From classical Mechanics to science of the engineer

Robert Halleux e Diane Lalevitch

With the Scientific Revolution, the world of Aristotle - of elements and qualities - gave way to one where everything was done "by form and movement", which is to say, where natural phenomena were reduced to displacements of matter in space and time. If the world is written in the language of mathematics, then mechanics is the most mathematicized, and therefore the most perfect of the new sciences, the best meeting ground between theory and observation or experiment.

Classical mechanics as a whole, however, was far from springing forth, already complete, from the brains of Galileo and Newton. As Pierre Duhem wrote in his famous Origins of Statics:

each theorem (...) was composed slowly, after an enormous amount of research, trials, doubts, discussions, contradictions. In this combination of efforts, no attempt was in vain; everything contributed to the outcome; each had its part to play, principal or secondary, in the formation of the definitive doctrine; even error was productive: (...). And meanwhile, as all these efforts were contributing to the advance of a science that we see today in all its perfection, none of those who made the efforts suspected the magnificence or the shape of the monument he was building. (...). Neither Bernoulli nor Lagrange could guess that their method of virtual displacement would one day be admirably suited to tackling electrical equilibrium and chemical equilibrium; they could not predict Gibbs, even though they were his forerunners. Skilled masons shaping a stone and cementing it, they were working on a monument the design of which had not been revealed by the architect.

From this groping in the dark, scientific historiography - which emphasises only the successes of science - associates Galileo with the law of falling bodies, uniform accelerated motion, parabolic tra-jectory of projectiles; Stevin, Roberval, Varignon with the parallelogram of forces; Jean Bernoulli with the principle of virtual work; and Leibniz with the rules of collision of bodies.

These various acquisitions were gradually co-ordinated during the 17th century, until the appearance of Newton's Principia (1687).

Replacing the Dutch, Voltaire and Madame du Châtelet gradually accustomed minds to Newtonism which was to overthrow the Cartesian system, in teaching.

The 18th century was the era of major syntheses of rational mechanics, already very similar to ours. D'Alembert's Treatise on dynamics (1743), and the Mechanics of Leonard Euler (1736), paved the way for the celebrated Analytical mechanics by Lagrange (1788).

The Gabinete in Coimbra reflected the typical course curriculum in the 18th century. Significantly, the mechanics course began with dynamics (instruments 4 and 5) with Galileo's immortal experiment of the inclined plane. This is a good point to quote the famous text of the Discorsi (1638):

In the thickness of a rule, that is, a plank of wood about 12 cubits long, half a cubit wide and three fingers thick, a groove a little over the thickness of a finger was cut. It was quite straight and, in order to be highly polished and smooth, it was completely covered with a sheet of parchment, as glossy as possible. A highly polished, hard round bronze sphere was allowed to run down the length of this groove. The rule, constructed as just described, had one end raised one or two cubits, arbitrarily, above the horizontal plane. The sphere was allowed to roll down along the groove, as already mentioned, and the time taken to complete the path was measured in the way I will describe below; this procedure was repeated many times to check the reliability of the time measured. During these repetitions, there was never any difference greater than one tenth of a pulse beat. Once this operation was accurately confirmed, we allowed the sphere to roll down one quarter of the length of the groove; the time taken was measured and was always equal to half the other... This experiment was repeated a hundred times, and it was found that the length run was in direct ratio to the square of the time, regardless of the inclination of the plane, that is, of the groove down which the sphere rolled. We also observed that the time taken to roll down differently inclined planes was in proportion to what [our demonstrations] attributed to it. As regards measuring the time, a large pail of water was suspended in the air; a small hole in its bottom permitted a trickle of water to escape, and this was collected in a small vessel throughout the time the sphere was rolling down the groove, or part of it; the amounts of water thus collected were weighed in a sensitive balance and the differences and relations of the weights gave the differences and relations of the times with such accuracy that, as stated above, these procedures, repeated countless times, never showed any appreciable difference.

The Father Mersenne balance (instrument 58) belongs to this Galilean perspective. It is an ingenious device for determining the "living force" acquired by a weight as it falls.

