galileo theory proyectil

The University of Chicago Press and The History of Science Society are collaborating with JSTOR to digitize, preserve and extend access to Isis.

http://www.jstor.org

Galileo's Theory of Projectile Motion Author(s): R. H. Naylor Source: Isis, Vol. 71, No. 4 (Dec., 1980), pp. 550-570Published by: on behalf of The University of Chicago Press The History of Science SocietyStable URL: http://www.jstor.org/stable/230500Accessed: 06-05-2015 07:02 UTC

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

This content downloaded from 163.178.101.228 on Wed, 06 May 2015 07:02:37 UTCAll use subject to JSTOR Terms and Conditions

Galileo's Theory of

Projectile Motion

By R. H. Naylor*

ALILEO'S EARLIEST WORK on projectile motion dates from the time J of his professorship at Pisa in 1589-1592. Late in life he was to recount

that the first objective of his study of motion was to discover the curve described by a projectile. However that is not the impression given by his essay De motu, written around 1592, in which the projectile trajectory figures as but one of many problems of motion and where no particular importance is attached to it. Only in the period 1604-1608 did he identify the trajectory as parabolic and, as a result of further considerable effort, come to recognize the great theoretical significance of his discovery. It was quite natural for Galileo, when he looked back on his research of 1604-1608, to regard it as the culmination of his mature creative work on motion. This article studies the development of Galileo's theory of projectile motion. Further, it argues that not only were Galileo's experiments closely connected with his theoretical work, but that they were designed primarily to investigate theoretical problems.

It is clear from his essay Le meccaniche, written around 1596, that Galileo had adopted the principle of horizontal inertia only a few years after coming to Padua in 1592. By 1602 he had discovered the isochronism of the pendulum and had developed two or three basic theorems relating to motion on inclined planes. In 1604 he wrote to Paolo Sarpi telling him of the law of fall. Though convinced that the law was valid he was at that time attempting to justify it, by deducing the law from a more fundamental empirical principle. He believed that the percussive effects of falling bodies showed that instantaneous velocity increased with distance in free fall, and he attempted to deduce the s oc t2 law from that assumption. A draft of such a proof survives on one of his manuscript notes (fol. 128); there he argues that average velocity increases with the square of the distance fallen. This meant that average velocity was proportional to the square of the final instantaneous velocity.

By 1609 Galileo's theory had changed considerably. He had rejected the idea that velocity increased with distance and believed instead that velocity increased with time in free fall. In addition to this new principle Galileo also saw instantaneous velocity and average velocity as increasing steadily with time. That is to say, by 1609 Galileo saw average velocity as proportional to instantaneous velocity. This process of theoretical evolution was a direct result of Galileo's work on the projectile trajectory.

*School of Humanities, Thames Polytechnic, London SE18 6PF, England. The completion of the research involved in the preparation of this paper was made possible by the

generous support of the British Academy and the Krieble Delmas Foundation.

Isis, 1980, 71 (No. 259) 550


GALILEO'S THEORY OF PROJECTILE MOTION 551

I. THE ESTABLISHMENT OF THE FORM OF A PROJECTILE TRAJECTORY AS PARABOLIC (FOLIOS 81r AND 107r)

The motion of projectiles interested Galileo from his earliest work on motion, De motu, written around 1592.1 There he advanced a theory similar to earlier sixteenth-century discussions of projectile motion-notably that proposed by Niccolo Tartaglia in his Nova scienza of 1537.2 Galileo's interest in this subject was shared by his patron Guidobaldo del Monte, who some time before 1600 devised an experiment revealing important features of the trajectory, the account of which survives in a manuscript preserved in the Bibliotheque Nationale, Paris:

