# The catenary resists

And now it’s tim for the catenary:

To admire the form of this curve, all we have to do is attach a chain, a string or a wire to two points in a constant gravity field. The wire or similar will adopt the form in which it only supports its own weight and no other additional tension. This is the situation in which it is most at rest, that of minimum rigidity. The urban landscape is full of catenaries, but the most interesting are, without doubt, the inverted catenaries that Gaudí used in many arches. The most notable difference between a Gothic cahedral and the Sagrada Familia in Barcelona is that Gaudí’s church rises to the same height without the need for buttresses. Question: are shells and the skeletons of large animals inverted catenaries?

Another example of the use of the catenary in the architecture is the modernist Masia Freixa (1907-1910) by Lluís Muncunill:

The first man who tried to find the solution to the problem of the hanging chain was probably Leonardo da Vinci (1452-1519), who draw some similar situations in his papers. Galileo Galilei (1564-1642) studied the problem and said that the chain should adopt a parabolic form in his *Discorsi e Dimostrazioni Matematiche, intorno a due nuove scienze attenenti alla meccanica & i movimenti locali *(1638):

SALVIATI: In this case of the rope then, Sagredo, you cease to wonder at the phenomenon because you have its demonstration; but if we consider it with more care we may possibly discover some correspondence between the case of the gun and that of the string. The curvature of the path of the shot fired horizontally appears to result from two forces, one (that of the weapon) drives it horizontally and the other (its own weight) draws it vertically downward. So in stretching the rope you have the force which pulls it horizontally and its own weight which acts downwards. The circumstances in these two cases are, therefore, very similar. If then you attribute to the weight of the rope a power and energy [possanza ed energia] sufficient to oppose and overcome any stretching force, no matter how great, why deny this power to the bullet?Besides I must tell you something which will both surprise and please you, namely, that a cord stretched more or less tightly assumes a curve which closely approximates the parabola. This similarity is clearly seen if you draw a parabolic curve on a vertical plane and then invert it so that the apex will lie at the bottom and the base remain horizontal; for, on hanging a chain below the base, one end attached to each extremity of the base, you will observe that, on slackening the chain more or less, it bends and fits itself to the parabola; and the coincidence is more exact in proportion as the parabola is drawn with less curvature or, so to speak, more stretched; so that using parabolas described with elevations less than 45° the chain fits its parabola almost perfectly.

SAGREDO: Then with a fine chain one would be able to quickly draw many parabolic lines upon a plane surface.

In 1669, the German mathematician Joachim Jungius (1587-1657) demonstrated that the form adopted by the chain wasn’t a parabola and one year later, Jakob Bernoulli (1654-1705) proposed a contest looking for the first mathematician who could find out the real forma of a hanging chain. The problem was solved by Johann Bernoulli (1667-1748), Christiann Huygens (1629-1695) and Gottfried W. Leibnitz (1646-1717) each independently. All three solutions were published in the German *Acta Eruditorum* in 1691. One year later, Johann Bernoulli wrote his *Lectures on the Integral Calculus* compiling his lessons to Guillaume Marquis de L’Hôpital (1661-1704). In this work we can read:

The importance of the problem of the catenary in Geometry can be seen from the three solutions in the

Actaof Leipzig of last year (1691), and especially from the remarks that the renowned Leibniz makes there. The first to consider this curve, which is formed by a free-hanging string, or better, by a thin inelastic chain, was Galileo. He, however, did not fathom its nature; on the contrary, he asserted that it is a parabola, which it certainly is not. Joachim Jungius discovered that it is not a parabola, as Leibniz remarked, through calculation and his many experiments. However, he did not indicate the correct curve for the catenary. The solution to this important problem therefore remained for our time. We present it here together with the calculation, which was not appended to the solution in theActa.There are actually two kinds of catenaries: the common, which is formed by a string or a chain of uniform thickness, or is of uniform weight at all points, and the uncommon, which is formed by a string of non-uniform thickness, which therefore is not of uniform weight at all points, and certainly not uniform in relation to the ordinates of any given curve. Before we set about the solution, we make the following assumptions, which can easily be proven from Statics:

1) The string, rope, or chain, or whatever the curve consists of, will be assumed to be flexible and inelastic at all of its points, that is, it undergoes no stretching as a result of its weight.

