IN PROGRESS: Issues raised by Cusanus‘ De quadratura circuli in Codex Yale 24

 

Among the thousands of volumes held and once held, at the Baroque library in the Bendictine monastery at Melk, is a bound assortment of manuscripts which is now in New Haven at Yale University. This transfer was the result monasteries in Austria being caught between diminishing financial returns on the land they owned and rising costs. Between the two world wars, many were forced to sell precious manuscripts from their libraries to dealers who then sold them on to libraries and museums in the United States. In 1937, the codex known as Codex Melk 794, was acquired by the neurosurgeon and book-collector, Harvey Cushing and in the same year, was bequeathed by him, along with the rest of his collection to Yale University. Now known as Codex Yale 24, the volume is a part of the Harvey Cushing/John Hay Whitney Medical Library and may be seen online at: https://findit.library.yale.edu.

Consisting of an assortment of manuscripts that date from the Renaissance, the texts are mostly concerned with astronomy, the calendar and on how to make and use instruments such as the quadrant and the astrolabe. All this fits in with the monastery at Melk having had connections both to the University of Vienna and to the First Viennese School of Astronomers and Mathematicians that included such high-calibre astronomers and mathematicians as Georg von Peuerbach and Regiomontanus whose works are among the forgotten gems of Codex Yale 24. One such is an illustration of the classification of comets drawn up by Peuerbach: https://findit.library.yale.edu.

Between 1453 and 1460 the monks at Melk were also following an exchange of letters and articles known as the Controversy of the Doctrine of Ignorance, in which Nicholas of Cues, otherwise known as Cusanus (1401-1464), played a prominent part. Although the theological aspects of the debate are well-known, there was also a mathematical aspect to the controversy and the presence of Cusanus‘ Quadratura circuli, or On Squaring Circles, among the manuscripts of Yale 24 shows that the monks at Melk followed the debate both theologically and mathematically. On Squaring Circles dates from 1453 and the Yale version is one of only two manuscript copies to have survived: https://findit.library.yale.edu.

In mathematics, Cusanus is important as he was the first to realise that π is neither a rational number, nor is it an irrational number but rather, is what mathematicians call a „transcendental number“. Associated with irrational numbers are infintessimals. This is because between any enumeration of an irrational number or a transcendental number and its approximation, there is a difference that as the accuracy of the approximation increases becomes infintessimal. Infintessimals were first discovered by Zeno of Elea who found them so contradictory that he used their existence to argue that change and movement in the universe were impossible. Infintessimals involve acknowledging that between any two points on a line or between any two numbers in a series, an infinite number of divisions can be made. Following the discovery and acceptance of irrational numbers, the infitessimals that are associated with them were however seen with suspicion as there was reluctance to postulate that the universe was inherently infinite. Yet Cusanus saw things differently and argued that the infinite nature of the universe was a reflection of the infinite nature of God. In his Doctra ignoranti of 1440, Cusanus developed a form of speculative theology based on the general characterists of numbers and geometrical figures.

In Quadratura circuli, Cusanus was concerned with developing a geometrical method by which the circle could be approached by means of straight lines that denoted margins within which π lay. Two hundred years before Leibniz and Newton, he was trying to develop a mathematics that via successive increments, could reach into and partially quantify the infinite and transcendent nature of π and the circle. Translated into English by Alexander Curtis, the text shows how Cusanus was concerned with a rigorously demonstrating how a „geometry of the infinite“ could be derived from conventional Euclidian geometry. Masterfully elegant, from a line, a,b, Cusanus constructs an equilateral triangle whose sides added together give the length of a,b. He then goes on to construct a square whose sides are the length of a,b divided into four. Comparing these figures and the circles contained within them, Cusanus postultes a circle from whose radius π may be extracted.

Cusanus saw that for each regular poygon there is, in addition to an enclosing circle and an enclosed circle, a third circle whose circumference is equal to the perimeter of the polygon. As regular polygons acquire ever more sides, so both the enclosed and the enclosing circles approach the dimensions of this third circle. In the case of a polygon with an infinite number of sides, the polygon is no longer a polygon but rather has become this circle. Thus P → C. As C = 2πr, so π = C/2r and if r = 1, then π = C/2. On the diagram below, the large circle has a radius twice that of the smaller circle and as radius and circumference are linearly proportional, the circumference of the larger circle is also twice that of the smaller circle. Taking the radii  found, dividing by 2 and putting the results into the diagram as the length of C/2 would thus enable π to be extracted. The problem is that although r = 1, all the other lengths involve π, which however is precisely what is being sought. With r = 1, the formula πr2 means that the area of the smaller circle is π2 with the sides of the square that squares it being √π. The square is defined by the semi-circle of radius 2r which passes through the corner of the square and designating its corner also indicates the end of the 1 x π rectangle that connects the square and the first of the small circles.