If one throws a ball with a catapult or with artillery, or by hand, or by some other means, above the horizontal line, it will take the same path in falling as in rising, and the shape is like that which, when inverted under the horizontal line, a ropes makes which is not pulled, being composed of both the natural and the forced, and it is a line which in appearance is similar to the parabola and hyperbola.... 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 see clearly that as it rises so it descends, and it is reasonable since the force it has acquired in its ascent operates so that in falling it goes in the same way, overcoming the natural movement in coming down so that the force that it overcame from B to C, conserving itself, operates so that from C to D it is equal to CB, and the force, descending and gradually lessening, is such that from D to E it is equal to BA, seeing that there is no reason to show that from E towards DE the force is at all expended that, although it lessens continually towards A, yet continues to be the reason why the weight never travels in a straight line towards E.3

This experiment and its analysis represented an important advance in the under- standing of the trajectory insofar as it recognized for the first time that the trajectory -whatever its precise geometrical form- was symmetrical about its vertical axis. Previous analyses of projectile motion, like those of Tartaglia and Galileo, had been primarily concerned with the trajectories of cannon shot. For this reason it had been

C

B D

A E

Figure 1. Trajectory as traced by Guidobaldo del Monte.

argued-on the basis of observation- that missile trajectories were not symmetrical in form. The argument advanced by Guidobaldo was a direct result of his carrying out the experiment on a small scale, thus avoiding interference from air resistance, and of his obtaining a permanent trace of the projectile trajectory, thus avoiding the difficulties of relying on direct observation of the moving projectile.4

ILe Opere di Galileo Galilei, ed. Antonio Favaro, 20 vols. (Florence, 1890-1909), Vol. I, p. 337; translation in I. E. Drabkin and Stillman Drake, Galileo Galilei on Motion and on Mechanics (Madison: University of Wisconsin Press, 1960), pp. 110-114.

2Niccol6 Tartaglia, Nova scienza (Venice, 1537), pp. 39-43; translation in Stillman Drake and I. E. Drabkin, Mechanics in Sixteenth Century Italy (Madison: University of Wisconsin Press, 1969), pp. 84-85.

3Quoted in G. Libri, Histoire des sciences mathematiques en Italie, 4 vols. (Paris, 1838-1841), Vol. IV, pp. 387-398. My translation; italics added.

4For details of the reconstruction of Guidobaldo's experiment and its relationship to Galileo's


552 R. H. NAYLOR

No doubt Guidobaldo kept Galileo abreast of these new ideas and methods of research. There is evidence of this exchange of ideas in Galileo's letter to Guidobaldo of 29 November 1602.5 Unfortunately earlier letters of this corres- pondence do not survive, but it is clear that they had already exchanged letters concerning the subjects under discussion. Moreover this letter, when compared with De motu, reveals that Galileo had adopted a radically different view of the relationship between experiment and theory, one rather similar to the view expressed by Guidobaldo when discussing his experiments on the projectile trajectory. Both that discussion and Galileo's letter reveal a new awareness of the significance of close agreement between experiment and theory in carefully controlled or selected instances.

One other piece of evidence that can be dated with certainty-and that reveals the same attitude toward experimentation-is the letter Galileo wrote to Paolo Sarpi on 16 October 1604.6 In it Galileo explicitly states his view that when a projectile is thrown vertically upward and thereafter falls along the same vertical line, the velocities at each point on that line are the same for upward and down- ward motion. The letter also shows that Galileo had proposed extending his theory to cases in which missiles were projected at an angle to the horizontal. In these cases he conjectured that the speed of the projectile was the same at equal distances from the highest point on the trajectory. Thus, according to Galileo's theory, a stone or arrow projected at an angle to the horizontal should return to the level of projection with a -speed equal to that with which it had been projected.

Now whereas this theory does explain instances in which air resistance is very small, like that described by Guidobaldo in his experiment, it does not explain many other phenomena for which air resistance is significant. This point was made by Sarpi in the letter to which Galileo was replying, written to Galileo on 9 October 1604.7 Sarpi referred to the case of an arrow shot from a strong Turkish bow and pointed out that the arrow exerted greater force and moved most rapidly immediately after being released. The speed of the arrow, it seemed, continually decreased during its trajectory. From this it appeared that Galileo's theory did not apply to arrows or firearms.