2) If the catenary ABC is held fixed at any two points A and C, then the necessary forces at points A and C, are the same as those which support a weight D, that is equal to the weight of the chain ABC and is located at the meeting point of two weightless strings AD and CD, that are tangent to the curve at points A and C. The reason for this is clear.

Because the weight of the chain ABC exerts its action at A and C in one direction at each point, namely, in the directions of the tangents AD and CD, and the pull of the same or equal weight D at A and C likewise goes in the directions of AD and CD. Therefore the necessary forces at points A and C must also in both cases be the same. Accordingly, one obtains the necessary force at the lowest point B, when one seeks the force that the weight E [Figure 2] exerts at the same point, when it is held by two weightless strings, one of which is tangent to the curve at B, and therefore is horizontal, while the other is tangent curve at B, and therefore is horizontal, while the other is tangent to the curve at point A:

3) When a chain fastened at points A and C is then fastened at any other point F, so that one could remove the portion AF, the curve represented by the remaining piece of chain FBC does not change, that is, the remaining points will stay in the same position as before the fastening at F. This needs no proof, because Reason advises it and experience lays it daily before our eyes.

4) If we retain the previous assumptions, then before and after the fastening at F, the same (that is, the original) force must obtain at particular positions on the curve, or, what amounts to the same thing, a point will be pulled with the same force after the fastening [at F] as before it. This is nothing but a corollary of the preceding number. Consequently, as one lengthens or shortens the chain BFA, that is, wherever one chooses the fastening point F, the force at the lowest position B neither increases nor decreases, but always remains the same.

5) The weight P, which is held by any two arbitrarily situated strings AB and CB, exerts its forces on the points A and C in such a relation, that the necessary force at A is to the necessary force at C (after drawing vertical line BG), as the sine of angle CBG is to the sine of angle ABG, and the force of the weight P is to the force at C as the sine of the whole angle ABC is to the sine of the opposite angle ABG. This is proven in every theory of Statics.

With these assumptions, we find the common catenary curve in the following manner. Let BAa be the desired curve; B, its deepest point; the axis or the vertical through B, BG; the tangent at the deepest point, which will be horizontal, BE; and let AE be the tangent at any other point A. Draw the ordinate AG and the parallel EL to the axis:

Let x = BG, y = GA, Gg = dx and dy = Ha, and the weight of the chain, or, since it is of uniform thickness, the length of the curve BA = s. Since at point B, an ever constant force will be required (by assumption 4), whether the chain be lengthened or shortened, that force, or the segment C = a expressing it, will therefore be a constant.3 Imagine now that the weight of the chain AB is concentrated at and hangs at the meeting point E of the tangent strings AE, BE; then (by assumption 2) the same force is required at point B to hold the weight E as was required to hold the chain BA. But the weight E (by assumption 5) is to the force at B, as the sine of the angle AEB, or as the sine of its complementary angle EAL is to the sine of angle AEL, that is, as EL is to AL. Wherever on the curve one chooses the fixed point A (the curve always remains the same, by assumption 3), the weight of the chain AB is to the force at B [which force equals the constant a], as EL is to AL, that is:

s : a = EL : AL = AH : Ha = dx : dy or dy : dx = a : s

Hence it follows that the catenary BA is the same as that curve whose construction and nature we have given above, by the method of inverse tangents, where we first converted the proportion dy : dx = a : s to the following:

dy= a · dx / √(2ax + x

^{2})at which point the curve was constructed through the rectification of the parabola as well as through the quadrature of the hyperbola.

Then, Johann Bernoulli explained how to find dy:

To find the nature of the curve so created that DC : BC = E : AD:

Let AC = x, CD = y, AD = s. By assumption, dy : dx = a : s [we have a constant segment E = a]. Therefore: dy = a dx : s.

