Evangelista Torricelli

How to use this course

Evangelista Torricelli is remembered for one experiment: a glass tube of mercury, inverted in a bowl, that showed the weight of the air and produced the first sustained vacuum in a laboratory. He was primarily a geometer. In his one published book he showed that an infinitely long solid can have a finite volume, measured the cycloid, and gave twenty proofs of an Archimedean theorem. He extended Galileo’s theory of projectiles and founded the quantitative study of jets of water. He did all this in about six productive years, and died at 39.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Torricelli wrote in the geometric language of his day, in ratios and proportions. The course states his results in modern notation and marks where the modern form adds something he did not have.

Timeline

Faenza and the Jesuits, 1608–1626

Evangelista Torricelli was born on 15 October 1608. His family came from Faenza in the Romagna, then part of the Papal States, and most biographies, including the Dictionary of Scientific Biography, place his birth there. Some recent accounts place it in Rome instead, and older and newer sources also disagree on his mother’s name. The undisputed facts are these: he was the eldest of three children, his father Gaspare was a textile worker of modest means, and the family identified with Faenza, which honours him as a native son with a statue raised in 1868.

The family was poor, and the boy’s education was entrusted to his father’s brother Jacopo, a monk of the Camaldolese order. Jacopo taught him humanities and saw him into a Jesuit college, where he studied mathematics and philosophy from about 1624 to 1626. Most historians place this school in Faenza. His progress was striking enough that the family decided he should continue in Rome, under the care of a Camaldolese connection of Jacopo’s.

Rome with Castelli, 1626–1632

In Rome, Torricelli became the pupil of Benedetto Castelli, a Benedictine monk who had studied with Galileo Galilei at Padua and now taught mathematics at the Sapienza, the papal university. Castelli was a pioneer of the science of flowing water: his Della misura dell’acque correnti (1628) dealt with the flow in rivers and canals. There is no record that Torricelli enrolled at the university. He studied privately with Castelli and in return served as his secretary, deputizing for him in teaching when Castelli was away.

Through Castelli, Torricelli entered the circle of Galileo’s followers. Bonaventura Cavalieri, another of Castelli’s pupils, was then developing the method of indivisibles, and Torricelli became his ally and correspondent. Michelangelo Ricci, a younger Roman mathematician who later became a cardinal, became his closest friend.

The letter of 1632

On 11 September 1632, Torricelli answered a letter from Galileo to Castelli. It is his earliest surviving letter. He described himself as the first in Rome to have studied Galileo’s works minutely, and as a Copernican by conviction, having read the newly published Dialogue Concerning the Two Chief World Systems. Within a year Galileo had been condemned by the Inquisition, and Torricelli never again wrote openly on cosmology.

The Ciampoli years and Arcetri, 1632–1642

The next decade is poorly documented. Torricelli appears to have served as secretary to Giovanni Ciampoli, a prelate and friend of Galileo who had fallen from papal favour and was sent to govern small towns of the Papal States. The places suggested include Montalto, Norcia, San Severino and Fabriano. In these years he wrote most of what became Opera geometrica, mastering Archimedes and the new indivisibles, and composed a treatise on motion that extended Galileo’s Two New Sciences (1638).

Galileo’s last assistant

In April 1641 Castelli, passing through Florence, showed Torricelli’s treatise on motion to Galileo, then old, blind and under house arrest at his villa in Arcetri. Galileo was impressed and invited the author to come and help him prepare further dialogues for the Two New Sciences. Torricelli arrived on 10 October 1641. He joined Vincenzo Viviani, who had lived with Galileo since 1639. For three months the two young men took Galileo’s dictation and discussed geometry with him. Galileo died on 8 January 1642.

Torricelli was preparing to return to Rome when Grand Duke Ferdinando II de’ Medici asked him to stay in Florence as court mathematician, with lodging in the ducal palace. He did not receive Galileo’s second title of philosopher. A Latin anagram of his name, published in 1715, read “En virescit Galileus alter”: here flourishes another Galileo.

Indivisibles, straight and curved

For the general reader. To compare two shapes, slice both into infinitely thin strips. If corresponding strips are always equal, the shapes have equal area. Cavalieri made this rule the basis of a method; Torricelli used it with more freedom, slicing along curves as well as straight lines, and he often checked his results with Archimedes’ older, rigorous method.

In modern terms the principle is the statement that \(\int\ell_1(t)\,dt=\int\ell_2(t)\,dt\) when \(\ell_1=\ell_2\). It is a theorem of integration, but Cavalieri treated it as a principle about the “collection of all lines” of a figure.

