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Galileo's early discoveries in projectile motion, specifically his difficulties in reconciling the parabolic shape of projectile trajectories with traditional understanding of natural and forced motion. Cavalieri's connection of Galileo's claims to experiments they conducted together is also explored. The document also touches upon Galileo's discussions with Guidobaldo del Monte and Paolo Sarpi, and their common interests in motion-related topics.
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M A X - P L A N C K - I N S T I T U T F Ü R W I S S E N S C H A F T S G E S C H I C H T E Max Planck Institute for the History of Science
P R E P R I N T 9 7 ( 1 9 9 8 )
I SSN 0948-
J ü r g e n R e n n , P e t e r D a m e r o w, S i m o n e R i e g e r, a n d M i c h e l e C a m e r o t a H u n t i n g t h e W h i t e E l e p h a n t W h e n a n d h o w d i d G a l i l e o d i s c o v e r t h e l a w o f f a l l?
Jürgen Renn, Peter Damerow, Simone Rieger, and Michele Camerota
Mark Twain tells the story of a white elephant, a present of the king of Siam to Queen Victoria of England, who got somehow lost in New York on its way to England. An impressive army of highly qualified detectives swarmed out over the whole country to search for the lost treasure. And after short time an abundance of optimistic reports with precise observations were returned from the detectives giving evidence that the elephant must have been shortly before at that very place each detective had chosen for his investigations. Although no elephant could ever have been strolling around at the same time at such different places of a vast area and in spite of the fact that the elephant, wounded by a bullet, was lying dead the whole time in the cellar of the police headquarters, the detectives were highly praised by the public for their professional and effective execution of the task. ( The Stolen White Elephant , Boston 1882)
In spite of having been the subject of more than a century of historical research, the question of when and how Galileo made his major discoveries is still answered insufficiently only. It is mostly assumed that he must have found the law of fall around the year 1604 and that only sev-
(^1) This paper makes use of the work of research projects of the Max Planck Institute for the History of Science in Berlin, some pursued jointly with the Biblioteca Nazionale Centrale in Florence, the Istituto e Museo di Storia della Scienza, and the Istituto Nazionale die Fisica Nucleare in Florence. In particular, we have made use of results achieved in the context of a project dedicated to the development of an electronic representation of Ga- lileo’s notes on mechanics (together with the Biblioteca Nazionale Centrale and the Istituto e Museo di Storia della Scienza, both in Florence), of results achieved in a study of the time-sequence of entries in Galileo’s manu- scripts by means of an analysis of differences in the composition of the ink (together with the Biblioteca Nazi- onale Centrale, the Istituto e Museo di Storia della Scienza, and the Istituto Nazionale die Fisica Nucleare, all in Florence), and finally of results achieved in the context of a central research project of the Max Planck Insti- tute for the History of Science, dedicated to the study of the relation of practical experience and conceptual structures in the emergence of science. We would especially like to acknowledge the generous support of sev- eral individuals involved in these projects: Jochen Büttner, Paolo Galluzzi, Wallace Hooper, Franco Lucarelli, Pier Andrea Mandó, Fiorenza Z. Renn, Urs Schöpflin, Isabella Truci, and Bernd Wischnewski. From several other individuals we received helpful suggestions acknowleged at appropriate places throughout the paper.
The Standard Dating of the Discovery of the Law of Fall and of the Parabolic Trajectory
convinced that he had found such a proof and intended to make it a core topic of the Fifth Day of the Discorsi. Only his declining health hindered him to realize this plan and the Fifth Day remained unwritten.
Our primary concern is to show that this puzzling course of his discovery is not an exceptional comedy of errors but rather the normal way of how scientific progress is achieved. We shall argue that scientific knowledge generally does not develop as a sequence of independent dis- coveries accumulating to a new body of knowledge but as a network of interdependent activities which only as a whole make an individual step understandable as a meaningful “discovery.“
The question when and how Galileo made his celebrated discoveries of the law of fall and of the parabolic shape of the projectile trajectory has been extensively discussed in the last one hundred years by historians of science. Contrary to the testimony of Galileo’s disciple Viviani,^2 who ascribed such discoveries already to the young Galileo, it is widely accepted today that these discoveries date into the late Paduan period. Most writers date the discovery of the law of fall to the year 1604 and assume that Galileo discovered the parabolic shape of the projectile trajectory even some years later. In his influential Galileo Studies , Alexandre Koyré lapidarily affirmed:^3
The law of falling bodies—the first law of classical physics—was formulated by Galileo in 1604.