However, to be able to eliminate the letter s (which is always necessary in the determination of curves), one must proceed thus:

dy

^{2}= a^{2}· dx^{2}: s^{2}Therefore:

ds

^{2 }= dx^{2 }+ dy^{2}= (s^{2}· dx^{2}+ a^{2}· dx^{2}): s^{2 }⇒ ds = [dx · √(s^{2}+ a^{2})] : sTherefore:

dx = (s · ds) : √(s

^{2}+ a^{2})and the integral thereof:

x = √(s

^{2}+ a^{2})From this is obtained s = √(x

^{2}– a^{2}) and ds = (x · dx) : √(x^{2}– a^{2}) = √(dx^{2}– dy^{2}). If the equation is simplified one obtains:x

^{2}dx^{2}– a^{2}dy^{2}= a^{2}dx^{2}And finally: dy = (a · dx) : √(x

^{2}– a^{2}).

We can say that Johann Bernoulli calculated the equation of the catenary in a such original way. Nowadays, we write:

*y* =* a* cosh (*x/a*) =* a* (*e ^{x/a} + e^{-x/a}*)/2

I think that that’s enough for today. Thanks Leibnitz, thanks Johann and Jakob Bernoulli and thanks to all the men who made possible this beautiful curve!

# A Roman abacus in London

This is a replica of an ancient Roman abacus which we can find in the great Science Museum of London (there are some abacus like this in other museums as the British Museum or the National Library of Paris). It consists in a small metallic plate with nine parallel slits: the right slot is related with ounces and next to it, from right to left, the other slots are for units, tens, hundreds, thousands units, thousand tens, thousand hundreds and millions, with its corresponding figures:

The seven left slots are divided in two different sections: the upper one have one moving pieces and the lower have four moving pieces. The Roman represented the units of each power of 10 putting the lower pieces near the centre of the abacus if they were less than 5. When they need to represent 5 units, they only moved one upper piece to the centre. Thus, number 6 was one upper piece and one lower piece in the centre, 7 was one upper piece and two lower ones,… For example, the abacus located in the British Museum represents number 7.656.877 (or a similar number):

The two first right slots corresponded to the ounces: the Roman unit (= *as*) were divided in 12 equal ounces. The slot marked with the symbol O had five moving pieces in the lower slot and it was used to count the multiples of 11 and 12 ounces. The first slot was divided in three parts with four moving pieces:

If the upper piece was next to the “pound” symbol (in the left), the piece value was 1/2 ounce or 1/24 of *as* (= *semuncia*); if it was next to the symbol in the middle of the slot (= *sicilius*), the piece value was 1/4 ounce or 1/48 of *as*; and if it was next to the “number 2” (in the right), the piece value was 1/72 of *as* (=* duella*).

This kind of abacus were very popular as small fast calculators among the Romans as we can see in a marble sarcophagus in the Capitoline Museums of Rome: there is a young boy standing at his feet holding an abacus (he is probably counting the money which the deceased is holding in a money purse):

Location: Science Museum at London (map)

# The spiral packs

The sphere isn’t the only figure exhibited in Cosmocaixa. “The spiral packs” and we find it in the nature and in our quotidian life:

The spiral is a circumference that twists away on the plane that contains it. It is the best way of growing without occupying too much space. It is frequently found in animals when there exists the contradictory need for something massive, voluminous, broad or long that does not affect mobility (horns, tails, tongues, trunks, shells, etc.) and in plants to grow something that will subsequently be unrolled. If we unrolled all the spirals we have at home (kitchen and toilet paper, audio and video tapes, adhesive tape, records, springs,…) we would be forced to leave the house, as we would not all fit.

One of the best examples of a spiral in the nature is the the Nautilus shell:

Apart of some different shells, we also find vegetables and fossil:

Are there more shapes to study?

# A Latvian Möbius strip

When you go to Riga from the airport by car, you go through Kārļa Ulmaņa gatve and at the beginning of Lielirbes iela you can see these three Möbius strips. The Möbius strip was discovered independently by Johann Benedict Listingand (1808–1882) and August Ferdinand Möbius (1790-1868) in 1858 and it’s a very curious non-orientable surface. It’s a good welcome sculpture to the city!

