See also: Module

A module is a conventional Measuring unit adopted to regulate various the Proportion S of a unit (Construction, Machine…). It corresponds to the smallest commune measures which dimensions of the elements must have entering the composition of this unit so that they can be superimposed, to combine or be juxtaposed without final improvements. In Greek, the module is indicated by τόυος, the your ; the Latin origin modulus , of modus indicates the gives rhythm , the measurement. The term can also be employed in the direction of standard , of gauge or of gauges. Lastly, by extension it indicates also a element, a unit constitutive of a unit.

If it is difficult to precisely know the origin of this concept which is not simple to conceive, its constancy through the times, in variable forms, and especially the remarkable use which was made by it, invite to study it.

Principle of the module

That is to say $M\; \backslash ,$ the module, conventional measuring unit. Starting from $M\; \backslash ,$, one determines various dimensions $Di\; \backslash ,$ of a unit. For Vitruve, the modular rate/rhythm includes/understands

• the symetriae : $Di\; =\; nor.\; M\; \backslash ,$

For example, in the Doric order, the Entablement is worth 3,5 modules, this one corresponding to the semi-diameter of the columns which is measured with the birth of the was.

• the proportion : $Di/Dj\; =\; M\; \backslash ,$

For example in the Doric order, “the entablature is with the column like 1 to 4” When $M\; =\; \backslash \; varphi$ (Golden section), the two sizes $Di\; \backslash ,$ and $Dj\; \backslash ,$ is known as in the divine proportion.

Consequently there is nothing astonishing with what is called module , the ratio of two sizes, like characterizing physical properties or mechanical:

1. Quotient of the diameter by the number of teeth = module of gears
2. Relationship between the pressure which is exerted on a body and the reduction in the unit volume which results from it = module from compressibility
3. annual Medium flow (liter/second) by km = specific or relative module of a catchment area
4. mechanical Contrainte $\backslash \; sigma\; \backslash ,$ which would generate a lengthening $\backslash \; varepsilon\; \backslash ,$ of 100% the initial length of a material = Modulus Young noted $E\; \backslash ,$ and such as $\backslash \; sigma\; =\; E.\; \backslash \; varepsilon\; \backslash ,$ and which characterizes the rigidity of a material
5. comparative Diamètre = module of the medals or currencies

There is nothing of astonishing either to find this term to indicate a component of a unit: Lunar module, Module of order, module of a Formation or plug-in of a Software etc

The Colorado beetle of Polyclète

In sculpture, the gun is a whole of rules being used to determine the ideal proportions of the human body.

The theory of the gun of Polyclète is one of the bases of the Greek classicism: it applied it to its virile statues like the Diadumène and the Doryphore with which Polyclète had undertaken to show, by a “ statue of which all the parts would be between them in a perfect proportion ”, which are the reports/ratios of size in which nature placed the perfection of human forms. It achieved so well its goal which the statue which it gave like example and as model was regarded as a chief of undeniable work. In this work, the head between on the whole seven times in the body, twice between the knees and the feet, twice in the width of the shoulders and twice in the height of the chest.

Greek mechanics: establishment of the proportions to the module

Genesis of a technical corpus

If one loses oneself in Conjecture S about the origin of the technique, the origin of the Technologie seems to correspond to the advent of the treated which supposes a beginning of rationalization to transmit a know-how. From the VI E century before J.C a filiation is established, a tradition which makes it possible to transmit the technical asset of one generation to the other. In this diagram, individual know-how is gradually integrated in a Corpus which becomes accessible to all.

Formation of the technical spirit

In the formation of the scientific spirit , Gaston Bachelard distinguishes three decisive stages which can apply more or less to the techniques Greek:

1. the primitive stage in which the observations can lead to nothing because they butt against insurmountable difficulties
2. the intermediate stage in which one knew to distinguish from the basic elements. It is precisely the stage which makes it possible to release either a module for construction , or a applicable formula.
3. the decisive stage in which the technical fact is connected to an abstracted scientific system. The Greek science will be primarily axiomatic and will not allow to reach the dream mechanics to give him a true mathematical formulation.

Turns of Dyads

Diadès was pupil of Polyeidos and engineer of Alexandre Large the. He composed a treaty of machines of war lost today and gave himself like the inventor of transportable turns as well as various machines the such Trépan, the corbel, the flying bridge. The turns were height variable and made up of several stages bordered of a covered way, and of which the height was decreasing. The tradition affirms that one took care to always preserve same the proportions of dimension, of materials in the turns various heights. After “experiment” and reflections, one had thus arrived at quantified notations, applicable to all the cases. For each machine one had tables of proportions which had to be followed rigorously: it is there the first known use of the principle of the module .

