Archimedes & Apollonius

5.1 Archimedes of Syracuse (c. 287–212 BCE)

Archimedes of Syracuse is widely regarded as the greatest mathematician of the ancient world — and one of the greatest of all time, ranked alongside Newton and Gauss. Born around 287 BCE in the Greek colony of Syracuse on the island of Sicily, he was the son of the astronomer Phidias and may have been related to King Hieron II, the ruler of Syracuse. As a young man, he studied at Alexandria, the intellectual capital of the Hellenistic world, where he likely encountered the works of Euclid and the mathematical traditions of the great Library.

Returning to Syracuse, Archimedes spent most of his life in the service of King Hieron, producing a body of work that spanned pure mathematics, mathematical physics, engineering, and astronomy. His mathematical achievements include the computation of areas and volumes using proto-calculus techniques, the most accurate ancient approximation of $\pi$, the foundations of hydrostatics and statics, and a number system capable of expressing astronomically large quantities.

The “Eureka” Story

The most famous anecdote about Archimedes, told by Vitruvius, involves King Hieron's golden crown. The king suspected that his goldsmith had secretly alloyed the crown with cheaper silver, but could not prove it without melting the crown down. He challenged Archimedes to determine the truth.

While bathing, Archimedes noticed that the water level rose as he lowered his body into the tub. He realized that the volume of water displaced equals the volume of the submerged object. Since gold is denser than silver, a crown of pure gold would displace less water than an equal-weight crown alloyed with silver. Overcome with excitement, he leapt from the bath and ran through the streets of Syracuse shouting “Eureka!” (“I have found it!”).

War Machines and the Siege of Syracuse

During the Second Punic War, the Roman general Marcellus besieged Syracuse (214–212 BCE). Archimedes designed ingenious defensive machines that held the Romans at bay for over two years. Ancient sources describe giant catapults that could hurl massive boulders, cranes (the “Claw of Archimedes”) that could lift enemy ships out of the water and capsize them, and — most controversially — burning mirrors that could focus sunlight to set ships on fire.

When Syracuse finally fell in 212 BCE, Archimedes was killed by a Roman soldier, reportedly while engrossed in a geometric diagram drawn in the sand. His last words, according to tradition, were “Do not disturb my circles” (Noli turbare circulos meos). Marcellus, who had given orders that Archimedes be taken alive, was said to be greatly distressed by his death.

5.2 The Method of Exhaustion

Archimedes's most profound mathematical achievement was his use of the method of exhaustion— developed by Eudoxus — to compute areas and volumes that had eluded all previous mathematicians. By inscribing and circumscribing sequences of polygons or other known shapes, he “exhausted” the region in question, trapping the true area or volume between ever-tighter bounds.

The Quadrature of the Parabola

In his treatise Quadrature of the Parabola, Archimedes proved one of his most beautiful results: the area of a parabolic segment (the region between a parabola and a chord) is exactly$\frac{4}{3}$ times the area of the inscribed triangle with the same base and vertex.

Archimedes's proof proceeds by repeatedly subdividing the parabolic segment. Start with triangle $\triangle ABC$ inscribed in the segment. Then, in each of the two remaining parabolic segments (between the parabola and the sides $AC$ and $BC$), inscribe new triangles in the same way. Archimedes showed that each new pair of triangles has a combined area equal to $\frac{1}{4}$ of the previous triangle.

Let $T$ denote the area of $\triangle ABC$. After the first step, the total inscribed area is $T$. After the second step, it is $T + \frac{1}{4}T$. After the third step,$T + \frac{1}{4}T + \frac{1}{16}T$. Continuing this process indefinitely:

$$\text{Area}(S) = T + \frac{T}{4} + \frac{T}{16} + \frac{T}{64} + \cdots = T \sum_{k=0}^{\infty} \frac{1}{4^k}$$

Summing the Geometric Series

Archimedes needed to evaluate the geometric series $\sum_{k=0}^{\infty} \frac{1}{4^k}$. He did this by a clever algebraic identity. Consider the finite sum:

$$S_n = 1 + \frac{1}{4} + \frac{1}{4^2} + \cdots + \frac{1}{4^n}$$

Archimedes established the identity:

$$1 + \frac{1}{4} + \frac{1}{4^2} + \cdots + \frac{1}{4^n} + \frac{1}{3} \cdot \frac{1}{4^n} = \frac{4}{3}$$