# The Sphere protects

We can define the three-dimensional sphere as the set *S*^{2} = {(*x*,*y*,*z*) ∈ ℝ: *x*^{2}* + y*^{2}* + z*^{2}* = R*^{2}}. It’s one of the most famous shapes studied in schools and we find it in the nature very easily. In the Cosmocaixa of Barcelona there is a special exhibition about shapes and the sphere is one of the main protagonists of it. The sphere protects and the reason hat we can read in one of the explanations is:

In the inert world, the sphere emerges easily in isotropic conditions, that is, when no one direction in space has priority over another. That is why stars and planets are round. That is why, too, if we blow air into a liquid, the bubble adopts this form, the minimum that contains a given volume. Natural selection favours circular symmetry in living beings (eggs, medusas, sea urchins, fruit, seeds, etc.) for two reasons: a) because it occupies the minimum surface, the sphere is the frontier with the exterior that loses heat most slowly and b) because, not having any edges, it is the most difficult shape to catch or bite. In both cases, then, the sphere protects.

We can also see different examples of this fact like these four examples: 1) These radial aggregates of pyrite crystals from Peru:

2) These Marcasite nodules (FeS_{2}):

3) These Dinosaur eggs (*Titanosauridae*) from China:

4) This colonial coral (*Scleractinia*) from the Pacific Ocean:

The exhibition is very interesting and it tries to answer the question “Why are some forms more common than others?”:

What do a star like our Sun, a planet, a fish egg, an orange, a bubble and the point of a pen have in common? All these objects, inert, living or manmade, share the same form: they are spheres. Why are there so many spheres, circles and circumferences? Does the fact that something is spherical help in some way? Spheres are found more frequently than other forms. What are the forms that we are most likely to find in nature? Does being circular, spiral, hexagonal or fractal in shape serve any prpose? Do objects with the same form share anything else apart from their shape? Perhaps they share the same function, that is to say, the property that helps the object in question, wether inert, living or manmade, to persist in nature.

As you can imagine, I must talk about spirals, hexagons,… but I’ll do it in next posts!

**Location**: CosmoCaixa at Barcelona (map)

# The Tro-Cortesianus Codex

In 1880, Léon de Rosny figured out that the two extant parts of the Tro-Cortesianus Codex (also know as the Madrid Codex) were a single codex and Spaniards were very lucky to have one of the most important codices of the Maya civilization because it’s the longest of the surviving Maya codices. The Museo Arqueologico Nacional acquired it from book-collector José Ignacio Miró in 1872 who had purchased it in the Spanish region called Extremadura a few years before. The director of the Museum decided to name the Cortesianus Codex after Hernán Cortés (Medellín, 1485 – Castilleja de la Cuesta, 1547) supposing that he had brought the codex to Extremadura.

The Codex is dated in the period 1250-1500 and it consists in 56 sheets painted on both sides to produce a total of 112 pages. Each page measures roughly 23.2 by 12.2 centimetres (9.1 by 4.8 in). The content consists of horoscopes, astronomical tables and almanacs used by the priests in the performance of their ceremonies and rituals. The Mayans reached a precise idea of the movements of the Sun, the Moon and the planets and they estimated at 584 days the synodic revolution of Venus (the cycle is really equal to 583,92 days). They also realized that the solar year consisted of approximately 365,242 days. So the synchronization error between the Venus cycle, the solar year and the liturgical year was only one day every 6,000 years!

We can also see a lot Mayan numbers so we must notice how Mayans wrote their numbers. Mayan numbers consist in series of points and horizontal (or vertical) lines. Each line represent 5 and each point represents 1. For example, in the next page:

The page is divided in four images and in the lower one we can see numbers 3, 7, 11, 8, 6, 8, 13 and 1. Now you can explore the other parts and say which numbers are in each one!