But Galileo was less interested in exploring projectiles of all sorts than in obtaining precise information that could be related to a mathematical model of the trajectory. From the surviving manuscript notes it is possible to establish that he studied the projectile trajectory by means of small-scale experiments that provided, in permanent form, details of the trajectories. His approach was thus similar to Guidobaldo's.

In one such experiment, an early one dating from about 1605, Galileo used a grooved plane inclined at a small angle to the horizontal. A metal sphere was released from a point on the inclined plane elevated some distance above ground level and allowed to roll off the end and complete a free trajectory.

views in De motu and the Discorsi see R. H. Naylor, "The Evolution of An Experiment: Guido- baldo del Monte and Galileo's Discorsi Demonstration of the Parabolic Trajectory," Physis, 1974, 16:323-346.

5Opere, Vol. X, pp. 97-100. 6Ibid., pp. 114-116. 7Ibid., p. 113.



Table 1. Comparison of data on folio 81r and results of the reconstructed experiment (RE). Data for the reconstructed experiment are expressed in millimeters. Galileo's measurements, on folio 81r as elsewhere, are likely to have been expressed in punti, a unit of distance (= nearly 0.95 mm) found on the linear scale of his proportional compass. Vertical drop below Value of horizontal projection shown as an

end of groove increase over previous columns

folio 81r RE folio 81r RE folio 81r RE folio 81r RE

329.5 329.5 250 250 250.0 250.0 250 250.0 183.5 183.5 170 171 177.5 178.0 178 179.0 106.0 106.0 121 124 130.5 130.0 131 132.0 53.0 53.0 81 80 87.5 88.5 89 89.5

Points on this trajectory were located by placing a horizontal plane at various measured distances below the point of projection. The point at which the sphere hit the plane was located using some substance that bore an impression of im- pact-perhaps clay or some suitably treated paper. The data on folio 81r, which represent the measurements Galileo obtained in carrying out this form of experiment, are transcribed in Figure 2. If the experiment is reconstructed with a plane inclined at 20?30' to the horizontal, it is possible to obtain values for the projections at four horizontal levels below the point of projection very similar to those indicated against the smaller curve on folio 81r. With the plane inclined at 7? and 3.50, two further sets of projections similar to the values adjacent to the other two curves on folio 81r may be obtained (see Table 1). Such an experiment would have provided Galileo with a means of establishing the geometrical form of the trajectories, revealing that, with due allowance for the rather small experimental errors, the trajectories were parabolic in form.8

After this Galileo progressed to other experimental investigations and to mathematical analyses of the trajectory, attempting to establish the mechanical principles that would account for the phenomena. One manuscript, folio 107,

89 87. 5 81

/ 131 / 130-5 / 121

178 1775 170

146

c 250 250 250 b

Figure 2. Transcription of detail from folio 81r.

8For full details of the reconstruction of this experiment see R. H. Naylor, "Galileo: The Search for the Parabolic Trajectory," Annals of Science, 1976, 33:153-172 (a photograph of folio 81r appears in that paper).


554 R. H. NAYLOR

* *. * L'-,'T'" ::''* -

*. -* ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ & . /~~~ .. . .....

-. -4

* * .... .

,. * * * 4

.

j .

. . ...

-~~~~~~~~~~~~~~~ ,

FigUre 3. Folio 107r (Vol. 72, MSS Galileani; courtesy of the Biblioteca Nazionale di Firenze). reveals evidence of both activities. The object of the analysis on folio 107r is unclear, but an apparently similar type of investigation on folio 117r provides just sufficient detail for the former analysis to be reconstructed. This will be discussed presently.

Folio 107r (see Fig. 3) bears evidence of Galileo's study of two curves. One curve is sketched as a dotted line and the second and continuous curve is drawn adjacent to it. Both lines have a common axis and apex and meet at one other point. Both curves are catenaries. The resemblance between the form of a projectile trajectory and the curve formed by a hanging rope or chain was noted by Guidobaldo in the passage quoted above. But as Galileo was to point out later in Two New Sciences,9 there are in fact differences between these curves.