As in the diagram above, a,b is the length of the perimiter of the successive polygons generated, the length of the circumference of the final circle and of the larger half circle and has a length of 2πr, one can say that π = a,b/2r. The problem is that although, starting from initial radii of 7 and 14, it is not known by what factor the radius arrived at is to be reduced, so that again one finds oneself invoking something that is unknown. If however one takes the lower radius used of 7 and inserts it into π = a,b/2r and rearranges, one arrives at π = r/3.5. This rule of thumb is then improved upon by Cuanus, who introduces a progression into the diagram above that relates r, √π, π and 2πr and results in the formula π = r/3.555… This is achived via the geometric mean, 16/9, which relates the numbers to one another as the terms of a successive series. In the case of the first two terms, the relation is only approximate but in the case of 2πr:π, it is much better. Dividing 11.18 and 11.78, which are the maximum and minimum radii arrived at, first by 2 and then by 16/9 results in 3.194 ˂ π ˂ 3.36.  Although this is incorrect and even the lower limit lies outside the limits given by Archimedes, this is not the point, as Cusanus‘ main purpose was to establish from first principles, a general method by which ever improved results could then be generated. As a Neo-Platonist, Cusanus saw the transcendent nature of the circle and equilateral triangle as something that pointed, all be it indirectly, towards the divine nature of God. The geometry he developed in Quaradtura circuli was thus not only a method by which ever improved approximations to π could be arrived at, it was also intended as a means by which the divine and transcendental nature of God could be stumbled upon, apprehended and marvelled at.

The Yale maunscript includes a figure that does not occur in the principes. This shows the radii and chords of the triangles associated with the equilateral triangle and its enclosed and enclosíng circles. For the reader, the production of numerous such diagrams are part and parcel of understanding a Renaissance text on geometry, as steps are frequently left out and there is often much that is not explained. This the reader is left to ponder and think through. While this makes for slow reading, once an issue has been grasped, there is a deeper understanding that is not easily forgotten. See: https://findit.library.yale.edu.

Four years later, in his A Dialogue on Squaring Circles of 1457, Cusanus arrived at an approximation to π that was an improvement on Archimedes‘ approximation of π > 3 1/7 and < 3 10/71. As in Quaradtura circuli however, he made the mistake of assuming that as polygons acquire ever more sides, they approach the circularity of a circle in a linear fashion, when in fact they follow a curve that leads to a line that is almost straight.

Rectifying this oversight involves utilising the differential calculus developed by Leibniz and Newton and the technique they developed can be seen as continuing the task begun by Cusanus, with algebraic techniques succeeding where Cusanus‘ geometry stalled. Yet even here there are discrepancies, for the sides of the regular solids, whilst they follow the curve shown above, do not do so exactly.

The why’s and wherefore’s of this are currently being worked on by Alexander Curtis.

When Cusanus sent a copy of A Dialogue on Squaring Circles to Peuerbach, the latter, not sure what to make of it, sent it on to Regiomontanus. A third text, On a method of measuring curves with rectangles, was written by Cusanus in 1462. In 1463, in Venice, Regiomontanus finally got around to formulating a response to Cusanus work. Dying unexpectedly three years later, his responses were only published posthumously in 1533, as an appendix to a work on spherical trigonometry. Here the whole debate is presented, with Regiomontanus‘ responses being preceded by the texts that had provoked them. Both Cusanus‘ later texts are awaiting translation by Alexander Curtis. For two articles on Cusanus and the discovery of the infinite see: https://intergralscience.wordpress.com and  https://archive.schillerinsitiute.com. In the second article, an explaination is given on how Cuanus arrived at the value for π given in his A Dialogue on Squaring Circles. 

