FROM THE WACHAU TO CENTRAL ITALY AND BACK

The following is a mathematical, geographical and historical journey back in time that presents the backgrounds of a number of projects worked on by Alexander Curtis. The departure point is a codex of astronomical texts once held by the monatersy at Melk with a selection being introduced in a reader-friendly manner that culminates with a mysterious painting, a venonous controversy and an accusation of murder. As much of the story is told with reference to astrolabes, towards the end, instructions are given on how, using a ruler and compass, an astrolabe for any given latitude may be constructed.

Among the thousands of volumes held and once held, by the Bendectine monastery at Melk, is a bound assortment of manuscripts known as Codex Yale 24. Once Codex Melk 794, the volume is now in New Haven and belongs to Yale University. This change of ownership was precipitated by monasteries being caught between diminishing financial returns on the land they owned and rising costs. In order to remain financially afloat, between the two world wars, many monasteries were forced to sell precious manuscripts from their libraries to dealers who then sold them on to libraries and museums in the United States. Summarising the situation, the London dealer, E. P. Goldschmidt, recalled:

„In those years Austria was my principal happy hunting ground, and the financial straits in which the ancient monasteries and abbeys there found themselves afforded extradorinary opportunities for buying from them books which had stood on their shelves for centuries“.

In 1937, Goldschmidt bought Codex Melk 794 and sold it on to the neurosurgeon and book-collector, Harvey Cushing. In the same year, Cushing bequeathed the volume to Yale University, along with the rest of his not unsubstantial collection. Becoming Codex Yale 24, the collection of manuscripts is a part of the Harvey Cushing/John Hay Whitney Medical Library and may be seen online at: https://findit.library.yale.edu.

The assorted manuscripts of Codex Yale 24 date from the Renaissance and are mostly concerned with astronomy, the calendar and on how to make and use instruments such as the quadrant and the astrolabe. The monastery at Melk had connections to both the University of Vienna and the First Viennese School of Astronomers and Mathematicians. This latter included the important figures, Johannes of Gmunden (1380/1384-1442), Georg Aunpekh, generally known as Georg von Peuerbach (1423-1461) and Johannes Müller of Königsberg, known as Regiomontanus (1436-1476). Of the thirty-two works in the codex, two are by Johannes of Gmund, eight are by Peuerbach and two are by Regiomontanus. Towards the end of the codex there is a text by the theologian, philosopher and mathematician, Nicholas of Cues, who, otherwise known as Cusanus (1401-1464) was a cardinal and the Bishop of Brixen. For a partial index see: https://findit.library.yale.edu.

I. JOHANNES VON GMUNDEN

Johannes of Gmunden was the first Professor of Mathematics, Physics and Astronomy at the University of Vienna and wrote on the making and use of a variety of astronomical instruments. This was an important part of the task of re-connecting astronomy with observation and empirical measurement which, essential to the scientific endeavour, had been relegated to second place during the Medieval Ages. Although he mainly wrote on astronomical instruments as opposed to making them himself, an instrument that he did make is a quadrant. Dated to 1438, this can be seen at the Kunsthistorisches Museum in Vienna: https://www.khm.at and was made for Emperor Friedrich III who was interested in astronomy and for whom Johannes of Gmunden was court astrologer.

With a quadrant, the height of the sun in the sky can be measured and converted into the time of day. This is achieved with reference to an elevation of the geocentic model of the sun rotating around a point that represents the position of an observer on Earth. On the diagram „Z“ stands for „zenith“ and indicates the point in the sky directly above the observer, whilst „H“ indicates the horizon with the grey area being the surface of the Earth below which the sun sets during the night. As the sun rises from below the horizon on the left, it ascends up into the sky to obtain a maximun height at midday on the right and then follows exactly the same path back down towards a nadir that represents midnight. The Eliptic, which contains both these paths of rising and setting and the sun’s oscillations as it rises higher in the sky during summer and lower during winter, is shown in turquoise, with the red and blue lines at the top and bottom indicating the paths followed by the sun on the days of the summer solstice (red) and the winter solstice (blue). The yellow lines inbetween show the paths followed on the days when the sun enters a new sign of the Zodiac.

On the diagram, both the annual and the daily movement of the sun are accounted for by circles, the larger circle being a circle of hours and the smaller circle a circle of the signs of the Zodiac. In this way, the turquoise lozenge which contains the sun’s movement can be calibrated both horizontally and vertically. As can be seen however, the horizontal and vertical axses of the Eliptic are tilted with respect to the observer, with the amout of tilt being dictated by the the latitude, L, of the observer. This pushes the Eliptic down towards the observer’s horizon. The higher the latitude, the more the Eliptic is pushed down, so that at the North Pole (90°), the Eliptic is on the same horizontal plane as the observer’s horizon with the summer sun providing light 24 hours a day, while in the winter, the sun is invisible for months on end. At the Equator meanwhile, the situation is the opposite and the Eliptic stands vertically, resulting in day and night being of equal length and there being no seasons. The observer’s latitude read from the zenith dowards indicates the path of the sun at the time of the equinoxes, when the sun’s path passes through the centre point of the main circle. The extent of the Eliptic is then established by reading 23.5° in both directions away from this centre.

Section I in Progress

 

On this form of quadrant known as an „old“ quadrant the scale, the daylight hours of summer are the same length as the night time hours of winter with there being accordingly a large discrepencies between the duration of an hour during the night and an hour during the day. Despite this, the old quadrant was able to hold its own against the „new“ quadrant devised by Profatius in Spain during the thirteenth century. This is because its construction provides the basis for the construction of the cylinder and ring forms of sundial and provides the means by which a shadow square may be constructed so that the quadrant may be used for surveying purposes.

 

On the new quadrant, time is read off on a scale that is constructed so as to divide the hours of the day into 24 equal periods which are shown as segments on the dial. In this way time is standardised and the daylight hours of summer are exactly the same length as those of winter. developed by Nevertheless it was only gradually that unequal hours were replaced by equal hours during the fourteenth and fifteenth centuries as clocks were slowly introduced.

On the old quadrant shown below the square scale shows two scales of shadows, the umbrae vertae and the umbra rectae.

 

 

These are the beginnings of the dial’s construction and are the means by which the scales of equal and unequal hours are arrived at. C The shadow scales are also the means by which from an angular measurement, the height of a building may be ascertained and so are basis of the instrument’s use in surveying.