Several instruments (instrument 54) compare the descent of a weight along an inclined plane and along a cycloid. The cycloid, formerly known as a roulette, is a curve arising from a point on a circumference which rolls on its plane over a fixed straight line. The mathematical study of the cycloid was very fashionable in the 17th century. Pascal had calculated its centre of gravity and its area; Wren, its length; Roberval, its tangent. Christiaan Huygens calculated the movement of a material point along a cycloid.

The cycloid had, moreover, another application which we see in the pendulum clock (instrument 50): Galileo had established the isochronism of pendulum oscillations. In a clock the pendulum replaced, to advantage, the primitive balance arm as regulator. This was the invention described by Christiaan Huygens in his Horologium Oscillatorium of 1657. But the isochronism obtained was not perfect because the period of the pendulum was not completely independent of the oscillation amplitude. Huygens, in 1659, conceived the idea of restricting a pendulum to describing a cycloid, which could be done if its suspension cord was blocked in its oscillatory movement by stops, causing it to describe cycloid arcs.

The teaching of dynamics was completed by an ingenious device (instrument 52) designed to demonstrate another major Galilean achievement, the parabolic trajectory of projectiles.

Finally, the parallelogram of forces and the law of composition of movements was illustrated, leading directly to statics, with machines still used in physics courses today: various kinds of levers, straights and angulars (instruments 10 to 12), pulleys, blocks and tackle, endless screws. The multiplying effect of composite levers (instrument 13), applied later in bascules, was observed; likewise with simple machines, the conditions of equilibrium on an inclined plane (instrument 45), thanks to another Galilean apparatus, and finally the calculation of the centre of gravity (instruments 6-8).

The triumph of the mechanist paradigm and the mathematization of the world inevitably led to the development of measuring techniques, not only for space and time, but also for heat, cold, dryness and humidity, domains which had only been qualitative until then.

Balances had been constructed long before the theory of levers was completely understood. The balance with a pointer on arms of equal length and weights (instruments 14 to 16) comes from ancient times. From the third century BC it competed with the "Roman" balance which had a sliding pointer on a graduated shaft (instruments 18 and 19). The two kinds are known as drop-pan balances since the pans are below the pointer. It was Gilles Personne de Roberval who, in 1669, devised a balance with raised pans, kept horizontal by an articulated parallelogram. It was not until the 19th century, however, that Roberval's balance became widely used in shops and homes.

The Coimbra collection includes a "fraudulent" balance with arms of unequal length (instrument 17). This, indeed, was a domain with which physics was very much involved. The Roman balances, since their accuracy depended on the divisions on the shaft being equal, were especially blamed for all the cheating (to say nothing of the variations in the standard weights), but there would have been no progress without them. They came in all sizes, from small apothecaries' balances to huge machines for weighing pig iron.

But the small balances or precision balances used by precious metal assayers demanded much greater accuracy. Since the 15th century they had been "under the lantern", that is to say, kept under a glass bell jar to guard against dust and air currents. The balance developed by Magellan (instrument 22), with its regulating screws and air bubble level is a perfect example of this. The increasing precision of experimental balances was to be a crucial factor in the work of Joseph Black (1728-1799) and Lavoisier (1743-1794), thus permitting the chemical revolution and the major ponderal laws.

On the other hand, the principle of multiplication of forces by a coupled series of levers, described in treatises on mechanics since the 16th century, led to the first bascule bridge, designed by John Wyatt in 1741, and to the 19th century industrial bascules. Model n.º 20 is a very early example.

The collection in Coimbra has several other specimens of measuring equipment: an odometer, for measuring distance (instrument 48); a hygrometer (instrument 72); a pyrometer, a dilatometer (instruments 85 and 86), and a Desaguliers tribometer.

The odometer, based on the reduction principle of a set of toothed wheels, had already been described in the first century by Vitruvius. It could measure distance travelled by counting the rotations of the wheels on a carriage, or any other vehicle, but it was also used by surveyors. The hygrometer shows the level of dryness or humidity in the air. It was first developed in the 17th century and is based on the phenomenon of absorption, that is, on the expansion produced by humidity on certain organic substances. Any substance that swells or shrinks as a result of dryness or humidity can be used as a hygrometer; hemp, tripe, etc. have all been used. The increase in weight of hygroscopic bodies could also be measured. But, as d'Alembert remarked: "All that can be expected of a rope hygrometer is that it will show if there is more or less humidity in the air in comparison with the previous day, and this can be known from other signs, so it is useless to make a machine that improves nothing. It would be more interesting to know by how much the humidity or dryness has increased or decreased from one moment to another and to be able to make these instruments comparable. But it seems very difficult to make hygrometers that have this advantage...". The apparatus loses its sensitivity as it ages.