SOON TO BE PUBLISHED: An Article on Art and Mathematics in Krems and Borgo Sansepolcro

 

In Krems at Obere Landstrasse 5, a Late Gothic doorway forms the entrance to the Church of the Burgher Hospice. Behind the church, there lies the complex of the hospice itself. At the beginning of the sixteenth century, a sculpture was commissioned showing Saint Vitus, the patron saint of apothacaries and dancers who was accordingly also invoked in cases of dancing mania, epilepsy and fever. Carved by an anonymous master and dated to 1520, the sculpture is a masterpiece of Northern Renaissance art. According to Christian tradition, Saint Vitus was martyred as a young man by being placed in a vat of boiling oil and the sculpture made for the Citizens‘ Hospice shows the saint praying to God for strength that he might withstand the torment to come. His prayers answered, Saint Vitus was saved from his fate by a host of angels who, descending down from on high, rescued him and carried him up to heaven.

In an article in the final stages of completion, Alexander Curtis shows that the work draws upon a geometrical construction that occurs in the mathematical writings of Piero della Francesca. This leads to the Tuscan town of Borgo Sansepulcro and to Piero della Francesca’s The Resurrection, described by Aldous Huxley as „the best picture the world“.

Not only a brilliant painter, Piero delly Francesca was also an outstanding mathematician and wrote two books on mathemtics and geometry, Trattato d’Abaco and Libellus de quinque corporibus regularibus. In between he wrote De Prospectiva Pingendi, in which he established projective geometry as a completely mathematical discipline, ruthlessly worked out from first principles. As Alexander Curtis shows, there is a connection between the Saint Vitus sculpture and Piero’s Resurrection and his mathematical work and this all comes together in an extremely accurate, yet hands-on method of squaring the circle.

Unlike his Trattato d’Aboco which consists of arithmetical and geometric exercises, Piero’s Libellus is devoted solely to geometry and to fully substantiate his article is currently being translated into English by Alexander Curtis. In Book I, Piero covers plane geometry, in Book II, the Platonic Solids enclosed in spheres are examined. Book III features Piero the Platonic Solids as they occur, enclosed not in spheres but in each other.

In Book IV Piero turns his attention to semi-regular and irregular solids. The semi-regular solids were first investigated by Archimedes, only by the Renaissance, all that remained of his treatise was description of the thirteen solids with nothing being known how they were to be constructed. In all, Piero succeded in reconstructing six of the lost Archimedian Solids, with five being featured in the Libellus.

After the Archimedian Solids a number of irregular and distorted solids are examined such as a truncated pyramid, a cylinder that has been hollowed out and a sphere from which a cylindrical core has been removed.

These innocent Cinquecento investigations anticipate other forms, some of whose volumes and areas were only quantified following the invention of differential calculus two hundred years later.

Although these forms look innocuous enough, from them, other more sinister forms may be derived.

During the Second World War, both Piero’s The Resurrection and the Saint Vitus piece in Krems came close to being destroyed by such devices. In the case of Piero’s The Resurrection, this was only saved from being shelled to pieces due to the insubordination of the British artillery officer, Anthony Clarke, who ignored orders to bomb Sansepolcro as some years before, in a travel essay he had read how Aldous Huxley had praised the painting in no uncertain terms. Using shells such as depicted above centre and left, the bombing of Sansepolcro was intended to „soften up“ German resistance and when Clarke’s commanding officer established radio contact, the insurbordinate captain played for time and said that he could see no obvious targets. This saved the day, as it was soon established that the retreating Germans had already left Sansepolcro and had he obeyed orders, Clarke would have been shelling an abandoned town.

Meanwhile in Krems, the Saint Vitus sculpture and two paintings from the Renaissance had been approrited by Hiltler for his art museum in Linz and along with thousands of other art works from all over Europe, were stored in a salt mine at Altaussee. In a specially adapted gallery that can still be seen, Saint Vitus shared a space with Michaelangelo’s Brugges Madonna, Leonardo da Vinci’s Leda and the Swan and Vermeer’s The Artist in his Studio. During the last days of the war, a further eight boxes, branded „Careful, marble, do not drop“, were delivered to the mine. However, far from containing freshly looted works art, the newly delivered crates contained 1,100 pound bombs of the type shown left in the illusration above, even though these being bombs that were ususally dropped from the air.

With these, the fanatical governor of the region intended to play his part in implementing the Führer’s infamous Nero Decree, according to which anything of value to the allies was to be destroyed. Although Hilter’s deputy sent a communication to say that this did not apply to the works of art at Altaussee, the governor reused to accept this, as the order had not been signed by Hitler personally. Citing an example from the First World War, the director of the mine said that in order to do the job properly, they would have to seal the entrance to the mine and as time was running out, recommended that they do this by blasting with smaller charges and then using fuses to detonate the 1,100 pound bombs inside. To this the governor agreed, not realising that it was a bluff with which the manager intended to save the priceless collection of art from destruction. Secretly, a team was assembled and overnight the bombs were removed. A few days later, the entrance to the mine was detonated and the collection saved. When the governor found out he was furious and ordered all involved to be shot but by then it was too late, for commands were no longer being obeyed.

