The collection of Mathematical Machines of the University Museum
Marcello Pergola

The collection of mathematical machines, stored in the Museum, is in progress. They models have been built on the base of descriptions from the scientific and technical literature from the classical Greece to the start of the XX century and after a series of experiments about a possible didactical use of them. The didactical purpose has determined some features of the collection, that is, first of all, an anthology from a much larger set. There are several advantages in the didactical use of these artefacts : they arouse interest ; they reinforce intuition and imagination ; they deepen the relationship between mathematical models and reality ; they foster the search and the production of proofs ; they address new or uncommon issues, related to movement ; last but not least, they lead in a spontaneous and natural way the users (before all, but not only, teachers and students) to immerse themselves in the historical dimension and to reflect on the relationships between mathematics, society and culture. On the one side, the users avoid the risk of belittling history and of destroying their own past ; on the other side, they meet several problems, that will be outlined below.

Because of their mathematical nature, the machines of the Museum could be substituted with virtual models (i. e. computer simulations). Yet the situation is deeply changed : from an epistemological perspective there is a shift from the relationship between concrete and mathematical models to the relationship between two different kinds of mathematical models. Besides, the physical manipulation of concrete threedimensional objects is much richer and more suggestive than the mouse manipulation of a virtual object : a real (or imagined) physical manipulation is almost always the base on which the computer simulation is built. It is much better to place physical and virtual models side by side and to experiment with both.

As every science, mathematics addresses some field already available in a prescientific form of access and relationship. It remains 'captive' of that field, but in a continuos comparison with other sciences and activities, it progressively differentiates, states, enriches themes, methods and languages and gives sense to its own research programs.

The artefacts of this collection offers a meaningful example of that. Whether used by practitioners or by theoreticians, they have had complex relationships with forms of conceptualisation and contents of mathematical knowledge. Though deeply different (because of their concreteness) from mathematical objects, they are close and develops side by side with them. Symmetrically the communities of mathematicians are always well distinguished from the community of technicians (artisans, engineers, artists, merchants, and so on) ; yet both are rolled up by a dense net of communications and exchanges.

Machines and instruments constitute one of the contact (or friction) plane between science and technology : there is always the tendency to find an equilibrium point, by reducing each one to a language ; yet there is always the possibility that non scientific realities influence on the formalised scientific thinking.

Surely a collection centred on geometry isolates - in the immense universe of machines - very particular regions ; yet it illustrates - in this microworld - much more general historical events. It point out some of the threads by means of which the mathematical thinking - that cannot be completely reduced to sensible world - has been always linked to concrete activities, actions and operations : it is true also today, because of the strong interlacement between instrumental and theoretical aspects.

The models come from a time range that is too wide to allow a detailed individual observation. In a few exceptional cases they are production tool, because their main purpose is to embody a law or an abstract scheme : yet some important changes that have characterised the development of the concept of machine are represented by them. We refer to :

a) the shift from the machines meant as artificial organisms (sometimes with magic connotations) to the machines where the difference between natural and artificial is smaller (like in Descartes) ;

b) the shift from the machines designed as a whole and built as unique pieces by skilful and cultured artisans to the machines decomposed analytically into elementary parts (like in the careful drawing of Renaissance engineers) ;

c) the discovery of the contradiction between the precision needed to build machines and the machines needed to build precise pieces ;

d) last, but not least, the separation between physical and conceptual structure of machines (at the beginning of the XIX century).

In Euclid's geometry the straightedge and the compass are the more precise calculating instruments for problem solving. The theory develops a mathematical model of the activity with these instruments. There are strict criteria: the constructibility by straightedge and compass becomes the existence criterion for geometrical objects ; the concept of infinity is admissible only when it can be reduced to well controlled iterative procedures. However this criterion of constructibility concerns only the existence of the object : its properties - i. e. the geometrical 'truths' are found and not constructed (this believe is maintained by mathematician for a long time even in modern age). We know that the Greek geometry has known also other instruments ; yet the solution found by means of them have been considered as provisional. In this theoretical choice we see surely the scarce consideration, if any, that the Greek philosophers had for practical activities and the influence of Aristotle : the prohibition of mixing different genera has been an epistemological obstacle for a long time, in those problems that required the comparison between space and time or the ratio between non homogeneous magnitudes or the fusion (composition in a modern sense) of different movements. In this way the separation between geometry and mechanics, typical of the Greek thinking, is coded. This can explain why in the theory of conics the method of sections is preferred against other generation techniques (that were known however) ; why optics is conceived as a pure geometrical theory ; why statics is given a great privilege.

