The Mathematical Achievements and Methodologies of Archimedes

 

	Volume of the cylinder = ´ Volume of the sphere
Surface area of cylinder (including both ends) = ´ Surface area of sphere

Summary

Archimedes is regarded as one of the three most important mathematicians of ancient Greece. His numerous contributions to the to the study of mathematics also earned him the title of being one of the three most important mathematics ever. It was two thousand years ago that he started his study in problems involving perimeters, areas and volumes. Moreover, his methodologies in thinking ( for example, using concepts of differentials, mechanics and solutions to cubic equations) were unique and pioneering.

The aim of this article is to discuss the mathematical achievements and methodologies of Archimedes, and each of these topics is given a separate section below. The writer hopes that through quoting Archimedes’ proofs to some of his statements, the reader will learn more about the beautiful mind of the mathematician and become better able to understand his thoughts.

In each of his books, Archimedes took a look at some interesting mathematical problems, which in turn led to even more problems. In this way, he made his contribution to may of the branches of mathematics. In the section ‘Mathematical Achievements’, the reader will be given a detailed discussion of Archimedes’ books and achievements. To investigate the philosophy of Archimedes’ studies, the ‘Methodologies’ section will deal with his thinking methodologies. Summarizing the chapters on his mathematical achievements, this section should serve as the center of interest of this article.

As this is an article on mathematical history, there will be some mathematical terms and concepts in the discussion. The writer has tried his best to make the article easy to understand for his readers, whether or not they are keen or good in mathematics.

 

I. The Bibliography of Archimedes

Archimedes (287 – 212 B.C.) was born at Syracuse of Sicily as a son of the astronomer Pheidias. It is said that Archimedes was a relative of Hieron, the king of Syracuse.

Archimedes learnt from the disciples of the mathematician Euclid when he was young. When he was learning at Alexandria, he made friends with Conon, Eratosthenes and many others. He kept in touch with the academics in Alexandria even when he got back to Syracuse.

Archimedes was most famous for his contributions in mechanics. This is probably due to his many amazing stories [3]. He said that he could move the Earth just if he was given a fixed point — and that was his theory of leverage. In fact, he did move a ship all by himself using a lever. Another story is that he discovered buoyancy when he was having a bath. He was so excited about his discovery that he ran naked on to the street, shouting “Eureka!”, which meant “I’ve got it”.

When his country was attacked by Romanians, he put all his studies and researches away and successfully invented a number of powerful weapons. His enemy was quoted to have said, “He (Archimedes) sat at the seashore, and easily threw our boats to and fro as if he was just playing with some coins…..he is even more powerful than the monsters in the legends as he can throw so many bombs at us.” ([6], p.68) in the end, some Romanian soldiers got into Archimedes’ camp secretly and killed him.

However, the achievements of Archimedes in mathematics were no less important. When faced with difficult problems, he gave innovative ideas and logical proofs. That is why he is regarded as one of the three most important ancient Greek mathematicians with Euclid and Apollonius of Perga, as well as one of the three most important mathematicians ever with Newton and Gauss.

 

II. Mathematical Achievements

According to “The Works of Archimedes” [6], Archimedes’ works included “On the Sphere and Cylinder”, “On the Measurement of a Circle”, “On Conoids and Spheroids”, “On Spirals”, “On Plane Equilibriums”, “The Sandreckoner”, “On Quadrature of the parabola”, “On Floating Bodies”, “Book of Lemmas” and “The Method”. Each of his works was focused on some particular problems in mathematics or physics. Very often, he was directed to other interesting problems involved in solving the aforementioned ones. Therefore, his contributions are far greater than the titles of these books might suggest. We will discuss his achievements just in mathematics below. In order to highlight his contributions in the many branches of mathematics, the classification of his achievements may not follow from the works in which the discoveries are noted.

2.1 Sphere and Cylinder

One of the most important of Archimedes’ achievements is his studies on spheres and cylinders. Although there was not much known about the value of the ratio between the circumference and diameter of a circle, and it was impossible to evaluate ratios using irrational numbers with the number system at that time, Archimedes was still able to point out the relationships between the volumes and areas of various geometric shapes. His works nearly gave the exact formulae for volume and surface area for spheres. Related materials can mostly be found in the first two sections of “On the Sphere and Cylinder”.