This dating is primarily based on the few contemporary documents by Galileo himself which provide clues to his knowledge of the law of fall and the form of the trajectory. In particular, two letters by Galileo stand out because of the testimony they offer to his knowledge at certain precise dates. The first letter is directed to Paolo Sarpi and dated 16 October 1604;^4 it provides clear evidence that, at this point in time, Galileo knew the law of fall. The second letter is di- rected to Antonio dei Medici and dated 11 February 1609.^5 It shows that Galileo, by that time,
(^2) According to Viviani, Galileo discovered the isochronism of the pendulum already as a student in Pisa around 1583; see Viviani’s letter to Leopoldo dei Medici, Galilei 1890-1909, XIX: 648; see also his biography, Galilei 1890-1909, XIX: 603. He claims furthermore that Galileo performed experiments on free fall already between 3 1589 and 1592 when he was professor in Pisa; see Viviani’s biography of Galileo, Galilei 1890-1909, XIX: 606. 4 Koyré 1966, 83. 5 Galileo to Paolo Sarpi, October 16, 1604, Galilei 1890-1909, X: 116. Galileo to Antonio de Medici, February 11, 1609, Galilei 1890-1909, X: 228.
Hunting the White Elephant
knew that projectiles reaching the same height take the same time to fall down, a property that follows, from a modern point of view, from the decomposition of the parabolic trajectory into its horizontally uniform and vertically accelerated components.
Of course, these documents provide at most a terminus ante quem for Galileo’s discoveries. The hesitation of modern Galileo scholars to follow Viviani’s testimony and to accept an earlier date is, partly at least, due to the study of his early manuscript De Motu which in spite of its anti- Aristotelian tendency shows him as deeply influenced by the categories and assumptions of me- dieval Aristotelian physics.
The cornerstones 1604 and 1609 define a scaffolding for more or less speculative stories about what really happened at those times. It is, indeed, customary to find these dates in reconstruc- tions of the sequence of Galileo’s discoveries, even though the claims of what made up these discoveries widely diverge among different authors – some even maintain that the law of fall was only discovered in 1609 and the parabolic trajectory perhaps even later. As an example we quote the succinct reconstruction of the sequence of Galileo’s discoveries by the influential German historian of science Friedrich Klemm:^6
Attempts showing him that the difference in the speed of fall of bodies of the same size with different specific weights becomes smaller the more the medium is dilut- ed, suggested to Galileo around 1600 to assume that the speed of fall in vacuum is going to be of equal magnitude for all bodies. (...) The next step is now to give up all considerations about the cause of the growing speed of fall and to limit himself to treating mathematically the motion of fall taking place for all bodies in a vacuum in the same way. In 1604 Galileo makes the as- sumption, convinced that in nature everything is constituted as simply as possible: the speed of fall in vacuum increases with the distance of fall traversed. This ap- proach leads him into contradictions. Finally in 1609 he comes to recognize the pro- cess of fall as a uniformly accelerated motion, that is, he comes to the insight that the velocity grows with time. Starting from here he obtains by using developments of the late middle ages: the distances are to each other as the squares of the times. This again gives him the possibility to verify the hypothetical approach by the experiment (the fall trench experiment).
This reconstruction is obviously guided by the supposition that Galileo proceeded methodically and step-by-step from one discovery to the next. Naturally, such a view of Galileo’s progress must remain speculative as long as it is not supported by a detailed analysis of contemporary sources.