Finally, we can talk about the Mayan calendar from two pages of the codex and the representation of some Mayan numbers and the famous glyphs. We can see in the middle of the picture a torch inside a square decorated with the 20 glyphs corresponding to the fundamental set of 20 successive days:* Imix, Kimi, Chuwen, Kib, Ik, Manik, Eb, Kaban, Akbal, Lamat, Ben, Etsnab, Kan, Muluc, Ix, Kawak, Chikchan, Ok, Men* and *Ahaw.*

Each day was represented by a different glyph and they were related to gods and sacred animals and objects: *Imix* was related to the crocodile and the nenuphar, *Kimi* to the Death’s divinity,… :

In the liturgical calendar, the 20 days used to be also related to the 13 first natural numbers in a cyclical period. Firstly, the first day was related to 1, the second to 2, the third to 3,… and the thirteenth to 13. Then, the fourteenth was related to 1 again, the fifteenth to 2,… and the twentieth to 7. The first day of the second cycle was related to 8 and so on. After 13 complete cycles (= 13 · 20 days = 260 days) the first day of the cycle was related to 1 again.

Outside the square, the upper section of the picture represents the East (*Lakin*), the right part is the North (*Xaman*), the left part is the South (*Nokhol*) and the lower part is the West (*Chikin*). We can see perfectly five pairs of glyphs representing the days in each of the four corners and there also are 20 sets of 13 points representing the different cycles of the Mayan calendar. All the glyphs have the number 1 or the number 13 with them. So we are reading a Mayan liturgical calendar through the 260 points representing the 260 days!

Location: Museo de América at Madrid (map)

# I Saw the Figure 5 in Gold

The American poet and pediatrician William Carlos William wrote the descriptive poem “The Great Figure” in 32 words in 13 verses:

Among the rain

and lights

I saw the figure 5

in gold

on a red

fire truck

moving

tense

unheeded

to gong clangs

siren howls

and wheels rumbling

through the dark city.

The poem describes a red firetruck with the number 5 and it consists in one large sentence distributed in 13 short lines. Probably, William Carlos Williams saw this red firetruck with its great number 5 painted in it in a few seconds among the rain and this image inspired him to combine the words in this odd poem (there is a complete analysis of “The Great Figure” in this web).

Between 1005 and 1908, the versatile painter Charles Demuth met Williams in Philadelphia and twenty years later (1928), he dedicated one of his eight abstract portraits of his friends to Williams. This portrait entitled “I Saw the Figure 5 in Gold” pays homepage to William’s poem associating an accumulation of images related to him. For example, the names Bill and Carlos and the initials WCW are clearly distinguished. We can also see a big golden 5 in front of another 5 and red rectangles in the middle of a mess of lights and chaotic grey colors. I think that it’s easy to imagine the big 5 painted in the firetruck, isn’t it?

**Location**: The Metropolitan Museum of New York at New York (map)

# A Parisian Arithmetic

This medieval Flemish tapestry is entitled *L’Arithmétique *and it is part of an anonymous series on the theme of the seven liberal arts. The composition is organized around a young woman standing behind a table. This woman is teaching some traders and bankers how to manage with the Arabic figures. The inscription says:

Monstrat ars numeri que virtus possit habere. / Explico per que sit proportio rerum.

I should look for the Geometry and the Astronomy to complete this allegorical series, don’t I?

**Location**: Musée of Cluny at Paris (map)

# A Runic calendar in Tallinn

A Runic calendar is a perpetual calendar based in the 19 year Metonic cycle of the Moon. The Greek astronomer Meton of Athens (Vth century BC) observed that a period of 19 years was equal to 235 synodic months and 6.940 days which is almost equal to 19 solar years except for a few hours. This cycle was used in the Babylonian calendar and Meton computed all the necessary parameters and the intercallary months to adjust the periods of the Sun and the Moon.

Runic calendars were written on parchment or carved onto staves of wood (as the one of the Estonian History Museum), horn or bone. It appears to be a medieval Swedish invention and the Nyköping staff, believed to date from the 13th century, is the oldest one which is preserved. The Runic calendar preserved in the Museum is dated in 1819 and its first line is made up of the first seven letters of the Runic alphabet (runes). 52 weeks of 7 days were laid out using 52 repetitions of this first seven runes and each rune corresponded to each weekday varied from year to year. On another line, many of the days were marked with one of the 19 symbols representing the 19 possible positions of a year in the Metonic cycle (called “Golden Numbers”).