2.1.1 “On the Sphere and Cylinder”

Archimedes started with six definitions and five hypothesis. His fifth hypothesis later became the famous Axiom of Archimedes, “For any two line segments with length a and b, if a < b , then there exists a natural number n such that na > b .” There were altogether 44 statements in the word. As in his letter to Dositheus, there were three major results, as explained below.

2.1.1a Statement 33: The volume of a sphere is equal to 4 times the area of the largest circle it contains.

Statement 34 states that, ” the volume of a sphere is 4 times the volume of the cone with height equal to the radius of the sphere and the base identical to the largest circle the sphere contains.”. This seems to imply that statement 34 is an extension of statement 33. Yet, according to “The Method” ([4], page 339 – 340), statement 33 actually came from statement 34. This means that the order in which the statements appeared in the work might not be that of the discoveries.

2.1.1b Statement 42 and 43: The curved surface are of a spherical cap is equal to the area of a circle with radius equal to the distance between the vertex at the curved surface and the base of the spherical cap.

Curved surface area (shaded) = Area of circle of radius r

Statements 42 and 43, discussing the situations where the spherical cap is greater than and less than the semi-circle respectively.

2.1.1c Deduction from statement 34: The cylinder with the large circle of a sphere as base and diameter as height has a volume times of the sphere.

	Volume of the cylinder = ´ Volume of the sphere
Surface area of cylinder (including both ends) = ´ Surface area of sphere

This is perhaps the discovery Archimedes felt proudest of: he dictated that this statement to be craved on his gravestone. In fact, the Romanian invaders buried him according to his wish.

2.1.1d Proofs of the Statements

As discussed above, each of the statements in Section I started from five hypotheses. Using the second hypothesis, i.e., “For any two plane curves with common endpoints, if they are not identical and have the same convexity, with one enclosed in the other or partially enclose and partially overlapped, then the enclosed curve is the shorter one”. From this hypothesis we get statement 1, “A polygon inscribing a circle has a larger perimeter than the circle.” This means that PA + AQ is longer than curve PQ.

Statement 2 states that “Given any two unequal numbers, we can find two unequal segments such that the ratio of the lengths of the segments is equal to that of the numbers”. Due to limited space here, the writer will not provide the proof of statement 2. The key point here is that statement 3, which states that “Given two unequal numbers and a circle, we can construct polygons inscribing and inscribed by the circle such that the ratio between the perimeters of the polygons equal to that between the two numbers”, follows easily from statements 1 and 2. As the two numbers are arbitrarily chosen, we can approximate the circumference of the circle to an arbitrarily determined precision, thereby proving statements on the areas of circles. The method of proving statements is called method of exhaustion, and we shall discuss this method in the section “Methods”.

The approaches of other statements are similar. These statements describes the volumes and other properties of spheres, and some properties of sectors and spherical cap.

2.1.2 “On the Sphere and Cylinder” II

There are nine statements, six of which have the form of problems. These problems, based on chapter one, aim to discuss the properties of spherical cap and methods of construction. The problems are listed below:

1. Statement 1: Find a sphere with an equal volume as a give cone or cylinder.
2. Statement 3: Construct a plane which divides a given spheres into two spherical cap, such that the ratio of the surface areas is as given.
3. Statement 4: Construct a plane which divides a given spheres into two curved surfaces of spherical caps, such that the ratio of the volumes is as given.
4. Statement 5: Construct a spherical cap surface so that it is similar to another spherical cap surface and an equal volume as a third spherical cap surface.
5. Statement 6: Construct a spherical cap surface so that it is similar to another spherical cap surface and an equal surface area with a third spherical cap surface.
6. Statement 7: Construct a plane which cuts a given sphere to give a spherical cap surface which has the same base and height with a cone of volume at a given ratio to that of the spherical cap surface.

Statement 4 is of particular historical value as it is the first statement to arouse people’s interest in cubic equations. We will deal with the topic of cubic equations in more depth in a later chapter.

2.2 Measurement of a Circle

Nearly every ancient civilization would came across the problem of circles. On this topic Archimedes wrote a book named “On the Measurement of a Circle”. however, much of the contents of this book was lost ([2], page 50) and only three of the statements remains.

2.2a Evaluation of the Areas of Circles

Statement 1 states that ” The area of a circle is equal to a right-angled triangle with the circumference and the radius as straight edges”. In “The Nine Chapters of Mathematical Art”, there was also a statement meaning ” the product of the semi-circumference and the radius of a circle equals its area” ([6], page 80). Although both statements gave the formula for calculation of a circle’s area, the discovery by Archimedes might be earlier than that in China.