Philon of Athens and proportions of the temples

Philon of Athens lived at the end of the 4th century. According to Vitruve, he would have worked with the temple of Cérès and Proserpine, with Eleusis. He would have written a treaty of Poliorcétique and a treaty on the proportions of the temples now lost and which one finds the mention in the work of Vitruve.

The Archimedes' screw

Born and died in Syracuse, Archimedes does not belong in a strict sense to the school of Alexandria to which it is however close as well by the problems studied as by the methods employed. At the time of the author, the Archimedes' screw existed probably for a long time and attribution with Archimedes is due to an erroneous comment of Commandin at the 16th century.

This mechanism provides us another example of the use of the module by the Greek mechanics: in the construction of the screw, one must respect proportions which are stated on the basis of module which, here, is the length of the screw. The diameter of the screw represents 1/16e of module, the step of the propeller 1/8e, the diameter of the cylinder envelope is equal to the step of the Hélice. The slope of the unit must be 3 heights for 4 basic what represents the triangle pythagorician

Philon de Byzance and machines of jet

Philon de Byzance, which lived with Alexandria and Rhodos, is the first mechanic whose work mainly reached us. Its treaty of the machines of jet clearly shows the evolution towards the nevrobalistic artillery. He teaches us that the first engineers who dealt with improving these machines acted only by Empirisme because “ the old ones had only conceived the form and the general provision of these machines, they did not obtain remarkable ranges because the proportions which they used were not well adapted. Their successors, removing die, adding die, made these instruments harmonious and effective ”

These first technicians did not have what Philon de Byzance called the element first and which made it possible to determine dimensions of each machine component, thus ensuring the best proportions to him. Thus measurable basic elements, comptabilisables appear and which represent the base of a conceptualization. One then attends the passage of the exceptional machine to the rational, standardized, indefinitely reproducible and finally banal machine.

With this intention Philon de Byzance establishes an elementary relation between the energy available, i.e. produced by the beams of fibers elastic, and the weight of the ball. To determine energy, it is based on the bore by which pass the elastic fiber beams. The cubic root of the weight in drachmas of the projectile, increased by a tenth, represents the bore of the frame expressed in fingers (measuring unit). An algebraic translation (the Greeks did not control the algebra) would give $d\; =\; 1,1\; \backslash \; sqrtp$ with D the bore of the frame and $p\; \backslash ,$ the weight of the projectile. A table allowed the use of the module , that Philon de Byzance call the your : each part then represented a multiple or a fraction of the module. Thus the Euthytone (catapults) was traced in a square whose sides had 16 modules and the Palintone (Baliste), in a Isosceles triangle of 19 basic modules and height. If we do not have the tables associated with these machines, Philon delivers some examples to us:

• For a weight of ten mines (that is to say 1000 Drachma S), the diameter of the attic windows was to be of eleven fingers
• For a weight of a Talent, it was to be of 21 fingers.

It was the same for the machine components, the Péritrète, the Barillet, the thickness of the Moyeu, the Hypothème (support), the arms, the length of the cord archère, which was double among that of the arms. The module becomes here measuring unit and Philon de Byzance will specify that “ no one did not dare to deviate from the form ”. These machines become easy to dismount, to store, see to repair because their mode of design allows the appearance of the spare parts.

Héron of Alexandria or the tradition moving

The treaty of the machines of war of Héron of Alexandria distinguishes two types of machines of jet:

• the machines euthytones which launched only features (scorpions)
• the machines palintones (triggerfishes) which launched features or balls of stone (lithoboles)

Héron of Alexandria takes again exactly the formula of Philon de Byzance for the calculation of the module from which all the machine must be built. It adds to it the formula for the machines which launch features: in this case, the bore must be equal to the ninth length of the feature. Thus for a feature of 3 bent, the module will be of 8 fingers.

As at Philon de Byzance, one will find in his treaty a graphic solution of the famous problem of the Duplication of the cube being used for here to calculate the scale of proportion of two machines whose balls have their weight in a given report/ratio. Thus, for a machine which would launch balls twice heavier, that is to say $p\text{'}\; =\; 2p\; \backslash ,$, one obtains $d\text{'}\; easily\; =\; D\; \backslash \; sqrt2\; \backslash ,$, which corresponds precisely to the problem of the duplication of the cube, problem which consists in multiplying a dimension by $\backslash \; sqrt2$

From now on all is defined, all is dimensioned, all is put in tables that nobody can nor does not want to modify: this technique is now saturated.

Vitruve the compiler

The module was clearly described by Vitruve and it is in its writings that for the first time the term appears. However Vitruve seems to be only the agent of an already old tradition.

