Ptolemy's Construction for the Side of a Pentagon as a Formula

Ptolemy shows how to construct the length of the side of a pentagon in Fig. 1-1 on page 48 of Toomer's Ptolemy's Almagest. It is BE in the figure below but Ptolemy does not explain how this was arrived at. There seems to be another missing key needed to understand his methods.


One possibility is that he used two procedures relating the chords of a pentagon to solve the equivalent of two equations with two unknowns. Ptolemy's theorem can be used to relate the sides of an inscribed quadrilateral and for the pentagon there are only two unknown chords. Applying Ptolemy's theorem to three neighboring sides of a pentagon and the larger chords needed to complete the quadrilateral results in a quadratic equation involving the two unknown chord lengths. The quadratic equation can be solved for the ratio of the lengths of the two sides which turns out to be the golden ratio.


One can then use the formula for finding the chord of half an angle to obtain another relation between the two chords. Substituting the golden ratio into this formula simplifies it and reduces it to an equation involving one unknown which can be easily solved for the length of the side of a pentagon.


The figure above is a copy of the one in Ptolemy's Almagest. Ptolemy assumed that the diameter of the circle had 120 parts but in what follows it is assumed that the radius of the circle is 1.


Following the construction through one step at a time and converting the lengths involved into numbers we end up with the same formula obtained by the algebraic solution above. Ptolemy follows the steps of the construction computing the lengths and sides of the triangles in the figure and thus obtains the chord of 72°.

Ptolemy's method for finding the chord of half the angle given the chord of an angle is algebraically equivalent to the formula used above. He also uses the proportion of the golden ratio in his proof. So it is not unlikely that the solution of these equations was involved in finding the formula for the side of the pentagon.

Half-Angle Chord Formula

Why is Ptolemy's chord table more precise than Hipparchus'? The 7 1/2° steps suggest that he arrived at the angles by successive division 60° by 2 (7 1/2° = 15°/2). Subdivisions of 72° do not exist so he appears not to have know the chord of 72° which is the side of the pentagon.


As mentioned before, one can use the golden ratio and a method for the chord of a half angle to find the side of a pentagon. In the figure above c_1 is the chord of the angle and c_2 that of half the angle. The sides, x and y, are the required sides of the right triangles. Here the circle is assumed to have a unit radius.


One can then use the Pythagorean Theorem to find a formula for the chord of the half angle in terms of the chord of the original angle. This along with the golden ratio will allow us to find an expression for the side of a pentagon.

Knowledge of Euclid's Elements is Assumed by Ptolemy

One can find the theorems concerning inscribed angles in Euclid's Elements which would explain why Ptolemy assumes that they are known. They are Propositions XX through XXII of Book IV and roughly state the following.

Prop. XX The central angle of a circle is double the angle at the circumference.

Prop. XXI Inscribed angles in same segment of a circle are equal.

Prop. XXII The sum of opposite angles of a quadrilateral inscribed in a circle is two right angles.

One also wonders how he arrived at a value for the chord of 120° which is the side of a pentagon. The Pythagoreans knew how to construct a pentagon from a isosceles triangle with base angles of 72° and a peak angle of 36° (see Elements, Book IV, Prop. XI). But that doesn't tell how Ptolemy arrived at a magnitude for the side of the pentagon. One possibility is the use of the golden ratio which is found in Euclid's Elements (Book VI, Prop. XXX) and what is known as Ptolemy's theorem or some other method to find the chord of half the angle if the chord of an angle is known.

Ptolemy was not the first to construct a table of chords. The credit for that goes to Hipparchus who constructed a 7 1/2° chord table which Ptolemy later improved on with a 1/2° table. Hipparchus anticipated much of what Ptolemy did concerning astronomy. He is believed to have traveled to Alexandria. Hipparchus made astronomical observations at Rhodes where the Antikythera mechanism is believed to have come from.

More on the Inscribed Angle Theorem

The inscribed angle theorem is a prerequisite needed to follow Ptolemy's proofs. It is a key for gaining an understanding of the heavens as he saw them. So it would not hurt to do a little review concerning inscribed angles. To prove the inscribed angle theorem we need the following figure.


In the figure the inscribed angle is angle ACB, its central angle is θ and the corresponding elements are indicated in green. Inside the circle are lines from its center to the ends, A and B, of the arc of the central angle and an additional line from the center to point C is needed to form two isosceles triangles inside the circle. In the following the inscribed angle is φ = γ + δ.