Here is Archimedes’ proof of the statement. Let A be the area of the circle ,C the circumference and T the area of the aforementioned triangle. Assume A > T, we can construct a regular polygon P with a sufficient number of sides such that P is inscribed by the circle and

A – P < A – T，

That is, P > T.

However, this is not possible because we can divide the regular polygon into congruent triangles with h shorter than r, and the perimeter of P is shorter than C implying P < T, contradicting the above. Similarly, we cannot have A < T. And therefore A = T. This method of proving statements, “prove by contradiction’, is now a very common tool. We will discuss more about this in the section “Methodologies”.

2.2b Circumference-Diameter Ratio

In “Elements”, Euclid discussed many properties of circles, but not those involving the evaluation of the ratio between the circumference and diameter, area of circle and length of circumference ([6], page 81). Yet, Archimedes pioneered to use bounding techniques to approximate a quantity and evaluate error terms.([6], page 81). All these works are noted in Statement 3 ── “The ratio of the circumference to the diameter of any circle is between and “.

The proof of Statement 3 is given below ([6], page 82). Archimedes gave an approximation to the value of in his proof.

As in the figure,

（），

.

(Nobody knows how Archimedes get this approximation of , and we will discuss this when we summarize his achievements in arithmetic.)

Adding these two equations, we have

.

It is easy to prove that . Hence, the area of DOBC equals to and respectively. Thus,

.

Adding 1 to both sides,

Rearranging the terms and using the inequality above, we get

or

and this gives the upper bound of the diameter of the regular hexagon. Archimedes also calculated the dimensions of regular 12-gon to regular 96-gon using the same strategy. Similarly, he evaluated the lower bounds and got the bounds in the statement.

 

2.3 Quadrature of the Parabola, Conoids and Spheroids

Apollonius of Perga is a mathematician of the same historical period as Archimedes. He was famous for his studies in conics. His method of coordinates inspired the mathematician Fermat to develop the field of analytical geometry ([8]). Comparing with Archimedes’ studies of parabola, solid bounded by a paraboloid or hyperboloid and a plane, and spheroids, the former is certainly better known. However, the use of proportions and mechanics in Archimedes’ studies was also distinguished ([4], page 27 – 34, 332 – 338). Many of these works were done with the help of the method of exhaustion. We will discuss the results of his studies here and his methodologies later.

There are a total of 24 statements in his work “On Quadrature of the Parabola” and the most influential one is Statement 24── “The area of the parabolic segment of a parabola is equal to times that of the triangle with the same base and height.”

In another book “On Conoids and Spheroids” he gave 32 statements. A conoid is the volume formed by revolving a parabola or a hyperbola. Likewise, a spheroid is formed by revolving an ellipse. Archimedes’ studies focused on the volumes of these forms and here are two of the more important statements in his book:

1. Statements 21 and 22: The volume of any segment of a paraboloid is times that of a cone (or a segment of a cone) with the same base and axes.
2. Statement 24: The ratio of the two segments formed by cutting a solid bounded by a paraboloid with two planes in an arbitrary way is equal to that of the squares of the lengths of their axes.
   

   Segment APp : Segment AP‘p‘ = AN2 : AN‘2

 

2.4 Spirals

A characteristic of ancient Greek geometry studies is the objects were usually limited to those that can be drawn. This implies that some curves with peculiar shapes were often missed. It was not until the Alexandrian period that this ‘rule’ was broken, and Archimedes was one of these rule-breakers ([5], page 125). Hw discussed, in his work “On Spirals”, a special type of curve — the locus of an endpoint of a uniformly-rotating line segment which has the other endpoint (origin) fixed and its length increasing at a uniform rate. This is what we call a spiral or, Archimedes’ spiral. In polar form, the equation of a spiral is r = aq.

It is not known why Archimedes drew this type of curve. A popular suggestion for the reason is that he used the concept of mechanics and combine two velocities vectorically to get the spiral. This is the first concept of differential in history ([1], page 150). The second time this concept appeared was in 1629, introduced by Fermat. Hence, it was indeed amazing that Archimedes could propose in “On Spirals” some techniques of constructing tangents to a spiral.