When, in the XVI and XVII centuries, the mathematical studies are reconsidered by the humanists, the cultural space is still characterised by forms of thinking rooted in the medieval tradition. Some important novelties arise from outside the institutions of the official culture and introduce further tensions and contrasts, forcing deep transformations of the scientific (and mathematical) thinking:

At that time, the geometry is reconsidered with deeper links with concrete reality than the Greek geometry. It is difficult to understand that Euclid geometry is a model ; it is difficult to accept Euclid's constructivism. In particular it is belittled Euclid foundation of the concept of ratio of magnitudes. In this context the meeting between geometry and algebra happens. Algebra is developed in the field of the new commercial activities rather than in the universities and studied by practitioners rather than by theoreticians. Algebra carries into mathematics several linguistic and technical innovations and a more unbiased mentality and a different way of structuring reasoning.

On the one side, algebra offers a powerful means to simplify the exposition of proofs, when scientists are engaged in showing their superiority to the ancients and their innovative capability. On the other side, algebra contributes to the slow maturation of the awareness that mathematical expressions have a formal character, that mathematics is an artificial language, a net of relationships rather than a set of objects together with their absolute properties. Surely the imaginary of mathematicians is still (and for a long time) geometrical : yet the process of arithmetisation of theories is - after Descartes, quick enough. The analytical style, first complementary to or integrated with the synthetic style, later becomes autonomous and elaborates its own criteria of rigour. The use of machines - especially curve drawing devices - has played a fundamental role in the theoretical and historical development of analytic geometry, of algebraic symbolism, of calculus, of the concept of function.

Another important novelty is linked to the techniques for plane representation of the threedimensional space (i. e. perspective and graphical methods for cutting stones), studied and developed in the XV and XVI century in artist workshops and in building and military sites or shipyards. The success of perspective in the art of the XVI century, its importance and the enthusiasm aroused depend on reasons deeply connected to the culture of the time. In projection, the image is determined by the distance and the position of a point of view : this perfectly corresponds with the vision of the world in a period that had introduced an historical distance - comparable with the perspective distance - between itself and the classical past and that had posed the human mind in the centre of the universe, as the perspective posed the eye in the centre of the graphical representation. The frontal position of the eye is recommended as opportune in most of treatises, because it highlights the mathematical organisation of space through the harmony and the beauty of proportions ; besides it points out the presence of a subject who, through the laws that regulate such harmony and being himself part of that unitary harmony, acts on reality. The power of the artist is similar to the one of the Prince, who knows how to construct and to maintain his domain by means of the scientific knowledge of vices and virtues of human beings and of the ways how the Fortune may be directed for his advantage. It is a secular power, that belongs to the earth, but is comparable to God's one, because it can create. In this way, in the Renaissance works the subject and the object imply each other or rather, the object (in particular the object of science) is a function of the subject. The formulation of rules implies transgression. In the rule itself there is the possibility of a licence. In the Renaissance painting, the man is in the centre ; anamorphoses and eccentric perspectives (where the artist's eye is lateral) are not only an exercise, but also a sign of a philosophical crisis. The same law that rule harmony and beauty may hide them. If the representation of an object must be different in order to give the observer the right image from a given point of view, how could we trust what we see ? Even if the new science allows us to gain partial truths (such as the unity of conics, that are anamorphoses of the circle), who, but God, could grasp the whole (Pascal)? In such a destructured and homogeneous space, how could the subject overcome his solitude, while searching the foundations, the system of reference, the origin of every science (Descartes) ?