The dotted line on folio 107r has a number of figures marked against it along one side. They appear to represent values obtained from the study of distances related to points on the dotted ink line. It is very clear that Galileo was interested in the differences between the figures associated with particular points on that curve, that is, in the differences between the successive distances. The figures written alongside it are 25'/2, 40, 54, 67, 79, 91, 1021/2, 1131/2, and 123'/2. At the side of the manuscript is a list of the differences between these figures: 141/2, 14, 13, 12, 12, 111/2, 11, 10.

Below the curves can be seen three series of numbers. The upper series is spaced out just above the edge of the manuscript at distances equally spaced from the apex; the figures of the series, the squares 1, 4, 9, 16, 25, 36, 49, 81, 100, 121, 144, are spaced at intervals of 10 punti along the line at right angles to the axis. They represent the distance from the x axis of points on the parabola spaced at 10 punti intervals. Thus the spacing and the figures provide the

90pere, Vol. XliI, pp. 186, 310; translation in H. Crew and A. de Salvio, Dialogues Concerning Two New Sciences (New York: McGraw Hill, 1963), pp. 143, 278.



Table 2. Figures found on folio 107v. The empirical data in the third column are compared to the figures in the first column and to the data of the reconstructed experiment.

Data on folio 107v Analysis of folio 107v data

Percentage Data of the Column Column Column Column 1 x 33 difference reconstructed

1 2 3 (= column 4) between experiment columns 3 and 4 in punti

1 1 33 33 0 33 4 2 130 132 -1.5 133 9 3 298 297 +0.3 296

16 4 526 528 -0.4 530 25 5 824 825 -0.1 828 36 6 1192 1188 +0.4 1190 49 7 1620 1617 +0.2 1615 64 8 2123 2112 -0.5 2101

equivalent of the equation y cc x2 and indicate that Galileo was comparing these curves with a parabola.

Folio 107v bears further signs of an investigation related to the parabola. The same figures appear representing the distances that define a parabola, but in this case there is no sketch. Instead there appears a third column of figures, clearly obtained by measurement. The three series of figures are shown in Table 2. The fourth column in the table shows the measurements obtained by the author in conducting a reconstructed experiment on a projectile trajectory (described below). The similarity between the third and fourth columns shows that the third series probably records an experiment aimed at establishing the form of the trajectory.

The very presentation of the data on folio 107v is consistent with the supposi- tion that Galileo conducted the experiment to establish whether or not the trajectory was parabolic. That is, the successive values of the coordinates placed the controlled variable (the x coordinate) next to the measured variable (the measured y value). Galileo knew the ratios to be expected between the coordinates if the trajectory was parabolic along its entire length. Folio 81r records this form of investigation, and it would of course have been necessary for Galileo to establish the form of the trajectory before he conducted his final experiment on folio 116v. Moreover, the catenary on folio 107r suggests that Galileo may have possibly compared the two curves as an independent means of assessing the folio 107v experiment.

The folio 107v experiment, like Galileo's earlier investigations, used a grooved inclined plane, in this case one that curved at its lower end (see Fig. 4). A small metal sphere about 1 centimeter in diameter was released at A to run down a smooth straight wooden groove to B where the groove bends smoothly to become horizontal at C. The sphere therefore leaves the groove at C traveling horizontally and completes its trajectory at F, where it strikes a horizontal board. The figures in the third column on folio 107v correspond to the measured values obtained in its experiment, while the figures in the first and second columns represent points on a parabola.