Curved indivisibles

Torricelli allowed the indivisibles to be curves. A disc of radius \(r\) is the union of concentric circles of radius \(\rho\), \(0\le\rho\le r\). A right triangle with legs \(r\) and \(2\pi r\) is the union of segments parallel to the long leg, of length \(2\pi\rho\) at distance \(\rho\) from the apex. Each circle equals the corresponding segment in length, so the disc equals the triangle: Archimedes’ Proposition 1 of Measurement of a Circle in one line.

A warning

Torricelli knew that careless use of indivisibles gives contradictions, and he collected examples. Draw the diagonal of a rectangle that is not a square. Match each horizontal line in one triangle with the vertical line in the other triangle that meets it on the diagonal. The matched lines always have lengths in a fixed ratio different from 1, yet the two triangles are plainly equal. The paradox arises because the “spacing” of the indivisibles differs on the two sides. Torricelli concluded that indivisibles must be compared with equal thickness, an insight later made precise by the Jacobian factor in a change of variables.

The acute hyperbolic solid

For the general reader. Spin the curve \(y=1/x\) about the \(x\)-axis. The result is a trumpet that goes on forever, yet Torricelli proved that it holds only a finite amount of space. Contemporaries found this astonishing. It was the first clear case of an unbounded figure with a finite measure, and it fed a long philosophical argument about the infinite.

Proof by cylindrical shells

Torricelli sliced the solid into coaxial cylindrical surfaces. The surface at radius \(y\), \(0<y\le 1/a\), runs from \(x=0\) to the hyperbola at \(x=1/y\), so its area is

the same for every \(y\). Each such surface equals a disc of radius \(\sqrt2\). Stacking the discs at heights \(y\in(0,1/a]\) gives a cylinder of radius \(\sqrt2\) and height \(1/a\). By the principle of indivisibles, the two solids are equal. Torricelli also supplied a proof in the style of Archimedes, by inscribed and circumscribed figures, to satisfy readers who distrusted indivisibles.

In calculus, the tail alone has volume \(\int_a^\infty\pi x^{-2}\,dx=\pi/a\), and the added cylinder \(\pi(1/a)^2a=\pi/a\), total \(2\pi/a\).

The surface

The lateral surface area of the tail is

The paradox of a trumpet that can be filled with paint but not painted belongs to later commentators; Torricelli’s own emphasis was on the finite volume. The solid is now often called Gabriel’s horn.

The cycloid

For the general reader. A point on the rim of a rolling wheel traces an arch called the cycloid. Galileo had tried to find its area by cutting shapes out of sheet metal and weighing them. Torricelli computed it exactly: three times the area of the wheel. He also found its centre of gravity and its tangents. The French mathematician Roberval had found the area first, and the resulting quarrel outlived Torricelli.

The cycloid generated by a circle of radius \(r\) rolling on the \(x\)-axis is

Proof with indivisibles

Consider half an arch, from \(t=0\) to \(t=\pi\), inside the rectangle \([0,\pi r]\times[0,2r]\) of area \(2\pi r^2\). Introduce the companion curve \(x=rt\), \(y=r(1-\cos t)\). At each height \(y\), the horizontal distance between the cycloid and the companion is \(r\sin t\), which is the half-chord of the generating circle at that height. By Cavalieri’s principle the region between the two curves equals a semicircle, area \(\tfrac12\pi r^2\). The companion curve, a sine curve, cuts the rectangle into two congruent halves by central symmetry, each of area \(\pi r^2\). So the region above the half-arch has area \(\pi r^2-\tfrac12\pi r^2=\tfrac12\pi r^2\), and the region under it \(2\pi r^2-\tfrac12\pi r^2=\tfrac32\pi r^2\). Doubling gives \(3\pi r^2\).

Tangents by composition of motions

Torricelli, like Roberval, found tangents by treating the curve as the path of a point with two simultaneous motions. The tracing point moves with the centre of the wheel (velocity \(r\omega\) horizontally) and turns about it (velocity \(r\omega\) tangent to the wheel). The tangent to the cycloid is the diagonal of the parallelogram of these two velocities. Since both have the same magnitude, the tangent bisects the angle between them, and one finds that it passes through the top of the rolling circle.

Projectiles and Torricelli’s principle

For the general reader. Galileo showed that a cannonball follows a parabola. Torricelli asked what happens when the gun is fired at every possible angle with the same charge. The trajectories fill a region bounded by another parabola, which he called the parabola of safety: outside it, no shot can reach you. He also stated a principle about centres of gravity that later became a pillar of statics.