The interest in perspective has conditioned the development of pure geometry in the XVII century. The mathematical conceptualisation of the graphical activity starts very early (Piero della Francesca). It becomes autonomous very quickly, with the study of the empirical practices and of mathematical instruments used in artists workshops. The function of machines has been twofold : a direct one, because the analysis, the design and the construction of 'automatic' instruments for drawing is strictly interlaced with the early rigorous formulations of projective geometry ; an undirect one, because, the need of describing the machines has induced the scientists to develop techniques for plane graphical representation, that is much more effective than word description.

The third event is the progressive refusal of the Aristotelian concept of science, that belittled the mechanical arts. This very event is actually the condition for the expression of the two above. The revaluation of mechanical arts (that starts in the XV century) is related to the growing social importance of technicians and to the birth of a new figure of intellectual, i. e. the artist-engineer. A great role has been played in this process by the Renaissance courts (where a lot of technicians worked together), the commercial economy (which required the production of navigation instruments, astronomical observations and techniques for calculating), the exchange of information between artisans, artists, technicians and scientists. There is the tendency to melt artistic and technical activity with scientific knowledge and to overcome the contrast between active and contemplative life. In this period the interest in machines is manifold : they are means to master nature ; they are intelligence and ingenuity tests ; they are status symbols ; they are tools within abstract theoretical constructions.

This interest, that concerns the whole society, contributes to the process of mathematisation of nature. This process has two main phases. The first concerns the Renaissance naturalism, where mathematics is not only a human activity but also the language of reality. The second concerns the refusal of the dualism between the physics of the earth and the physics of the heaven and the mechanistic image of reality. The machines play a role in both phases : in the former they maintain a symbolic and magic meaning ; in the latter they realise mathematics, offering it the motivation or the final goal.

The small collection of geometrical machines has a peculiar role. The expert eye of the user can conceive Euclid's drawings as machines, by moving them in the mind : this activity suggests new constructions and new mechanisms. On the one side, the machine may come before the theory, offering the way to overcome the prohibition and the obstacles coming from the cultural tradition ; on the other side it may condition the development of the theory, changing the meaning and the functions of the law that control the object (examples will de discussed below). The machine may be real or mental. The process of fusion between mathematics and mechanics is so advanced that the destiny of the machines and of the mathematical objects is shared : while the cultural space changes, both machines and mathematical objects (even if they maintain their sensible aspect, if any) change in the same time.

The machines of our collection that have been built until now may be divided into five classes (with wide overlapping):

1) The Geometry of Conic Sections ;

2) Projection and Perspective ;

3) Transformations ;

4) Curvigraphs ;

5) Mechanical Solution of Problems.

A first group of models illustrates the classical theories of Menaechmus and Apollonius. They differ in two aspects. Menaechmus uses only cones obtained by rotation of suitable right-angled triangles ; the cut is done perpendicurlarly to one of the sides of the axial triangle ; hence to obtain all conics, different kinds of cones are needed. Apollonius on the contrary uses a generic cone and cut it by means of planes with different inclination ; hence from the same cone we can obtain all types of conics. Symptoms (i. e. characteristic properties) are obtained by both authors with reasoning in threedimensional space, but Apollonius interprets them in the plane by means of application of areas : in this way the standard names (ellipse, hyperbola and parabola) are invented. Several problems remain open : the identity between the sections of cylinder and ellipses was questioned ; the two branches of hyperbola were not conceived as parts of the same curve and so on. The models of this group are statical but we, on the basis of century tradition, can transform them into mental machines in order to introduce movement.

A second group of models contains machines that draw curves in the plane. The threedimensional ones are a direct mechanisation of the classical definition. The others are based on the knowledge of the symptom to be used. The machine is built in order to obey the symptom. In this way the symptom changes its status : the symptom is no more a static truth but it works and constructs the conic. Besides every relationship with the underlying cone is cancelled : the points of the curve are positioned in the plane with respect to the fixed part of the mechanisms, that has the function of a system of reference. In some cases (like in the machine of Paciotti), the same machine can be used to draw all types of conics by means of small continuous movements. In this way two important issues are focused : the unitary nature of conics and the importance of the intuitive concept of continuity, that will be used widely in the XVII century.