There are 28 statements in “On Spirals”. The first 9 was about the proportions between circles an their tangents and statements 9 to 12 was on arithmetic progression, including

to facilitate the discussion on area of spirals afterwards. Statements 13 to 20 studies the tangents to spirals. In particular, in Statement 20, a construction method of the tangent was noted: for any point P on the first revolution, construct OT perpendicular to OP. Then the tangent at P would cut OT at T. If we use O as the center and OP as radius, the circle drawn would cut the pole at K, then the length of OT equals to the arc length between K and P. And then he discussed the techniques to use for a point on the nth revolution, for an arbitrary n.

He discussed the matter on area using method of exhaustion from Statement 21 onwards. In Statement 21, he pointed out that:

If a spiral has the form r = aq, then the area bounded by the first revolution of the spiral and the pole is = .

 

2.5 Notations for Numbers

There was a misperception among some ancient Greeks that the number of sand grains in the world was infinite and the number could not be represented by a number. To remove this misperception, in his only work on arithmetic, “Arithmetic on Sand Grains”, Archimedes proposed a new way to denote large numbers and calculated the number of sand grains in the world. At that time, the largest unit in the number system was “M” which meant ten thousands. To represent twenty thousands, the used , where b represented 2. Thus, the largest number that could be represented was ten thousands times ten thousands i.e., 10⁸. The new system used the concept of index: the numbers from 1 to 10⁸ was put in the first class, the numbers from 10⁸ to 10¹⁶ the second class, …, and the numbers from to was put in the (10⁸)th class. And all the numbers above were considered to be in the first period. was denoted by P. In the second period, the first class ranged from P�1 to P�10⁸, and the (10⁸ )th class ranged from P� to P�. In this way, the first class in the (10⁸)th period ranged from �1 to �10⁸, and the (10⁸ )th class from � to .. Archimedes hypothesized on the size of the Earth, the distance between the Earth and the Mon, the size of the sum and the Space and the number of sand grains in a seed, and determined that the number of sand grains in the world could just be 10⁵¹, which was far less than �. However, the mathematician was not satisfied by these: he could have further developed the system so that it could represent an arbitrarily large number. Yet, having seen that the misperception was gone and probably the Romanians were going to invade the country, Archimedes did not proceed further ([6], page 90).

 

2.6 Approximation of a Square Root

There are two major achievements. One is the approximate value of , and the other was the approximate values of the square roots of some large numbers.

2.6a The Approximate Value of

In 2.2b the writer mentioned that Archimedes found an approximation of when he was studying the circles. Actually, the approximation was:

Interestingly, these two fractions are the closest approximations to with denominators no greater than 153 and 780 respectively ([6], page 81). There were a lot of suggestions to the way by which he obtained these approximations. Many believe that he had used the inequality

.

It is because his friend Heron was known to give an approximation to in this way. Alkarkhi (11th century), an Arabic, used the approximation after consulting some Greek materials ([4], page 68).

Most people believe that Archimedes began the approximation with the value and get . Comparing with , we have . Substituting into we get . Using this as the new value for a and repeating the above process, we get in the end. Using similar techniques, we will get and therefore .

In fact, he could have used instead of as the initial value for a, and he could have done more repetitions to get a better approximation. However, he did not do so. This may be because he thought his approximation was close enough to and it would be easier to extract common factors in further calculations using the above values ([4], page 70).

2.6b Approximate values of some large numbers

In his work “On the Measurement of a Circle”, Archimedes wrote down the following approximations([[4], page 70):

These values are mainly obtained through the Euclid’s theorem (a + b)² = a² + 2ab + b² and fractions with 60 as denominator.

 

2.7 Cubic Equations

Mathematicians in Ancient Greece sometimes discussed cubic equations, but only those in the form y = ax³. This is also reflected by the fact that in Statements 1 and 5 of “Spheres and Cylinders” Archimedes dealt with the equation a² : x = x : b. However, in Statement 4 he talked about the following ratio:

,

where a is the radius of the sphere, m : n is a given ratio between two line segments with m > n, and x the height of the longer segment. In his solution, he generalized the problem into:

(a – x) : c = b² : x²,

in which the segment a is divided into the two parts (a – x , a), and b and x are considered to be the areas of squares. He solved the problem by finding the intersection of the parabola and hyperbola. Furthermore, he pointed out the conditions in which there are zero, one or two roots between 0 and a ([4], page 116 – 117).

Apollonius of Perga also encountered this problem when he was investigating the number of normals that could be drawn through a given point on a conic. Nevertheless, he tried to solve the problem geometrically instead of by dealing with cubic equations. hence, Archimedes’ results offer a more general solution ([4], page 119).