To see why this holds, note that each term satisfies $\frac{1}{4^k} = \frac{4}{3} \cdot \frac{1}{4^k} - \frac{1}{3} \cdot \frac{1}{4^k}$. Summing telescopically:

$$S_n = \frac{4}{3}\left(1 - \frac{1}{4^{n+1}}\right) = \frac{4}{3} - \frac{1}{3 \cdot 4^n}$$

As $n \to \infty$, the remainder term $\frac{1}{3 \cdot 4^n} \to 0$, so:

$$\sum_{k=0}^{\infty} \frac{1}{4^k} = \frac{4}{3}$$

Therefore, $\text{Area}(S) = \frac{4}{3} T$. Archimedes then completed the proof rigorously using the method of exhaustion (double contradiction), showing that the area can be neither greater nor less than $\frac{4}{3}T$.

5.3 Archimedes' Calculation of $\pi$

In his treatise Measurement of a Circle, Archimedes established the most accurate ancient approximation of $\pi$ — a result that would not be surpassed for several centuries. His method was beautifully simple in concept: he trapped the circle between inscribed and circumscribed regular polygons, computing their perimeters to obtain lower and upper bounds for $\pi$.

The Method of Inscribed and Circumscribed Polygons

Consider a circle with diameter 1 (so circumference $= \pi$). A regular $n$-gon inscribed in this circle has perimeter:

$$p_n = n \sin\!\left(\frac{\pi}{n}\right)$$

A regular $n$-gon circumscribed about this circle has perimeter:

$$P_n = n \tan\!\left(\frac{\pi}{n}\right)$$

Since the circle lies between the two polygons: $p_n < \pi < P_n$ for all $n$. As $n$ increases, both bounds converge to $\pi$.

Archimedes started with hexagons ($n = 6$) and doubled the number of sides five times, reaching 96-gons ($n = 96$). He didn't use trigonometric functions (which hadn't been developed yet), but instead used a recursive geometric relationship. If $s_n$ is the side length of the inscribed $n$-gon in a circle of diameter 1, then the side of the inscribed $2n$-gon is:

$$s_{2n} = \sqrt{\frac{1 - \sqrt{1 - s_n^2}}{2}}$$

Similarly, if $S_n$ is the side of the circumscribed $n$-gon, then:

$$S_{2n} = \frac{S_n}{\sqrt{1 + S_n^2} + 1} \cdot 2$$

The upper bound $\frac{22}{7}$ became the standard approximation of $\pi$ used for centuries. It is accurate to about 0.04%. The method itself could be extended to arbitrarily many sides for greater accuracy; later mathematicians (notably the Chinese mathematician Zu Chongzhi, c. 480 CE) used Archimedes's approach with polygons of up to 12,288 and 24,576 sides to obtain$\pi \approx \frac{355}{113} = 3.1415929\ldots$, correct to six decimal places.

5.4 Archimedes' Principle of the Lever

In his treatise On the Equilibrium of Planes, Archimedes established the mathematical foundations of statics — the study of bodies in equilibrium. His treatment of the lever is considered the first-ever example of mathematical physics: physical laws derived from axioms through rigorous logical deduction.

Archimedes famously declared: “Give me a place to stand and I will move the Earth” (Dos moi pa sto, kai tan gan kinaso). This was not empty boasting — it is a direct consequence of the lever principle. With a sufficiently long lever and a fulcrum, any weight can be moved by any force, however small, simply by making the lever arm long enough.

Archimedes also computed the centers of gravity of various figures: triangles (intersection of medians, at $\frac{1}{3}$ of the way from each side), parallelograms (intersection of diagonals), and parabolic segments. The center of gravity of a parabolic segment lies at$\frac{3}{5}$ of the distance from the vertex to the midpoint of the base — a result that required the full power of his exhaustion methods.

5.5 The Sand Reckoner

In The Sand Reckoner (Psammites), Archimedes set himself an audacious challenge: to show that a number can be named that exceeds the number of grains of sand required to fill the entire universe. This was not merely an exercise in large numbers — it was a fundamental contribution to the theory of numeration and a demonstration that mathematics can handle quantities of any magnitude.