In Codex Yale 24, in an unillustrated tract, Johannes of Gmunden discusses an instrument known as a torquetum: https://findit.library.yale.edu. This combines a projective map of the heavens made from the North Pole with a second projective system which shows the lines and circles of ascention and declination for an observer. Where the one plate is inclined according to the tilt of the ecliptic the other is inclined according to the latitude of the observer. At the top of the instrument there is a dial from which the angular height of a heavenly body may be read off.

With this instrument information concerning the movements of the sun and stars relative to the observer can be computed, such as rising and setting times and periods of visibility.

In Codex Yale 24 the second tract by Johannes of Gmunden is Composito et usus cuiusdam instrumenti ad faciliter et breviter invenienda varia loca omnium planetarum. This is an unillustrated tract on a form of equitorial dial that was discussed by Campanus of Novara (around 1220-1296) and which is the fore-runner of an instrument known as an albion: https://findit.library.yale.edu. In terms of projection, the albion is similar to an astrolabe, nevertheless where astrolabe performs calculations relating to the sun and stars, the albion is concerned with the positions of planets. The only surviving example of an albion was made in Vienna during the fifteenth century (see below). Appropriately, in Codex Yale 24, Johnnnes of Gmunden’s treatise is followed by a table that details the movements and changing positions of the planets: https://findit.library.yale.edu.

II. CUSANUS‘ DE DOCTRA IGNORANTI

Between 1453 and 1460 the monks at Melk followed an exchange of letters and articles known as the Controversy of the Doctrine of Ignorance, in which Cusanus played a prominent part. This was precipitated by Vincent of Aggsbach, a monk and former prior of the Aggsbach Chartehouse which was only a few miles down river from Melk. The background to the controversy was a work that Cusanus had written thirteen years before. Following its appearance Cusanus‘ De doctra ignoranti had repeatedly prompted theologians to write replies and formulate theior own positions and it was Vincent’s of Aggbach’s reply to one of these other tracts that initiated the controversy. In De doctra ignoranti Cusanus had developed a form of speculative theology based on the general characteristics of numbers and geometrical figures which he then extrapolated out to the physical world and all that is and exists within it. Arguing that although the infinite, all-enclosing oneness of God could not be reduced to or equated with conventional geometry, aspects of the sphere and circle could nevertheless be used to show how God transcended human rationality. For Cusanus in approaching God, it was important to realise that one knew nothing. In one of a number of geometrically derived exercises, associating that which was divine and eternal with that which was circular and that which was straight and countable with that which was mortal, he argued that any polygon, regardless of how many sides it might have, would always be different from the circle that enclosed it. Conversely should a polygon have an infinite number of sides, it would in fact be a circle and would no longer be no longer a polygon. This was because a polygon is a many-sided regular figure whose sides can always be divided by two, so that the total number is automatically doubled. With a circle this is not possible as the number of sides is already infinite and cannot be increased any further. Therefore, as soon as one says that the process of doubling can no longer be carried out, the polygon has ceased to be a polygon and has becone a circle and in becoming so has transcended the bounds of that which is straight.

Dubbed by Pope Pius II as „the Hercules of Popes“, Cusanus was an papal diplomat who frequently travelled on church missions and negotiated at councils. Interested in astronomy, he never missed an opportunity of using his travels as a means of furthering his collection of manuscripts and astronomical instruments. On the occassion of a Reichstag in 1444, when the Holy Roman Emperor, Friedrich III and his Electors met in Nuremberg, he recorded:

„I bought there a very large globe (showing the celestial sphere), an astrolabe and a torquetum,  the book (by Ptolomy) of the Almagest and fifteen other books for 38 Rhein Gulden.“

These instruments can be seen at the Cusanus Monastery in Bernkastel-Kues. Like a torquetum an astrolabe is an instrument that enables the positons of the stars and sun to be ascertained. Effectively the astrolable works by shrinking the inclined plates of the torquetum onto a single plane. This means that in order for it to be used at different latitudes, instead of adjusting the angle of inclination of the plate showing the observer’s lines of ascention and declination, different plate must be used. Given the otherwise highly compact nature of the instrument, this however is a small price to pay and usually astrolabes are equipped with a number of different plates that can be inserted into the enclosing mater.

Image coming soon

On his astrolabe, from the smallest circle inscribed on a plate, he will have seen how the circles got progressively larger and larger until, instead of being shown as a circle or an arc, one was shown as a straight line.

Quite possibly prompted by this, in one exercise of De doctra ignoranti, Cusanus argued that the circumference of a circle of infinite radius is a straight line. He then went on to argue that, as the absolute largest thing that can be, is reached by a process of infinite expansion it is equatable with the smallest thing that can be, which is also reached by an infinite process, all be it one of reduction rather than expansion. In the case of an infinitely large circle, the centre is everywhere and nowhere, for the line of the circumference is infinite and straight and if the centre is at an equal distance from all points on the line, then it must in a sense be everywhere and nowhere. In the case of an infinitley small circle, the same holds true as the circle is so small that it can no longer contain the point that is its centre. In these and other ways, Cusanus showed how the circle and sphere were capable of containing that which was straight, whilst that which was straight could never attain the status of that which was curved. He thus used rational thought to show how when it comes to the infinite, that which is inherently finite will always fall short and the only way that we can intellectually grasp the infinite, is through the use of deductive reasoning that exposes the paradoxical nature of our attempts to comprehend that which cannot be comprehended. Cusanus‘ position is thus Socratic as, when it comes to God, the divine and the world as whole, like Socrates, Cusanus holds that all we really can know is that we know nothing. This however does not mean that a rational way of looking at things cannot lead to God but rather that it peters out in dilemmas and paradoxes that dumbfound and baffle us. From this form of learned ignorance, we can then reach out towards God. Carrying his reasoning over into the physical world, Cusanus argued that the universe must infinite and that accordingly the Earth could not be at its centre. Following on from this, he deduced that there must be other life forms in the universe and that the Earth must be in motion even though we do no direct impressions of this. From the shadows cast during eclipses of the sun and moon, he observed that we know that the Earth is larger than the moon and smaller than the sun. Aided in this way by astronomy and geometry, we can contemplete the world and arrive at deductions and realisations that lead to feelings of awe and wonder and so, through the recognition of the limitations of our powers of reason, help point towards the divine. Taken by himself, Cusanus saw man as nothing and it was only through God and his creation that answers to questions could be found and self-recognition and the beginnings of an understanding of God and the divine could be attained. Nevertheless, as something that is arrived at through intellect, Cusanus‘ position is diametrically opposed to that of Vincent of Aggsbach, who denied that the intelectualised form of ignorance advocated by Cusanus could be of any assistance in the search for God. This was because Vincent of Aggsbach saw the rational as something that had to be abandoned completely and which was not even fit to be used as a starting point. For more on this position, see the article, Where God whiles …, on the Reloading Humanism „News and Wachau Tips“ page.

