Archimedes of Syracuse (287 – 212 BC) is known as the greatest mathematician of his time and is considered to be one of the greatest of all time. He dominated Greek maths in the third century BC despite not being a native of the city of Alexandria, the centre of mathematical activity. 1 The son of an astronomer, Archimedes is credited with many great discoveries in mathematics, mechanics and engineering. During the second Punic war Syracuse was besieged by Romans and we are told that Archimedes invented war machines such as catapults, ropes and pulleys, and devices to set fire to the ships to keep the enemy at bay.
1 Archimedes did not think much of these inventions but it meant that mathematics and science were brought “more within the appreciation of the people in general”. 2 Archimedes’ work was both productive and thoroughly detailed and he was never reluctant to share his methods of discovery. What was different about Archimedes compared to other mathematicians of his time was the fact that his work illustrated his method of discovery of a theorem prior to presenting a rigorous proof.
This was to stop people claiming his work to be their own and he is quoted as saying, “those who claim to discover everything but produce no proofs of the same, may be confuted as having pretended to discover the impossible”. 2 The Method is a treatise containing a collection of Archimedes methods of discovery which was unexpectedly found in Jerusalem in the late nineteenth century. 3 The Method includes Archimedes’ methods of discovery by mechanics of many important results on areas and volumes.
It is the quadrature of the parabola, meaning to find the area of a segment of a parabola cut off by a chord4, which forms the subject of the first proposition of The Method. Archimedes derives the result in two ways, firstly mechanically and secondly purely geometrically. In fact Archimedes devoted a separate treatise on the mathematical proof of this theorem. 5 The geometrical proof is based on Euxodus’ method of exhaustion technique which means to calculate an area by approximating it by the areas of polygons. In Archimedes’ proof the polygons he uses are triangles.
Archimedes cut a parabola with a chord BC creating a parabolic segment and then drew a triangle ABC whose base was the length equal to that of the chord BC. The triangle ABC leaves two segments in which Archimedes adds another two triangles ABD and ACE. Again these two triangles create four more segments in which Archimedes constructs four more triangles. Archimedes continued this process realising the more triangles he constructed, the more closely the sum of these areas were nearing the area of the parabolic segment.
He also noted that the total area of the triangles created at each stage is a quarter of area of the triangles constructed in the previous stage. 3 Archimedes used this relationship to show that the area of the parabolic segment could be given by the sum of the infinite series, X/4 + X/42 + X/43 + … + X/4n, which is clearly (4X)/3, where X is the area of the initial triangle ABC. 1 However, infinite processes were frowned upon in his day1 so Archimedes needed to prove this in another way.
He completed the argument through a method called double reductio ad absurdum, which is Latin for “reduction to the absurd”, and is also known as proof by contradiction. Archimedes assumed that Y = (4/3)X, X being the area of the triangle ABC, is not equal to the area of the segment, Z, so therefore Y must be greater or less than Z. Archimedes then proceeds to rule out both of these possibilities. Firstly if Y is less than Z then triangles can be drawn in the segment, with total area T, giving Z – T < Z – Y. But this would imply that T > Y which is impossible because the summation formula shows that T < (4/3)X = Y.
And secondly if Y > Z, n is determined so that ((1/4)n)X < Y – Z. Since also Y – T = (1/3)*((1/4)n)X < ((1/4)n)X, it follows that Z < T, which is again impossible. Hence Archimedes proved by double reductio ad absurdum, a very common method of proof in his time, that Z cannot be more or less than Y = (4/3)X meaning in fact that Y = (4/3)X = Z. 3 An important lemma to this proof of Archimedes’ shows how to find the sum of a geometric series and because Archimedes had no notation to express a series with arbitrarily many terms his result was given for a series of five numbers.
However Archimedes’ method can easily be generalised to adopt a more modern notation with n denoting an arbitrary positive integer. 3 To summarise, Archimedes combined the method of exhaustion with a deep geometric understanding and a clever summation of a series of terms with identical successive ratios to demonstrate that the exact area of a parabolic segment is 4/3’s of the area of the initial triangle inscribed into that arc. Archimedes’ work on the quadrature of the parabola was both long and detailed.
It is this work by Archimedes that is considered a forerunner to modern methods of integration. 6 On this method of integration by Archimedes, Chasles said, “it gave birth to the calculus of the infinite conceived and brought to perfection by Kepler, Cavalier, Fermat Leibniz and Newton”. 2 For all this work how much did Archimedes actually accomplish? It is true that the method of exhaustion is a work of a creative genius, but it did have two major flaws. 7 Firstly, it was not general.
For each different problem, a different ingenious way of drawing triangles or some other polygon needed to be devised. Archimedes apparently was unable to find the area of a general segment of an ellipse or hyperbola. 1 The analytic approach of the modern era is completely general to the point that we do not necessarily use numbers. The second, and larger, flaw was that the method of exhaustion was not at all rigorous by modern standards. Quite simply, there was no inclusion of a limit concept. Archimedes took a segment of a parabola and filled it with some large, but finite number of polygons.
The sum of the areas of these polygons would converge to the area of the figure in an easy to work with geometric series. But because of a lack of a concept of infinity Archimedes did not consider this as a series, meaning it would have been impossible for him to make the method of exhaustion at all rigorous. It is because of these flaws in the method of exhaustion, the Greeks refusal to accept the concept of infinity, that it is easy to ignore Archimedes’ work. But he did come extremely close to discovering the integral.
This method of exhaustion used by Archimedes is very similar to the modern method of approximating areas of curves with simple shapes such as rectangles and trapezoids. After Archimedes it would be over 2000 years before a suitable and rigorous method of integration that was devised. It wasn’t until the early 17th century that further developments in calculus were achieved and a modern and more rigorous calculus proof of finding the area under a curve was born from ‘first principles’. A method is outlined and illustrated below.