Cavalieri’s name is virtually unknown and his childhood gives no clues as to his actual mathematical contribution. Cavalieri joined an Italian religious order when still a boy, but it was recognised at an early stage that his real talent was for mathematics.
In 1629, supported by the church, he became chair of mathematics at the University of Bologna. He was inspired primarily by Galileo. He regarded himself as a disciple of his and corresponded with him. Characteristically for the time, his work looked at what we would now regard as mathematics and physics.
In the latter he studied optics, developing the design for a reflecting telescope; he also studied astronomy and astrology. A major problem in astronomy is the awesome calculations that need to be made from countless observations.
This can be very time consuming and the slightest error means that vital clues may be missed, and opportunities for preditions and theories lost. Because of this, he became the first Italian to understand the powerful method of performing such calculations that logarithms provided and he published tables of logarithms of trigonomic functions for the use of astronomers.
But what the world now remembers Cavalieri for is his influential book Geometria indivisibilis continuorum nova published in 1635. This work developed and extended Archimedes’ method of exhaustion by making use of Kepler’s theory of the infinitesimal.
This device allowed Cavalieri to find the area and volume of a number of geometric figures. The essence of the method is to regard an area as made up of an indefinite number of lines or ‘indivisibles’. A solid volume is similarly composed of areas that are indivisible. By averaging these, and then using a multiplier, expressions for the area or volume emerge. The method is simple yet profound.
The book faced fierce contemporary criticism and was re-drafted and carefully revised in response. This paid off when it became a key source for later 17th century mathematicians.
It is now widely regarded as one of the first and decisive steps towards the integral calculus. This method of obtaining areas and volumes is still in use in schools as a means to motivate and facilitate the introduction of 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,
That’s Cavalieri’s method in action.