III. CUSANUS‘ DE CIRCULI QUADRATURA

After the sceptical position expounded in De doctra ignoranti, over the years Cusanus began to see that, by accepting the infinite nature of the universe and in mathematics, by postulating the infinite as something existed as a fundamental charactereistic of the world, he began to abandon his scepticism and found ways by which, in a step by step fashion, that which was circular could be approximated to and approached with reference to such things as the straight lines of regular polygons. This in turn influenced his position with respect to our knowledge of God so that even before the Controversy of the Doctrine of Ignorance had started, he was already refining and developing his views on how God could be approached. Called upon to voice his opinion, with respect to the theological aspects of the controversy Cusanus draughted two replies. In addition, between 1450 and 1457, he wrote seven tracts on the squaring of the circle and how the infinite could be approached mathematically. The first of these works was De circuli quadratura in which he argued that while π was something could be approached, it could still never be precisely attained. In accordance with this view he then gave an approximation that was an improvement on the value arrived at by Archimedes in his Measurement of a Circle. Investigating a  96-sided regular polygon, Archimedes had arrived at a value for π of between 3 1/7 and 3 10/71. Yet at the beginning of his On the Sphere and the Cylinder, Archimedes had stated that where there is a straight line that extends between two points, through a process of successive division, which in the extreme case may be continued ad infinitum, any point on the line designated by another means can be reached and thus defined. This Cusanus realised, was only true of intervals that were defined by straight lines and did not include intervals that were in any way derived from curves. Although with referance to any point, other points on a straight line can be defined by means of an arc that is swung over the line, at the point where the arc crosses the line the arc is seen in terms of a tangent and at this single point, the tangent is perpendicular to the line it crosses. This is important as in all other cases of arcs that intersect, either with each other or with straight lines, although the results are intervals that look angular, they are in Euclidian geometry, not seen as angles, for an angle is something that can only be formed from the intersection of two straight lines.

As the sides of a polygon contained within a circle are progressively increased, the cut-off left-overs between the polygon and the circle become progressively smaller and with them, the cornicular angles that pertain between the sides of the polygon and the circumference of the circle. In this way the polygon approaches the circle but can never reach it, as for any polygon defined or constructed there will always be one with a greater number of sides and thus with cornicular angles that are smaller than those generated by the preceding polygon. Cusanus realised that the cornicular angles associated with any approximation of a polygon to a circle were equivalent to the numerical amount by which the value for π arrived at using Archimedes‘ method differed from the actual value. Although these cornicular angles could not quantified they were real and their reduction in an approximation was an indication that a closer approximation had been arrived at than hitherto. Just as the polygon can never become a circle without ceasing to be a polygon, so any approximation to π will always be constrained by the logic of straight lines and so can never acquire the aspect of that which is circular. In mathematics Cusanus was thus the first person to realise that π is neither a rational number, nor is it an irrational number but rather, is what mathematicians now call a „transcendental number“. This means that there is no quadratic equation in terms of which the transcendental number can be expressed in a way that the elements of the equation can be arranged so as to make zero. Thus although the root of two is an irrational number, it is the only number that satisfies the equation: x2 − 2 = 0 and accordingly root two is not a transcendental number. Likewise, the equation x2 − x − 1 = 0 represents the number phi, which is associated with the Golden Section and which whilst irrational, is not transcendental. Like the square, when the sides of other polygons are assigned lengths that are characterised by rational numbers, their diagonals are either rational or irrational and so can be expressed by means of a quaratic equation. In the theoretical case of a polygon that becomes a circle, a transition takes place and the numbers characterising the polygon also change and ceasing to be either rational or irrational numbers become instead transcendental. As the name suggests and as the squaring operation that it contains implies, quadratic equations are associated with all that is linear, while transcendental numbers are associated with something else. In the case of π, this something else is rotational and is that which enables the radius of a circle, despite being straight, to sweep out angular motion and so describe a circle. Angular motion thus generates something that escapes the net cast by a quadratic equation. As Cusanus saw, associated with both irrational and transcendental numbers are differences known as „infinitesimal’s“. This is because between any enumeration of an irrational or transcendental number and its approximation, there is a difference that as the accuracy of the approximation increases becomes infinitesimal. In the case of a circle and a polygon, as seen, the unquantifiable cornicular angles are equilvalent to the infinitesimal differences between an approximation to π and the actual value. 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. Infinitesimal’s 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. This suggests that on such a line, any irrational or transcendental number can be arrived at simply by dividing the line often enough. While this true of irrational numbers, as Cusanus knew, despite the infinite amounts of division that may be invoked, this is not true of transcendental numbers which are a different species of number. Following the discovery and acceptance of irrational numbers, the infinitesimal’s that are associated with them were 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 and accordingly saw each improved approximation of π as an approach towards a facet of God.

Having established the unreachable and yet approachable nature of π, in De circuli quadratura, Cusanus considered an equilateral triangle together with two circles, one which it encloses and one which it is enclosed by. As the radius of the smaller circle is half that of the larger and as the smaller circle has a radius that is rational, it follows that both circles have radii that are rational. It also follows that between them there will be a circle whose circumference is exactly that of the perimeter of the triangle. The question is then of whether the radius of this third circle is rational/irrational or transcendental. Although transcendental, Cusanus shows  how it may nevertheless be approached.

To this effect, line b,c a quarter of the way in from c. This results in point k. If from the centre of the two circles, d, a line is drawn from b to k and if d,k is then extended by a quarter, the result is a circle with a circumference that is equal to the perimiter of the triangle.

Following the method used by Archimedes in his approximation to π , the radius of the enclosing and enclosed circles associated with any polygon are related to one another by a right-angled triangle which is formed from their radii. The two radii give the lengths of the shorter side and the hypotenuse and this enables the remaining length to be arrived at using Pythagoras‘ Theorem. The length of the side of the polygon concerned is then obtained when this length is multiplied by two. In the case of an equilateral triangle, if the radius of the enclosing circle is said to be 60 then that of the enclosed circle is 30 and this means that the longer side is √2700 in length. Inside the equilateral triangle, this 30,60,√2700 triangle, reflected about itself six times, forms the equilateral triangle. Half of √2700 is √675 and this indicates the quarter point on the side of the equilateral triangle. Meanwhile the distance from the centre of the circle to the quarter point is √1575.

