The Origins of Statics/Chapter 2
The commentaries of the Scholastics touching the Mηχανικὰ Pροβλήματα of Aristotle added nothing essential to the ideas of the Stagirite; in order to see these ideas push from new rootlings, and to bear new fruits, it is necessary for us to await the debut of the XVIth century.
- "If, to the countenance of these men placed as Colossi at the entrance of the XVIth century (1), one would dare to testify a preference, possibly the palm would be accorded to Leonardo da Vinci, genius sublime who enlarged the circle of all human knowledge. In the arts, Michelangelo and Raphael did not skim within an eclipse of his glory; his scientific discoveries, his philosophical researches, placed him at the head of the savants of his epoch. Music, military science, mechanics, hydraulics, astronomy, geometry, physics, natural history, anatomy, were perfected by him. If all his manuscripts were existing still, they would form an encyclopedia the most original, the most vast, that had ever been created by a human intelligence."
During his lifetime, Leonardo da Vinci published nothing. Diverse testimonies assure us that in dying he left in manuscripts certain finished treatises, notably a treatise on painting and a treatise on perspective; but no part of these works have reached us. The Trattato della pittura, published at Paris by Dufresne in 1651 and often republished since, the Trattato del moto e misura dell' acqua printed at Bologne in 1828, are not but rhapsodies more or less faithful. The veritable thought of Leonardo has to be found in the notebooks where he was noting his scarcely bloomed thoughts.
Of these notebooks, many have been lost; after a good many vicissitudes, many have been saved (2). An important collection of these writings has found itself at the Library of the Institute of France; diverse leaves, stolen by Libri and sold by him to Lord Ashburnam, have become, thanks to M. Leopold Delisle, the property of the National Library; other manuscripts find themselves in Italy; among those here, a place of choice must be reserved to the registry that the Ambrosian Library of Milan keeps under the name of Codex Atlanticus.
Under the auspices of the Ministry of Public Instruction and thanks to the minute cares of M. Charles Ravaisson-Mollien, all the manuscripts of Leonardo da Vinci existing in France have been published. This admirable collection gives, in six volumes in-folio (3), the photographic fac-simile of each one of the leaves darkened by Leonardo, the transcription of the phrases which are traced there and their translation in French.
The Italian government has undertaken to publish, under a form even more luxurious, all the papers of Leonardo that Italy possesses; from this collection a first volume has appeared (4).
One would not know how to defend oneself from a curiosity moved through leafing through these notes left by Leonardo da Vinci; all the thoughts, all the images which have presented themselves to the spirit of the grand artist find themselves there, testifying, by their diversity and even their disorder, of the universal genius who has conceived them.
Designs innumerable, to the plume or to the blood, representing figures of men or of animals, leaves, churches, machines, plans of monuments or of fortresses, waves or projections of water currents, geometrical rough drafts, entangling themselves with compact lines of a straight, regular hand-writing traced from right to left.
There is an extreme variety of subjects to which these lines correspond. Domestic accounts, painter receipts, personal souvenirs, anecdotes of coarse, salty Gallic wit, pieces of verse, neighbor with profound reflections on the arts and the sciences; these reflections themselves soon following one another in numerous pages, regular and ordered, rough draft already nearly achieved of a treatise on painting, of a treatise on hydraulics, of a treatise on perspective; sometimes they consist of short phrases of which the erasures, the repetitions, the contradictions, the non-achievements reveal the intense labor of the thinker in search of the truth.
Among these fragments more or less finished, it is a great number of them which concern the diverse branches of Mechanics, the science that Leonardo was cultivating with passion. "La mechanica," he said (5), "e il paradiso delle scientie matematiche percheche con quella si viene al frutto matematicho."
Now, in 1797, Venturi (6) signalled the extreme importance of many of these fragments. From their reading flowed the conclusion that Leonardo da Vinci, died May 2, 1519, was already in possession of some of the great truths of which one was attributing the invention to Galileo or to his immediate predecessors; of this number was the celebrated Principle of virtual velocities (7), become, since Lagrange, the foundation of all Mechanics.
Later, Libri (8), by extracts more extensive, completed and confirmed the discovery of Venturi. That it is possible for us today to be familiar in detail with a great part of the manuscripts left by Leonardo da Vinci, we ought to salute in him he who, pushing forward our understanding in Statics and in Dynamics beyond the point where Aristotle and Archimedes had led them, has determined the renaissance of Mechanics.
