Bonaventura ,

Cavalieri

“Rigor is the concern of philosophy not of geometry.” – BonaventuraCavalieri

Bonaventura Francesco Cavalieri

Italian priest and mathematician

Born at Milan in 1598,

died at Bologna on November 27, 1647.

He became a Jesuit at an early age;

on the recommendation of the Order he was in 1629 made professor of mathematics at Bologna; and he continued to occupy the chair there until his death. I have already mentioned Cavalieri's name in connection with the introduction of the use of logarithms into Italy, and have alluded to his discovery of the expression for the area of a spherical triangle in terms of the spherical excess.

He was one of the most influential mathematicians of his time, but his subsequent reputation rests mainly on his invention of the principle of indivisibles.

Born Francesco Cavalieri in Milan, he took the name of his father, Bonaventura, when he entered the Congregation of Hieronymites or Jesuati (not Jesuit) order when he was thirteen. The order was established in 1367 to care for and bury the victims of the Black Death, the plague that killed more than one-fourth of Europe’s population.

He received minor orders in 1615, and the next year he was transferred to the Jesuiti monastery at Pisa. There he studied philosophy and theology and was introduced to geometry, to which he devoted the rest of his life. He quickly absorbed the works of Euclid, Archimedes, Apollonius and Pappas. Cavalieri became an accomplished mathematician and one of the most illustrious disciples of Galileo.

He was one of the most influential mathematicians of his time, but his subsequent reputation rests mainly on his invention of the principle of indivisibles.

Indivisibles are difficult to explain precisely. We can think of them as in some sense the things from which continuous substances are constructed. A point was the “indivisible” of a line, that is, Cavalieri considered a line to be composed of an infinite number of points. A plane is composed of an infinite number of indivisibles, namely lines, and squares of plane circles were the “indivisibles” of a pyramid, cone, etc.

In Cavalieri’s treatment, a moving point generated a line; a moving line generated a surface; and a moving surface generated a solid. Cavalieri was not the first person to consider geometric figures in terms of the infinitesimal. It had already been incorporated into medieval scholastic philosophy. Cavalieri was not the first to use such a concept in computing areas and volumes, but he was the first one who published a work on the concept. His work provided a deeper notion of sets, namely that it isn’t necessary to assign elements to a set; it is enough that there exists some precise criterion for determining if an element does or does not belong to a set.

Directorium generaleuranometricum (A General Directory of Uranometry, 1632).

first to recognize and popularize the value of logarithms in his Directorium generale

uranometricum (A General Directory of Uranometry, 1632).

Uranometry is the science of themeasurement of the positions, magnitudes, ,

etc. of the stars. The tables of logarithms that he published included logarithms of trigonometric functions for use by astronomers. However, his greatest

was his principle of indivisibles – a procedure that further developed Archimedes’ method of exhaustion, although Cavalieri would not be aware of Archimedes’ work. Cavalieri announced the principle in 1629, but it did not appear in print until six years later in his treatise Geometria indivisibilibus continuorum.

Cavalieri's first book was Lo Specchio Ustorio, overo, Trattato delle settioni coniche, or The Burning Mirror, or a Treatise on Conic Sections.[3] In this book he developed the theory of mirrors shaped into parabolas, hyperbolas, and ellipses, and various combinations of these mirrors. The work was purely theoretical since the needed mirrors could not be constructed with the technologies of the time, a limitation well understood by Cavalieri.[4]

. One example will suffice. Suppose it be required to find the area of a right-angled triangle. Let the base be made up of, or contain n points (or indivisibles), and similarly let the other side contain na points, then the ordinates at the successive points of the base will contain a, 2a ..., na points. Therefore the number of points in the area is a + 2a + ... + na; the sum of which is ½ n²a + ½na. Since n is very large, we may neglect ½na for it is inconsiderable compared with ½n²a. Hence the area is equal to ½(na)n, that is, ½ × altitude × base. There is no difficulty in criticizing such a proof, but, although the form in which it is presented is indefensible, the substance of it is correct.

It would be misleading to give the above as the only specimen of the method of indivisibles, and I therefore quote another example, taken from a later writer, which will fairly illustrate the use of the method when modified and corrected by the method of limits.

Let it be required to find the area outside a parabola APC and bounded by the curve, the tangent at A, and a line DC parallel to AB the diameter at A. Complete the parallelogram ABCD. Divide AD into nequal parts, let AM contain r of them, and let MN be the (r + 1)th part. Draw MP and NQ parallel to AB, and draw PR parallel to AD. Then when n becomes indefinitely large, the curvilinear area APCD will be the the limit of the sum of all parallelograms like PN.

Nowarea PN : area BD = MP.MN : DC.AD.But by the properties of the parabola

MP : DC = AM² : AD² = r² : n²,and MN : AD = 1 : n. Hence MP.MN : DC.AD = r² : n³. Therefore

area PN : area BD = r² : n³. 