556 R. H. NAYLOR

/~~~~ H

Figure 4. Schematic representation of the experiment depicted on folios 114r and 1 16v.

h

F'4 - D v

Such an apparatus was built for the reconstructed experiment. In it the sphere was always released from the same point on the inclined groove AB. Measure- ments were made on a smooth, flat wooden plank. The plank had nine equally spaced lines marked upon it, which we assumed to be 200 punti apart, the first being positioned immediately below the point of projection C. The plank could be raised or lowered. Its position was first adjusted so as to make the sphere, when released, project to the second line; the plank was then lowered to the next position the sphere was expected to reach according to the equation y x x2. Slight adjustments were then made in order to make the sphere project to the third line. However, the sphere did not project the same distance from C each time it was released from the fixed point on the inclined groove AB. A scatter of values was obtained, and the plank was raised or lowered slightly to obtain the best distribution about the line on the plank. A spirit level was then used to ensure that the plank was level, and the vertical drop was found by measuring the distance of the top of the plank below the point of projection C. There is evidence on folio 107v that Galileo also measured distances to obtain the values in the third column. That he used a rule having major divisions of 60 punti is clear when measured values are converted into the values used in the third column.

It is worth noting that unlike other of Galileo's mathematical investigations, in which distances were measured on scale diagrams-as is probably the case for folio 107r-there would obviously be no point whatsoever in measuring the consecutive y values on a sketched parabola. Galileo did use measurements in mathematical investigations when doing so yielded a result directly or avoided lengthy calculation. But he already knew the mathematical characteristics of each point on a parabola; the question that interested him on folio 107v was how well his empirical data corresponded with a mathematical model.

The measured values were very close to the ideal values, as can be seen from Table 2. The variations of the measured values from the mathematical model are less than 2%. These variations, in a group of eight measurements, are less than those of the later folio 116v experiment, where a maximum variation of



3.6% arose in five measurements.'0 If one considers the average deviation from the ideal values, it is less than 0.4% in the case of the folio 107v data, whereas the average deviation of the data on folio 116v is more than 1.7%. The folio 116v experiment was evidently accurate, and there were two reasons for this: first, that only distances were measured, and second, that a mean value for the projected distance could be obtained from the observation of a number of projections. Probable sources for the variation between the different measurements recorded in the folio 116 experiment include errors in measuring the projection (D) and the vertical drop (H). A further source of variation might be some non-uniformity in the inclined groove. In the experiment associated with folio 107v any effects of the non-uniformity of the groove are very much reduced, since the sphere always rolls along the same section of the groove, Again, errors that arise in measuring the vertical drop (H) are there avoided, since the sphere is always released from the same point on the inclined groove. Finally, it seems possible that when Galileo was investigating the validity of a particular mathematical model his knowledge of the idea may also have influ- enced his measurements.

II. THE CONSERVATION OF HORIZONTAL INERTIA (FOLIO 175v)

In investigating projectile trajectories through such experiments, Galileo would have been refining further the technique devised by Guidobaldo. If taken up by Galileo during the early 1600s, the technique would quite logically have led to the experiment recorded on folio 81r and then to the folio 107 investigations. To this point the mathematically formulated experiments would have centered on issues of geometry. But as Guidobaldo's discussion of his original experiment shows, the symmetry of the trajectory about the vertical axis leads inevitably to the study of motion and mechanics, since it suggests that the motion itself is symmetrical and thus that momentum in the horizontal direction is preserved. As Guidobaldo implied, the form of the trajectory suggests that the velocity of the projectile is the same at points equally distant from the apex of the trajectory-a view, as we have seen, Galileo expressed in his letter to Sarpi in 1604.

We might then expect Galileo to use the knowledge he had gained of the geometrical form of the trajectory as he attempted to establish a mechanical explanation of projectile motion. Two manuscripts in particular show evidence of his taking such a course when analyzing the problems-folios 175v and 117r. These documents show that once he confirmed that the projectile trajectory was parabolic, Galileo became concerned about the motion along the trajectory. And once he reached the view that horizontal momentum was conserved without change and that motion tended to be conserved in any direction, as a result of his study on folio 175v, he then examined on folio 117r the way in which the velocity of the projectile changed.

On folio 175v Galileo used a theoretical construction in order to examine the form of the trajectory that would result according to a particular model of pro-

10For further details of the reconstruction of this experiment see R. H. Naylor, "Galileo and the Problem of Free Fall," British Journal for the History of Science, 1974, 7:105-134.