As the circumferance of a circle is given by the formular 2πr, π is found by dividing circumferance by diameter. As the circumferance is the same as the perimeter of the equilateral triangle and this is 6 x √2700, dividing this by 2 x 5/4 x √1575 results in a value for π of 3.1423376 which is an improvement on the approximation given by Archimedes of between 3.1428571 and 3.140845. That the method is an approximation can be seen by the fact that when a radius of exactly 2 1/2 is constructed using 1,1√2 triangles it emerges that it does not pass through k but through a point slightly beyond k.

IV. CUSANUS‘ QUADRATURA CIRCULI

The presence of Cusanus‘ next tract on the squaring of the circle among the manuscripts of Yale 24 is significant as it shows that in the Controversy of the Doctrine of Ignorance, the monks at Melk were following the debate not just theologically but also mathematically. Entitled Quadratura circuli, this work also appears to date from 1450 with the Yale version being one of only two manuscript copies to have survived: https://findit.library.yale.edu.

Where in the first part of De circuli quadratura Cusanus had argued deductively to establish that π was transcendental and had then given a specific method by which it could be approximated to, in Quadratura circuli he established the transcendental nature of π geometrically and then went on to give a generalised and likewise geometrical method by which margins could be identified within which π lay. Unlike the specific method given in De circuli quadratura, the „geometry of the infinite“ of Quadratura circuli, allows ever-improved approximations to be made. Cusanus first considers a line, a,b, from which an equilateral triangle is constructed whose sides are formed from a,b in a such way such that nothing is left left over. He then constructs a square whose sides, in like manner, are a,b divided into four. Comparing the two figures, as in De circuli quadratura he observes that in both cases and indeed in any regular polygon, in addition to an enclosing circle and an enclosed circle, there is also 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 and these piecemeal steps towards the third circle, are real steps even though the circle itself can never be attained. Considering the triangle, Cusanus observes that if the radius of the containing circle is 14, then that of the contained circle is 7. Seven is also the mean between 0 and 14 and on the diagram above is taken as the length of f,n and h,t, while aa,p and t,o are given by the differences between the two circles associated the triangle and the two circles associated with the square. These result in triangles g,aa,p and h,t,o which are then enlarged to form triangles g,bb,q and h,s,q with q designating the radius of the circle whose circumference is that of the line a,b. As this is however hypothetical, the triangle g,bb,q is in fact triangle g,bb,u with there being a gap between q and u with q giving the upper limit and u giving the lower limit. That g,pq and b,t,q and thus also g,p,u and h,t,q, are straight lines reflects the fact that Cusanus sees the progression towards the sought circle as being linear with relations between analogous parts being characterised by a single proportion.

At the beginning of his Measurement of a Circle, Archimedes had stated that the area of a circle is equal to that of a right-angled triangle whose height is the radius of the circle and whose width is the circumference of the circle. Adapting this by saying that the area of a circle is equal to a rectangle formed by the radius of the circle and half the circumference, Cusanus develops a diagram which shows that, given the radius of a circle, everything else that one wants to know is related to π which however is unknowable.

Nevertheless, as in De circuli quadratura, Cusanus supplies a rule of thumb by which the process of aproximation can be initiated. This is the ratio 16:9 which relates the lengths of r,q to s,u and s,u to to r,s and r,s to a,b. As the radii of the containing and contained circles of the equilateral triangle are 7 and 14 then it it follows that the half-lengths of its sides are √147. This multiplied by 9/16 makes √(82 11/16) which is both the length of the half side of the square and the radius of the contained circle. Again using Pythagoras‘ Theorem, the root of twice this amount, in other words, √(165 6/16), gives the square’s diagonal and the radius of the circle that contains the square.

If from √(82 11/16) 7 is subtracted this gives the difference between of the radius of the circle enclosed in the square and the radius of the circle enclosed in the triangle which is a difference of, as Cusanus says, „a little bit more than two“ (9.093 – 7 = 2.093). If from 49, √(165 6/16) is subtracted, the difference of „a little more than one“ (14 – 12.86 = 1.14) will be arrived at between the radius of the circle enclosing the triangle and the radius of the circle that encloses the square. On the first of the three diagrams above, these form aa,p and t,o and if the lines of the hypotenuses of g,p and h,o are extended out towards s,r, they will approach q. This is done duplicating the triangles g,aa,p and h,t,o and attaching them to the original triangles so that the lines of the hypotenuses are continued. This results in triangles g.bb,q and h,s,q. The new radii are then found by adding 2.093 to 9.09 which makes 11.186 and subtracting 1.14 from 12.86 which leaves 11.72. As the circumference of the circle is the same as the perimeters of the equilateral triangle and the square which are 6 x √147 and 8 x √(82 11/16) which both make 72.746, this figure, divided by the two diameters obtained specifies the bounds within which π lies. This is because the extended triangles have fallen short and do not meet at a single value. Where 11.72 gives a value of 3.1034, 12.86 gives a value of 2.828 and this results in a  range which is wider than that given by Archimedes. To improve upon this, the triangles are again duplicated and then reduced prior to the lines of their hypotenuses being extended as before. As it is only the height of the triangles that is of interest, this is achieved by taking the difference betwen 11.186 and 11.72 and multiplying it first by 2.093/(1.14 + 2.093) and then by 1.14/(1.14 + 2.093) to result in intervals of 0.345 and 0.188 which respectively are added onto 11.186 and subtracted from 11.72 to result in 11.532 and 11.531. The circumference of the circle is then divided by two times the mean radius to result in a value for π of 3.154. Although this lies outside the range established by Archimedes, Cusanus‘ primary purpose was to establish from first principles, a general method by which ever improved results could then be generated. This he saw as being possible through the application of the same technique to polygons with more sides that the equilateral triangle and the square. In this way, Cusanus saw the regular plane figures as pointing towards the divine transcendence of a circle even though they could not attain it.

The Yale manuscript of Quadratura circuli includes a figure that does not occur in the principo precepts. This is equivalent to the left-hand diagram above: https://findit.library.yale.edu. For the reader, the production of such diagrams is 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 work out 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. It also underscores the Renaissance view of the world as a labyrinth that one must be read and work one’s way through. For more on this, see the article, The World’s a Labyrinth! on the „News and Wachau Tips“ page.

