the barycentre of meridian surface has as a dimension $z\_G$ checking

$\backslash \; int\_a^b\; F\; (Z)\; dz\; \backslash \; times\; z\_G\; =\; \backslash \; int\_a^b\; zf\; (Z)\; dz$

And the G' barycentre of volume has as a dimension $z\_\; \{G\text{'}\}$ checking

$\backslash \; int\_a^b\; f^2\; (Z)\; dz\; \backslash \; times\; z\_\; \{G\text{'}\}\; =\; \backslash \; int\_a^b\; zf^2\; (Z)\; dz$

Had Torricelli known the theorems of Guldin of 1643? In any case, their research was supplemented.

Torricelli cinematician, raises of Galileo

The Cycloid

Admiror of Galileo, raises gifted of Cavalieri, it will continue ideas which existed already at Roberval.
Jean Itard (historian of the Koyré Center, Paris) thoroughly carried out the survey about the squaring of the cycloid one:
Galileo would have answered that he had vainly sought, including by cutting out an owner out of paperboard and weighing it. The Roberval-Torricelli competition is tighter, and it is difficult to see there clearly, because at the time, one stated to have found, but one did not publish, for fear the other doubles you: the questions of priority will be the nightmare of XVIIe.
But the result of the investigation is there: roberval priority, without ignoring the merits of Cycloid Torricelli.cf. .

time Diagram

On the other hand, he is really the inventor of the time Diagramme and the Diagramme of spaces: he very clearly says in any general information, then on simple examples, that if the speed of the mobile is v (T) , its X-coordinate will be $\backslash \; int\_0^t\; v\; (T)\; dt$, in our modern notations which go back to Leibniz ( Lettre in Oldenburg of Oct. 29, 1675).

And reciprocally, Torricelli speaks about the “theorem about inversion”: if one has X (T) , the “tangent” will give speed:

$v\; (T)\; =\; \backslash \; frac\; \{X\; (T)\}\{TT\_1\}$,

where $TT\_1$ is the subtangent, T is the point of the x-axis of X-coordinate T , $T\_1$ is the point of intersection of the tangent to the curve at the point of X-coordinate T with the x-axis.

It is thus on the same level as Barrow in England. In any case, James Gregory, raises in Bologna, with the continuators of the work of Cavalieri, could benefit from it.

Diagram of spaces: the formula $\backslash \; frac\; \{1\}\; \{v\; (X)\}\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{2gx\}\}$ does not frighten him any more, whereas Galileo hesitated, not wanting to use work of Cavalieri.

Parabola of safety

• See detailed article: Parabola of safety
  

It is him which, for the first time invents the concept of envelope , and finds the total solution of the freefall “with violence”, and the complete description of the parabola of safety, via a method little connue.
Unfortunately, it did not supplement its work: without introduction of the resistance of the air, the concept of asymptote (cf Ballistic external) does not exist; and its work is risée of the artillerists (bombardieri).

polar Coordinates

In addition, by its method of the indivisible curves , it will be able to deal with the problems of curves in polar coordinates, at one time when Descartes had just spoken about Cartesian Coordonnées: it can integrate the surface and the length of the spirals R = $\backslash \; theta^k$; but especially, it will discover the cochlea (cf Spirale logarithmic curve) and its extraordinary properties ( Opere , III, p 368, p 477, Lettres in Ricci of 1646 and Cavalieri (1598-1647)):

That is to say a spiral logarithmic curve, the arc resulting from M is rolled up an infinity of time around O but its measurement is finished: to trace the tangent in Mr. Y to place the point C so that triangle OCM is isocèle.
The arc measures OC+ CM.
The surface swept by OM is equal to the surface of the OCM.
triangle Bernoulli would not have said better.

Torricelli, dynamician?

He is still well early, before 1647, of speaking about dynamics. Nevertheless, somebody studied proto-dynamics would certainly be interested by the evolution of the concept of percussion since Galileo about 1590 until the lessons of Torricelli in 1644-1646 in Academia del Crusco. A specialist would find there sentences disconcerting of the type:

“ gravity is a fountain from where momentum spout out continuously… They will be preserved and incorporated; when the low register comes to give the percussion on the marble, it does not apply any more its load of 100 pounds, girl of only one moment, but the forces omnes (summoned) ten moment old girls. ”

If one replaced the urgent word per duration, this text would be of an incredible modernity.

Tragedy destiny

October 5th, 1647, he wrote in Cavalieri: “I will write a book on “new lines”. ” The 25, typhoid carried it. And Cavalieri dies shortly after. Who recovers this “will fatras ideas”? A cassette, the famous cassette, bequeathed to Serenai? This cassette would contain also the receipt of manufacture of very good glasses of optics: Torricelli was one of the Masters.

John Wallis and Jacques Bernoulli would have been regaled of such a treasure. Will the searchs for filiation be undoubtedly to seek, that is to say towards the school of Tiny (Mersenne?), that is to say towards Stefano degli Angeli and its pupil James Gregory, but it is clear that this untimely death and the work of the Inquisition will weaken the Italian school. Florence remained active; nevertheless the flame will pass in flourishing England (Gregory, Wallis, Barrow), whereas Venice must count its under, for its war against the Othomans, and that Germany has just been devastated by the Thirty Year old war (1618-1648).

If heritage and continuation there is, it is perhaps at Christiaan Huygens (1629-1695) that it is necessary to seek. Filled teenager (he is the son of Constantijn Huygens, Prime Minister of Holland), he accepted from Mersenne, which knew it via Descartes, many problems transmitted by Torricelli, which were used to him as training as of the 16 years age (1645).

See too

Related articles

• Formula of Torricelli
• Principle of Torricelli
• Mathematical in Europe at the XVIIe century

External bond

• the trumpet of Gabriel

Sources

• De Gandt (François), ED., the work of Torricelli. Science galiléenne and new geometry . Nice, Publication of the Faculty of Arts and social sciences/Paris, Beautiful letters, 1987. ISBN 2-251-62032-X.
• De Waard, the barometric experiment , Thouars 1936.
• Fanton d' Andon (Jean-Pierre), Horror of the vacuum . Paris, CNRS 1978. ISBN 2-222-02355-6.
• Michel Blay, in F. of Gandt: the Work of Torricelli (Nice 1987).
• Mersenne: Opera (in particular the voyage in Italy 1644-1645).

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