When was bonaventura cavalieri born

Students will enjoy using this graphically illustrated, interactive learning tool. The timeline feature presents key events in the life of Cavalieri. Tapping on timeline points will bring up descriptions of historically significant events in the history of mathematics and the role that Cavalieri played.

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Read about some of the significant events in the history of mathematics and the important contributions that Bonaventura Cavalieri made during his lifetime. Now Available on the Apple App Store. Almost contemporaneously with the publication in of Descartes' geometry, the principles of the integral calculus, so far as they are concerned with summation, were being worked out in Italy.

This was effected by what was called the principle of indivisibles, and was the invention of Cavalieri. It was applied by him and his contemporaries to numerous problems connected with the quadrature of curves and surfaces, the determination on volumes, and the positions of centres of mass. It served the same purpose as the tedious method of exhaustions used by the Greeks; in principle the methods are the same, but the notation of indivisibles is more concise and convenient.

It was, in its turn, superceded at the beginning of the eighteenth century by the integral calculus. Bonaventura Cavalieri was born at Milan in , and died at Bologna on November 27, He became a Jesuit at an early age; on the recommendation of the Order he was in 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.

The principle of indivisibles had been used by Kepler in and in a somewhat crude form. This theory allowed Cavalieri to find simply and rapidly the area and volume of various geometric figures.

Howard Eves writes [ 26 ] :- Cavalieri's treatise on the method of indivisibles is voluble and not clearly written, and it is not easy to learn from it precisely what Cavalieri meant by an "indivisible. Similarly, it seems that an indivisible of a given solid is a planar section of that solid, and a solid can be considered as made up of an infinite parallel set of this kind of indivisible.

Now, Cavalieri argued, if we slide each member of a parallel set of indivisibles of some planar piece along its own axis, so that the endpoints of the indivisibles still trace a continuous boundary, then the area of the new planar piece so formed is the same as that of the original planar piece, inasmuch as the two pieces are made up of the same indivisibles.

A similar sliding of the members of a parallel set of indivisibles of a given solid will yield another solid having the same volume as the original one. This last result can be strikingly illustrated by taking a vertical stack of cards and then pushing the sides of the stack into curved surfaces; the volume of the disarranged stack is the same as that of the original stack.

These results give the so-called Cavalieri principles: 1. If two planar pieces are included between a pair of parallel lines, and if the lengths of the two segments cut by them on any line parallel to the including lines are always equal, then the areas of the two planar pieces are also equal.

If two solids are included between a pair of parallel planes, and if the areas of the two sections cut by them on any plane parallel to the including planes are always equal, then the volumes of the two solids are also equal. The method of indivisibles was not put on a rigorous basis and his book was widely attacked. In particular, Paul Guldin attacked Cavalieri [ 8 ] :- The debate between Cavalieri and Guldin is usually mentioned in connection with the objections made by Guldin to Cavalieri's use of indivisibles.

Although that is probably the main issue between Cavalieri and Guldin , a more careful reading of the debate will allow us to indicate the existence of other interesting issues The argument really centres around the fact that Guldin was a classical geometer following the methods of the ancient Greek mathematicians.

There is something in his argument relating to Kepler since in that work Kepler does regard a circle as an infinite polygon composed of infinitesimals. However, Cavalieri's indivisibles are different from Kepler 's infinitesimals. As to the reference to Sover, Cavalieri, in his defence, pointed out that he wrote his book before Sover's book was published. Guldin attacked Cavalieri's indivisibles by arguing that when a surface is generated by rotating a line about the axis, the surface is not just a set of lines.

He writes see [ 8 ] :- In my opinion no geometer will grant Cavalieri that the surface is, and could, in geometrical language be called "all the lines of such a figure"; never in fact can several lines, or all the lines, be called surfaces; for, the multitude of lines, however great that might be, cannot compose even the smallest surface.

As Mancosu writes [ 8 ] :- Guldin was a "classicist" geometer, steeped in the idea of explicit construction, sceptical of considerations of infinity in the domain of geometry, and wary of the risk of ending up with an atomistic theory of the continuum.

If one asks whether Guldin or Cavalieri is right, then the answer must be Cavalieri. However, a positive side to Guldin 's attack was that Cavalieri improved his exposition publishing Exercitationes geometricae sex which became the main source for 17 th Century mathematicians.

Cavalieri was also largely responsible for introducing logarithms as a computational tool in Italy through his book Directorium Generale Uranometricum. We mentioned above that his appointment to Bologna had been initially for a 3 -year period. This book of logarithms was published by Cavalieri as part of his successful application to have the position extended. The tables of logarithms which he published included logarithms of trigonometric functions for use by astronomers [ 31 ] :- The work is divided into three parts, devoted to logarithms, plane trigonometry, and spherical trigonometry.

In addition to noteworthy innovations in terminology, the work includes important demonstrations of John Napier 's rules of the spherical triangle and the theorem of the squaring of each spherical triangle that, attributed to Albert Girard , was later claimed by Joseph Lagrange.

Galileo praised Cavalieri for his work on logarithms, in particular the book he wrote entitled A hundred varied problems to illustrate the use of logarithms Cavalieri also wrote on conic sections, trigonometry, optics, astronomy, and astrology. He developed a general rule for the focal length of lenses and described a reflecting telescope. He also worked on a number of problems of motion.

Piero Ariotti writes that, regarding the reflecting telescope [ 13 ] Cavalieri's work of interest is his 'Specchio ustorio', printed in and reprinted in In this work Cavalieri concerned himself with reflecting mirrors for the express purpose of resolving the age-long dispute of how Archimedes allegedly burned the Roman fleet that was besieging Syracuse in B.

The book, however, goes well beyond the stated purpose and systematically treats the properties of conic sections, reflection of light, sound, heat and cold! Cavalieri's claim that one would obtain a telescope by combining concave mirrors with concave lens have led some historians to claim that Cavalieri invented the reflecting telescope before James Gregory or Isaac Newton.

He even published a number of books on astrology, one in entitled Nuova pratica astromlogica and another, his last work, Trattato della ruota planetaria perpetua in However, although these use the terminology of astrology, they are serious astronomical works. Cavalieri did not believe that one could predict the future from astrological considerations, and certainly did not practice astrology.

He states this clearly in his work. Cavalieri corresponded with many mathematicians including Galileo , Mersenne , Renieri, Rocca, Torricelli and Viviani.