V. PEUERBACH, REGIOMONTANUS AND CUSANUS

Even before he had graduated, Peuerbach was invited to give a series of lectures on astronomy at the University of Padua which was held in 1449/1450. In Italy Peuerbach encountered humanism and met Cusanus with whom he stayed for some months in Rome in 1451. He then returned to Vienna and completed his studies in 1453. Thereafter he taught classics at the University of Vienna and astronomy at the School of Saint Stephen’s. In 1453 Cusanus refined and brought together his thoughts on the squaring of circles and released De mathematicis complimentis which begins with the method of squaring a circle expounded in Quadratura circuli. Receiving this new work, the doctor and astronomer, Paolo dal Pozzo Toscanelli (1397-1482), wrote a critique, which Cusanus sent to Peuerbach. When in 1457 Cusanus released Dialogus de circuli quadratura, he also sent a copy to Peuerbach. Not sure of the significance of what Cusanus was getting at, Peuerbach draughted some comments and then left the matter at that. In 1461, when Peuerbach died, his papers, including Toscanelli’s critique and Cusanus‘ later dialogue were passed on to Regiomontanus. Then soon after, the young astronomer recieved another tract by Cusanus. This was De una recti curvique mensura, a copy of which Cusanus had sent to the exiled Greek cardinal and archbishop, Bessarion (1403-1472). Cusanus and Bessarion had been friends since 1437, when Cusanus had travelled to Constantinople as a papal diplomat and had then journeyed back to Italy the following year in company with Bessarion. Following Peuerbach’s death, Bessarion adopted Peuerbach’s star pupil as a protegé and Bessarion passed the text of De una recti curvique mensura on to Regiomontanus.

In De una recti curvique mensura, instead of squaring the circle directly, Cusanus gives three rule of thumb methods by which the curved arcs of a circle could be translated into straight lines of the same length and one method by which the area defined by a half circle could be translated into one defined by a rectangle. Meanwhile, as the name implies, in Dialogus de circuli quadratura, he refers indirectly to De circuli quadratura and once again considers an equilateral triangle whose perimeter is rational and which is enclosed by a circle, within which there is second circle whose circumference is the same length as the perimeter of the triangle and so is also rational. The question of De circuli quadratura is thus reversed and instead of going from the triangle’s enclosed and enclosing circles to a circle with a circumference the same length as the perimeter of the equilateral triangle, Cusanus goes from this circle, to the circle that encloses the triangle. As the circumferance of the first circle is rational, the questiion is then of whether this second circumferance is rational/irrational or transcendental. Establishing that, as before, although it is transcendental in length, this can nevertheless be approached, he then used a different rule of thumb to arrive at an approximation. Drawing a quarter of a square enclosed within the inner circle, Cusanus says that the length of the hypotenuse of the resulting triangle, a,b,c, added to one of the shorter sides approximates to the diameter, g,f, of the larger circle that encloses the equilateral triangle.

In this way Cusanus again made his point that π is something that can be approached but never exactly established. In Receiving De una recti curvique mensura from Bessarion, Toscanelli’s critique and  Dialogus de circuli quadratura from Peuerbach and Quadratura circuli from an unkown source, it was only in Venice in 1463, that Regiomontanus finally got around to formulating a response to Cusanus‘ work. Where Toscanelli had questioned Cusanus‘ assumption that as polygons acquire ever more sides, they approach the circularity of a circle in a linear fashion, Regiomontanus established this was not the case. When the differences between the perimeters of regular polygon and the circumferences of the circles that contain them are plotted on a graph they approach the circularity of a circle in a haphazard manner, that approximately follows a curve that eventually leads to the almost straight line of an asymtope.

In Quadratura circuli, had Cusanus worked through examples using polygons other than the equilateral triangle and the square, he would have obtained different and less favourable results. Thus where Peuerbach was puzzled and at a loss to see what his friend was getting at, in a letter to the mathematician Christian Roder, Regiomontanus described Cusanus‘ geometry as „ridiculous“ and dismissed him as an „ape of Archimedes“. While it is correct that Cusanus did make a false assumption, for all his mathematical brilliance, Regiomontanus failed to grasp not one but three essential points. On the one hand, two hundred years ahead of his time, Cusanus was feeling his way towards the differencial calculus that would later be developed by Leibnitz and Newton. Here, between incremental values which specify a required degree of accuracy, curves are reduced to straight lines. On the other hand, a full four hundred years ahead of his time, Cusanus had realised that π was a transcendental number that was to be seen as standing apart from the traditional categories of rational and irrational. In working his way towards these realisations, Cusanus, again four hundred years ahead of his time, further called into question the basic assumptions of Euclidean geometry. It is thus highly ironic that when Regiomontanus‘ De triangulis omnimodis was published posthumously in 1533 by Johannes Schoner, the motif for the frontispiece was the first of the three diagrams given above from Cusanus‘ Quadratura circuli. This tract, together with Dialogus de circuli quadratura and De una recti curvique mensura, Schoner had included as an appendix along to De triangulis omnimodis along with Toscanelli’s critique, Peuerbach’s notes and Regiomontanus‘ replies.

VI. PEUERBACH AND REGIOMONTANUS

Where Johannes of Gmunden only made a small number of instruments, Peuerbach and Regiomontanus were more prolific and made astrolabes, sundials and other instruments, a number of which have survived. In Codex Yale 24 there is an unillustrated treatise on the construction of astrolabes by Peuerbach: https://findit.library.yale.edu and an astrolabe that may well have been made by him is held by the German National Museum in Nuremberg.

At the small town Peuerbach in Upper Austria, which is Georg Aunpekh was born, a large-scale copy of the front side of this astrolabe adorns the gable of the town hall and inside there is a museum dedicated to the astronomer and his life.

A tract written by Regiomontanus on astrolabes is held at the National Library in Vienna as a part of Codex Vienna 5292 which also includes Canones albionis, a tract written by him in 1455 on the use of an instrument known as an albion. Originally invented by Richard of Wallingford during the Middle Ages, the only known example of an albion was made in Vienna sometime between 1450 and 1460. Not as yet subjected to detailed investigation, it is likely that it is based on the revised accounts of planetary motion that Peuerbach was developing during this time. There is also the question of the nature of the relation between it and the tract on the albion written by Regiomontanus in Codex Vienna 5292. At some unkown point in time, the viennese albion then made its way to Rome and is now at the Museo Astronomico e Copernicano.

On its reverse side the albion features a chart of the heavens.