De motu gravium (1644)

The second part of Opera geometrica, De motu gravium naturaliter descendentium et proiectorum, rebuilt Galileo’s theory of falling bodies and projectiles. Torricelli extended Galileo’s results to launches at any angle, noted that the motion of a projectile is reversible, and gave tables for gunners. For speed \(v\) at angle \(\theta\), with \(g\) the acceleration of gravity,

Torricelli’s principle

Galileo had used a related idea. Torricelli made it an explicit axiom and derived from it the law of the inclined plane. Christiaan Huygens adopted it in his work on the pendulum, and it is a special case of the principle of virtual work: at equilibrium, the potential energy is stationary.

The law of efflux

For the general reader. Punch a hole near the bottom of a full bucket. How fast does the water come out? Torricelli’s answer: as fast as a drop would be moving if it had fallen freely from the water’s surface down to the hole. Ernst Mach later called him the founder of hydrodynamics for this result.

Torricelli argued from observation: a jet directed upward rises nearly to the level of the water in the vessel, falling short only because of friction and air. By the reversibility of motion (Lesson 07), the jet must leave with the speed of fall from that height. Consequences: the speed varies as the square root of the depth, and the discharge through a hole of area \(a\) is \(a\sqrt{2gh}\) (reduced in practice by the contraction of the jet).

A century later, Daniel Bernoulli’s equation \(p+\tfrac12\rho v^2+\rho gz=\text{const}\) along a streamline gives the theorem at once: at the free surface and at the jet, the pressure is atmospheric and the speed at the surface is negligible, so \(\tfrac12\rho v^2=\rho gh\).

Draining a tank

For a vessel of horizontal cross-section \(A(h)\) at height \(h\) above a hole of area \(a\), continuity gives

The barometer and the sea of air

For the general reader. Pump makers knew that a suction pump cannot raise water more than about ten metres. Galileo blamed a limited “force of the vacuum”. Torricelli realized that the air itself has weight and presses on the water outside the pipe. Mercury, about fourteen times heavier than water, should then stand about fourteen times lower. He tested this with a tube of mercury and so invented the barometer.

The experiment

In 1643 or early 1644, in Florence, a glass tube about two braccia long, closed at one end, was filled with mercury, stoppered, inverted, and opened under the surface of mercury in a basin. The mercury fell until the column stood about one and a quarter braccia (roughly 76 cm) above the basin, leaving an apparently empty space at the top. Viviani is often named as the one who first carried out the experiment, at Torricelli’s suggestion.

Torricelli explained the result in a letter to Michelangelo Ricci of 11 June 1644, and answered Ricci’s objections in a second letter of 28 June. The first letter contains his most famous sentence: we live submerged at the bottom of an ocean of air. Two arguments stand out.

• Shape does not matter. Tubes ending in a large bulb at the top, with much more “vacuum” above the mercury, held the column at the same height. A force of the vacuum should be stronger when there is more vacuum.
• Weight explains everything. The column of mercury balances the weight of a column of air of the same base reaching to the top of the atmosphere. A column of water should therefore stand about 18 braccia, the pump makers’ limit.

He added that his aim had been to build an instrument showing changes in the air, sometimes heavier and sometimes lighter. He also tried, without success, to observe whether sound passed and insects survived in the vacuum.

Confirmation

Marin Mersenne saw the experiment in Italy in 1644 and spread news of it in France. Blaise Pascal repeated it with water and wine in long tubes at Rouen and reasoned that if the air’s weight was the cause, the column must be shorter on a mountain. On 19 September 1648 his brother-in-law Florin Périer carried a barometer up the Puy de Dôme and found the mercury several centimetres lower at the summit than at Clermont. In 1654 Otto von Guericke’s Magdeburg hemispheres made the force of the atmosphere spectacular. The unit of pressure torr, \(\tfrac1{760}\) of a standard atmosphere, honours Torricelli.

Minimum distances and spirals

For the general reader. Where should three villages build a shared well so that the total length of the paths to it is as short as possible? Pierre de Fermat posed the problem; Torricelli solved it. The answer is the point from which each side of the triangle is seen at 120°. Torricelli also measured the length of the logarithmic spiral, one of the first curves to be rectified exactly.

Proof by rotation

Rotate the plane by \(60^\circ\) about \(B\), sending \(A\) to \(A'\) and an arbitrary point \(P\) to \(P'\), with \(A'\) outside the triangle. Triangle \(BPP'\) is equilateral, so \(PB=PP'\), and \(PA=P'A'\). Hence \(PA+PB+PC=A'P'+P'P+PC\ge A'C\), a length that does not depend on \(P\). Equality holds when \(A',P',P,C\) are collinear, which forces \(\angle BPC=120^\circ\) and \(\angle APB=120^\circ\). The construction shows \(F\) lies on the line \(CA'\) joining \(C\) to the apex of the equilateral triangle erected outward on \(AB\). Torricelli’s construction uses the circumcircles of these equilateral triangles, which pass through \(F\).