(^6) Klemm 1964, 79-80.
v = a ⋅t
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1602 Galileo “begins studies of long pendulums and motion on inclines“, in 1604 he “discovers the law of pendulums from careful timings; finds the law of fall,“ and only in 1608 he “discov- ers the parabolic trajectory by measurements.“
It thus emerges as a peculiarity of recent Galileo historiography that the dates at which Galileo supposedly made his major discoveries have remained largely unchallenged, in spite of the rel- atively weak direct evidence available for this dating. This peculiarity is all the more surprising as the dating and the sequence of Galileo’s discoveries have been extensively and controver- sially discussed in the older Galileo literature around the turn of the century. The possibility that Galileo first discovered the parabolic form of the trajectory and only then the law of fall was, for instance, seriously considered by Emil Wohlwill, together with the possibility that the law of fall was discovered much earlier than is now commonly assumed.^10 But Emil Wohlwill’s substantial contributions had as little impact on the Galileo studies of the last fifty years as those of his eminent Italian contemporaries, Antonio Favaro and Raffaello Caverni. As we will see in the following, there are good reasons to take up the debate where it was left a century ago.
Concerning the sources from which Galileo derived his major discoveries there is much less agreement among recent historians of science than concerning the dating. The assumptions about his sources range in fact from pure empirical evidence achieved exclusively by means of careful experimentation and precise measurements, on the one hand, to predominantly theoret- ical speculation in direct continuation of scholastic traditions only scarcely supported by em- pirical demonstrations, on the other hand.^11
(^10) See Wohlwill 1993, I: 144-162 and Wohlwill 1899. In the latter article Wohlwill still argues that the law of fall was discovered shortly before he wrote his letter to Paolo Sarpi in 1604 in which the law is explicitly mentioned. In the first volume of his final work on Galileo published shortly before his death, however, he developed an ingenious argument for a quite different dating. Based on the sources available at that time and, in particular, based on the first publication of excerpts from Galileo’s notes by Caverni and Favaro, he developed a recon- struction of Galileo’s discovery which he qualified as “only the most probable” way of how Galileo might have found the law of fall; this reconstruction is essentially coherent with what is presented in the following. The new evidence provided here shows that Wohlwill’s conjecture was much more sound than the interpretations which 11 are currently^ en vogue. In discussions, these positions are still represented prototypically by Drake, on the one side, and Koyré, on the other side.
The Neglected Issue: Trajectory and Hanging Chain
In spite of the wide range of different reconstructions of the discovery process,^12 however, a simple fact has nearly been completely neglected both by the older and the more recent litera- ture: for Galileo, there exists a close connection between the parabolic trajectory and the cate- nary, that is, the curve of hanging chains. This neglect is all the more astonishing as the connection is explicitly made a subject of discussion in his final word on the matter, the Dis- corsi. In the course of the discussions of the Second Day, Galileo’s spokesman Salviati de- scribes two methods of drawing a parabola, one of them involving the trajectory of a projected body, the other one using a hanging chain:^13
There are many ways of drawing such lines, of which two are speedier than the rest; I shall tell these to you. One is really marvelous, for by this method, in less time than someone else can draw finely with a compass on paper four or six circles of different sizes, I can draw thirty or forty parabolic lines no less fine, exact, and neat than the circumferences of those circles. I use an exquisitely round bronze ball, no larger than a nut; this is rolled [ tirata ] on a metal mirror held not vertically but somewhat tilted, so that the ball in motion runs over it and presses it lightly. In mov- ing, it leaves a parabolic line, very thin, and smoothly traced. This [parabola] will be wider or narrower, according as the ball is rolled higher or lower. From this, we have clear and sensible experience that the motion of projectiles is made along par- abolic lines, an effect first observed by our friend, who also gives a demonstration of it. We shall all see this in his book on motion at the next [ primo ] meeting. To describe parabolas in this way, the ball must be somewhat warmed and moistened by manipulating it in the hand, so that the traces it will leave shall be more apparent on the mirror. The other way to draw on the prism the line we seek is to fix two nails in a wall in a horizontal line, separated by double the width of the rectangle in which we wish to draw the semiparabola. From these two nails hang a fine chain, of such length that its curve [ sacca ] will extend over the length of the prism. This chain curves in a parabolic shape, so that if we mark points on the wall along the path of the chain, we shall have drawn a full parabola. By means of a perpendicular hung from the center between the two nails, this will be divided into equal parts.