The one that Félix Ravaisson (9) has been able to justly name "the great initiator of modern thought" is, in Statics, a faithful disciple of Aristotle; his newest thoughts have their source in the meditation of Mechanical Questions posed by the Stagirite. He admitted, from the first, the law which serves as foundation to the peripatetician Statics; he expresses it with a great precision (10):
- "First: If a power moves a body for some time and some space, the same power will move half of this body in the same time two times this space.
- "Second: As well the same force will move half of this body, in all this space, in half of this time.
- "Third: And half of this force will move half of this body, in all this space, during the same time.
- "Fourth: And this force will move twice this moving body, in all this space, in twice this time, and a thousand times this moving body, in a thousand like times, in all this space.
- "Fifth: And half of this force will move all this body, in half of this space, in all this time, and a hundred times this body, in a hundredth of this space, in the same time.
- "Seventh: And if two separated forces move two separated moving bodies in so much time and so much space, the same united forces will move the same united bodies in all this space and all this time, because the first proportions stay always the same."
This law appears so essential to Leonardo da Vinci, that he formulates anew a little further on (11):
- "First: If a power moves a body in some space, in some time, the same power will move half of this body in the same time twice this space.
- "Second: If some force moves some moving body, in some space, in an equal time, the same force will move half of this moving body in all this space in half of this time.
- "Third: If a force moves a body in some time in a certain space, the same force will move half of this body, in the same time, half of this space. . . .
- "Sixth: If two separated forces move two separated moving bodies, the same united forces will move, in the same time, the two reunited moving bodies, the same space, because it stays always the same proportion."
However, to this expression, Leonard supplies now a correction; a very small force does not impress upon a very massive moving body a very small motion; the force does not disturb it at all. This result of our daily experiences, all the mechanicians of antiquity and of the Middle Ages admit, without analyzing it, as a first law of equilibrium and of motion; from that, comes the necessity of completing the preceding statements by the propositions that are seen here:
- Fourth: If a force moves a body some time, in some space, it is not necessary that one such power moves a double weight, in a double time, two times this space; because it would be possible that such a force would not be able to move this moving body.
- Fifth: If a force moves a body so much time, in so much space, it is not necessary that half of this force moves this same moving body in the same time half of one such space, for possibly the force would not be able to move the body.
These restrictions express the impossibility of certain motions not to clash with the axiom of Aristotle; they cause the anticipation of certain equilibria which do not flow from peripatetician Statics. We will see the bearing of it when we expose the ideas of Leonardo da Vinci touching perpetual motion. For the moment, let us hold up the consequences which draw themsleves from the antique Principle.
Among these consequences, it is suitable to cite at the first rank the one that Aristotle had already obtained, the law of equilibrium of the balance or of the lever; Leonardo da Vinci formulates it in his turn (12): "This proportion that will have the length of the lever with its counter-lever, you will find this proportion likewise in the quality of their weights and, similarly, in the slowness of the motion and in the quality of the path travelled through by their extremities, when they reach the permanent height of their pole." Or well again (13): "So much is added in accidental weight to the mover placed at the extremity of the lever as the moving body placed at the extremity of the counter-lever exceeds it in natural weight."
"And the motion of the mover is greater than that of the moving body by as much as the accidental weight of the mover exceeds its natural weight."
There are not at all there, elsewhere, any remarks particular to the lever; within the machines the most complicated, the axiom of Aristotle always affords room to compare the power of the mover to the resistance of the thing moved.
- "The more a force (14) extends itself out of a wheel across the wheel, out of a lever along the lever or out of a screw across the screw, the more the force is powerful and slow."
- "If two forces are produced by a same motion and by a same force, that which will consume the most time will have more power than any other. And a force will be more feeble than another by as much as the time between the one from the other."
These principles render account very easily of the properties of tackle-blocks; Leonardo da Vinci exposits with the greatest exactitude the properties of these mechanisms. Here is, for example, a figure traced by him (fig. 1), that these reflections accompany (15):
- "The powers that the cords interposed between the pulleys receive from their mover are between them in the same proportion as the one that there is between the velocities of their motions.
- "Of the motions made by the cords on their pulleys, the motion of the last cord is in the same proportion with the first that is the one from the number of the cords; that is to say that if they are 5, the first cord moving itself six feet, the last moving itself a fifth of six feet; and if they are 6, this last cord will have a motion of a sixth of six feet, and so on ad infinitum.