To see where the Saint Vitus sculpture was stored during the last years of the war go to: www.salzwelten.at. As hostilities came to an end ceased the collection was saved from further mis-appropriation through the efforts of „The Monuments Men“ an allied team of art historians who tirelessly worked to restore works of art to their rightful owners. Moved to a central depot in Munich, the Krems works of art were returned through the efforts of Theresia Mahrer, the Krems town councilor responsibble for culture and education. For articles on the gripping stories surrounding the Nazi’s art plundering operations and the restoration of thousands of works of art see: stillehelden-salzkammergut.at and www.smithsonianmag.com or read Lynn Nicholas‘ book, The Rape of Europa, or the later, The Monuments Men, by Robert Edsel, which was the basis for the 2014 film of the same name and which features a star-studded cast.

AN INTRODUCTION AND A RECCOMENDATION: The Planispheric Astrolabe and David King’s, Astrolabes and Angels, Epigrams and Enigmas

 

At Melk, the monastery owns two astrolabes (not on display) which while they may have been made elsewhere, could equally have been made by a monk as within the pages of Yale 24, in a text written by Christiannus de Prachatitz, all the information necessary for the construction of an astrolabe is given. This is given on pages 371-392 of the manuscript which correspond to 381-392 of the digital page finder. See: https://findit.library.yale.edu.

An astrolabe made by Alexander Curtis

In the stereographic projection on which the planespheric astrolabe is based, lines are projected from a point on the globe conceived of as a sphere where an observer is seen as standing and are followed through the sphere to a plane on the other side that is parallel with the sphere’s equator. On historical astrolabes this is the South Pole and the projection accordingly shows the North Pole as a point at the centre of the projection.

Meanwhile the South Pole, is flung out in all directions to infinity. While the great circles of longitude become straight lines that extend out from the North Pole in all directions, the circles of latitude that on the globe lie in-between the North and South Poles become circles, separated by ever increasing distances as the projection proceeds outwards. This first projection is an objective, global projection which is then juxtaposed and combined with a second system of projection that reflects the observer’s position on Earth. This second projection results in one line of longitude becoming a straight line whilst the rest become the segments of circles with large radii. The observer’s equivalent to lines of latitude are called „lines of altitude“ by astronomers and likewise become circles, a significant number of which are however shown in their entirety. Despite the distortions that occur as one moves out away from the centre of the projection, angles remain undistorted and it is this that enables the instrument to be used as a computing device that calculates angles and the differences between angles. To the see Prachatitz’s construction along with the resulting circles of latitude and altitude see: https://findit.library.yale.edu.

In Prachatitz’s text, following Medieval practice, the construction is conducted vertically, yet as the Patriarch of Aquileia, Danielle Barbaro, realised, this obscures an understanding of what is going on and so, in his masterfully instructive and wonderfully comprehensive work, The Practice of Perspective, the construction is presented horizontally.

From Danielle Barbaro’s La pratica della perspettiva of 1569, page 171. Within triangle •,A,B, the three dotted lines show the observer’s horizon (labelled as „Eorizon“), the vertical axis, marked „α“ and the line that culminates in the observer’s zenith which is marked „z“ and is labelled „Zenith“

In Barbaro’s diagram the globe is shown with the circles of altitude, a, b, c, et cetera, for an observer at a position 22° North of the Equator. These are then projected onto a line that includes the base of the triangle the •,A,B and result in points A, B, C, et cetera. As can be seen, •,A,B is a right-angled triangle, with angle •,A,B being 22° and angle •,B,A being 68°. The inclination of the 90° angle at • is thus that which indicates how the 180° of altitude are to be distributed. For a position 0° North of the Equator the right-angled triangle is symetrical and is tilted neither one way nor the other. This means that the triangle •,A,B spans is divided in half by the line marked „α“, that on the diagram lies between points M and K. Considering this shows how the so-called „tangent scale“ is intergral to the projection. On the tangent scale, the 180° that lie between any two diametrically opposed points on the observer’s horizon embraces are condensed into 90°. To construct the circles of altitude, the distance between A and B is divided in two and the point arrived at is used as the centre of a circle whose circumference passes through A and B. This circle defines the observer’s horizon. Smaller circles of altitude are arrived at by successively adding the desired number of degrees to 22 on South side and subtracting from 22 on the North side. As before, the distance between pairs then gives the diameters of the ever decreasing circles of altitude. Thus A is paired with B, C with D, E with F et cetera with J being missed out so that the pairing ends with R and S, between which there is Z. This latter is the observer’s zenith and is the point that lies directly above an observer and is found by subtracting the observer’s latitude from 90°. In this way, the Arctic Circle, the Tropic of Cancer, the Equator and the Tropic of Capricorn along with the observer’s circles of altitude and horizon can be constructed.