At a prominent place, the end of the Fourth Day of the Discorsi , Galileo returns to what he con- sidered a surprising fact, the parabolic shape of the catenary:^14
Salviati. (...) But I wish to cause you wonder and delight together by telling you that the cord thus hung, whether much or little stretched, bends in a line that is very close to parabolic. The similarity is so great that if you draw a parabolic line in a vertical plane surface but upside down—that is, with the vertex down and the base parallel to the horizontal—and then hang a little chain from the extremities of the base of the parabola thus drawn, you will see by slackening the little chain now more and
(^12) For the literature on this subject, see the references in section 3.3.1 of Damerow, Freudenthal, McLaughlin, and Renn 1992; see in particular, the ingenious reconstruction of Galileo’s inclined plane experiment, described in 13 Settle 1961. 14 Galilei 1974, 142f. Galilei 1974, 256.
Evidence I: Galileo Using Hanging Chains
Was it this approximation that Galileo had in mind when he identified the catenary with the pa- rabola? There is overwhelming evidence that this is not the case. In the finished parts of the Dis- corsi he clearly pointed out that he assumed not a fortuitous but a substantial relation between the parabolic trajectory and the catenary.^17
Salviati. Well, Sagredo, in this matter of the rope, you may cease to marvel at the strangeness of the effect, since you have a proof of it; and if we consider well, per- haps we shall find some relation between this event of the rope and that of the pro- jectile [fired horizontally]. The curvature of the line of the horizontal projectile seems to derive from two forc- es, of which one (that of the projector) drives it horizontally, while the other (that of its own heaviness) draws it straight down. In drawing the rope, there is [likewise] the force of that which pulls it horizontally, and also that of the weight of the rope itself, which naturally inclines it downward. So these two kinds of events are very similar.
This supposed theoretical relationship between the catenary and the trajectory will be exten- sively discussed in the following. Previously, however, another question has to be answered: did Galileo really produce, as he claimed, parabolic curves by means of projected bronze balls and hanging chains or did he merely invent these stories in order to give his idea of a theoretical connection between catenary and trajectory a lively illustration?
Fortunately, this question can easily be answered in the case of the first method of drawing par- abolic lines he describes. Among Galileo’s notes on mechanics, Ms. Gal. 72, there is a folded sheet of rough paper designated as folio 41/42 (see figure 1) that has obviously been used for drawing catenaries as Galileo described it in the Discorsi. The sheet was fixed at the wall by means of two nails; the holes in the sheet of paper, through which the nails were driven, are still visible.^18 Chains of different length where fixed at these two nails and their shape was copied to the paper by means of some needles. Finally, using this perforated sheet, the resulting curves were copied by letting ink sip through the little holes pierced into the paper along the hanging chains.
(^17) Galilei 1974, 256. (^18) The distance between the two suspension points is 443 mm. According to two notes, one on the left and the other on the right side of the curves, “total amplitude 465,” this distance was measured by Galileo as 465, prob- ably indicating 465 “points.” Thus, the size of the unit used by Galileo is 0.95 mm.
Hunting the White Elephant
Figure 1. MS. GAL. 72, FOLIO 41/42 WITH CURVES PRODUCED BY MEANS OF A HANGING CHAIN
Another folio page, 113 recto, shows a drawing containing curves which represent projectile trajectories of oblique gun shots projected under various angles. The curves consist of ink dots which are joined by faint lines. In addition to these representations of trajectories, the folio page contains several drawn or scratched auxiliary lines, such as straight lines representing the direc- tions of the shots or the levels of their maximum heights.^19
A comparison of the curves representing the trajectories has shown that they fit precisely the template represented by folio 41/42 (see figure 2). Thus folio 113 is a preserved example of the application of the very technique of drawing supposedly parabolic curves by means of a hang- ing chain which Galileo describes in the Discorsi.
(^19) Several uninked construction lines can be found on the folio page 113r. An analysis of these lines has provided evidence that a basic unit of exactly the same size was used on this folio as on folio 41/42. A set of parallel lines can be identified which are drawn vertically to the baseline of the parabolic trajectory in equal distances of pre- cisely 15 “points” measured in the basic unit of folio 41/42. The total distance measured along the baseline from the origin of the shots to the vertical representing the target is divided by the parallel lines precisely into 16 parts of 15 “points” each. Furthermore, an arc of a circle with a radius of 100 “points” can also be identified. At the vertical target line a number “100” is written exactly 100 “points” above the base line. But not only the small unit is common to both folios. Assuming that the distance between the parallel lines defines a higher unit of 15 “points,” the “total amplitude 465”on the template folio 41/42 measures precisely 31 of these units.