Not withstanding the fact that he served as court astrologer to King Ladislaus Posthumus of Bohemia and Hungary, Peuerbach saw the heavens and all that was therein contained as the manifestation of a system that was subject to physical laws that could be rationally formulated. He was thus saw comets as physial phenomena and rejected the view that they were portents of divine origin. In a report he wrote for Ladislaus on the comet of 1456 (now known as Halley’s Comet), he identified nine differet kinds of comet. A copy of the report is held by the monastery at Melk, while in Codex Yale 24, an illustration shows the nine different kinds of comet: https://findit.library.yale.edu. A year later, in the same year that Cusanus released Dialogus de circuli quadratura, as a part of the program of revising and checking over the ptolomaic account of planetary motion, together with Regiomontanus, Peuerbach observed an eclipse of the moon at Melk. The circustances of his being in Melk was the hospitality afforded by the monastery to King Ladislaus and his retinue as they journeyed from Mariazell to Prague where the king was due to be betrothed. As Peuerbach was Ladilaus‘ astronomical/astrological advisor, his presence at such an eventful time is to be expected. Although the royal visit and King Ladislaus‘ unexplained death soon after, ferature in the Melk chronicles, the presence of the astronomers left no trace. Evidence of the event however is provided by the meticulous records maintained by Regiomontanus of all the observations he made along with attendenant calculations and here, Melk is explicitly cited as the place of observation. These were discovered among Regiomontanus‘ papers by Schöner and published in 1544. In the chronological list of observations the Melk observation of 1457 is the first entry. Made on the Third of September, the astronomers may well have carried out their observetaions from one of two Medieval/Renaissance towers that survived the later Baroque rebuilding of the monastery.

Reconstructing the event, the astro-historian, Georg Zotti has confirmed that the data given by Schöner is completely in accord with Melk being the place of observation. Although it is not known what kind of instrument was used, an astrolabe is a distinct possibility and in the German edition of Hartmann Schedel’s Liber chronicarum of 1493 Regiomontanus is shown holding an astrolabe.

Eclipses of the moon are important for astronomers as they provide an exact alignment of sun, moon and Earth, so that from the time between two lunar eclipses, a very precise average may be obtained of the length of a lunar month. By the Renaissance it was becoming clear to mathematicians and astronomers that a calendar reform was necessary and this made the precise duration of a month one of a number of matters that were of pressing importance. In order to measure the duration of the event, Peuerbach and Regiomontanus measured the height of Alkione, which is one of the stars in the Pleiades. From the duration the event, they then extracted an exact time for the actual alignment.

As the Baroque mathematician and astronomer, Georg Matthias Bose, observed in Vienna at a celebratory speech made on the 300th anniversary of the event, 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 critically verifying the data given in tables. Drawing from the work of earlier Islamic astronomers and from a team of Jewish scholars that Alfonso X of Castile had assembled  around him in Toledo, a book of tables giving the positions of the sun, moon and other planets was complied. Known as the Alfonsine Tables, an edito precepts appeared in 1483. Nevertheless both this work and the Almagest, the catalogue of stars and planets along with their positions that had been drawn up by Ptolomy, diverged from what was observed when precise observations were made. This was due to  the „Procession of the Equinoxes“ a phenomenna which results in the so-called „fixed stars“ appearing to rotate through 360° over a period of 25,800 years. Although this was not understood at the time, when the results of observations were critically compared with what was given in the tables, divergences were discernable. In 1460, three years after the Melk observation, Peuerbach and Regiomontanus observed another lunar eclipse, again measuring the height of a star in order to ascertain the time of occultation. Although the time of the event lay within the bounds of acceptable accuracy, the locations given in the tables were at odds with what was observed and for Bose, the Melk observations were the beginnings of a new era:

„The lover of Astronomy, who in general is not completely ignorant of her history, will immediately agree that that this already well-described lunar eclipse is the beginning of a new epoch and blooming in Astronomy. Behold! The light that this eclipse has unveiled can never be honoured and emphasised enough. The light that had been driven away by the darkness into which had Astronomy fallen, now shines like the golden red of dawn with her pink hair, in the next instant ready to announce the rising sun of Apollo. This was achieved by GEORGIUS PEUERBACH who alone carried the light back up to the heavens.“ 

In astronomy, Peuerbach’s achievement is, with respect to the motions of the planets, to have significantly improved upon the ptolemaic system by shifting the points about which the planets and their epi-cycles rotated. Although he was not the first to do this, Peuerbach expounded his views with unprecedented clarity and made no bones about the fact that the ptolomaic system needed improving upon. He thus paved the way for the seminal moment when Copernicus realised that the different centres of rotation of the various planets could be reduced back to a single point and the notion of epi-cycles rendered unnecessary if the Earth was made to rotate about the sun and not the other way around.

The planet Mars with its eccentric circle of orbit and its epicycles

In Codex Yale 24 Peuerbach‘ Theoricae novae planetarum, complete with coloured illustrations, features prominently at the beginning of the volume and i splaced immediately after the list of contents and a printed calendar by Regiomontanus: https://findit.library.yale.edu. Introduced at the beginning of the work as „a celebrated astronomer“ a further indication of Peuerbach’s standing, both during his lifetime and thereafter, is given by the fact that his New Theories of Planets was used by astronomers up until the middle of the seventeenth century and between 1472 and 1563 fifty-six editions were printed.

VII. REGIOMONTANUS AND BESSARION

A year after the observations of 1460, Bessarion, appeared at the court of Friedrich III in Vienna. There he hoped to pursuade the Emperor to join him in petioning the Pope to lauch a crusade against the Turks. Although unsuccessful in mobilising help for the fallen Byzantine Empire and its beleaged church, Bessarion, who was interested in astronomy, had a second item on his agenda and was hoping to persuade Peuerbach to take on the task of translating and commentating upon excerpts of Ptolomy’s Almagest. This was to replace a similar work which Pope Nicholas V had commissioned from the Giorgios of Trebizond and which, completed in around 1450, was found to be deficient. Due to the flourishing university and the First Viennese School of Mathematicians and Astronomers, during the fifteenth century, Vienna was the European centre for astronomy and Bessarion knew that if the job of up-dating Ptolemy’s Almagest was to be done properly he would need Peuerbach or someone of Peuerbach’s calibre. Although Bessarion was able to persuade Peuerbach to undertake the task, the astronomer died before the year was out, yet Regiomontanus had sworn to complete the task should his master fail to complete what he had started. Cancelling a series of lectures that he had been planning to give on Virgil at the university in Vienna, in the autumn of 1461, Regiomontanus and Bessarion embarked on the journey to Rome. Passing through Bologna, they then went to Ravenna, where Bessarion became so ill that he nearly died. As pointed out by David King in his Astrolabes and Angles, Epigrams and Enigmas, like Bessarion, Regiomontanus was pius and used a schematic drawing of a hill with a cross on it and two stars as an emblem on the books he owned. Both on their journey and during Bessarion’s convolesence, the two will have had plenty of time to talk and it can be assumed that Bessarion will have told Regiomontanus of a relic of the cross that he had and of an artist he knew who was painting a revolutionary series of frescos in Arezzo. Known as The Story of the Finding of the True Cross, Bessarion will have told Regiomontanus that the artist, Piero della Francesca, was also an accoplished mathematician, who like Regiomontanus, was interested in Archimedes. King further argues that a Byzantine astrolabe dating from 1062 and which is now in the Santa Giulia Museo della Città di Brescia, was brought to Italy by Bessarion. Showing it to Regiomontanus, Bessarion will have sadly observed on how, based on Ptolomy, it was out of date and inaccurate.

