Go back to article: Giovanni Canestrini’s models of Leonardo da Vinci’s friction experiments

Discussion

It is clear from its origins, as discussed above, that the exhibit shown in Figure 1 should be treated as two distinct artefacts: Model A derived from one source, and Models B, C and D which were collectively based on another.

Model A

The ‘table’ of Model A is based on the two central sketches from Codex Arundel f. 41r (see Figure 3) and was first created (as Model A*) in 1939. These sketches date from 1500–1505[16] and show two separate and distinct pieces of apparatus. The notes written on the same page of the notebook, some of which extend on to the facing page (f. 40v), consist of a general statement on the origins of variation of friction, detailed quantitative attempts to evaluate the effects of friction on rotors carrying various hanging weights, and the statement ‘circular friction is equal to linear friction’[17]. But there is no text that relates explicitly or implicitly to the sketches on which Model A is based, and we must therefore interpret them in the context of what we now know about Leonardo’s investigations of friction (Hutchings, 2016).

Leonardo’s first definitive statement on sliding friction (in Codex Forster III f. 72r, 1493–4) pre-dates the diagrams of Figure 3 by some 7–12 years, and by the time he made the sketches on which Model A was based his understanding of friction was well developed. His earliest sketch of a friction ‘experiment’ in the Forster notebook, whether intended to represent a real arrangement or a thought experiment, appears to show a string passing over a pulley. So also do several other later diagrams, rather than the roller that is very clearly drawn in the left-hand central sketch of Figure 3 and embodied in Model A. There is no similar sketch of a string attached to a block and passing over a roller elsewhere in the notebooks, and one can only speculate as to Leonardo’s purpose in drawing the roller in this case. By changing the position of the roller on the slope, the height of the string above the horizontal plane could easily be adjusted to allow blocks with different heights to be accommodated, and this may well have been his intention; the two simpler sketches lower down the page, of blocks on planes with horizontal strings, do indeed show progressively thinner blocks, and are consistent with this interpretation. Alternatively, or possibly additionally, Leonardo may have wished to avoid the effect of friction from a pulley that would otherwise add to the tension in the string that he was trying to measure. The use of a roller on a horizontal plane would certainly have that advantage, but a roller on a sloping plane as shown in the sketch would also add a contribution to the string tension from the weight of the roller. It is not clear from the sketch whether the roller was intended to be a solid or a thin-walled cylinder; the diagonal stroke across the top-left corner of the complete circle could be construed as (inaccurately-drawn) completion of the distal end of an open tube, but the absence of any shading on the ‘inside’ of the tube, as used to show depth elsewhere in the drawing, argues away from that reading. Even if it were a tube with negligible weight, its introduction would have added considerable complexity to Leonardo’s analysis of the experimental results, and we know from his earlier sketches and statements (summarised in Hutchings, 2016) that in his previous investigations of friction he had almost certainly used the much simpler arrangement of a pulley.

The right-hand central diagram in Figure 3 shows an evidently solid cylindrical rotor, with its weight marked as ‘2’ units, supported in a half-bearing[18]. The sketch illustrates a conceptual problem related to the analysis of friction on the surface of a rotating axle that Leonardo tackled in his notebooks over many years from 1493–1495 onwards. He evidently found it challenging but returned to it repeatedly and doggedly in several variants; indeed, a second example is seen at the top of Figure 3. The problem was to find what weight hanging from the string would just overcome the friction between the cylinder and its support, and the difficulty came from the fact that the hanging weight itself added to the rotor’s own weight and therefore enhanced the frictional force. In other sketches and associated notes in Codex Arundel f. 41r and 40v, as well as in many other places, Leonardo made unsuccessful attempts to solve this and the related problem in which weights hang from both sides of the rotor. The methods he used for its analysis were sometimes convoluted and inconsistent, and although he did eventually devise a method to derive a good approximation to the weight needed to overcome friction for a rotor with a single hanging weight when he assumed the value for the coefficient of friction in advance, he cannot have used this geometry as his primary empirical method for investigating friction. What the right-hand central sketch in Figure 3 shows is a relatively well-developed perspective view of a thought experiment he had been drawing repeatedly, either as a flat two-dimensional diagram or with perspective, for perhaps ten years. It is not a representation of an experiment, real or imaginary, to investigate friction, although it does provide an illustration of Leonardo’s correct appreciation that ‘circular friction’ (i.e. the sliding friction between a rotating and a stationary body) is essentially the same phenomenon as linear sliding friction. The drawing does not involve rolling friction, which Leonardo was well able to distinguish from sliding (Hutchings, 2016).