Hunting the White Elephant
In the case of the second method for drawing parabolas which Galileo in the Discorsi claims to have been using, that is, the generation of parabolas by rolling a ball over an inclined mirror, the evidence of the truth of his claim is somewhat more indirect. Galileo’s copy of the first edi- tion of the Discorsi contains numerous corrections, notes, and additions mostly by the hand of his disciple Vincenzio Viviani. These notes were probably added in part still by Galileo himself for a revised edition, but apparently written by Viviani (and possibly by other disciples) because of Galileo’s progressive blindness during the last years of his life. Unfortunately, Galileo’s Dis- corsi were never published with the revisions according to these notes, most likely because it is impossible to distinguish which of these notes were authorized by Galileo and which of them were inserted by Viviani on his own account only after Galileo’s death. In any case, these notes provide striking insights into consequences of Galileo’s Discorsi which were either implicit but insufficiently expressed in the printed text of the first edition or could be achieved immediately by an elaboration of his practical and theoretical achievements.
With regard to the present problem of whether or not Galileo really used the second method of drawing parabolas, the inspection of his copy of the first edition of the Discorsi provided a sur- prise. On a sheet of paper^20 inserted close to Galileo’s description of the second method for drawing parabolas one finds two curves which show the typical characteristics of such a meth- od: the indications of the bouncing of the ball at the beginning and the slight deformation of the parabola at the end due to friction (see figure 3).^21 If one could be certain that it was Galileo himself who produced these parabolas, it would be thus clear that the description in the Discorsi refers to an experiment he had actually performed himself.
The claim that Galileo used this method receives strong confirmation by the analysis of another manuscript, although this manuscript is definitely not written by Galileo himself but by Guidobaldo del Monte, a correspondent, benefactor, and a close associate of Galileo in his early research on mechanics. This document nevertheless provides additional evidence and even al- lows the conclusion, as we will show, that Galileo was familiar with this method of drawing parabolas already long before he wrote the Discorsi.
(^20) Folio page 90v of Galileo’s copy of the Discorsi , Galilei after 1638. (^21) This judgement is based on a careful repetition of the experiment under controlled conditions with the help of equipment designed by Henrik Haak and realized by the workshop of the Fritz Haber Institute of the Max Planck Society. Henrik Haak has furthermore assisted us in the reproduction of the historical experiment; the results will be the subject of a forthcoming publication.
Evidence II: Viviani’s Addition to the Discorsi and Guidobaldo’s Protocol of an Experiment
Figure 3. SHEET OF PAPER FOUND IN GALILEO’S COPY OF THE FIRST EDITION OF THE Discorsi (MS. GAL. 79, FOLIO PAGE 90 VERSO ) CONTAINING TWO PARABOLIC CURVES GENERATED BY AN INKED BALL THROWN ALONG AN IN- CLINED PLANE , INSERTED NEAR TO THE PLACE WHERE THIS METHOD IS DESCRIBED
Evidence II: Viviani’s Addition to the Discorsi and Guidobaldo’s Protocol of an Experiment
At the end of a notebook of Guidobaldo there are two drawings which possibly depict an in- clined plane used for such an experiment, together with a protocol which is perfectly resembled by the description of Galileo’s second method mentioned in the Discorsi (see figures 4 and 5). A closer inspection of Guidobaldo’s drawings shows that they actually represent a roof which may have offered a convenient setting ready-at-hand for originally trying out a method similar to that described by Galileo on a scale comparable to that of ballistics, the usual context in which projectile motion was considered at that time.^22
Figure 5. GUIDOBALDO ’S SKETCH AND A CONTEMPORARY REPRESENTATION OF A ROOF WITH “CAPRIATA ”^23
Guidobaldo’s protocol not only describes precisely the experimental setting but also reports re- sults, such as the symmetry of the generated curve and the close relation to the curve of a hang- ing chain, that can be deduced from the observation that the curves in both cases result from the same configuration of forces. The protocol begins with a summary of consequences that can be drawn from the outcome of the experiment, followed by a description of the method applied. It ends with a theoretical interpretation of the symmetry of the trajectory:^24