Archimedes’ works in cubic equations inspired al-Khazin (? – 965?), Ibn al-Haytham (965 – 1040) and Abu’l Jud (~1000) to continue the studies. Eventually, the conic solutions to all types of cubic equations were summarised by Omar Khayyam (1048 – 1131), ([6], page 80).

 

2.8 Other Mathematical Achievements

In this chapter, we will discuss some of the important, though less well-known, achievements of Archimedes.

 

2.8a Formula for the area of a triangle

Area of a triangle =, where s is the semiperimeter, a, b and c are the side lengths of the triangle.

This formula is known as the Heron’s formula. But it was in fact discovered by Archimedes ([5], page 125).

2.8b Semi-regular Polyhedron

Semi-regular polyhedron is a solid formed from faces of regular polygons with different number of sides. In addition, for identical faces, the arrangements of neighboring faces are identical. There are a total of 13 possible types of semi-regular polyhedron ([7], page 14 – 15).

Source: Reference [7], page 15

2.8c Construction of Regular Heptagon

There was a note on the construction of regular heptagon by Archimedes in some a translation work of an Arabic mathematician ([4], page 380).

2.8d The Cattle Problem

Archimedes proposed the following problem when he visited Hieron, the King of Syracuse:

Suppose W and w are the number of white bulls and cows respectively,

X and x are the number of black bulls and cows respectively,

Y and y are the number of yellow bulls and cows respectively,

Z and z are the number of dappled bulls and cows respectively.

where:

Then, what are the values of the unknowns?

Hieron spent much effort to get the answer. The smallest number got 7 digits. Just then Archimedes add two more conditions:

W + X = Perfect square
Y + Z = Triangular number

Hieron were unable to solve this. Experts on the history of mathematics believe that Archimedes himself could not solve this problem either because in 1965, with the help of computers, the answer was found to be of 206500 digits ([2], page 97 – 98).

Yet, from the solution of the first part of the problem and the difficulty of the second part, we can infer Archimedes’ ability in algebra.

2.8e “Book of Lemmas”

In the Arabic book “Book of Lemmas”, there were 15 problems on circles. These problems were believed to be reorganized by the descendents of Archimedes. The statements in “Book of Lemmas” do not seem to concur to a central theme. Neither are they done with method of exhaustion — they are just small discussions on the properties of circles. Below low are a few examples:

1. Statement 4: If AB is the diameter of a semicircle, N an arbitrary point on AB. Then the sum of areas of the two semicircles drawn with AN and BN as diameters respectively will be equal to that of the circle with PN as radius, where PN is the perpendicular of AB from P on the semicircle.
2. Statement 9: If AB and CD are not diameters and they intersect at right angles, then(arc AD) + (arc CB) = (arc AC) + (arc DB).

 

III. Methodologies

In his studies, Archimedes made good use of the works of predecessors, proposed innovative ideas and insisted on strict proofs ([4], page 26 – 27). As in “Elements”, his works started with a list of definitions and hypotheses. This is followed by the derivation and proofs of his statements. We will discuss the means he used in his proofs below. Besides, from “The Methods”, ([4], page 339 – 340), we know that the order of appearance of his statements does not match the chronological order of their discoveries. This means that the discoveries did not followed from the deductions in his works. We will also discuss the routes he got his ideas in this section.

 

3.1 The Use of Proportions

Archimedes used proportions in his statements and proofs. In fact, ancient Greek Mathematics mainly involved the uses of areas and proportions in solving problems. Euclid used both methods, Apollonius of Perga mainly used areas and Archimedes usually used proportions ([4], page 27). For example, Apollonius interpreted y² as an area and pointed out the relationships of y² = px in parabolae and in hyperbolae and ellipse, where p is the parameter and d is the diameter; however, Archimedes viewed parabolae as ([2], page 139). Hence, they came up with diverse results in their studies.

 

3.2 Method of Exhaustion and Proof by Contradiction

In the second statement of the twelfth chapter of Euclid’s “Elements”, there was a use of the method of exhaustion. Archimedes also employed this method in proving many of his statements. One example is that in 2.1.1d. Choose two arbitrary values to give a ratio R and draw regular polygons inscribing and inscribed by a circle. Archimedes proved that if the polygons have sufficient numbers of sides, the ratio between their areas must be smaller than R. Hence if R is close to 1, then the ration of the areas of the polygons is also close to 1. In this way, we can determine the area of the circle.