The theory of conics is developed in several directions, by means of the new kinds of plane representations of threedimensional objects ; by the simplification of Apollonius' treatise ; by the recourse of the application of algebra to geometry. In this last trend, new properties are stated and written by equations : they are then transformed into laws that can force a writing point. Hence a process is started. Conicographs become mathematical machines in a threefold way : they embody a geometric property of the drawn object ; they may be mental ; they depend on the geometric theory. It happens very often that for every new theoretical 'discovery' a new instrument can be designed : some mechanisms that work in orthogonal conjugation are reconsidered in oblique conjugation ; the properties of directing circles suggest to build mechanisms that uses the rhombus properties. Sometimes old instruments are given a new theoretical status, that allows to understand better the relationships between each other. In the case of organic geometry (Newton) the genesis of conics as trajectories of the intersection point of two movable straight lines is better understood by means of projective concepts that have developed independently.

From the XVI century on, the entrance of movement into mathematics is very large : on the one side, mathematics and mechanisms are no more separated ; on the other side, mathematicians tends to become more specialised and to conceive machines as mental models, leaving to technicians the solution of practical problems in the construction.

Another group of models illustrates the theorems of Dandelin on conics. This author (in the XIX century) reconsiders the point of view of Greeks, putting back the conics on the cone and finding new results. He recurs to intuition and to the principle of continuity ; besides he generates an interesting family of continuous plane curves (the focal curves, studied by Quetelet and others). His studies confirm some previous observations : they suggest the construction of new machines and put in a wider theoretical frame some machines already known to generate some curves (e. g. the Newton's square).

When Dürer visited Italy in 1506, he was looking for a rigorous theory of perspective : the results of his studies and of the meeting with some artists was condensed in a few pages and four famous images of his treatise. The four machines by Dürer are reproduced in the first group of models, that contains mechanical instruments for the imitation of reality. A second group of models contains instruments that depend on geometrical theory and could not exist without it.

As far as the first group of models is concerned, the following observations can be made :

1) the technical variations between the instruments aim at overcoming practical difficulties and at reaching higher level of automatism, however, in spite of the increase of mechanical refinement, they were later substituted by dark rooms and similar that are more precise and easy to be used ;

2) the instruments for perspective maintain a magical status, due to the scarce knowledge, if any, of the mathematical laws that describe the degradation from the space to the plane of the picture ;

3) the instruments for perspective were used by amateurs rather than by professional painters. As amateurs required simple rules to be used, a lot of treatises of different levels - from very simple and prescriptive to very difficult and even abstruse texts - were produced.

For the second group of models, we can say :

1) a strict link exists between the manipulation of mechanical instruments and the theorem of Stevin : If the picture plane rotates through the ground line and if the observer rotates through his foot in the same direction being parallel to the plane, the perspective is not changed and will remain also when the picture plane is stretched out on the horizontal plane. The Stevin movement is used by De La Hire to flatten a cone (together with its section) on the plane of its circular base : hence conics are supposed to be transformed of this circle. The genesis of geometric transformation concerns always the set of points of particular figures and only later involves the whole plane ;

2) Lambert's machines offers a solution to the problem of representing the transformed of any plane figure in the easiest and most automatic way ; they are not useful in practice (because of their size and scarce precision, but are theoretically important, as they explain the influence of the mechanist theory of the XVIII century on the statement of the properties of plane homology.

A third group of models concern anamorphoses. The historians have analysed the links between the production of anamorphic images, the start of tendencies in painting, the love for automata and the cartesian philosophy. From a mathematical perspective, plane anamorphoses do not add anything to what is known about standard perspectives ; it is an exasperated use of the same laws. Yet, because of the lack of formalised algorithms, there is more and more need to refer to empirical procedures. The anamorphoses obtained by mirrors are completely new and very difficult to be studied : in this case too the use of mechanical models proves to be essential.

3.3. Transformations 

The importance of the concept of transformation in the XIX century is related to the development of projective geometry as an autonomous and organic research field. From the very beginning (Desargues, Pascal, De la Hire, and so on) the invariance of some properties of geometric configuration under projection is related to either practical problems or to movement and continuity. Several trends of research meet in the theory of transformations, such as Bravais' works on crystal structure, Jordan's work on the groups of movements, the study of the relationship between affine geometry (Euler), the mechanics of deformations and the barycentric calculus; Helmotz's and Lie's studies on the movement of rigid bodies. At a more elementary level, isometries have played a central (often implicit) role in geometry, from Euclid on; their group structure has been used in practice before the abstract notion of group has been clarified.