The Greek number system, like most ancient systems, had difficulty expressing very large numbers. The largest named Greek number was the myriad ($\mu = 10{,}000$). A “myriad myriad” was $10^8$. Archimedes needed to go far beyond this.

Archimedes then estimated the size of the universe (using Aristarchus's heliocentric model, which he mentions is the largest cosmos anyone has proposed) and the size of a grain of sand. He concluded that the number of grains of sand that would fill the universe is less than$10^{63}$ — a number easily expressible in his system, well within the first period.

The Sand Reckoner is remarkable for several reasons: it shows Archimedes's awareness of Aristarchus's heliocentric model (centuries before Copernicus), it essentially develops a system of exponent notation and the laws of exponents ($10^a \times 10^b = 10^{a+b}$), and it demonstrates the power of mathematical abstraction to transcend the limits of the physical world.

5.6 Surface Area and Volume

Archimedes's masterpiece, On the Sphere and Cylinder, contains results that he himself regarded as his greatest achievements. He proved exact formulas for the surface area and volume of a sphere, and discovered a beautiful relationship between a sphere and its circumscribed cylinder that he wanted inscribed on his tombstone.

The Cylinder–Sphere Relationship

Consider a sphere of radius $r$ inscribed in a cylinder (a cylinder with radius $r$ and height $2r$). Archimedes proved that the ratios of their volumes and surface areas are both $2 : 3$:

Archimedes was so proud of this result that he asked for a sphere inscribed in a cylinder to be engraved on his tombstone. The Roman orator Cicero, visiting Syracuse in 75 BCE — more than a century after Archimedes's death — reported finding the neglected tomb with this very emblem.

The Cone

Archimedes also used the elegant relationship between the cone, sphere, and cylinder. A cone with the same base and height as a cylinder has exactly one-third the volume:

$$V_{\text{cone}} = \frac{1}{3}\pi r^2 h$$

For a cone inscribed in the same cylinder as the sphere (radius $r$, height $2r$):

$$V_{\text{cone}} = \frac{1}{3}\pi r^2 \cdot 2r = \frac{2}{3}\pi r^3$$

Thus $V_{\text{cone}} : V_{\text{sphere}} : V_{\text{cylinder}} = 1 : 2 : 3$ — a strikingly elegant ratio.

5.7 The Archimedean Spiral

In his treatise On Spirals, Archimedes studied a curve that now bears his name — the first mathematical curve to be defined by an equation in what we would now call polar coordinates.

Archimedes proved two remarkable results about this spiral. First, he showed how to construct the tangent line at any point on the spiral — a result that anticipates differential calculus by nearly two millennia. If $P$ is a point on the spiral at distance $r$ from the origin, then the tangent at $P$ makes an angle $\psi$ with the radius vector such that:

$$\tan \psi = \frac{r}{b} = \frac{a + b\theta}{b}$$

Area Enclosed by the Spiral

Archimedes computed the area enclosed between the spiral $r = b\theta$ (with $a = 0$) and the initial ray after one complete revolution ($\theta$ from $0$ to $2\pi$).

The Archimedean spiral also provides a theoretical solution to two of the great problems of ancient geometry: the trisection of an angle and the squaring of the circle. However, since the spiral itself cannot be constructed with compass and straightedge, these solutions were not considered admissible by the strict standards of Greek geometric construction.

5.8 The Archimedes Palimpsest

In 1906, the Danish philologist Johan Ludvig Heiberg discovered a remarkable document in Constantinople (modern Istanbul): a 10th-century Byzantine manuscript of Archimedes's works that had been scraped clean in the 13th century and overwritten with Christian prayers — a palimpsest. Beneath the prayers, using a magnifying glass, Heiberg could still make out the original mathematical text.

The palimpsest contained seven treatises by Archimedes, including the only surviving copy of The Method of Mechanical Theorems — a work that had been lost for over a thousand years. This discovery revolutionized our understanding of Archimedes's thought process.