The blocks and plank lying on the table in Model A, and set up to form an inclined plane in Model A*, were evidently inspired by the two sketches of blocks at the bottom of Figure 3, which as we have seen were quite possibly intended to illustrate the effects of varying the thickness of the block. Canestrini appears to have interpreted the sketched line with a T-shaped termination as a rigid bar with a handle, which would be an implausible device with which to attempt to measure a frictional force. In fact, Leonardo sketched a line with a T-shaped end on several other occasions to indicate the direction of application of a tensile force in the same way that we would nowadays conventionally draw an arrow[19], and to construe it as a rigid handle, rather than as a flexible string or cord, is misleading. And it would be wrong to suggest that Leonardo investigated the laws of friction by experimenting with a block on an inclined plane, since despite his efforts this geometry lay outside his powers of quantitative analysis (Hutchings, 2016, p 64).

In using Leonardo’s drawings as his inspiration for the ‘bench’ or ‘worktable’ of Model A* and the subsequent Model A, Canestrini not only created a chimera that combined two quite different devices, but also made other significant changes. As we have noted above, the roller drawn in the left-hand diagram might possibly be solid or alternatively a thin-walled tube, while in the other sketch the axle is definitely solid; neither is a thick-walled cylinder as used in the models. And there is no evidence in the original sketches, or indeed elsewhere in Leonardo’s writing on friction, to support the use of a hollow box rather than a solid block in any sliding friction experiment.

By describing this model as a ‘bench for friction experiments’, Canestrini was following in the distinguished footsteps of Roberto Marcolongo (1862–1943), mathematician and professor of mechanics who carried out extensive research on Leonardo’s mathematical and mechanical investigations. Marcolongo was a member of the Reale Commissione Vinciana and was involved in the organisation of the 1939 Milan exhibition as a member of the scientific committee (Millán Gasca, 2007; Catalogo, 1939). In a lengthy essay on Leonardo’s work on mechanics published in 1933, Marcolongo had reproduced the Arundel drawings and described them as ‘illustrations…of the bench which had served Leonardo for his experiments’[20] (Marcolongo, 1933, p 107). Subsequent writers have echoed the views of Marcolongo and Canestrini, including Reti (1971, p 106), Dowson (1979) and several more recent authors (see Hutchings, 2016). But as we have seen, these sketches and the resulting model provide a substantially misleading impression of the methods by which Leonardo might have investigated sliding friction, and they do not relate to rolling friction at all.

Models B, C and D

These three models, which were first produced for the 1953 exhibition, are based on a group of sketches in MS L f. 11v (see Figure 4) dating from 1497–1502 that show a roller or disc supported in three different ways, with a hand-crank drawn in each case to indicate the roller of interest. They are accompanied on the page by a rhetorical question comparing the ease of rotation for the three designs: ‘which of these is of easier motion, and how much: a or b or c?’.

Model B is the simplest: the roller is attached to a thin axle that is supported in bearings, and the resistance to rotation originates in sliding friction on the axle surface. Leonardo had first studied the friction of axles and the effect of the axle diameter on the frictional resistance some ten years earlier and by the date of these sketches was able to analyse it quantitatively (Hutchings, 2016).

In Model C the axle of the roller is supported on ‘twin-disk’ bearings that Leonardo had previously used in several designs (e.g. in MS B f. 33v, 1487-90; Madrid I f. 12v, 1493-7), and which he knew provided much less frictional torque than the plain bearing used in Model B.