In the XIX century, the mechanical engineering became one of the dominant technologies: the attention was caught by the study of articulated systems and linkages, that realise the transmission of movements. Actually the abstract theoretical apparates (which in spite of the links with concrete, are intellectual inventions) renew or change the gaze that observes and describes reality. Hence the theory of transformations and invariants threw a new light on the analysis and the design of machines. Elementary examples are given by the Scheiner pantograph and the Peaucellier inversor. The study of linkages is still now a borderline topic between theory (algebraic geometry) and applications (robotics and computer science)

A part of the models shows the most elementary plane transformations (isometries, stretchings and homoteties). The corresponding points are represented by a directing point and by a tracing point, which both have two degrees of freedom.

On the one side, these instruments should be conceived as pieces of machines to assemble together in order to build more complex mechanisms; on the other side some of them are composed by more elementary linkages. Some instrument may be conceived as the particularisation or the generalisation of another.

All the mechanisms are 'local' instruments: they determine a correspondence between limited plane regions, whilst geometric transformations are defined, globally, for all the points of the plane. The geometric transformation determined by the instrument has no direct relation with the physical movement of the linkage; yet, by exploring the linkage, some conjecture about the type of transformation can be stated and later proved in a rigorous way. These features make linkages for transformations a suitable field of experience for students at the secondary and tertiary level.

Other models draw the attention on the strict links between the threedimensional space and the plane. For instance there are models which illustrate the relationship between stereographic projection and circular inversion; or between the natural phenomenon of sunshadows and affinities.

We have already quoted some conicographs. Now we refer to algebraic curves of any degree and to transcendental curves. The topic is immense. Algebraic curves have often very pleasant shapes, but, what is more important, they have constituted an exercise field for the genesis of several basic concepts (in geometry and calculus) and the invention of algorithms to solve difficult problems.

In order to draw mechanically, by continuous movement, arches of a curve, it is possible to use some property of the curve, that can be embodied by some instrument. Hence the instruments that draw the same curve might be considered equivalent to each other in some sense: in the classical perspective, the whole of instruments which draw the same curve characterises the 'nature' of the curve. This idea that each mathematical object has a nature is questioned only in the XIX century. Now we have an instruments, the computer, which can draw any real curve, leaving out all the geometrical properties and focusing only on the numerical relationship between sets of real numbers.

A curve might be obtained by applying a suitable transformation to a known curve. This is the case of the solution given by Peaucellier to the problem of designing an instrument which could draw a straight line. The reverse too is true: sometimes the study of curves and curvigraphs has lead to invent linkages for transformations.

There are other techniques to obtain a new curve from a known one, which can help to design mechanisms. For instance, the pedal curve, but several other examples are available.

The same curve can be obtained by others in several different ways: in this way a net of relationships between algebraic curves can be built.

In geometry a fundamental role has been played by 'mental machines', that are documented in Descartes Géomètrie too. A famous theorem that concerns this kind of theoretical instruments was proved by Kempe in XIX century and exposes a general method to describe plane curves of the nth degree by linkwork.

The analysis of this kind of mechanisms consists of two complementary activities: first, the comparison of mechanisms which describe the same curve, to discover hidden equivalence and to find the geometric properties of some elementary figures, that, by movements, become versatile elements of complex machines; second the study of the 'biography' of the mechanism, that partly overlaps the biography of the related mathematical object, even if the change in the theoretical system can change the status of both. The transcendental curves have a very interesting 'biography' that dates back to the classical Greece and that shows many interesting contribution in the development of calculus.

In the last group of models we have collected instruments which were designed to solve some important problem, that has been studied for centuries and has played an important role in the development of mathematics. We refer for instance to the problem of the trisection of angle and of the duplication of cube. Some models are prototypes of whole families of instruments aiming at the same goal. Other are more related to intellectual games and may well represent the cultural atmosphere of a certain age.

The List of References is appended to the Italian version.