558 R. H. NAYLOR

A

D

H

I

|K

SI~~~~~~~~~~ M

R, Q

Figure 5. Transcription from folio 175v, with added lettering; all points depicted appear on the manuscript except those on OV and the trajectory PX.

jectile motion: one in which horizontal and vertical motions were independent. The vertical motion was considered to be unaffected by the direction in which the projectile moved and to be determined solely by the s x t2 law of fall. What is interesting is that in Galileo's model the horizontal motion of the projectile steadily decreased.

On folio 175v, transcribed in Figure 5, are drawn three straight lines and a roughly drawn curve tangential to two of the straight lines. One of these lines appears intended to be almost vertical, while the other is inclined somewhat in relation to the third, apparently horizontal line, which seems to be present simply for reference. What looks like a sphere is represented on the vertical line and on the inclined line. It appears that folio 175v relates to a study of the motion of a sphere down a near vertical incline followed by its deflection by the curve, followed by a free flight. It is not immediately clear on first examination whether the sketch or diagram is

intended to represent some observation-perhaps with an experimental arrangement-or whether it refers to a purely theoretical problem.11

On the near vertical line on folio 175v (AO in Fig. 5) there are a number of carefully marked intervals (AB, BC, . . . ,NO in Fig. 5). The measured dimen- sions of these intervals in millimeters are indicated in column 1 of Table 3; the values expected for the intervals according to Galileo's s x t2 law of fall appear in column 3. Table 3 thus indicates that the intervals Galileo marked on the vertical line were first calculated and then carefully marked out.

Firsthand examination of folio 175v reveals that Galileo also carefully marked out intervals along the horizontal line (OX in Fig. 5). That a compass was centered on 0 to mark out an interval OP equal in magnitude to ON is clear on the manuscript itself. Galileo marked out other intervals along OX in a similar fashion; they are shown in Table 4. Immediately beneath for comparison are the corresponding series of distances that appear on the near-vertical line AO. Marked very faintly vertically below the intersections on OX at P0, Q0, Ro, and S0 are points-PF, Q1, R,, and S1. The distances from each of these points to OX is shown in Table 5, column 1. Finally, the distances between Galileo's apparently rather freely drawn curve and -the line OV inclined to the horizontal are provided for comparison in Table 5, column 2. It was not possible to discern points on OV or the trajectory OX, and thus the points P, Q, R, S, Pv, QV, RV,

IIStillman Drake has suggested that fol. 175 (and details on fol. 117r) represent an experimental arrangement; "Galileo's Experimental Confirmation of Horizontal Inertia: Unpublished Manuscripts," Isis, 1973, 64:291-305 (see p. 296). That paper provides photographs of folios 114r, 116v, 117r, and 175v.



Table 3. Comparison between the points marked out on the vertical line in Figure 5 and the lengths expected from Galileo's theoretical model.

Measured values to Values to nearest Length expected nearest mm punto according to s t2,

in mm

AB 1+ 1+ Base AC 5- 5+ 4+ AD 9+ 10- 9+ AE 17 18 16+ AF 26 27 25+ AG 37 39 36+ AH 51 54 49+ AJ 66 69 64+ AK 83 87 81+ AL 102 107 100+ AM 123 129 121+ AN 146 154 144+ AO 171 180 169+

and Sv used here do not appear on the manuscript, but are constructed from the verticals formed by POP,, QoQ1, and so forth.

As can be seen, the series of distances in Table 5, column 1, correspond with the first four distances in Table 3, column 1. The first three values for the distances in Table 5, column 2, correspond with the first three distances on Table 3 and Table 5, column 1. The correspondences reveal the following:

* Uniformly accelerated motion in the vertical (perpe-ndicular to OX) and near vertical (along AO); the odd-number rule for the distances traversed in successive time intervals.

* Continuity of motion on deflection (OPO = ON, OQ0 = OM, ORo -OL, OSO = OK); that is, motion after deflection is the reverse of uniform acceleration and follows the inverse odd-number rule.