To cheer his sad and displaced patron up, King argues that Regiomontanus deceided to make Bessarion a new, up to date astrolabe. On the reverse side instead of engraving a hoary quadrant as was often done, he engraved an organum Ptolemaei. This is a projection of the part of the celestial sphere that is bounded by the two solstial circles and by the celestial colure. Regiomontanus had written a treatise on the organum Ptolemaei about in around 1456. In Codex Yale 24, the second work by Regiomontanus is an unillustrated tract on the quadrant as combined with an organum Ptolemaei: https://findit.library.yale.edu. There are also two works which consider  the organum Ptolemaei alone. These are by anonymous authors with the first being unillustrated and the second featuring one diagram: https://findit.library.yale.edu and https://findit.library.yale.edu. As the inclusion of the organum Ptolemaei within the circle of the astrolabe leaves empty space, Regomontanus engraved in the space above, an angel and in the space below, a scroll and a dedicatory verse.

This latter translates as something approaching:

Under the protection of the divine Bessarion, called cardinal,

I arise in Rome as the work of Johannes in 1462

As the shaky Latin of the original is utterly at odds with Regiomontanus‘ consummate command of both Latin and Greek, following a series of insights made by Berthold Holzschuh, King argues that it is an acrostic, with the mistakes and out of place intervals deliberately being designed to prompt a multitude of interpretations. 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.

Bessarion was close to Frederigo da Montefeltro, the Duke of Urbino and was god-father to all of the Duke’s children, including Buonconte the son by his first wife. As over the years, Piero della Francesca executed a number of works for Frederigo, Bessarion will have certainly known of him and his work. Showing promise as a linguist, Buonconte had died of the plague in his teens and is one of the candidates proposed by King for the golden-haired youth shown in Piero’s painting. Yet Bessarion also had close ties to two other grieving fathers so that Buonconte is by no means the only feasible candidate and King sees this as being the point of the work. Like the dedicatory verse on the astrolabe that Regiomontanus made for his patron, Piero della Francesca’s painting is, King and Holzschuh suggest, an acrostic in which multiple interpretations are intended. These moreover follow a common scheme. Thus there are two grieving fathers in the foreground, mourning the loss of a boy, who with golden hair, has his attention firmly fixed on another world. Then there is the scene of the flagellation itself, with the figure of Pontius Pilate looking on, either unable or unwilling to intervene. With his back to the viewer there is a man with a turban, followed by a figure who is reaching towards Christ. Then there is Christ himself, followed by the man who is actually doing the whipping. Correlating the eight figures in the painting with the columns of letters in the acrostic results in a multitude of meanings, all coherently structured by the archetypical roles indicated by the figures in the painting. This interpretation explains why in art history, there are so many interpretations of what the painting represents, its purpose and who the portrayed figures are, for this was the whole point of the work, to create something that in its meaning was multi-layered and endlessly shifting.

While the figure of Pilate has the features of the Byzantine Emperor, the man in the turban would seem, in the first instance, to  to be the Ottoman Sultan. For King, the man reaching out towards Christ is Judas and in relation to this figure he finds a reading in the acrostic that points to Giorgios of Trebizond for whom Bessarion had particular contempt. Like Giorgios, Bessarian also came from Trebizond and up until 1461, this had been the last remaining bastion of the Byzantine Empire. Unlike Bessarion however, who had fled Turkish rule, Giorgios had changed his alleigances. Not stopping there, Giorgios had further pronounced Muhammad a second Plato, only more sensitive than the first. Not surprisingly, Bessarion saw his former friend and fellow countryman as a two-fold traitor and in 1464, declared him a caluminator of Plato. This initiated a bitter controversy between the two, in which Regiomontaus, loyal to his patron and as an astronomer also dismissive of Giorgios, also became embroiled, with his own work on Ptolomey’s Almagest playing a part in undermining Giorgios‘ credibility.

The frontispiece of the edito precepts of Regiomontanus‘ Epitome, printed in 1496. The two men below the armillary sphere are Ptolomy (left) and Regiomontanus (right)

In Rome, after finishing his Epitome of the Almagest, Regiomontanus wrote his work on spherical trigonometry and made a sundial for Pope Pius II.

Image coming soon

In 1467, he then went to Hungary where he worked for Matthias Corvinus and made an astrolabe for him which has since been lost. In 1472, Bessarion died and three years later, continuing the feud between his former patron and his renegrade countryman, Regiomontanus announced plans to publish a book exposing the flaws in Giorgios‘ „worthless“ commentary on Ptolemy. In the same year however, he was invited by Pope Sixtus IV to Rome, to work on proposals for a reform of the calendar. Travelling from Nuremberg in Rome, Regiomontanus either fell victim to a plague, or following the interpretation given by King, was posioned by the sons of Giorgios of Trebizond and died, only forty years old, in 1476. For the full story, either order the book or read a summary at: www.davidaking.org.

VIII. PIERO DELLA FRANCESCA  AND REGIOMONTANUS?

In Piero della Francesca’s The Story of the Finding of the True Cross, convincing likenesses are given of both the Byzantine Emperor Johannes VIII Palaiologos, who in the Battle between Constantine and Maximus is depicted as Constantine and of Bessarion, who in the Meeting of Solomon and Sheba, is shown as Solomon. Given the prominent nature of the theme and the acute crisis faced by the Church with respect to the falling Byzantine Empire, such references to the present are to be expected and underlie the fact that Bessarion was associate with the Bacci family who commissioned the frescos. Given the close links between Bessarion and Frederigo da Montefetro and the fact that Piero della Francesca does appear to have been in Rome and Urbino when Regiomontaus was in Italy, it is possible that they met. Independantly of this, on his last journey to Rome, Regiomontaus may have seen Piero’s work on perspective which, generally reckoned to date from the 1470’ies, was held at the Ducal Library in Urbino. Here he will have been able to read of how, using a properly constructed system of perspective an artist can reach into the world of Platonic Forms and can correctly render the appearance of an object from any desired viewpoint. He will also have read of a „magic figure“, a form of mathematical collage that summarised the system of projection used in a painting. Functioning as a form of key, this contained all the parameters necessary in order to establish a mathematically proven and true relationship between the space depicted, the pictorial plane on which the depiction was located and the viewer on the other side of the plane. On the diagram below, the green square formed from the lower edge of the picture, appears on the picture plane as a red trapezoid.