- "The proportion of the motion of the mover of the pulleys with the motion of the weight elevated by these pullies will be such as the weight elevated by these pullies with the weight of the mover."
Let us suppose that one possesses a cause of motion well-determined: for example, a quantity of water, immobile in a reservoir, expecting that one allows it to fall, from a given height, in a smaller canal. This cause of motion possesses a determined mechanical power; we would be able to diversify the employment of this power, but we would not increase the extent of it; we would be able to surmount upon it greater and greater resistances, but on the condition that the power displaces them more and more slowly:
- "If a wheel (16) is moved in a moment by a quantity of water and that this water be not able to add to itself, neither by current, neither by quantity, neither by a greater fall, the office of this wheel is terminated. That is to say that if a wheel moves a machine, it is impossible that without employing it one time more times, the wheel moves two of it; therefore the wheel does as much work in an hour as two machines with a second hour; thus the same wheel can make an infinite number of machines turn; but, with a very long time, the machines will not do more work than the first in one hour."
A given weight, falling from a given height, produces therefore a mechanical effect of which the magnitude is independent of the circumstances which accompany this fall; this magnitude remains the same, whether the fall be accomplished at once or whether it be fractioned into parts:
- "If someone descends (17) step by step in making a jump from the one step to the other, and that you add all the powers of the percussions and weights of such jumps, you will find that the powers are equal to the totality of the percussion and of the weight that one such man would give falling, by perpendicular line, from the head to the feet of said steps."
The passages that we have come to cite contain the statement of a principle which is, for the art of the engineer, of capital importance; but this principle is, in the last analysis, not but the logical abutment of the axiom posed by Aristotle. Not content to cause to carry some fruits to the seeds deposited by the peripatetician Mechanicist, Leonardo da Vinci approaches and resolves a difficulty which had made the Stagirite hesitate.
The extremity of a lever which supports itself on a horizontal axis describes a circle's circumference placed in a vertical plane; the path travelled through by this extremity is therefore not directed in the way of the weight of the load to lift, weight which pulls following a vertical right angle. By it, it results that the resistance that it is necessary to surmount to make to turn from a certain angle the lever arm depends on the initial position from this lever arm. It is so much the greater as the lever is closer to the horizontal position.
Following what law varies the power or the resistance of a given load, when one inclines the lever to the extremity from which it acts? To this question, Leonardo da Vinci responds in these terms (18):
- "Such is the proportion that the space mn (fig. 2) has with the space nb, such is the one that the weight descended in d has with the weight that this d has in the position b."
Thus, the heavy weight at the extremity of an inclined lever arm has the same action as if it were hanging at the extremity of a certain horizontal lever arm. The first is derived by projecting the point of application on the vertical line following which the weight exerts its traction. This horizontal lever arm, Leonardo names it the potential lever arm.
- "Always (19) the junction of the pendants of the balances with the arms of these balances is a potential rectangle and is not able to be actual if these arms are slanting (20)."
- "Always the actual arms of the balance are longer than the potential arms and so much the more as they are more adjacent to the center of the world (21)."
- "And never (22) will the actual arms of the balance have in themselves the potential arms (fig. 3) if they are not in the position of equality."
At the extremity of a lever, one can cause to act a force of which the direction is different from the vertical; it will suffice
to employ a cord extended in this direction, passing over a pulley and pulled then by a weight. A rule similar to the preceding will help to evaluate the motive power of a similar engine; here is how Leonardo expresses (23) this rule.
- "In order to know at each degree of movement the quality of the force of the power which moves and in the same manner of the thing moved."
- I effect therefore as you see (fig. 4) in mn (this is to say that from the suspension of the thing moved, one imagines a line which cuts at right angles the line of the power which moves) mn with fh.
This line mn, analogous to the potential lever arm considered an instant ago, Leonard names the true term of the balance or even spirited arm.
- "That one is said (24) to be the true term of the balance, which joining its right line with the straightness of the cord drawn by the weight, this junction will be made composing the right angle
- as one sees (fig. 5) in m with mA and in the same way pn with nA, spirited arm."
Then accordingly as a force draws a moving body around an axis perpendicular to this force—a circumvolubile according to the word that Leonardo da Vinci employs—it imports little, in regard to evaluating the mechanical effect of it, to look for the point of application of this force; two elements only are to be considered: the intensity of the force and the shortest distance from the axis of the circumvolubile to the direction of the force.