Therefore, ultimately,area APCD : area BD

= 1² + 2² + ... + (n-1)² : n³=     n (n-1)(2n-1) : n³

which, in the limit, = 1 : 3.

Cavalieri’s mathematics - Integral Calculus

Here is an example of its use. Think about a circle and cutting it up into a large number of segments which are then stuck down as

shown (top to tail):

The shape on the right is ‘almost’ a rectangle and the more segments that are taken, the closer it will be to a rectangle.

The areas of the circle and the rectangle are the same - they comprise the same parts. But the length of the rectangle is half

the circumference of the circle; the height of the rectangle is just the radius of the circle. So the area of the circle is given by the

area of the rectangle,

A=(21 2λr).r=λr2 

It is perhaps worth noticing that Cavalieri and his successors always used the method to find the ratio of two areas, volumes, or magnitudes of the same kind and dimensions, that is, they never thought of an area as containing so many units of area. The idea of comparing a magnitude with a unit of the same kind seems to have been due to Wallis.

It is evident that in its direct form the method is applicable to only a few curves. Cavalieri proved that, if m be a positive integer, then the limit, when n is infinite, of     is 1/(m+1), which is equivalent to saying that he found the integral of x to     from x = 0 to x = 1; he also discussed the quadrature of the hyperbola.

Geometria indivisibilibus continuorum.

Geometria indivisibilibus continuorum nova quadam ratione promota (Geometry, developed by a new method through the indivisibles of the continua, 1635). In this work, an area is considered as constituted by an indefinite number of parallel segments and a volume as constituted by an indefinite number of parallel planar areas. Such elements are called indivisibles respectively of area and volume and provide the building blocks of Cavalieri's method. As an application, he computed the areas under the curves    – an early integral – which is known as Cavalieri's quadrature formula.

His theory was spurred by Kepler’s Stereometria and by the encouragement of Galileo. The main advantage of the method of indivisibles was that it was more systematic than the method of exhaustion. In effect, Cavalieri found a result equivalent to evaluating the integral:

a∫ xndx as an + 1/(n + 1)

0

Kepler’s Stereometria

Kepler's Nova stereometria doliorum vinariorum Kepler reported his results on wine barrels in his 1615 book, Nova

stereometria doliorum vinariorum (New solid geometry of wine barrels). The word Stereometria is from the Ancient Greek stereos that means solid or three-dimensional and metron that means a measure or to measure. Stereometria then means the art of measuring volumes, or solid geometry. Doliometry is an old-fashioned word from the Latin dolium that means a large jar or barrel.

This book is a systematic work on the calculation of areas and volumes by infinitesimal techniques. Building on the results of Archimedes, it focuses on solids of revolution and includes calculations of exact or approximate volumes of over ninety such solids (Edwards, p. 102). Today we use integral calculus to solve these kinds of problems.

Cavalieri’s method of indivisibles forms a crude type of integral calculus in which an area is thought of as consisting of lines and that a solid’s volume can be regarded as composed of areas that are indivisible. With his theory he was able to solve many problems connected with the quadrature of curves and surfaces, finding of volumes, and locating centers of mass, all of which were superseded at cone has 1/3 the volume of the prism or cylinder of equal base and height. He didn’t actually find the area of a figure or the volume of a solid as being so many “square units” or “cubic units;” instead he determined the ratio between the required area or volume with that of some other easily calculated area or volume.

Cavalieri did not rigorously develop his theory of indivisibles, but he did not view this as an important defect. He was intent on finding some relatively simple practical method for finding areas and volumes. He was not concerned with the Zeno’s puzzling paradoxes.

This is usually put in the context of a race between Achilles (the legendary Greek warrior) and the Tortoise. Achilles gives the Tortoise a head start of, say 10 m, since he runs at 10 ms-1 and the Tortoise moves at only 1 ms-1. Then by the time Achilles has reached the point where the Tortoise started (T0 = 10 m), the slow but steady individual will have moved on 1 m to T1 = 11 m. When Achilles reaches T1, the labouring Tortoise will have moved on 0.1 m (to T2 = 11.1 m). When Achilles reaches T2, the Tortoise will still be ahead by 0.01 m, and so on. Each time Achilles reaches the point where the Tortoise was, the cunning reptile will always have moved a little way ahead.

The paradoxes of the philosopher Zeno, born approximately 490 BC in southern Italy, have puzzled mathematicians, scientists and philosophers for millennia. Although none of his work survives today, over 40 paradoxes are attributed to him which appeared in a book he wrote as a defense of the philosophies of his teacher Parmenides. Parmenides believed in monism, that reality was a single, constant, unchanging thing that he called 'Being'. In defending this radical belief, Zeno fashioned 40 arguments to show that change (motion) and plurality are impossible.

haunted inquiries into infinite processes.He and other mathematicians of the period ignored

the logical imperfections in his use of infinitesimalquantities. They developed methods whereby

whenever a quantity changed in value according to some continuous law, as most things in nature seemed to do, the rate of increase or decrease in such a change was measurable. Later, when these logical imperfections were removed, mathematicians developed infinitesimal calculus, enabling scientists to pry loose the secrets of nature that for so long had been a closed book.