The hygrometer made from hair, perfected by Saussure in 1783, used hair that had previously been degreased, which would swell under the influence of humidity. Its use spread as it was so convenient.

The pyrometer was derived from a series of instruments which indicated temperature by the expansion of various substances on heating. Thus, the modern concept of temperature replaced sensory impressions of heat and cold. The thermoscopes of Philo of Byzantium (3rd century BC) and Hero of Alexandria (1st century AD), based on the expansion of air on heating, extended to the air thermometers of Santorio (1561-1636), Galileo (1564-1642), Robert Fludd (1574-1637) and Cornelius Drebbel (1572-1663). In the 17th century, these instruments, which were as sensitive to changes in atmospheric pressure as they were to temperature variations, were replaced by ones using liquid, namely the alcohol thermometers of the Accademia del Cimento (between 1657 and 1667). Investigators directed their efforts along two paths. One was theoretical: the definition of the concept of temperature and the establishment of a rational and precise scale; the other was practical and aimed to improve reliability of the instruments. The works of Fahrenheit (1724), Réamur (1730-31) and Celsius (1742) reflected this dual direction. For higher temperatures, the action of fire on metals and other solids was used; Musschenbroek's pyrometer used the expansion of an iron bar acting on levers and a wheel toothed up to a quadrant that measured the thermic expansion of the bar. The problem was that solids have a molecular state that is easily changed by heat and does not necessarily regain the same volume when it returns to its initial temperature. Indeed, as Maurice Daumas noted, they do not measure the quantity of heat so much as the coefficients of linear expansion.


At least half of the mechanics apparatus in the Coimbra collection is devoted to what might be called applied mechanics: maquettes of hoisting equipment; civil engineering; bridges, and road vehicles.

Since the 17th century, authors of treatises on mechanics have tried to be practical. They sought to liberate the skills and arts of an empiricism founded on the more-or-less effective formulas inherited from tradition. They wanted to base them, henceforth, on a solid physical and mathematical foundation. Mersenne, translating the Mechanics of Galileo (1634), added "several rare and new additions, useful to Architects, Makers of jugs and pitchers, Philosophers and Craftsmen". Roberval had foreseen a summary of his treatise on mechanics for craftsmen. In 1675, Luis XIV and Colbert pressured the Academy of Sciences to write "a treatise on Mechanics, where Theory and Practice are explained in a clear manner, understandable to everyone; it should, however, separate from Theory everything that is too closely related to Physics, everything that might lead to dispersal; that should be collected in a kind of introduction to the whole work. Next there should be a description, in the work itself, of all the machines used in the Arts, whether in France or in foreign countries". The word "Art" is used here in its old sense; " Arts and Crafts" would today be called "Techniques". It was in this spirit that, in 1729, Bernard Forest de Bélidor (1697-1761) published his Sciences of Engineers which, continually updated, was the vademecum of several generations of engineers right up to the 19th century. The introductory remarks are quite revealing:

Since we have sought in Mathematics the means to improve the arts, such progress has resulted as we had not previously dared to hope for; but, since there is only a small number of people in a condition to judge where this science might lead, we find it difficult to convince ourselves that it might be capable of all the wonders that are ascribed to it, given that what is discovered to be most beneficial is exactly that which is furthest removed from the public, even from those who could be most usefully served by these discoveries, through their inability to grasp the principles that led to the investigation of countless useful things unless they are instructed in the matter and are themselves in a position, so to speak, to make discoveries: besides, the opinion that the practice alone can lead them to these objectives is another obstacle, and one no less difficult to overcome. It is indeed true that experience contributes greatly to the supply of new knowledge, and that it daily gives to the most able people topics for reflection which would not have been perceived if it had not caused them to be born. But it is essential that this experience should be clarified, otherwise it might only produce highly confused ideas on everything it is offering. It follows from this that many imperfect things remain forever in the same state; they are transmitted from one generation to another with the same defects. And if perhaps someone thinks to call attention to it, all those in that profession will immediately protest against the innovation; it is hard to understand that those who have never in their lives worked on certain things, may yet be able to consider them fairly.