If one throws a ball with a catapult or with artillery or by hand or by some other instrument above the horizontal line, it will take the same path in falling as in rising, and the shape is that which, when inverted under the horizon, a rope makes which is not pulled, both being composed of the natural and the forced, and it is a line which in appearance is similar to a parabola and hyperbola. And this can be seen
better with a chain than with a rope, since [in the case of] the rope abc , when
(^22) Henrik Haak, who constructed the apparatus for our reproduction of the historical experiment, has directed our attention to the fact that the inclined planes depicted by Guidobaldo immediately before and almost immediately 23 behind the protocol represent a roof construction. 24 del Monte ca. 1587-1592, 237, and an illustration by Palladio, reproduced from Tampone 1996, 71. del Monte ca. 1587-1592, 236; a transcription of the text has been first published by Libri 1838, IV: 397f. Its significance for dating Galileo’s early work on motion was first recognized by Fredette 1969. The experiment described by Guidobaldo has been extensively discussed in Naylor 1974. Naylor concludes that Galileo could not have been convinced by the outcome of this experiment alone of the parabolic shape of the trajectory and that it was only in 1607 that he arrived at this conviction. Naylor thus agrees with the standard dating of this discovery, a conclusion that we will attempt to refute in the following. The crucial significance of the experi- ment for Galileo’s discovery of the law of fall was first suggested by Damerow, Freudenthal, McLaughlin, and Renn 1992, 336f.
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ac are close to each other, the part b does not approach as it should because the rope remains hard in itself, while a chain or little chain does not behave in this way. The experiment of this movement can be made by taking a ball colored with ink, and throwing it over a plane of a table which is almost perpendicular to the horizontal. Although the ball bounces along, yet it makes points as it goes, from which one can clearly see that as it rises so it descends, and it is reasonable this way, since the violence it has acquired in its ascent operates so that in falling it overcomes, in the same way, the natural movement in coming down so that the vio- lence that overcame [the path] from b to c , conserving itself, operates so that from c to d [the path] is equal to cb , and the violence which is gradually lessening when descending operates so that from d to e [the path] is equal to ba , since there is no reason from c towards de that shows that the violence is lost at all, which, although it lessens continually towards e , yet there remains a sufficient amount of it, which is the cause that the weight never travels in a straight line towards e.
The similarity of this protocol of Guidobaldo’s experiment and Galileo’s description of his sec- ond method to draw parabolas raises, of course, the question of what relation exists between these two reports. Did Guidobaldo and Galileo independently make the same observation? If not, who of them did the experiment and who only heard of or reproduced it? We will show in the following that not only are both referring to the same experiment, but that, moreover, Gali- leo was even present when this experiment was performed. Previously, however, the issue has to be analyzed in some more detail.
When has the technique of tracing the trajectory of a ball,which both Galileo and Guidobaldo described, been developed? A first clue to the answer to this question is provided by the out- come of Guidobaldo’s experiment itself. This outcome, as it is reported in Guidobaldo’s note, was in one respect incompatible with the view of projectile motion prevailing at the time of the young Galileo. In the Aristotelian tradition, projectile motion was conceived of as resulting from the contrariety of natural and violent motion, the latter according to medieval tradition act- ing through an impetus impressed by the mover into the moving body. According to this under- standing of projectile motion, the trajectory cannot not be symmetrical because the motion of the projectile is determined at the beginning and at the end by quite different causes. At the be- ginning it is dominated by the impetus impressed into the projectile, at the end by its natural motion towards the center of the earth.
Hunting the White Elephant
According to Tartaglia’s theory, the trajectory of a projectile consists of three parts. It begins with a straight part that is followed by a section of a circle and then ending in a straight vertical line (see figure 6). This form of the trajectory also corresponds to Tartaglia’s adaption of the Aristotelian dynamics to projectile motion in the case of artillery, a case that was, of course, much more complicated than what was traditionally treated in Aristotelian physics.