Sir George Gabriel Stokes, First Baronet,

Click icon to add picture

, physicist and mathematician, was born on 13 August 1819 in Skreen, County Sligo, Ireland. He was the youngest of eight children born to the rector of Skreen, Gabriel Stokes (1762–1834), and Elizabeth Haughton, the daughter of John Haugton, the rector of Kilrea, County Londonderry.

was an Irish mathematician and physicist who made many important contributions to fluid dynamics, optics, and mathematical physics. Together with James Clerk Maxwell and Lord Kelvin, he was a major contributor to the fame of the Cambridge school of mathematical physics during the mid-nineteenth century.

Stoles exerted unusual influence beyond his direct students through extending assistance in understanding and applying mathematics to any member of the university. He served in many administrative positions, including for many years as secretary of the Royal Society. He held strong religious convictions and published a volume on Natural Theology.

George Gabriel Stokes was the youngest of eight children of the Reverend Gabriel Stokes, rector of Skreen, County Sligo, and Elizabeth Haughton. Stokes was raised in an evangelical Protestant home.

Stokes was first tutored by a church clerk, but at the age of 13 was sent to a school in Dublin for a more formal course of education. Stokes's father died in 1834, but his mother secured the financing to send him to Bristol College. His mathematics teacher there was Francis Newman, the brother of Cardinal Newman.

In 1837, Stokes transferred as an undergraduate to Pembroke College at the University of Cambridge, where his brother William, breaking with the family tradition of attending Trinity, had studied. On graduating as "senior wrangler" and first Smith's prizeman in 1841, Stokes was elected to a fellowship at the college.

THE MOTION OF LIGHT

In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of adiffusing viscous term (proportional to the gradient of velocity) and a pressure term - hence describing viscous flow.The equations are useful because they describe the physics of many things of academic and economic interest. They may be used to model theweather, ocean currents, water flow in a pipe and air flow around a wing. The Navier–Stokes equations in their full and simplified forms help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled withMaxwell's equations they can be used to model and study magnetohydrodynamics.

Properties of lightPerhaps his best-known researches

are those that deal with the wave theory of light. His optical work began at an early period in his scientific career. His first papers on the aberration of light appeared in 1845 and 1846, and were followed in 1848 by one on the theory of certain bands seen in the spectrum. In 1849, he published a long paper on the dynamical theory of diffraction, in which he showed that the plane of polarization must be perpendicular to the direction of propagation.

Fluorescence

In the early 1850s, Stokes began experimenting with filtered light. He passed sunlight through a blue-tinted glass, and then shone the beam through a solution of quinone, which has a yellow color. When the blue light reached the quinone solution, it produced a strong yellow illumination. Stokes tried the same experiment with the solutions of different compounds, but found that only some showed an illumination of a color different from that of the original light beam. Stokes named this effect fluorescence.

The phenomenon of fluorescence was known by the middle of the nineteenth century. British scientist Sir George G. Stokes first made the observation that the mineral fluorspar exhibits fluorescence when illuminated with ultraviolet light, and he coined the word "fluorescence". Stokes observed that the fluorescing light has longer wavelengths than the excitation light, a phenomenon that has become to be known as the Stokes shift. In Figure 1, a photon of ultraviolet radiation (purple) collides with an electron in a simple atom, exciting and elevating the electron to a higher energy level. Subsequently, the excited electron relaxes to a lower level and emits light in the form of a lower-energy photon (red) in the visible light region.

SPECTROSCOPYSpectroscopy /spɛkˈtrɒskəpi/ is the study of the interaction between matter and radiated energy.Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, e.g., by a prism. Later the concept was expanded greatly to comprise any interaction with radiative energy as a function of its wavelength or frequency. Spectroscopic data is often represented by a spectrum, a plot of the response of interest as a function of wavelength or frequency.

An excellent example is his work in the theory of spectroscopy. In his presidential address to the British Association in 1871, Lord Kelvin (Sir William Thomson, as he was known then) stated his belief that the application of the prismatic analysis of light to solar and stellar chemistry had never been suggested directly or indirectly by anyone else when Stokes taught it to him in Cambridge some time prior to the summer of 1852, and he set forth the conclusions, theoretical and practical, which he had learned from Stokes at that time, and which he afterwards gave regularly in his public lectures at Glasgow.

These statements, containing as they do the physical basis on which spectroscopy rests, and the way in which it is applicable to the identification of substances existing in the Sun and stars, make it appear that Stokes anticipated Kirchhoff by at least seven or eight years.