- "There is always (25) one same power and resistance
- in whatever place one has attached the cord on the line abc (fig. 6) and in the same way en-dessus [French preposition meaning "the face of a yarded sail which is turned toward the back] on the line mn."
- "In whatever part that the cord nc (fig. 7) be linked from the part ac, that does not make a difference, because one always employs a line which falls perpendicularly from the center of the balance to the line of the cord, that is to say, to the line mf."
These diverse passages show that Leonardo da Vinci had conceived in the neatest manner the notion of moment of a force in regard to an axis, at least in the case where the force is situated in a plane perpendicular to the axis; that he knew to formulate, for a solid mobile around an axis and subjected to similar forces, the condition of equilibrium.
It did not appear that between this theory of moments and the axiom of Aristotle, he had looked to establish any link. One such link exists however; the notion of moment appears successively if one takes for measure of the motive power that exerts a load hung at the extremity of an oblique lever arm, not the product of this load and the speed with which the extremity of the lever turns, but the product of this load and the speed with which
it lowers itself. This modification to the statement of the axiom of Aristotle would fully bring into accord, elsewhere, with the idea, expressed by Leonardo in a passage that we have cited, of taking the vertical distance of the fall of a weight as measure of the mechanical effect produced. But to catch sight of this link between the axiom of Aristotle and the notion of moment, it is necessary to make appeal to the definition of the instantaneous speed of the movement of the load; now, this notion, which was being obliged to play such a great role in the development of infinitesemal analysis, was still well confused in the intellect of Leonardo and his contemporaries.
If it is a mechanical problem which often presented itself to the meditations of the great painter, this is assuredly the study of the weight of a heavy object which slides on an inclined plane; one cannot leaf through his manuscripts without encountering at each instant, with variant particulars, a same design: on one pulley, a cord is stretched by two weights which slide on two planes unequally inclined.
The research of the laws which preside at the equilibrium of one such mechanism has certainly solicited the incessant efforts of Leonardo; from the first attempt, he has recognized that a weight sliding on an inclined plane pulls on the cord which sustains it less strongly than if it were descending in free fall and as much less strongly as the plane is less inclined; but this qualitative account would not satisfy the geometer, who exacts a quantitative relation.
To obtain this relation, Leonardo da Vinci multiplies and varies the trials; here is one of them which, by some considerations somewhat a little strange, gives us a result which approaches some truth.
He proposes to himself to compare the speeds with which one same sphere falls on some planes diversely inclined. He remarks that when the sphere is in equilibrium on a horizontal plane, the center of this sphere is on the vertical of the point where it touches the plane; the distance from the center of gravity to this vertical increases with the inclination of the plane and, in the same time, increases the speed with which the sphere, left to itself, descends this plane. He supposes, at length, that there is a proportionality between the speed of the descent and the distance from the center of gravity to the vertical of the point of support; from there, he draws without trouble this conclusion: the speed with which a sphere falls on an inclined plane is to its speed in free fall in the same proportion as the height of fall to the the length of the line of greater slope described by the body in motion. Elsewhere, for Leonardo da Vinci as for Aristotle, the intensity of a mechanical action is proportional to the speed that it communicates to a given moving body; the preceding proportion is therefore equal to the proportion of the weight of the sphere descending the inclined plane to its weight in free fall.
Here is the passage (26) where this curious solution is summarized:
- "The spherical and heavy body will take a more rapid movement so much as its contact with the place where it runs is more distant from the perpendicular of its central line. As much as ab (fig. 8) is less long than ac, so much will the ball fall more slowly through the line ac, and so much more slowly as the part o is smaller
- than the part m, because p being the pole of the ball, the part m being above p would fall with a more rapid movement, if there were not this bit of resistance that makes in counter-weight to it the part o; and if there were not the said counter-weight, the ball would descend through the line ac so much more quickly than o goes into m, that is to say that if the part o goes into the part m a hundred times, the part o, still lacking in the rotation of the ball, it would descend more quickly some hundredth of the ordinary time on n and the central line; and p is the pole where the ball touches its plane, and the more there is space between n and p, the more its course is rapid."
Leonardo could not declare himself satisfied with one such method; he attempted therefore to begin by a more rational way the problem of the inclined plane.