Circle, inscribing polygon, inscribed polygon

It was originally difficult to evaluate the area bounded by a curve. By constructing the polygons, we can make our estimate of the area arbitrarily close to the actual value. In fact, this method can be applied to the areas bounded by conics and the volumes of the solids of revolution. In “On the Sphere and Cylinder”, “On Measurement of a Circle”, “On Quadrature of the parabola”, “On Conoids and Spheroids” and “On Spirals”, Archimedes often use the method of exhaustion to solve the problems.

We need not construct both the inscribing and inscribed polygons in using the method of exhaustion: sometimes we need only one of them. One example is Statement 24 of “On Quadrature of the parabola” as mentioned before, it stated that “The area of the quadrature of any parabola is equal to times that of a triangle with the same base and height.”

The proof ([5], page 119 – 120) involves using a series of triangles to approximate the parabola. Firstly, we determine the vertex, P, of the parabola and construct DPQp. Then we find the vertices R and r between PQ and Pq, and construct DPQR and DPqr. Note that the preceding statements in the book has proved that

Area of DPQR = Area of DPqr = DPQp.

Hence, the shaded area = DPQp + DPQp.

Then he proved that if we repeat the above process in DPQR and DPqr, we can form a series of triangles such that the sume of areas differs from that of the actual segment area by an arbitrarily small value. Hence we have

Parabola PQp = DPQp + DPQp + DPQp+ DPQp+�+ DPQp

The method of exhaustion gives us the formulae for calculating areas. However, as the formulae involved infinite sequences and there was not an infinity concept, the problem was not yet completely solved. To finish the problem, we need to prove by contradiction.

In this method, we first propose a value to be the answer and hypothesized on the contrary. If we can prove that there will be a contradiction if the true value is smaller or greater than the proposed value, then we are done. Now as Archimedes proposed that the shaded area is DPQp, we may assume that

area of the parabolic segment > DPQp.

Since the difference between the area of the parabola and the sum of areas of the triangles can be made arbitrarily small, we have

area of the parabolic segment > sum of areas of the triangles > DPQp,

Yet, according to Statement 23 [Note], if we sum the series to the mth term (m > n),

sum of areas of triangles + (DPQp)= DPQp

and this would lead to a contradiction because it implies

sum of areas of triangles < DPQp.

Similarly we can prove that

area of the parabolic segment < DPQp

would also lead to a contradiction. Thus, the value of the area should be as proposed.

From the above, we can see that method of exhaustion and proof by contradiction are

 

3.3 The Use of Mechanics

Up to this point, you may become as amazed as those who studied Archimedes’ works before 1906: how could Archimedes came up with so many innovative ideas? Indeed, Archimedes’ works were so neat that one could only find his definitions, hypotheses, statements and proofs, but not his routes to gather his ideas. Now we will take a look at these ‘hidden’ routes.

In 1906, J. L. Geiberg (1854 – 1928) found in Turkey a book containing a lot of Archimedes’ works, including “The Methods”, which was written by Archimedes to describe where his beautiful ideas came from. Most of the 15 statements in the book We do not focus on the proofs here, but on something of even greater value.

Archimedes is famous for his studies in mechanics. In fact, mechanics led him to many of is discoveries in mathematics. As an example, the first statement in “The Methods” was on the areas bounded by parabolae, and the proof was discussed in the previous chapter. In the book, he wrote down how he discovered the relationship:

Figure duplicated from page 29 of Reference [2]

In the above figure, BD is the diameter of the segment, CF is the tangent at C and P is an arbitrary point on the chord of the parabola. Note that AKF and OPNM are parallel to BD, C, B, N, K and H are collinear and KH and KC have equal lengths. As B is the vertex, using the previous theorems, we have EB = BD, MN = NP, FK = KA. In Statement 5 it is also proved that

MO : OP = CA : AO = CK : KN = HK : KN

Now assume that the segments have weights proportional to the lengths. View K as the pivot of HC and put a line segment TG of equal weight as OP at H. Since the center of gravity (midpoint) of MO, N, lies on KN, the two sides of the pivot K is balanced.

Since P is arbitrary, OP can be any parallel line between the parabolic segment ABC and AC and thus MO can be any parallel line in DAFC with the aforementioned relationship with OP.

