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Cavalieri (Francesco Bonaventura, 1598-1647) was a well-known Italian mathematician, born in Milan in 1598. In 1629, the great scientist Lu Liluo recommended Cavallelli as a professor of mathematics to the University of Pollenia. At the same time, Cavallelli presented his manuscript of Geometry and a booklet on conic curves and its application to optics to the chief election officer to prove that he was qualified for the job. Unsurprisingly, Cavalieri was the first professor at the University of Pollenia among many applicants. Since then, he has been teaching and researching at the University of Pollenia until his death in 1647. He has published 11 books, including the famous "Geometrics", "One Hundred Different Problems" and so on.

Cavallelli

In the history of world mathematics, Cavallelli is known mainly for his insignificant methods. This method considers that a line is composed of an infinite number of points, a surface is composed of an infinite number of lines, and a volume is composed of an infinite number of surfaces. Points, lines, and areas are indispensable components of lines, areas, and volumes, respectively. Cavalieri obtains the relationship between the area or volume of two planar or three-dimensional figures by comparing the relationship between the non-components of two planar or three-dimensional figures. This is the famous Cavalieri theorem: sandwich between Two plane figures between two parallel straight lines are intercepted by any straight line parallel to the two straight lines. If the length of the obtained two section lines is equal, the area of the two plane figures is equal; sandwiched between two parallel figures The two three-dimensional figures between the planes are cut by any plane parallel to the two planes. If the areas of the two cross-sections obtained are equal, then the two three-dimensional figures have the same volume. This theorem is known in China as the "Zu Zhe principle", and the latter part of it is consistent with the claim that the famous power mathematician Zu Zhe in the North and South dynasties, who calculated the volume of the sphere, had "the same power potential, but the product cannot be different." . Cavalieri used the above principles to obtain the area of many flat figures and the volume of three-dimensional figures, which are the basic prototypes of the current three-dimensional geometry textbooks for seeking geometric volume. Cavalieri also proved the formula equivalent to the definite integral of the power function we see today by using the non-component method: and the Gilding theorem: the volume of a solid figure obtained by rotating a plane figure around a certain axis is equal to the center of gravity of the plane figure The product of the circumference of the circle and the area of the plane figure.
Although Cavalieri's indivisible method lacked scientific evidence at the time, it was able to quickly and correctly obtain results that were not obtained by his predecessors. As the historian of mathematics Smith puts it: "Only from a mathematical perspective, the Italian mathematician who had the greatest influence on science in the 17th century was Cavalieri." Unfortunately, he died early and died at 49.
Bonaventura Cavalieri was born in Milan in 1598. He became a Jesuit monk at the age of fifteen. He studied at Galileo. From 1629 until his death at the age of 49 in 1647, he was a professor of mathematics at the University of Bologna. He was one of the most influential mathematicians of his time and wrote many books on mathematics, optics, and astronomy. Most of them first introduced logarithms to Italy. But his greatest contribution was a paper on method of indlisibles published in 1635: "Indivisible Geometry", although this method dates back to Democritus (circa 410 BC) and Azerbaijan. Kimid (circa 287-212 BC); perhaps Kepler's efforts to find certain areas and volumes directly inspired Cavalieri.
Cavalieri's dissertation is well written and unclear, and it is difficult to know exactly what the so-called "inliable" is. An "indivisible element" of a given plane slice seems to refer to a chord of the slice; an "indivisible element" of a given cube refers to a plane section of the cube. A plane piece is regarded as consisting of an infinite set of parallel chords, and a solid is regarded as consisting of an infinite set of parallel plane sections. Cavalieri then said: If we slide each element in the set of parallel strings of some given planar piece along its own axis, so that the endpoints of these strings still draw a continuous boundary, then this is formed The area of the new flat sheet is the same as that of the original flat sheet. The plane section of a given solid is similarly slid to generate another solid with the same volume as the original solid. (This final result can be given a compelling illustration as follows: take a pile of thick cardboard placed vertically, and then make the sides of the pile a curved surface. The volume of the heap is the same.) These conclusions, slightly generalized, give the so-called
Cavalieri's fuzzy concept of indivisible elements is a bit like a tiny part of a figure, which has caused quite a lot of theory, and has been influenced by some researchers on this subject [especially the Swiss goldsmith and mathematician Paul Guldin , 15771642)] severe criticism. Cavalieri tried to completely reform his discourse to deal with these objections without success. The French mathematician Roberba mastered this method proficiently and claimed to be an independent inventor of the concept. Toricelli, Fermat, Pascal, Saint Vincent, Barrow and others effectively used the indivisible method or Something similar to it. In the course of these people's work, we obtained the results equivalent to the integrals of expressions such as xn, sin, sin2, and sin.

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