He recognized that the weight which pulls the moving body towards the center of the earth could be decomposed into two forces, the one normal to the inclined plane upon which the weight slides, the other tangent to this plane; it is this last which pulls along this moving body:
- "The uniform weight which descends obliquely, said he (27),
- divides its weight into two different aspects. One proves it. Let ab (fig. 9) move according to the obliquity abc; I say that the weight of the heavy body a b shares its gravity in two aspects, that is to say according to the line b c and according to the line n m; why and how much the weight is greater for one than for the other aspect and which obliquity is the one which shares the two weights in equal parts, will be said in the book On weights."
This decomposition will need to be employed in some varied circumstances. If, for example, a weight, hung by a cord at the extremity of a lever arm, oscillates in the manner of a pendulum, it will not weigh at each instant on the lever but by the vertical component of its weight; it will seem therefore as much less heavy as the cord to which it is suspended will be removed from the vertical (28).
In the same way, a heavy body sustained by two divergent cords shares its weight between these two cords.
Following which rule is the decomposition of a weight in two different directions made? It does not appear that Leonardo had suspected the rule of the parallelogram of forces on which depends the solution of the problem posed; after several re-examinations, he announces an erroneous solution. Here is a passage (29) where this erroneous solution is very explicitly formulated:
- "The weight which suspends itself in the angle will give from it some weight at the sides of this angle which will be between them in the same proportion as is that of the obliquity of their sides. Or: one such weight will distribute itself between its supports in the same proportion as is that of the two angles born of the division of the angle where this weight sustains itself, division of angle which is made by the vertical which descends in the center of the suspended heavy body; thus the angle abd (fig. 10) being cut by the line eb and the angle ebd being 9/11 of the angle abc, the angle abe is 2/11; ab is 9/11 of the weight and db 2/11."
This rule for decomposing a weight according to two directions finds itself repeated in another passage (30):
- "If the angle created by the concourse made by two oblique cords which descend to the suspension of a heavy body is divided by the central line of the heavy body, then this angle is divided into two parts between which there will be the same proportion that is that in which the aforesaid heavy body divides itself between the two cords."
The figure joined to this statement shows to us that in this passage as in the preceding, Leonardo takes for the relationship of the two partial angles what he considers the relationship of the lengths that they intercept on an even horizontal; in other terms, the relationship of the trigonometric tangents of these angles.
Sometimes (31), elsewhere, an analogous rule seems to him to define the relationships of two weights sustained by two unequally inclined planes and drawing the two extremities of a cord which rolls on a pulley; he thinks that these weights ought to be in inverse ratio to the obliquities of these planes, and he takes for the relationship of these obliquities the relationship of tangents of the angles made with the horizon.
Did Leonardo constantly hold himself to this incorrect rule on the decomposition of forces? It is probable that he did not content himself with it; that his spirit, always at work, looked for better, and it seems that he has, on this point, caught a glimpse of the truth; this is, at least, what we believe to be able to conclude from a note (32), scanty and unfinished, that we are going to analyze.
On a pulley, mobile around the axis d (fig. 11), rolls a cord pmonq that holds the two weights p and q; these slide on two unequally inclined planes da, dc; the two bits mp, nq of the cord are held in such a way that they are respectively parallel to the planes da, dc. Moreover, the figure is made in such a way that the projection de of the spoke dn on the horizontal hf is two-thirds of the spoke of the pulley, whereas the projection dg of dm on the horizontal hf is equal to one third of the
same spoke. It is in question to evaluate the component of the weight q in the direction of nq or dc and the component of the weight p in the direction of mp or da; here is, on the subject of this evaluation, what Leonardo writes:
- "The weight q, because of the right angle n, above df, is two thirds as heavy as its natural weight, which was three pounds, which remains with the power of two pounds, and the weight p which, to it also, had three pounds, remains with the power of one pound, because of m rectangle of the line hd, at the point g; therefore we have here two pounds against one pound".
What principle suggests to Leonardo da Vinci this correct assertion? It is difficult to declare it with entire certitude. Nevertheless, the lines that we have just cited seem to indicate to us that the rule to which he has made appeal, from a manner more or less conscious, is not at all the rule of the parallelogram of forces, but what amounts to this proposition, which to him is equivalent: the moment of a resultant of two forces is equal to the sum of the moments of the components.
Has Leonardo therefore attained to familiarity with this important theorem? In those of his manuscripts which have been published, we have not turned up from them any trace other than these which come to be related. The manuscripts still unpublished, these in particular which compose the celebrated Codex Atlanticus, do they enclose some passages capable to confirm this opinion? It is permitted to hope and, consequently, to wish the prompt publication of these precious relics.