Example: Finding the Volume of a Sphere

Consider a sphere, a cone, and a cylinder, all with the same radius $r$ and the cylinder having height $2r$. At height $x$ above the center:

• The cross-section of the sphere is a circle of radius $\sqrt{r^2 - x^2}$, with area $\pi(r^2 - x^2)$
• The cross-section of the cone is a circle of radius $|x|$, with area $\pi x^2$

Adding these at each height:

$$\pi(r^2 - x^2) + \pi x^2 = \pi r^2$$

This is exactly the cross-section of the cylinder! By Cavalieri's principle (or Archimedes's mechanical reasoning), the volume of the sphere plus the volume of the cone equals the volume of the cylinder:

$$V_{\text{sphere}} + V_{\text{cone}} = V_{\text{cylinder}}$$

$$V_{\text{sphere}} + \frac{2}{3}\pi r^3 = 2\pi r^3$$

$$V_{\text{sphere}} = 2\pi r^3 - \frac{2}{3}\pi r^3 = \frac{4}{3}\pi r^3$$

Archimedes considered this mechanical method heuristic, not rigorous — it was his way of discoveringresults, which he then proved using the method of exhaustion for publication. He wrote to Eratosthenes: “I thought it worthwhile to write out the method and publish it, partly because I have already spoken of it, and partly because I am convinced that it will prove very useful for mathematics; for I suspect that some of the theorems which have not yet been proved could be discovered by this method.”

5.9 Apollonius of Perga

Apollonius of Perga (c. 262–190 BCE) was the greatest geometer of the ancient world after Archimedes. Born in Perga (modern Turkey), he studied at Alexandria and spent time at Pergamum. His masterwork, the Conics (eight books, of which seven survive — four in the original Greek, three in Arabic translation), is the definitive ancient treatise on conic sections and remained the standard reference for nearly two millennia.

The Conic Sections

A conic section is the curve obtained by intersecting a cone with a plane. Depending on the angle of the plane relative to the cone, one obtains three fundamentally different types of curves. Apollonius was the first to show that all three can be obtained from a single double-napped cone (two cones joined at their apex), and he gave them the names we still use:

The terminology comes from the ancient Greek method of “application of areas.” For a parabola, the rectangle “applied” to the latus rectum exactly matches the square on the ordinate. For an ellipse, it “falls short” (elleipsis); for a hyperbola, it “exceeds” (hyperbole).

Focal Properties

Apollonius discovered the remarkable focal properties of the conics. Every ellipse has two foci, $F_1$ and $F_2$, such that for any point $P$ on the ellipse:

$$|PF_1| + |PF_2| = 2a \quad \text{(constant)}$$

For a hyperbola:

$$\big||PF_1| - |PF_2|\big| = 2a \quad \text{(constant)}$$

For a parabola with focus $F$ and directrix $d$:

$$|PF| = \text{distance from } P \text{ to } d$$

Tangent Constructions

Apollonius developed sophisticated methods for constructing tangent lines to conics. For a parabola$y^2 = lx$, the tangent at the point $(x_0, y_0)$ has the equation:

$$yy_0 = \frac{l}{2}(x + x_0)$$

He also proved that the tangent at the vertex of a parabola is perpendicular to the axis, and that the subnormal (the projection of the normal onto the axis) is constant and equal to $l/2$.

Nearly two thousand years after Apollonius, Johannes Kepler discovered that planets move in ellipses with the Sun at one focus (Kepler's First Law, 1609). Isaac Newton then proved that any body moving under an inverse-square gravitational force follows a conic section — an ellipse, parabola, or hyperbola depending on its energy. The purely geometric investigation that Apollonius had pursued for its own sake turned out to describe the fundamental orbits of celestial mechanics.

5.10 Legacy

Archimedes is universally regarded as the greatest mathematician of antiquity and one of the greatest of all time. His work anticipated integral calculus by nearly two millennia, founded the science of mathematical physics, and demonstrated a level of mathematical ingenuity that was not matched until Newton and Leibniz in the 17th century.

“Give me a place to stand, and I will move the Earth.” — Archimedes (attributed by Pappus)

The work of Archimedes and Apollonius represents the pinnacle of Greek mathematical achievement. After them, Hellenistic mathematics entered a period of consolidation rather than breakthrough, though important contributions would continue to be made by Diophantus, Heron, Pappus, and others. The methods of exhaustion and the theory of conics would remain the most advanced mathematics in the world until the Islamic Golden Age and, ultimately, the invention of calculus in the 17th century.

Video Lectures

These video lectures demonstrate Archimedes' method for computing $\pi$ using inscribed and circumscribed polygons — one of the greatest achievements of ancient mathematics.