Since the inscribed angle, φ, is independent of α and β its value is the same for all inscribed angles on the same side of the of the chord, AB. The same half angle formula works for the remaining arc of the circle and its inscribed angle represented by the yellow lines. In this case the inscribed angle is again drawn on the side of the common chord opposite to that of the arc. The values of the two inscribed angles are not necessarily the same but since the sum of the two arcs of the circle is 360° the sum of their corresponding inscribed angles is 180°.

Ptolemy's theorem used to establish an upper bound for a chord

In the Almagest Ptolemy constructed a table of chords and to do this he used a theorem to establish an upper bound for the ratio of two chords. This was needed since he started with the known chords corresponding to the sides of an equilateral triangle, a square and a pentagon with central angles of 120°, 90° and 72°. He knew how to find the chord for the difference and sum of two angles from the chords for those angles. The greatest common denominator of the initial angles is 6° so he could find the chord of that angle. He also knew how to find the chord of half an angle from the chord of an angle and so he was also able to find the chords of 3° and 3/2°. But he wanted the chords for the multiples of 1/2° and there was no way to find the chord of 1/3 of an angle whose chord was known. He used an upper and lower bound of the ratio of two chords to show that his value for the chord of 1/2° was accurate to the desired precision.

On page 54 of Toomer's Ptolemy's Almagest he wants to prove that the ratio of a larger chord to a smaller one is less than the ratio of their corresponding angles. He starts with the chords joining the points A, B and C on the circle in the figure below.


He simply states that if the line BD bisect angle ABC then lines AD and CD are equal. This isn't exactly obvious and something appears to be missing from the argument. He seems to assume that everyone knows this and it is an example of why at times one needs to emulate a regenerative receiver in order to recover more detail.

What is missing is that the inscribe angle of a chord is the same for all inscribed angles on the same side of the chord. It is half its central angle. This is a generalization of Thales' theorem which states that for a triangle inscribed in a circle with the diameter as one of the sides the angle opposite the diameter is a right angle. Using the inscribed angle theorem the proof of Ptolemy's statement is actually quite simple. Possibly people in Ptolemy's time could do that in their heads.


(One can reverse the argument to show that if AD = CD then the line BD bisects angle ABC.) Ptolemy then goes on to show that the ratio of the chords is less that that of the angles or CB/BA < arc CB/arc BA.

We need change but what is best?

The parties in Aristotle's Politics, the democrats and the oligarchs, both have the desire to better themselves. This is good in itself but when one side or the other tries to better itself at the expense of the other then there is injustice. There is a need to compromise and the patriots of the American Revolution expressed as a desire to work for the common good.

Those in power need to exercise moderation and balance. One cannot just represent a faction and claim to be working for an entire nation. It is not winner-takes-all. When political parties become too divisive and single minded the political system fails and changes need to be made.

An arbitrary change is not likely to produce the desired effect since economics tries to optimize the overall gain from one's efforts. A small change in any direction reduces the total profits and one ends up worse off than where one started. The change has to go from one global maximum to a better one. The change has to be fundamental and of an evolutionary nature. One needs to be sure about what one is doing. Tinkering with "the system" can lead to catastrophic failures. One needs only to look at what happened to Chernobyl to be reminded of this. Good intentions do not always guarantee good results.

We need to equate success with good conduct. Just trying to tear down a bad system is not enough. We have to replace it with something better and stop repeating the mistakes of the past. This probably will require a better methodology than that currently in use.

An Extract from Book V of Aristotle's Politics

"Democracy, for example, arises out of the notion that those who are equal in any respect are equal in all respects; because men are equally free, they claim to be absolutely equal. Oligarchy is based on the notion that those who are unequal in one respect are in all respects unequal; being unequal, that is, in property, they suppose themselves to be unequal absolutely. The democrats think that as they are equal they ought to be equal in all things; while the oligarchs, under the idea that they are unequal, claim too much, which is one form of inequality. All these forms of government have a kind of justice, but, tried by an absolute standard, they are faulty; and, therefore, both parties, whenever their share in the government does not accord with their preconceived ideas, stir up revolution."

"That a state should be ordered, simply and wholly, according to either kind of equality, is not a good thing; the proof is the fact that such forms of government never last. They are originally based on a mistake, and, as they begin badly, cannot fail to end badly. The inference is that both kinds of equality should be employed; numerical in some cases, and proportionate in others."

"Still democracy appears to be safer and less liable to revolution than oligarchy...And we may further remark that a government which is composed of the middle class more nearly approximates to a democracy than to oligarchy, and is the safest of the imperfect forms of government."

"Inferiors revolt in order that they may be equal, and equals that they may be superior. Such is the state of mind which creates revolutions."

for more see Aristotle, Politics, Book V