Almost contemporaneously with the publication in 1637 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 It was, in its turn, superceded at the beginning of the eighteenth century by the integral calculus.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 1598, and 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.
The principle of indivisibles had been used by Kepler in 1604 and 1615 in a somewhat crude form. It was first stated by Cavalieri in 1629, but he did not publish his results till 1635. In his early enunciation of the principle in 1635 Cavalieri asserted that a line was made up of an infinite number of points (each without magnitude), a surface of infinite number of lines (each without breadth), and a volume of an infinite number of surfaces (each without thickness). To meet the objections of Guldinus and others, the statement was recast, and in its final form as used by the mathematicians of the seventeenth century it was published in Cavalieri's Exercitationes Geometricae in 1647; the third exercise is devoted to a defence of the theory. This book contains the earliest demonstration of the properties of Pappus. Cavalieri's works on indivisibles were reissued with his later corrections in 1653.
The method of indivisibles rests, in effect, on the assumption that any magnitude may be divided into an infinite number of small quantities which can be made to bear any required ratios ( ex. gr. equality) one to the other. The analysis given by Cavalieri is hardly worth quoting except as being one of the first steps taken towards the formation of an infinitesimal calculus. 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 , 2 a ..., na points. Therefore the number of points in the area is a + 2 a + ... + na ; the sum of which is 1/2 n 2 a + 1/2 na . Since n is very large, we may neglect 1/2 na for it is inconsiderable compared with 1/2 n 2 a . Hence the area is equal to 1/2( na ) n , that is, 1/2 x altitude x 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 n equal 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 . Now
area PN : area BD = MP . MN : DC . AD .
But by the properties of the parabola
MP : DC = AM 2 : AD 2 = r 2 : n 2 ,
and MN : AD = 1 : n . Hence MP . MN : DC . AD = r 2 : n 3 . Therefore area PN : area BD = r 2 : n 3 . Therefore, ultimately,
area APCD : area BD = 1 2 + 2 2 + ... + (n-1) 2 : n 3 = 1/6 n (n-1)(2n-1) : n 3
which, in the limit, = 1 : 3.
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 (1 m + 2 m + ... + n m )/ n m+1 is 1/( m +1), which is equivalent to saying that he found the integral of x to x m from x = 0 to x = 1; he also discussed the quadrature of the hyperbola.
Background and Author Introduction
This text introduces the pioneering work of Bonaventura Cavalieri, an important figure in the history of mathematics during the early 17th century. Born in Milan in 1598, Cavalieri was a Jesuit priest and a professor of mathematics at Bologna. His work laid foundational stones for integral calculus, a branch of mathematics that deals with summing infinitely many small quantities to find areas, volumes, and other quantities. Cavalieri’s principle of indivisibles was a revolutionary idea that helped mathematicians move beyond the ancient Greek methods of exhaustion, offering a simpler and more flexible approach to calculating areas and volumes.
Understanding the Principle of Indivisibles
Cavalieri’s principle states that a line is made up of infinitely many points, a surface of infinitely many lines, and a volume of infinitely many surfaces. This idea might sound abstract or even confusing at first, but it is a key step toward the concept of integration in modern calculus. By imagining shapes as composed of infinitely thin slices or points, Cavalieri could calculate areas and volumes by comparing these slices between different shapes.
For example, to find the area of a right-angled triangle, Cavalieri imagined the base as made up of many points and the height as containing a proportional number of points. By summing these points, he arrived at the familiar formula for the area of a triangle: one-half the base times the height. Although his method lacked the rigor we expect today, the underlying idea was correct and paved the way for future mathematicians.
Significance and Impact
Cavalieri’s work was significant because it introduced a new way of thinking about geometry and measurement that was more intuitive and less cumbersome than previous methods. His principle of indivisibles anticipated the integral calculus developed later by Newton and Leibniz. This method allowed mathematicians to solve problems involving curves and surfaces that were previously very difficult or impossible to handle.
His work also influenced the study of parabolas, spheres, and hyperbolas, expanding the understanding of these shapes and their properties. Cavalieri’s approach helped bridge the gap between geometry and algebra, leading to the powerful mathematical tools used in science and engineering today.
What Students Can Learn
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Mathematical Creativity and Innovation: Cavalieri’s story shows how new ideas often build on old ones. He took the ancient Greek method of exhaustion and improved it by imagining shapes as made of indivisible parts. This teaches students the value of creative thinking and looking at problems from new perspectives.
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Foundations of Calculus: While calculus might seem complicated, Cavalieri’s principle provides a simple introduction to the concept of summing infinitely many small parts to find a whole. Understanding this principle helps students appreciate the origins and importance of calculus.
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Historical Context: Learning about Cavalieri helps students see how mathematics developed over time and how different cultures contributed to knowledge. It also shows how science and religion coexisted, as Cavalieri was a Jesuit priest and a mathematician.
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Problem Solving: The examples given, such as finding the area under a parabola, demonstrate how mathematical reasoning can solve practical problems. Students can learn to apply logical steps and use approximations to approach complex questions.
Applying These Lessons in Life and Learning
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In School: Students can use Cavalieri’s principle as a stepping stone to understand integration in calculus classes. It encourages breaking down complex problems into smaller, manageable parts, a useful skill in any subject.
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In Daily Life: The idea of summing small parts to understand a whole can be applied in budgeting, cooking, or planning projects. For example, managing time by dividing tasks into smaller segments mirrors the indivisible approach.
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In Social Situations: Cavalieri’s dedication to both faith and science shows the importance of balancing different aspects of life and respecting diverse fields of knowledge. Students can learn to appreciate multiple viewpoints and collaborate across disciplines.
Cultivating Positive Traits from Cavalieri’s Work
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Curiosity and Open-mindedness: Cavalieri’s willingness to explore new ideas encourages students to stay curious and open to learning, even when concepts seem difficult or unfamiliar.
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Perseverance: His work was initially criticized and not fully accepted, but he continued refining his ideas. This teaches the value of persistence in the face of challenges.
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Analytical Thinking: The method of indivisibles requires careful analysis and logical reasoning, skills that are valuable in academics and everyday decision-making.
Conclusion
Bonaventura Cavalieri’s principle of indivisibles is more than just a mathematical technique; it is a story of innovation, perseverance, and the evolution of human understanding. For students, it offers a glimpse into the origins of calculus and the power of thinking differently. By studying his work, young learners can gain insights into problem-solving, the history of science, and the importance of combining creativity with logic. These lessons extend beyond mathematics, encouraging a mindset that is curious, persistent, and analytical—qualities that will serve them well in all areas of life.