The logarithmic spiral

In work left unpublished at his death, Torricelli showed how to rectify the spiral \(r=ae^{b\theta}\) (which he described as cutting all radii at a constant angle), and noted that the arc from any point inward to the pole, though it winds round infinitely often, has finite length.

Mathematician to the Grand Duke, 1642–1647

Torricelli’s Florentine years were short and productive. He lived in the ducal palace with a salary from Ferdinando II, who, with his brother Leopoldo de’ Medici, was a keen patron of experiment. In 1642 he gave an inaugural lecture to the Accademia della Crusca, the Florentine academy of language, and his later lectures on the force of impact, on wind, and on lightness and heaviness were published posthumously as Lezioni accademiche (1715). In them he explained winds by differences in the temperature and density of air between regions.

Opera geometrica

In 1644 he published his only book, Opera geometrica. It had three parts: De sphaera et solidis sphaeralibus, on solids generated by rotating regular polygons, in the manner of Archimedes; De motu gravium, on falling bodies, projectiles and efflux; and De dimensione parabolae, which gives twenty proofs of Archimedes’ quadrature of the parabola, some classical and some by indivisibles, as a showcase of the two methods side by side. The volume also contained the acute hyperbolic solid and the cycloid. It made his reputation across Europe.

Lenses

Torricelli was also a skilled lens grinder. He kept his technique secret, made telescope objectives judged superior to those of the professional makers, and built simple microscopes. The lenses brought him income and gifts from the Grand Duke. He is often credited with improving Galileo’s air thermoscope into a sealed liquid thermometer, which the Medici circle developed further.

Disputes, death and legacy

Torricelli’s publication of the cycloid area in 1644 provoked Gilles Personne de Roberval, who had found it around 1634 without publishing, and who held his results back to defend his chair at the Collège Royal. Roberval accused Torricelli of taking the proof from French papers seen in Galileo’s circle. The evidence suggests Torricelli found the results independently. The quarrel was unresolved at his death and was revived in 1658 by Pascal, whose history of the cycloid sided with Roberval.

In October 1647 Torricelli fell suddenly ill, probably with typhoid fever, while writing a defence of his priority against the French. He died in Florence on 25 October 1647, aged 39, and was buried in the church of San Lorenzo; the exact place of his tomb is not known. He left his manuscripts in the care of his friend Lodovico Serenai. Ricci declined to edit them, and Viviani, who took up the task, did not finish it. The papers were published in full only in the Opere of 1919–1944.

Legacy

• Physics: the barometer and the first sustained laboratory vacuum; the weight of the air; the law of efflux and the founding of hydrodynamics; Torricelli’s principle, which Huygens used and which leads to virtual work.
• Mathematics: the first unbounded solid of finite volume; the cycloid; kinematic tangents; the Fermat–Torricelli point; a free and productive use of indivisibles, which, with the work of Cavalieri, Fermat, Roberval and Wallis, prepared the calculus of Newton and Leibniz.
• Scientific style: Torricelli joined the geometric rigour of Archimedes to Galileo’s use of experiment, and his letter to Ricci is a model of an argument from experiment.
• Honours: the unit torr, the asteroid 7437 Torricelli, a lunar crater, the Torricelli Mountains in New Guinea, and a statue in Faenza (1868).

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Evangelista Torricelli, MacTutor History of Mathematics.
2. M. Gliozzi, Torricelli, Evangelista, Dictionary of Scientific Biography.
3. Evangelista Torricelli, Wikipedia.
4. G. Loria and G. Vassura (eds.), Opere di Evangelista Torricelli, 4 vols., Faenza, 1919–1944.
5. W. E. Knowles Middleton, The History of the Barometer, Johns Hopkins Press, 1964.

Mathematics and physics

1. E. Torricelli, Opera geometrica, Florence, 1644.
2. E. Torricelli, Lezioni accademiche, Florence, 1715.
3. P. Mancosu and E. Vailati, Torricelli’s infinitely long solid and its philosophical reception in the seventeenth century, Isis 82 (1991).
4. P. Mancosu, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century, Oxford University Press, 1996.
5. K. Andersen, Cavalieri’s method of indivisibles, Archive for History of Exact Sciences 31 (1985).
6. E. Torricelli, letters to M. Ricci, 11 and 28 June 1644, in Opere, vol. 3.