Leonardo da Vinci and Perpetual Motion Allan A. Mills CELEBRATING LEONARDO DA VINCI CELEBRATING umankind has always sought to reduce the the same wheel, then the machine H ABSTRACT need for its own manual labor. Draught animals were one might turn “forever.” solution, but much more beguiling was the concept of ma- A number of empirical attempts Leonardo da Vinci illustrated chines that would work “by themselves,” with no obvious prime to achieve such a hydraulic chimera several traditional forms of mover [1]. Even in ancient times, it appears that at least two perhaps were made but went un- “perpetual-motion machine” in categories had been proposed: the self-pumping waterwheel recorded because they never worked. small pocket books now known and the mechanical overbalancing wheel. None of course Drawings and plans of self-pumping as the Codex Forster. He was well aware that these designs, worked, and as science and technology progressed it became wheels persisted into the 18th cen- based on waterwheel/pump apparent that any such device was theoretically impossible. tury and even into modern times combinations, mechanical However, before this understanding was fully achieved and as amusing artifacts of linear per- overbalancing hammers or became well known, many technologists and hopeful inven- spective [3]. Progress in the under- rolling balls, would not—and could not—work. tors [2] felt obliged to devote time to this hoary problem. standing of efficiency, friction and Among the former was Leonardo da Vinci. Well aware of the the conservation of energy gradu- futility of all suggestions for achieving “perpetual motion,” he ally vindicated the practical knowl- simply recorded—and refuted—ideas that were prevalent in edge that, no matter how ingenious, his time. such devices simply could not work. THE SELF-PUMPING WATERWHEEL Leonardo’s Contribution Wheels turned by flowing water were well known in the an- The papers of Leonardo da Vinci bear witness to the semi- cient and medieval world and were sometimes used to drive quantitative understanding of machines that was emerging in bucket wheels to raise water for irrigation. The Archimedean the late 15th century. Perhaps the least known of his extant screw was another form of pump. It must therefore have oc- works are three tiny notebooks collectively referred to as the curred to many speculators that, if only the two could be com- Codex Forster, now in the Victoria and Albert Museum in Lon- bined, with the waterwheel driving a pump to supply water to don. They are believed to have been compiled between 1495 and 1497. A facsimile edition has been published, accompa- Allan A. Mills (retired university physics lecturer), 31 Thatch Meadow Drive, Market nied by a transcription of Leonardo’s mirror-image text in its Harborough, Leicestershire LE16 7XH, U.K. E-mail: <[email protected]>. archaic Italian [4]. (The facsimile drawings are in their origi- Fig. 1. Leonardo’s pump/waterwheel combination in Codex Forster I, folio 42v. Fig. 2. The “overbalancing wheel” of Villard de (Giunti facsimile) Honnecourt (13th century). ©2008 ISAST LEONARDO, Vol. 41, No. 1, pp. 39–42, 2008 39 Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/leon.2008.41.1.39 by guest on 26 September 2021 nal mirror-image form.) Unfortunately, no complete English translation of this difficult material is available, although ex- cerpts and discussions have appeared in various works on Leonardo. Codex Forster I, folio 42v, contains the drawing reproduced here as Fig. 1. The accompanying text says that “Screw A lifts water to screw B, and screw B with the same water moves screw A.” Leonardo CELEBRATING LEONARDO DA VINCI CELEBRATING emphasizes that this is impossible in still (“dead”) water and heaps scorn on the designers of such machines. The best that can be said in defense of the latter is that the arrangement would demonstrably work for a while if a holding tank were in- corporated at the top of the pump [5]. All that it would require would be to elim- inate friction and improve the efficiency of the machine to 100%! Fig. 3. Overbalancing hammer wheel shown in Codex Forster II, folio 90v. (Giunti facsimile) THE OVERBALANCING WHEEL The first recorded representation of Fig. 4. A more complex hammer wheel shown in Codex Madrid I, folio 145. the popular “overbalancing wheel” ap- Leonardo’s drawing has been digitally contrast-enhanced and rectified. pears in the sketchbook of Villard de Honnecourt, an enigmatic figure who traveled extensively in Europe in the 13th century [6]. In Fig. 2, it can be seen that he is struggling to depict a wheel mounted upon an axle turning within a stoutly braced frame. Around