(1) Libri, Histoire des Sciences mathématiques en Italie, since the Renaissance of Letters until the end of the XVIIth century. Paris, 1840, v. III, p. 11.
(2) One will find the detailed history of these manuscripts at the head of the first volume of the beautiful publication of M. Charles Ravaisson-Mollien: Les Manuscrits de Léonard de Vinci. Paris, A. Quantin, 1881.
(3) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien. Paris, A. Quantin, v. I (1881): Ms. A. from the Library of the Institute; v. II (1883): Ms. B from the Library of the Institute; v. III (1888): Mss. C, E and K from the Library of the Institute; v. IV (1889): Mss. F and I from the Library of the Institute; v. V (1890): Mss. G, L and M from the Library of the Institute; v. VI (1891): Ms. H from the Library of the Institute and Italian Mss. n° 2037 and n° 2038 from the National Library (Acq. 8070, Libri).
(4) I Manoscritti di Leonardo da Vinci. Codice sul volo degli uccelli e varie oltre materie. Publicato da Teodoro Sabachnikoff. Transcrizioni e note di Giovanni Piumati. Traduzione in lingua francese di Carlo Ravaisson-Mollien. Parigi, Edoardo Rouveyre, editore, MDCCCXCIII.
(5) "Mechanics is the paradise of the mathematical sciences, for it is by her that these sciences attain the mathematical fruit." (Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. E from the Library of the Institute, fol. 8, verso. Paris, 1888).
(6) Venturi, Essai sur les ouvrages de Léonard de Vinci. Paris, 1797.
(7) Venturi, loc. cit., pp. 17 and 18.
(8) Histoire des Sciences mathématiques en Italie, since the Renaissance of Letters until the end of the XVIIth century, v. III, pp. 10-60. Paris, 1840.
(9) Félix Ravaisson, La Philosophie en France au XIXth siècle, p. 5 Recueil de Rapports sur les progrès des lettres et des sciences, 1868).
(10) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. F of the Library of the Institute, fol. 25, recto. Paris, 1889.
(11) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. F of the Library of the Institute, fol. 51, verso. Paris, 1880 [sic].
(12) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. A of the Library of the Institute, fol. 43, verso. Paris, 1881.
(13) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. E of the Library of the Institute, fol. 38, verso. Paris, 1888.
(14) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. A of the Library of the Institute, fol. 33, verso; entitled: De la disposition de la force pour bien tirer et pousser. Paris, 1881.
(15) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. E of the Library of the Institute, fol. 20, recto. Paris, 1883.
(16) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. A of the Library of the Institute, fol. 30, recto. Paris, 1881.
(17) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. I of the Library of the Institute, fol. 14, verso. Paris, 1889.
(18) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. E of the Library of the Institute, fol. 72, verso. Paris, 1883.
(19) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. E of the Library of the Institute, fol. 64, recto. Paris, 1888.
(20) The text says, through error, are not slanting.
(21) That is to say more adjacent to the vertical.
(22) Léonard de Vinci, ibid., fol. 63, verso.
(23) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. I of the Library of the Institute, fol. 30, recto. Paris, 1889.
(24) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. M of the Library of the Institute, fol. 40, recto. Paris, 1890.
(25) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. M of the Library of the Institute, fol. 50, recto and verso. Paris, 1890.
(26) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. A of the Library of the Institute, fol. 52, recto. Paris, 1881.
(27) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. G of the Library of the Institute, fol. 75, recto. Paris, 1890. Cf. Ibid., fol. 76, verso.
(28) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. G of the Library of the Institute, fol. 76, verso; fol. 77, recto. Paris, 1890.
(29) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. E of the Library of the Institute, fol. 6, recto. Paris, 1888.
(30) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. G of the Library of the Institute, fol. 39, verso. Paris, 1890.
(31) Les Manuscrits de Léonard de Vinci, published by Charles Ravaisson-Mollien; Ms. G of the Library of the Institute, fol. 1, verso. Paris, 1888.
(32) I Manoscritti di Leonardo da Vinci. Codice sul volo degli uccelli e varie altre materie. Publicato da Teodoro Sabachnikoff. Transcrizioni e note di Giovanni Piumati. Traduzione in lingua francese di Carlo Ravaisson-Mollien. Parigi, Edoardo Rouveyre, editore, MDCCCXCIII, fol. 4, recto.