In previous works on perspective this two way relationship had not been fully worked out and in his De prospectiva pingendi, Piero established what is now known as projective geometry. The diagram given by Piero above thus assumes a viewer at a distance, d, away from the painting and with an eye, V, that is at a height, h, above the plane that reaches from the lower edge of the paining back to the horizon with a vanishing point, U.

Each painting has its own key with the magic figure for The Flagellation showing how the eye of the beholder is intended as being much lower down than in the example given above.

Nevertheless the key is an abstract summary and has no direct relation to the appearance of the picture or its elements. Philosophically however, it is of immense importance as it summarises the proof given by Piero that in creating a perspective system, the painter does reach into the world of Ideal Forms described by Plato. The painter who understands perspective is thus on the same level as a philosopher or a mathematician. As partial embodyments of the Good, the Ideal Forms lead towards God, yet they are not God and cannot be God in his all emcompassing, unifing aspect. This leads back to Piero della Francesca’s The Flagellation whose original title was, Convenerunt in unum, or „Coming together in unity“ with the subject matter being of how through the various roles of the figures in the picture, the Christan story and mystery unfolds. This, the picture suggests, extends down into the present with inaction and betrayal, whilst being contemptable, in some way also being necessary. Similarly, the loss and suffering of loved ones is also part of some divine, greater mystery that we can only partially, if at all, apprehend. This partial apprehension leads back to Cusanus who in his revised positions concerning the gulf between that which is circular and all-encompassing and that which is straight, did find ways by which the transcendent nature of the circle could be grasped in finite, step by step ways. Yet Piero also had a way of squaring of the circle and moreover used it in his art. This has been empirically established by Perry Brooks (in an article and in his doctral thesis) and with respect to Piero’s The Resurrection has been proved by Alexander Curtis (article forthcoming). In The Flagellation,  this is also a theme and when the perspective system used by Piero is reconstructed and a plan is made of the architecture and figures, Christ will be found to be standing at the centre of a large circle surrounded by large squares formed by patterns of square tiles.

A plan view of Piero’s The Flagellation as reconstructed by B. A. R. Carter in 1953

From this, an indication of a squaring of the circle can be extracted which, given the other circles that in his paintings that Piero did square, suggests that this is what Piero had in mind. In all cases the squaring of the circle is symbolic and is implimentated so as to create a sacred space in which the representation of Christ or the saint concerned, can be seen as forever reaching up to God on High.

As discovered by Holzschuh, the distance between the two outermost columns in the left hand side of the painting is cut by the column to which Christ is tied, by the Golden Section and as established by Alexander Curtis, this is how Piero squared the circle in his The Resurrection. Where the magic figure of De prospectiva pingendi gives access to the world of Ideal Forms and metaphorically speaking, opens the door, it does not show the way thereafter and so does not provide a direct path to God. Symbolically and geometrically however, the way is indicated through the Golden Section and had Regiomontanus and Piero della Francesca met, they may well have talked about this and how the divine could be approached through art and astronomy. That such themes were talked about, accounts for the otherwise unexplained presence of Cusanus‘ Quadratura circuli amid a collection of works concerned with practical astronomy, the calendar and the making of instruments.

IX. FROM CENTRAL ITALY BACK TO KREMS AND THE WACHAU

In the decade following his De prospectiva pingendi, Piero wrote Libellus de cinque corpri regularibus, or „Booklet on the Five Regular Bodies“. This was dedicated to Frederigo’s second son, Guidobaldo. In the dedication he expressed the hope that it might be placed in the studiolo next to the work he had written on perspective.

This was inlaid on all sides with intarsia. Depicted are the objects and insruments of humanist learning as well as such things as the helmet worn by a Christian knight and exotic things such as paraquetes in a cage, all shown as a trompe l’oeil in stunningly realistic perspective.

In his De prospectiva pingendi, Piero had shown had a mazzochio was to be drawn and in the Duke’s studiolo there is also a mazziochio. This was a tube that was made from sown together squares or rectangles of cloth and which stuffed and closed around upon itself to make a circle, was the basis of a Florentine form of hat that for artists was particularly challenging to draw.

As establised by Euclid, the five regular Platonic Solids are the only regular solids that can be and from this it follows that all other geometric solids must derive from them, including the thirteen semi-regular solids known as the Archimedian Solids. As Archimedes‘ treatise on these solids had become lost, all that was know about them was their number and appearance. However in his Libellus de cinque corpri regularibus, or „Booklet on the Five Regular Bodies“, Piero showed how five of these lost solids of Archimedes could be constructed from out of the Platonic Solids and this again is something that he may well have discussed with Regiomontanus had they met.

In making Archimedean Solids from Platonic Solids, Piero cuts bits off the edges of the latter using a variety of techniques, one of which is division by the Golden Section. Yet nowhere in the Libellus does he say how a division according to the Golden Section is to be achieved. In a stunningly consumate mathematical bon mot however, that is replete with unspoken conotations of symbolic significance, Piero gives, without specifically mentioning it, the method of constructing the Golden Ratio that he used in The Ressurrection in order to square the circle. Written in Italian and then translated into Latin, the Libellus was translated back into the vernacular by Piero’s townsman and fellow mathematician, Luca Pacioli. Printed in Venice, at some stage, under what appear to be irregular circumstances, the Libellus was bundled together with Pacioli’s Divina Proportione of 1509 and in this unusual and ad hoc form it would appear to have made its way North to Krems. There, in around 1520, when the burgesses of Krems decided to comission a sculpture for the hospice that they ran for the poor, the artist chosen for the commission used the geometry of Piero’s bon mot as a way of structuring the work, not only in its frontal elevation but also in plan.

X. THE PLANISPHERIC ASTROLABE

During the fifteenth century, Vienna was not only the European centre for astronomy but was also the place where the best astronomical instruments could be obtained. From the period, over forty instruments have survived including some twenty astrolabes. 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 available.

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.

In the stereographic system of projection on which the planispheric 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 symmetrical 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 integral 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. 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 calibration 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.

An astrolabe made by Alexander Curtis