the cir- cumference of the wheel are pivoted an odd number (here seven) of mallets hang- ing from their rigid wooden handles. The lowermost mallets hang vertically, but the upper ones rest against the rim of the wheel. The implication is that the four mallets on the left of a vertical through the center of the wheel will always over- balance the three on the right, causing the wheel to perpetually turn in a coun- terclockwise direction. It must have been thought that the blows of the descending mallets would assist this motion. The simplest model soon proves that, as depicted, the uppermost mallet falls over too soon: a manually started wheel rapidly comes to rest in equilibrium. To- day, we understand that the turning ef- fect (“moment”) of any mallet is given by the product of its mass times the distance of its center from a vertical through the center of the wheel. The wheel comes to rest with the sum of the moments on the left equal to those on the right, and no external work can be done. It makes no difference if there are an odd or an even number of mallets. Da Vinci’s Hammer Wheels Codex Forster II, measuring only 9.5 x 7.0 cm, depicts on folio 90v the overbalanc- ing hammer wheel shown in Fig. 3. The circumference of the wheel bears ratchet 40 Mills, Leonardo and Perpetual Motion Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/leon.2008.41.1.39 by guest on 26 September 2021 rectified and is copied here as Fig. 4. The arcs upon the face may represent initial attempts to construct the loci of the cen- ters of gravity of hanging weights as the wheel turns, but they neither are sym- metrical nor serve to position the ham- mers around the rim of the wheel rather than hanging vertically—as shown once they leave the lowest point. Simple peg- like stops are the simplest way of ensur- ing this, as shown in the model illustrated LEONARDO DA VINCI CELEBRATING in Fig. 5. The most significant part of Fig. 4 is the way in which Leonardo has com- pounded the moments in mobile-like [8] chains to prove that, as drawn, the wheel is resting in equilibrium about an imagi- nary vertical line through its axis. Many examples of this diagrammatic construc- tion are to be found in this codex, which has an emphasis on mechanics. A Model Hammer-Wheel For this model (Fig. 5), I fastened 12 brass weights to phenolic-fiber arms. The as- Fig. 5. A model of the “overbalancing hammer wheel.” (© Allan A. Mills) semblies pivoted at equal intervals around a 20-cm-diameter disk. Protruding pegs were positioned near each arm so as to limit its swing. The resulting wheel was teeth, and a pawl permits counterclock- stable, stationary equilibrium just like de supported by a low-friction axle through wise rotation only. The pivoted hammers Honnecourt’s mallet wheel. its center and could be rotated manually have therefore been positioned around Another drawing in Codex Forster II (fo- in both clockwise and counterclockwise an inner disc, where coarser teeth sur- lio 90r) is of a hammer wheel with addi- directions. Either way, it soon came to rounding the pivots prevent the rigid tional arcs upon its face. This particular rest in equilibrium. As the weights were handles from hanging vertically except drawing is cramped and unclear but is re- equal, it was expected—and confirmed— toward the bottom right-hand side. Ca- peated in Codex Madrid I, folios 144 and that the sum of the distances of the cen- sual inspection of the drawing might im- 145. (Dated to 1490–1496, this work is ters of the five weights on the left from a ply that the wheel will continuously rotate now available in facsimile with an ac- central vertical was equal to the sum of counterclockwise, but Leonardo is well companying commentary by Ladislao the corresponding distances of the seven aware that it will, in fact, soon assume a Reti [7].) The second drawing has been weights on the right. Fig. 6. Leonardo, rolling ball wheel of Codex Forster II, folio 91r. Fig. 7. Leonardo, rolling ball wheel of Codex Forster II, folio 91v. (Giunti facsimile) (Giunti facsimile) Mills, Leonardo and Perpetual Motion 41 Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/leon.2008.41.1.39 by guest on 26 September 2021 del Saber [9]. Such a construction, how- ever, would be no better as a perpetual motion machine! Oh speculators on perpetual motion, how many vain projects you have created in this search. Go and be the companions of the seekers for gold! —Leonardo da Vinci [10] CELEBRATING LEONARDO DA VINCI CELEBRATING Acknowledgments I am grateful to Pauline Mills for translations of rel- evant passages in the Codex Forster. References and Notes 1. Hele Shaw, “Perpetual Motion,” Nature 37 (1888) p.