The first part of the trajectory was conceived by Tartaglia as reflecting the initially dominant role of the violent motion, whereas the last straight part is in accord with the eventual domi- nance of the projectile’s weight over the violent motion and the tendency to reach the center of the earth. The curved middle part might have been conceived of as a mixed motion compounded of both violent motion in the original direction and natural motion vertical downward. But, since violent and natural motion were supposed to be contrary to each other, this conclusion ap- peared to be impossible to Tartaglia. He claimed instead the curved part to be exclusively due to violent motion as is the first straight part of the trajectory. He proved the proposition:^29
No uniformly heavy body can go through any interval of time or of space with mixed natural and violent motion.
Figure 7. COMPARISON OF TARTAGLIA ’S CONSTRUCTION OF PROJECTILE TRAJECTORIES (LEFT FIGURE ) WITH GA- LILEO’S FIGURE IN De Motu (RIGHT FIGURE )^30
Tartaglia’s construction of the trajectory was influential throughout the 16th century, although it could not be brought into agreement with the simple explanation for the obvious fact that non- vertical projection is never perfectly straight, and, in addition, corresponds only roughly to the visual impression of the motion of projectiles, and certainly could not be justified by precise
(^29) Drake and Drabkin 1969, 80. (^30) The first figure is taken from Tartaglia 1984, 11. The second figure has been redrawn on the basis of a microfilm reproduction of the original manuscript.
How Can the Aristotelian View of Projectile Motion Account for a Symmetrical Trajectory?
observations of their trajectory. The simple shape of Tartaglia’s trajectory, however, immedi- ately allows one to draw a number of conclusions about projectile motion by geometrical rea- soning that seemingly were theoretically convincing and practically useful.
It is well known that Galileo originally also adhered to this theory. In his early manuscript De Motu , written about 1590, he contributed to this theory by proving at the end of his treatise the proposition that objects projected by the same force move farther on a straight line the less acute are the angles they make with the plane of the horizon. At that time, he obviously had no doubt that the traditional view of a straight beginning of the trajectory was correct, adding an own con- tribution to further developing this theory (see figure 7). He tried to explain the different lengths of the straight parts of the trajectories of bodies projected under different angles by arguing that different amounts of force are impressed into the body according to the different resistances if the angle of projection is varied. In the course of the proof of this proposition, however, he de- veloped Tartaglia’s theory further in the direction already taken by Tartaglia himself. In Tarta- glia’s late publication, the Quesiti , he again, possibly under the pressure of Cardano’s criticism at his claim that natural and forced motion cannot act simultaneously,^31 stated even more clearly than in his Nova Scientia that the trajectory is in no part perfectly straight. Galileo elaborated the theoretical explanation for the curvature of the trajectory given by Tartaglia even further. This theoretical explanation brought him into conflict with an argument he had developed ear- lier in order to explain acceleration in free fall. Galileo argued that only in the case of vertical projection violent and natural motion due to their contrariety cannot act together at the same time, whereas in the case of oblique projection the trajectory may well be explained by the si- multaneous effects of both.^32
When a ball is sent up perpendicularly to the horizon, it cannot turn from that course and make its way back over the same straight line, as it must, unless the quality that impels it upward has first disappeared entirely. But this does not happen when the ball is sent up on a line inclined to the horizon. For in that case it is not necessary for the [impressed] projecting force to be entirely used up when the ball begins to be deflected from the straight line. For it is enough that the impetus that impels the body by force keeps it from [returning to] its original point of departure. And this it can accomplish so long as the body moves on a line inclined to the horizon, even though it may be only a little inclined [from the perpendicular] in its motion. For at the time when the ball begins to turn down [from the straight line], its motion is not contrary to the [original] motion in a straight line; and, therefore, the body can change over to the [new] motion without the complete disappearance of the impel- ling force. But this cannot happen while the body is moving perpendicularly up- ward, because the line of the downward path is the same as the line of the forced motion. Therefore, whenever in its downward course, the body does not move to-
(^31) See Drake and Drabkin 1969, 100-104. On Cardano’s criticism see Arend 1998. (^32) Galilei 1960, 113.