Archimedes believed that as the numbers (OP and MO) of line segments in the parabolic segment ABC and DAFC are equal, for the pivot K, the parabolic segment ABC formed from all OP will balance the DAFC formed from all MO, and the center of gravity ofDAFC is the point W on CK, where CW = 2WK.

Hence,

parabolic segment ABC： DAFC = WK : KH = 1 : 3 (∵CK = KH).

Therefore,

parabolic segment ABC = DAFC = DABC. (The previous statement proved the relationship of 4 times)

The theme of “The Methods” is to divide the unknown quantity into many small quantities and use the lever to determine the proportional relationships of these quantities with other small quantities. The sum of the latter group of quantities is smaller to calculate ( like the triangleDAFC above. This allows us to find the unknown ([6], page 74 -750). Note that this is a preliminary concept of integration, and Archimedes successfully combines the studies of mechanics and mathematics.

This method is indeed marvelous. But Archimedes pointed out that this is just a way to provide guesses, not proofs. This means that we still need other means (the method of exhaustion and prove by contradiction) to prove the results obtained. Actually, sometimes the method of using mechanics fails. Consider the figure below ([6], page 75 – 76):

DC is an altitude of the triangle. If EF // AB, EG AB and FH AB, then GE = HF. As EF goes from AB to C, at each height, we can find the corresponding EG and FH. This leads to the conclusion that the areas of DACD and DBCD are equal, which may not be true.

 

IV. Conclusion

Archimedes, Euclid and Apollonius of Perga are the three greatest mathematicians in ancient Greece. Also Archimedes is also considered one of the three greatest mathematicians ever, as with Newton and Gauss. Just like Newton, Archimedes is famous for his achievements in mechanics. Yet, his achievements in mathematics may not be neglected.

Many works were written by Archimedes, including “On the Sphere and Cylinder”, “On the Measurement of a Circle”, “On Conoids and Spheroids”, “On Spirals”, “The Sandreckoner”, “On Quadrature of the parabola”, “Book of Lemmas” and “The Methods”. These works are mainly on geometry topics: in two-dimensional plane, he worked on circles, parabolae, hyperbolae and tangents to spirals; in three-dimensional space, he discussed the areas and volumes of cylinders, cones, spheres, spheroids and conoids and the approximation of Pi. Other achievements include Heron’s formula, construction of regular heptagon, semi-regular polyhedron, and many more.

Archimedes seldom touch on the topic of arithmetic, as reflected by the fact that he had only one piece of work, “The Sandreckoner”, on the topic. He invented a new system to represent large numbers. Some of his contribution to the field arises from the need of solving geometry problems. For example, he gave the approximations of and square roots of some large numbers, insights on the solutions of cubic equations and arithmetic series.

In terms of methodologies in research, Archimedes is also outstanding. His use of proportion in his statements and mechanics in discovering relationships was pioneering. The invention of the concept of integration and the use of the method of exhaustion and proof by contradiction rendered him invaluable tools in proofs.

In conclusion, Archimedes is well deserved to be named as one of the greatest mathematicians.

 

V. References

[1] John Stillwell (1989), Mathematics and Its History, Springer.

[2] T. L. Heath (1965), A History of Greek Mathematics, Volume II, Oxford.

[3] E.T. Bell (written in 1937),Jing Zhujun (translated in 1998).Da shu xue jia (translated from Men of mathematics). Publisher: Chiu Chang Publishing Company.

[4] T. L. Heath (written in 1912), Zhu Enkuan, Li Wenming (translated in 1998). Ajimide quan ji (translated from The Works of Archimedes). Publisher: Shanxi ke xue ji shu chu ban she (Shanxi).

[5] Morris Kline (written in 1908), Lin Yanquan, Hong Wangsheng, Yang Kang Jingsong (translated in 1983). Shu xue shi : shu xue si xiang de fa zhan (translated from Mathematical thought from ancient to modern times). Publisher: Chiu Chang Publishing Company.

[6] Xie Enze, Xu Benshun (1994). Shi jie shu xue jia si xiang fang fa, p.61-99.Publisher:Shandong jiao yu chu ban she (Jinan).

[7] Cai Zhiqiang, Sun Wenxian(199?). Shu xue li ti mo xing zhi zuo : fu : duo mian ti yan jiu. Publisher: Chiu Chang Publishing Company.

[8] Hong Kong University of Science and Technology. http://www.edp.ust.hk/math/history/5/5_5/5_5_34.htm