The man registered in a circle and a square, realized on the same drawing by Léonard de Vinci, illustrates a passage of the book “Of Structured” of Vitruve (Marcus Vitruvius Pollo, first century BC, credit under Jules César and Auguste) that the Renaissance republished and adulated.

The proportions of the man relate to only one relatively short passage (781 Latin words) in chapter 1 of book III. An extract of paragraph 2 clearly indicates the implementation by the artist of a modular rate/rhythm :

§. 2 “nature indeed ordered the human body according to the following standards: the face, since the chin until the top of the face and with the root of the hair is worth the tenth its height, just as the hand opened, since the articulation of the wrist until the end of the major one: the head, since the chin until the top of cranium, is worth a eighth; top of the chest measured at the base of the neck to the root of the hair one counts a sixth; medium of the chest at the top of cranium, a quarter. As for the face, the third its height measures base of the chin at the base of the nose; the nose, of the base of the nostrils until the medium of the line of the eyebrows, is worth as much of it; of this limit to the root of the hair one defines the face which constitutes the third third thus. The foot corresponds to a sixth height of the body, the front armlever with a quarter, as well as the chest. The other members also have specific proportions, which make them commensurable between them…. ” “The proportion is the report/ratio that all work has with its parts, and that they have separately, compared to the whole, according to the measurement of a certain part. Because, just as in the human body, there is a relationship between the elbow, the foot, the palm of the hand, the finger and the other parts, thus in the works which reached their perfection, a member in particular made judge size of all work” Chapter II Of what architecture consists

“The ordinance of a building consists in the proportion which must be carefully observed by the architects. However, the proportion depends on the report/ratio which the Greeks call analogy; and, by report/ratio, it is necessary to hear the subordination of measurements to the module , in all the whole of the work, it by what all the proportions are regulated; because never a building could not be well ordered if it does not have this proportion and this report/ratio, and if all the parts are not, the ones compared to the others, like are those of the body of a well trained man”

If thus nature so much composed the body of the man, that each member has a proportion with the whole, it is not without reason which the old ones wanted that in their works this same relationship of the parts with the whole was observed exactly.

But among all the works whose they regulated measurements, they mainly attempted to determine the proportions of the temples of the gods, in whom what there is of good or made evil is exposed to the judgment of the posterity. The division and even the nomenclature of all measurements for the various works were taken on the parts of the human body; thus there were the finger, the palm, the foot, bent, etc, and these divisions were reduced to a perfect number, that the Greeks call telion. ” Chapter first of book III

Thus in ancient and traditional architecture, the module is the common conventional measurement of a ordinance corresponding in general to the semi-diameter of the barrel of the column in its low part.

Among the original elements noted in the work of Vitruve, one notices a remarkable extension of the practice of the module. For the machines of jet, the formula of the mechanics of Alexandria is adapted to the Roman Measuring units. The Temple S are built starting from modules with the definition of the architectural orders (ionic, doric). It is not to the ship where it is not question of module and which consists here of the interval of the ankles on which the oars take their support.

Illustration of the architectural orders in the encyclopedia of Diderot and D' Alembert. To note the reference to the module like scale of measurement:

. In 97, one entrusts to him the administration of the service of water of Rome and it will write a treaty of the acqueducs of Rome. All that interested the gauge and the calibration of the conduits related to with the first chief the administration: “ any gauge is determined by its diameter, or its perimeter or the measurement of its section ”. The table of Frontin is built starting from the quineria , pipe of 5, to the pipe of 125 which allowed a classification of the gauges: one then speaks about quinary module .

Thereafter, Frontin will note that the model is not necessarily homothetic, i.e. the reduction on a same all element scale of the unit. Thus for the module of the pipes of water conveyance , the scale of the modules as for the bore functions by arithmetic progression of the 5 to the 20. To the top, it proceeds like the series of the square roots of the terms of an arithmetic progression. Centuries later, James Watt will still observe it in the small-scale model of the machine of Thomas Newcomen.