Bélidor also saw the need to give two explanations, one mathematical, another that does not use mathematics.

The construction of bridges (maquette 42) perfectly illustrates this interference between applied mechanics and artisan empiricism. The industrialization which occurred in the 18th century required major civil engineering works, especially bridges. The whole arch, inherited from the Romans, was largely supplanted by the segmented arch, which had the advantages of lowering the level of the walkways, eliminating bulging shapes, and widening the water drainage exits. But the calculation of the vaults and the springers of their arches, the abutments and piers of a bridge continued to rely on traditional knowledge. For instance, Leone Battista Alberti's rule proposed that the length of the springer should not be less than one fifteenth of the arch span, and the pier should be a fifth of the width of the arches. Mathematicians endeavoured to calculate, from a theoretical base, the span of the springers in the vaults' arches, the thrust of the bridge's arches, the resistance of the supports, the thickness of the vaults' abutments, the width of the springers. Bélidor, along with la Hire, thus regarded the arches' springers as a set of angles whose equilibrium, from the keystone of the vault to the abutments, is maintained by cancellation of their reciprocal pressures. The scale model in Coimbra was designed to show this, having first been constructed on a vaulted arch which was later removed.

To construct the piers, the engineers of the 18th century remained faithful to the box pile method to prevent the entry of water. They drove the piles into the river bed with a kind of crane (instrument 37), which was known in France as a ram or bell because the noise it made when working was reminiscent of the sound of bells. It was a heavy block, raised up by means of a rope and then released; a fine application of the fall of bodies. It was struck with all the pile driver's might. The sole source of energy was human strength; a team of fifty men was needed for the pile driver. This was a traditional machine, only improved in the details of hooking up the pile driver. For underwater work, the diving bell (instrument 2), already described by medieval authors, was used. It was widely used in Holland to recover items from the river or sea bed after shipwrecks, or lost for any other reason. With lead as ballast, an experienced worker could stay down for half an hour.

The same mixture of tradition and slight improvement is found in hoisting devices, tripods, cranes and capstans. The energy was still supplied by men who pulled the rope or winch, turned the windlass, or walked in a "cage". The frame was still made of wood. Advances were made to pulleys and toothed wheels, such as the ratchet wheels (instrument 39) or la Garoust's (la Garousse) "toothed wheel lever". This is an engagement wheel with teeth which transforms alternating circular movement into continuous circular movement. The lever has a double tooth; when one is locked on to a tooth on the wheel, the other is installed on a new tooth. La Garouste is thus the father of our "racagnac" [rack-and-pinion].

Finally, throughout most of Europe, the growth of traffic contrasted with the abysmal quality of the road network and vehicles. Wrought iron axles, rolled iron blade springs, hot-worked, with bows, the hoops of wheels, all began to appear on private vehicles. The problem was not resolved for heavy transport. The improvement of facings and carriages also seemed to fall within the scope of mechanics. In France, the Ècole des Ponts et Chaussés, founded in 1747, was dedicated to training engineers capable of studying these problems mathematically. For carriages, efforts were directed largely at calculating traction, or the work supplied by one horse, and the resistance associated with vehicle weight, the radius of the wheels, thickness of the hoops, road surface, etc. ... D'Alembert, therefore, in his article, "Chariot", in the Encyclopédie, propounded the theory of the wheel which he regarded as a lever whose fulcrum is the lower end that rests on the ground; he showed that large wheels roll more easily, but with less stability. The study of centres of gravity found an application here. These endeavours explain the attempts at models of lowered platforms. The issue was clearly current in the laboratory in Coimbra. The latest developments in research was followed, such as Catherine François Boulard's work (instrument 46). He was an architect from Lyon (died 1794) who had studied the form and nature of hoops (1781), and the carriage that was "lightest, most easily rolling, least likely to spoil roads" (1788).

As in theoretical Mechanics, the collection of applied Mechanics also faithfully reflects the most recent developments in technological research. But it is a technology of wood and animal strength, taken to perfection, or, in other words, to saturation point. The steam engine and advances in metal construction seen in the following decades were much needed to clear the way forward.