From Danielle Barbaro’s La pratica della perspettiva of 1569, p. 172

In his Planispherium, Ptolemy recommends terminating a stereographic projection at the largest invisible circle which on a projection of the Northern Hemisphere is the Antartic Circle. This however is impractical and all known astrolabes terminate the projection at the Tropic of Capricorn. With reference to horology, Ptolemy’s Organum is briefly discussed by two anonymus authors on manuscript pages 267-272 and 446-448 of Codex Yale 24: https://findit.library.yale.edu and: https://findit.library.yale.edu. A reconstruction by Alexander Curtis of the Organum described by Synesius of Cyrene, shows how the useful part of the plate shrinks in relation to the rest and how the constellations become progressively distorted as one moves away from the centre.

The observer’s equivalent of lines of longitude are called by dialists „gores of azimuth“ and are what astronomers call, „lines of ascension“. On the diagram below, the lines of ascension are shown for an observer 22° North of the Equator. For Prachatitz’s version of the same see: https://findit.library.yale.edu.

From Danielle Barbaro’s La pratica della perspettiva of 1569, p. 174

The lines of ascension are arrived at by finding the centre of the arc that links the points where the Equator crosses the observer’s horizon. These points represent East (left) and West (right) and through them a circle is drawn. This is then divided into eighteen equal segments and a line is drawn from the centre that is parallel to the East-West axis. Using one half of the circle, lines are drawn that connect the zenith with the nine divisions of the circle. The points where these lines cross the line that runs through the circle’s centre are marked off and are then used as the centres of circles that pass through the zenith.

Finally there is the rete, which is an objective projection of the positions of the stars as seen from the pole of the observer. As the stars appear to move over the observer’s head, the rete is moveable and can be set to give the positions of the stars and the sun at any given time.

From Danielle Barbaro’s La pratica della perspettiva of 1569, p. 170

As the sun appears to have an additional form of motion, its position is read off from an eccentrically placed inner circle on the rete. This represents the Elliptic which lies with one point on the Tropic of Cancer, whilst diametrically opposite, another point grazes the Tropic of Capricorn. Although each sign of the Zodiac is allocated 30°, the callibration is constructed and read off from the centre, with the eccentric positioning of the Elliptic accordingly resulting in an uneven scale. For different latitudes, different plates were made and these were made so as to be exchangeable. To this effect, astrolabes generally feature a backing plate or mater, with a frame or limb that encloses the exchangeable plates. The limb is calibrated so that with a sight or alidade mounted in the middle, the heights of heavenly bodies in the sky can be measured. This is shown by Prachatitz at the beginning of his treatise:  https://findit.library.yale.edu. The alidade also functions as a rule that enables values to be read off at various places on the rete, plates and limb. For Prachatitz’s diagram of the rete and limb see: https://findit.library.yale.edu. In this way, the planispheric astrolabe combines objective and subjective points of view, so that the positions of stars and the sun at any given time can be ascertained or, from the height of a given body, the time may be read off. Among the pages of Codex Yale 24, there are is a second, unillustrated treatise by Prachatitz on the construction of the astrolabe (manuscript pages 275-281) which is followed by a treatise on the use of the instrument (281-310). See: https://findit.library.yale.edu. There is also an unillustrated treatise on constructon by Peuerbach (221-230): https://findit.library.yale.edu.

On 3rd September, 1457, Peuerbach and Regiomontanus observed an eclipse of the moon at Melk. To do this they may well have used an astrolabe and in the German edition of Hartmann Schedel’s Liber chronicarum of 1493 Regiomontanus is shown holding an astrolabe. In making the Melk observation, Peuerbach and Regiomontanus may well have used one of two Medieval/Renaissance towers that survived the Baroque rebuilding.