Modulate and Arab penmanship: formula of workshop to the plays of the spirit

The true successors of the Greek mechanics were certainly the Arab which translated the Greek treaties before making use of it as bases for their own work. In addition, it seems natural that any civilization develops a “esthetic system” founded on the love of the harmony and who can cover a great diversity of forms. One of the great principles of esthetics, already stated by Plato, is that of “the harmony of the parts and the whole by which the unit of this last is essential on the multiplicity parts”. Contrary to arts of the western world, heirs to Plato and Aristote, the art of the Muslim world hardly shows interest for the study of the proportions of the human body. If, in Coran, “God created the man harmoniously” (XXXII, 9), the concern of the Moslem artists will not be precisely to compete with this divine accuracy. On the other hand, a field of art completely specific to the Muslim world is provided by the Arab penmanship. This one is initially developed for the copy of the Coran and the uses of the chancellery califale and will be codified in a “proportioned writing” (Al-khatt Al-mansûb) allotted to the Vizier Ibn Muqla (885/886-940). In his “treaty on the writing and the calame” (Risâlat Al-khatt wa-l-qalam), the author gives the bases of a system of proportions based on the letter Alif, in the shape of vertical pole and which is registered in a circle being used as standard (module). Each letter is then formed starting from this circle what will give the six styles of penmanship classical Arabic (naskhî, muhaqqaq, thuluth, riqa', rayhânî and tawqî), each one being characterized by the proportion of the letters compared to the alif one.

This very intellectualized design of penmanship will be taken again then by all the universities of penmanship. At the beginning of the 14th century, at the Mamelukes of Egypt or at the Mongolian of Iran, one observes in the production of manuscripts of prestige a development of the preoccupation with a page layout. Thus the format of the layers of paper often presents remarkable proportions: most frequent are has (1 X 1,414), the double rectangle of Pythagore (1 X 1,5) and more rarely, the right-angled gold (1 X 1,618).

Moreover, the field of the page is divided between the calligraphic rectangle, or spaces writes, and the margin, both answering precise reports/ratios. Written surface is divided by the ruling, the width of the page divided by an integer gives the number of lines then. Later with Villard de Honnecourt, or like showed it Rosa Viro with the bible of Gutenberg, the typographers will follow regulating layouts or ways of calculating to determine empagement work, in order to ensure a coherent distribution of the white and printed surface

Modulate, figures of reference and regulating layout

Many a constructions with the rule and the compass leads to the implementation of standard geometrical figures, such as the pentagon, figures associated to they-even with particular modules (Golden section in the case of pentagon).

In architecture it is possible to build volumes according to all kinds of proportions . The classical architecture gave three “averages proportional” to find the height of a part starting from its base:

*la mean arithmetic: $h=\; \backslash \; frac\; \{l+L\}\; 2$

*la mean geometric: $h=\; \backslash \; sqrt\; \{l*L\}$

*la mean harmonic: $h=\backslash frac\{2lL\}\{l+L\}$

In the theories of the Rebirth, the base will have proportions ranging between 1 and 2 while passing by 7 intermediate positions: 1/1, 4/5, 3/4, 1/√2, 3/2, 1/Φ, 3/5, 4/7 and 1/2. These proportions rise directly from Pythagore and Plato.

However these calculations are often only theoretical. Thus Andrea Palladio published the description of the buildings which it built in its treaty of architecture I quattro libri dell' will architettura (Four Books of architecture). In this treaty, he explains why he used these methods to dimension the parts and he dimension his drawings, but a precise measurement of the buildings shows that there can be rather important differences between the theory and reality. The plan of the Villa Rotonda is built according to the system of the geometric mean. More close to us, an architect Le Corbusier, who worked on the concept of traced regulating , invented a system of measurement based on the Suite of Fibonacci: the Modulor. If one takes an example, the Maisons Jaoul, of which the frontages are made up according to the Modulor , one realizes that actually theoretical measurements all are difficult to find in the building.

The module seen by the encyclopedists of the Lights

The encyclopedia of Diderot and Alembert devotes an article to the module . The three principal meanings are those relating to the medals and currencies, architecture and mathematics with the module of a Logarithme: ==MODULE== MODULATE, S. Mr. ( Alg. & Géom. ) Some authors thus call the line which one takes for subtangent logarithmic curve in the calculation of the logarithms. See . Thus, in the logarithms of Neper, the modal is 0,434294; &, in the logarithms of Briggs, it is the unit. When it is said that a line is it logarithm of the report/ratio of has with B, C being taken for module , that wants to say that this line is the X-coordinate of a logarithmic curve of which under-tangense is C , this X-coordinate lying between two ordinates equal to & with B has. Mr. Côtes, in his Harmonized mensurarum (commented & developed by dom Walmesley in its Anasyse of the reports/ratios), frequently employs this expression of module which besides is not extremely used. ( O ) ===Module=== , ( Art numism. ) term borrowed of architecture by Médaillistes, to fix by given sizes their medals, & to compose some various continuations in the médailliers; thus they reduced all the sizes of the medals of bronze to three modules , which they name of the parts of large, of means, & petie bronzes, & one writes by abbreviation G.B. Mr. B.P.B. (D.J. ) ===Module=== , ( Architedure. ) measurement taken at will to regulate the proportions of the columns, & symmetry or distribution of the édifice.