Eudoxus of Cnidus Invented Forerunner of Calculus With Method of Exhaustion

Eudoxus of Cnidus Invented Forerunner of Calculus With Method of Exhaustion

By Jen Breitegan

Eudoxus of Cnidus (pronounced Ny-dus) was a Greek mathematician who lived in Asia Minor from 390-337 BCE. His work, called Method of Exhaustion, is regarded by modern mathematicians to be the forerunner of calculus today. 

This distinction contradicts assumptions that were held for many years about who invented calculus. It was long believed that 17th-century mathematicians Isaac Newton and Gottfried Wilhelm Leibniz were the fathers of modern-day calculus. 

However, research now shows the true inventor was Eudoxus, born centuries before.

Discover why Eudoxus of Cnidus is considered among the greatest of classical Greek mathematicians, and how his work led to calculus as we know it!

Eudoxus of Cnidus (390-337 BCE), Greek Mathematician

Unfortunately, Eudoxus’s original writings have been lost to the ravages of time, as is the case for many early mathematicians. 

The good news is that his work was so influential that it was cited by famous math scholars and historians who followed him. 

Third-century historian Diogenes Laertius wrote short biographies of famous philosophers and mathematicians. We have a brief understanding of the life of Eudoxus thanks to these writings. 

According to Laertius, Eudoxus attended lectures at Plato’s Academy in Athens, Greece at the age of 23. 

He then spent 16 months teaching in Egypt. In his spare time, he studied astronomical events from an observatory. He returned to Asia Minor to teach at the Platonic Academy for a period of time, and then finally went back to his birthplace Cnidus. 

Eudoxus lived out his remaining years in Cnidus. He became a legislator and continued academic research until his death in 337 BCE.

Eudoxus’s Method of Exhaustion

The Method of Exhaustion was a mathematical technique invented by Eudoxus. It finds the area of a shape by inscribing polygons inside of it, with an increasing number of sides. 

Eventually, the areas of the successive polygons merge to equal the area of the original shape. 

Renowned mathematicians Euclid and Archimedes were both influenced by Eudoxus’s method. It was referenced in Euclid’s Elements and Archimedes’s On the Sphere and Cylinder and Method.

This method is considered the forerunner of integral calculus. Integral calculus deals with lengths, areas, and volumes of shapes. 

The Method of Exhaustion was given its name after the Renaissance by European scholars. It described both Eudoxus’s original technique as well as more modern proofs where mathematicians “exhausted” the area of a shape with a succession of polygonal approximations.

Related: Archimedes: A Mathematician With An Obsession

Eudoxus’s Additional Contributions to Math & Astronomy

Eudoxus’s cited works in mathematics went beyond the Method of Exhaustion. He is credited by fellow mathematician Euclid with creating a method of comparing ratios that made irrational numbers measurable. 

An example of his method:

Assume  ab = cd 

Multiply the numerators by a number “x” and the denominators by another number “y.” 

According to Eudoxus, if ax > yb, then ac > yd must also be true. 

Since there is no requirement that a, b, x or y be rational, this method works for both rational and irrational numbers. Eudoxus was the first known scholar to measure or compare irrational numbers. His work was credited by 19th-century German mathematician Richard Dedekind as inspiration for Dedekind’s famous definition of the real numbers, known as a Dedekind cut. 

Related: Euclid of Alexandria, An Influence That Spans Millenia

Eudoxus is also credited with the theory of irrational magnitudes and with solving a problem that involved constructing a cube with two times the volume of a given cube.

Eudoxus had a passion for astronomy in addition to math. His biggest claim to fame in this field was his geometric model for the movements of the sun, moon, and planets (there were five known planets in his time). He was the first known astronomer to make the attempt.

Rendering of Eudoxus's model of planetary motion, via Wikimedia

Aristotle himself endorsed the basic principles of Eudoxus’s model. This influenced astronomers’ continued interest in the work through the Renaissance period.

Eudoxus is also credited with a schematic description of constellations as well as phases for fixed stars and the weather during those phases. 

Eudoxus’s Enduring Legacy

It’s important to note that although Eudoxus’s original work has been lost to time, his ideas have lived on for centuries in the work of mathematicians who came later and added to his creativity and brilliance with their own theories. 

This is a testament to the idea that human knowledge and learning build upon the ideas of our ancestors. One idea born ages ago may become the foundation upon which an entire field of study is built. 

This is certainly the case for Eudoxus of Cnidus, a mathematician who continues to inspire others, and a reason why many believe he was the most innovative math scholar of his time.

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