In Model D the roller of interest is supported by two lower rollers analogous to the one in Model B, and drives them by friction. Leonardo was already fully familiar with the concept of a roller or disk driving others by friction, for example using the principle to reduce the effect of frictional torque on an axle in a stack of 24 disks (Madrid I 103r, 1493–7). The way in which the weight of the top roller leads to loads on the bearings is perhaps something that would have concerned him, although there is no evidence for this from the sketches and notes on this page.

In summary, Leonardo already knew in broad terms the answer to his question, and it is likely that his sketches represented a ‘thought experiment’ in which the advantages of these different designs are being compared rather than any real experimental apparatus. The sketch concerns the relative merits of three designs for bearings, and does not represent a method for investigating the friction that would act on the axle shafts or between the rollers. The crank-handles in the drawings are an artistic device to indicate which component is to be turned, rather than suggesting that the torque on it might be measured in some way.

Canestrini’s own descriptions of the sketches behind Models B, C and D varied in his different publications: ‘studies of rolling friction’; ‘axles rotating on rollers’; and ‘studies of friction in axles’. None of these is accurate, as Leonardo’s own words that accompany the sketches make clear: what he was depicting was a comparison between three methods of supporting a roller, and he was asking which method caused the least resistance to motion. As with Model A, these sketches and the models derived from them do not show plausible methods for studying friction, either in sliding or rolling.

Model E

Although Model E was presumably destroyed during the Second World War together with the other models from the 1939 exhibition (Giorgione, 2015; Landrus, 2013) and was not subsequently reconstructed, it provides a further example of a model presented as a ‘friction experiment’, being described in this way by Canestrini (1939b, p 160) and in the exhibition guide and catalogue (Guida, 1939; Catalogo, 1939). Leonardo’s sketch of 1499–1500 (Codex Atlanticus f. 1081v, see Figure 7) on which this model was based clearly shows two lower conical rollers rather than the three used in the model (although three supporting rollers are shown in the neighbouring plan view and four in another sketch at the bottom on the same page). The diagram shows a design for a bearing to support a vertical rotating shaft with a conical end carrying a vertical downward load, and is similar to several earlier designs (1493–1497) in Codex Madrid I (ff. 102v and 113v) which also have two conical supporting rollers. The design is apparently ingenious and superficially attractive, using the cones to support the end of the shaft, combined with horizontal disk bearings to react against the sideways force exerted on the cones. In his accompanying note Leonardo is speculating on the sideways forces that will result from the ‘wedging’ action of the cones. Although the sketch is of a bearing that is intended to have a friction-reducing function, it certainly does not represent any kind of experimental investigation of friction.

In fact, though, the sketch on which Model E was based incorporates a fundamental flaw that would make it completely unsuitable as a bearing, in that the motions of the conically-ended vertical shaft and the two (or three) supporting cones are kinematically incompatible. For pure rolling to occur, which is essential for the device to act as a low-friction bearing and is evidently what Leonardo intended in this and many other bearing designs, the surface speeds of the rolling elements in contact must be the same. With this particular design it is possible to achieve this only at a single height within the device (i.e. on a single plane perpendicular to the axes of the cones); at any higher or lower level there will be slippage, and therefore sliding friction, between the surfaces of the cones. So if the cones make contact over any significant length, as is evidently intended from the diagram, the device will rotate only with considerable effort and would be completely unsuitable for use as a bearing. This is clearly an example of a ‘thought experiment’ rather than an actual practical design.

As we have seen above, Canestrini’s written descriptions of the drawing associated with Model E are reasonably accurate: ‘thrust rollers’ and ‘system of rotating bearings with thrust roller on spindles’. The drawing does indeed show a thrust bearing, in which the end of a vertical shaft is supported on rotating conical rollers, albeit, as we have seen, one that would not actually work. But the description of Model E in the 1939 exhibition guide and catalogue, and also in his book (Canestrini, 1939b), is quite different and erroneous: ‘model of experimental apparatus for the study of the action of forces and friction in a rotary system’. One must wonder whether this change of description might perhaps have been prompted by the discovery, once the model had been built, that it did not perform as either Leonardo or his twentieth-century interpreter had originally envisaged.

Component DOI: http://dx.doi.org/10.15180/160602/003