This kind of calendars were used until mid-19th century.

# Estonian History Museum

The Estonian History Museum is one of the most interesting attractions of Tallinn. We can read in the official web site that the story of the Museum began in 1802, when Tallinn’s town hall pharmacist, Johann Burchard** **(1776–1838), started a collection called *Mon Faible* (“My weakness”). The first exhibit was a Chinese opium pipe and Burchard put on the exhibition “Antiquities and rarities” at the House of the Brotherhood of the Blackheads in 1822. Twenty years later, the Estonian Literature Society was founded in Tallinn and one of its aims was to establish a museum “to broaden our knowledge of this country by studying its history, art, manufacturing, technology and nature”. Extensive collections were compiled over the following twenty years, which formed the basis of the Provincial Museum of the Estonian Literature Society, founded in 1864 at the house of St. Canute’s Guild.

In 1911, the Estonian Literature Society purchased premises on Toompea at 6 Kohtu Street, where the museum’s innovative activities could flourish. As the only museum in the city, it became an important focal point in Tallinn’s cultural life with educational lectures and exhibitions. The Museum retained its important position through its valuable collections although the Estonian National Museum in Tartu (founded in 1909) became the most important museum in the Republic of Estonia. In 1940 Estonia was incorporated to the U.S.S.R. and The museum was nationalized and the History Museum of the Estonian Soviet Socialist Republic was established in its place. Overbearing ideological pressure ruined the museum in the following years. In addition to subjugating the museum employees, items that were deemed harmful were eliminated, which meant destroying everything that reminded people of the republic of Estonia. The Museum preserved the greater part of the main collection and it moved to its current location at the Great Guild Hall in 1952. Finally, in 1989 the Museum was renamed the Estonian History Museum and many important exhibitions that introduced the contemporary history of Estonia were held in the late 1980s and early 1990s.

Estonia is a very young country and in the main exhibition we find a lot of references to its former background and to the Soviet occupation. Among all these interesting object there also are some references to mathematical objects. For example, we can see different scholar material as two geometrical figures…

and a mathematical ruler:

The this section of the exhibition about the schools we can read that the first educated Estonians were said to be monk Nicolaus, who was appointed by the Pope to carry out missionary work in Estonia in 1170, and the parish priest Johannes, who worked with the Livonians at the beginning of the 13th century. In the mid-13th century cathedral schools were established in Tartu, Pärnu, Tallinn and Haapsalu. Next to the monasteries, there were monastic schools, and in the 15th and 16th centuries, town schools were founded. The 17th century was of great importance to Estonian cultural history because it was when that high schools, along with print shops, were first founded in the country. The University of Tartu was founded in 1632. In the 1680s a network of village schools was created but regular school education in rural areas only took hold in the early 19th century. Peasant schools were mandatory, and students were taught in the native language; however, most of the children were allowed to study at home. Writing and arithmetic were taught starting in the mid-19th century. The 1897 census revealed that 94% of Estonians were able to read, which was the highest percentage among the nations of the Russian Empire. The Imperial University of Tartu reopened in 1802 and became a centre of science and intellectuals in the Baltic provinces. In 1803 the university employed an Estonian and a Latvian-language lecturer. On December 1, 1919, the University of Tartu of the Republic of Estonia was opened, marking the beginning of regular higher education in the Estonian language.

Another interesting object is in the lower lever: it’s an abacus which was taken from Ivan Mazepa’s (1639-1709) tent after he Battle of Poltava in 1709. The wooden box has a framed mirror and an abacus with beads made of white and red bone. Mazepa wanted to unite the Ukranian terriories into a single state. During the Great Northern War he initially supported Russia but later changed sides and lost the Battle of Poltava alongside Swedish troops.

Finally, there is a three-dimensional wooden puzzle made in 1920s or 1930s. It consists in 16 pieces. Different wooden puzzles were made in Estonia already in the 19th century:

As we can see, the Estonian History Museum is very interesting for a Mathematical lover. Furthermore, there is another object which must be mentioned but I’m going to talk about it in the next post.

**Location**: Estonian History Museum at Tallinn (map)