In order to establish the time of the event, Peuerbach and Regiomontanus measured the height of Alkione, which is one of the stars in the Pleiades. As the Baroque mathematician and astronomer, Georg Matthias Bose later observed, Peuerbach and Regiomontanus‘ measuring the height of Alkione was the first time since Antiquity that astronomers had used time as a means of pin-pointing an eclipse and the significance of the event lay in their using the observed time as a means of verifying the calculations summarised in tables. In his article on the observation, Bose quotes from a now lost chronicle at Melk.

The Melk observation was the first in a series of observations carried out with the express purpose of checking what had been theoretically calculated. Three years later, checking the information given in a table of eclipses which had been published a year before, Peuerbach and Regiomontanus observed another lunar eclipse, again measuring the height of a star in order to ascertain the time of occulltation. Although the time of the event was within the bounds of acceptable accuracy, the locations given in the tables were at odds with what was observed. In the so-called „Alfonsine Tables“ as well as in the Almagest, the catalogue of stars and planets along with their positions that had been drawn up by Ptolomy, there were also divergences. Thus in 1461, at the suggestion of the exiled Greek cardinal, Bessarion (1403-1472), Peuerbach began the immense task of translating and updating Ptolomy’s Almagest – only to die before the year was out. Bessarion therefore invited Regiomontanus to accompagny him to Rome where, at Bessarion’s expense, he could complete what Peuerbach had started. Two years later, the newly updated and translated Almagest was presented. In Astrolabes and Angles, Epigrams and Enigmas, David King reconstructs the circumstances and events relating to an astrolabe that Regiomontanus made for Bessarion and tells a moving and inspiring story of friendship, art and mathematics. The story can be read in detail in book form or as a summary at: www.davidaking.org.

Bessarion is known to have owned an old Byzantine astrolabe which, based on Ptolomy, was out of date and inaccurate. King argues that the astrolabe Regiomontanus made was a present designed so to cheer his sad and displaced patron up and convincingly interprets the otherwise puzzling text that Regiomontanus engraved on the reverse side of the instrument.

Not stopping there, King goes on to raise the possibility of there being a connection between the astrolabe that Regiomontanus made for Bessarion and Piero della Francesca’s enigmatic painting, The Flagellation. Whilst this is speculative, it is in tune with Renaissance thinking and shows how art and mathematics can used to overcome grief and traumatic up-heaveal’s. In seeing the work as a consolation piece, King’s view is related to the interpretation arrived at by Marylin Aronberg Lavin using art historical analysis, suggesting that somewhere, something is ringing true.

After Rome, Regiomontanus went to Hungary where he worked for Matthias Corvinus and also made an astrolabe for him. In 1475, he was again invited to Rome, this time by Pope Sixtus IV to work on proposals for a reforming of the calendar. Travelling to Rome, Regiomontanus either fell victim to a plague, or following the interpretation given by King, was poisoned and died, only forty years old, in 1476.

According to Euclid, an angle is an extent that pertains between two straight lines that meet at apoint. As a straight line can graze a curve at a tangent and at any point on a curve, there can only be one tangent, this means that although between a tangent and the curve grazed there is an extent that looks like an angle, it cannot be an angle as only one straight line is invloved. Yet for Cusanus it was an angle, the reason being that as the number of sides of a regular polygon increases, so this angle-like extent decreases, this allowing many-sided polygons to approach the perfection of the circle and ever improved approximations to π to be generated. When two curves meet, they also form something angle-like that Euclidian geometry denies is an angle. That Regiomontanus and Peuerbach made and used astrolabes and yet were critical of Cusanus, is ironic, as astrolabes use angle-like extents that are not really seen as existing. At best, as flattened versions of spherical angles, the angles marked out and used on astrolabes have a sort of „stand-in“ status. While this supposedly provsional mode of invocation suffices for mathematicians, it is not enough for the metaphysically rigorous philosopher for whom, when something is invoked and used, it is also seen as existing. Granting such angle-like extents an existential status opens doors to new worlds and entails the abandonment of the Euclidian postulate that parallel lines can never meet, for when space is warped or bent, parallel lines can meet. The warping of space also means that the shortest distance between two points is not necessarily a straight line and while this may strike one as counter-intuitive, it is not contradictory or illogical and moreover, is precisely what happens when something is sucked towards the central singularity of a black hole. In the curved spaces of hyperbolic geometry, the circle can be squared, as the hyperbolic square, like the circle, consists of curves, so that in squaring a circle, like is being compared with like. The potential for dialogue that takes place when a Euclidian view of the world is juxtaposed with a non-Euclidian one is currently being investigated by Alexander Curtis with a view to putting certain philosophical problems in context.