Table 4. Distances on the horizontal and vertical lines in Figure 5 compared. Length expressed to the nearest millimeter.

OP0 25 ON 25 P0QO 23 NM 23 Q0R0 21 ML 21 R0S0 19 LK 19

Table 5. Comparison of the relations between points on the curves in Figure 5 and points on the straight lines OV and OX. Length expressed to nearest millimeter.

Column 1 Column 2

Pop, 1+ PPV 1+ QoQI 5 QQV 5- RoRl 10- RRV 10+ SoSz 17 SSv 20


560 R. H. NAYLOR

* The principle of superposition (P0P, Q0Q, ROR, SOS correspond closely to AB, BC, CE, DE for a case of fall in the near vertical AO).

* A steadily decreasing horizontal motion. * A resulting trajectory that is not symmetrical about the vertical: the trajec-

tory is not a parabola. The lower series of points Pl, Q1, Rl, S, does not represent a parabola.

Similarly if OX is rotated to coincide with OV and Pl, Q1, R1, and S1 are relocated the same distances vertically below the new positions of Po, Q0, Ro, and S0 as they fall below the old positions on folio 175v, then the series of points Pl, Q1, Rl, S1 will still not lie on a parabola. The curve joining them will be closer to 0 than P, while P1 is the same distance below P0 as P is below PV. Similar considerations show Q1, R1, and S, in their new locations to be below the curve OPQRS. Thus the curve obtained in this case of the continuation of the motion along OV according to the inverse odd-number rule yields a less symmetrical trajectory than PQRSX and accordingly one that is less like a parabola. There is an alternative lower tail to the curve sketched on folio 175, which may simply show that Galileo entertained this possibility.

The form of OPQRS suggests that Galileo may have also considered the possibility that for motion in the direction OV horizontal momentum decreases according to the inverse odd-number rule. This appears to provide the best fit for the curve OPQRS, though the curve does not seem to have been drawn with care. But since 0, PI, Ql, RI, S, do not lie on a parabola, then P, Q, R, S, X also cannot if these points are also related to Pv, Q,, RV, and Sv by the odd- number rule. Even if this was not immediately obvious, it becomes clear after two or three points are plotted. This could explain why Galileo did not take great care in plotting this curve.

It is also true that if, after the projectile was deflected in the direction of OV, the horizontal component of its motion decreased according to the inverse odd- number rule, then the speed of the projectile would have to become infinite as the incline OV approaches the vertical. Galileo would have been bound to recognize this in time. From these geometrical constructions it was plain that if the momentum of a projectile decreased after deflection according to the inverse odd-number rule, then the trajectory could not be parabolic.'2

The data on folios 81r, 107v, and 175v allow the reconstruction of the ways in which Galileo related his investigations of the projectile trajectory to his pre- 1604 theory of motion. They show that he was able to account for its parabolic form in terms of the two major components of that theory: the law of fall and the principle of the conservation of momentum. Folio 175v indicates that Galileo could well have formulated this principle as he did as a consequence of his study of the trajectory. Folio 175v also indicates the relationship between the principle of superposition and the parabolic trajectory. It suggests that the principle of superposition as expounded by Galileo in Two New Sciences was known to be

12Winifred Wisan discusses a number of the draft proofs, etc., of Vol. 72 of the Galilean MSS and makes a quite different interpretation of Galileo's use of experiment and mathematics in "The New Science of Motion: A Study of Galileo's De motu locali," Archive for the History of Exact Sciences, 1974, 13:103-306, on pp. 227-228. She has drawn my attention to the discussion of the decrease of momentum on the inclined plane in the Discorsi, pointing out that it appears to be a remnant of an early consideration of this problem. See Opere, Vol. VIII, pp. 243-244. (Crew and de Salvio, Dialogues, pp. 206-207.)



Table 6. Comparison of figures on folio 114r to results achieved with the reconstructed experiment.

Figures on folio 114r Reconstructed experiment from folio 116v D H D

in punti in cm in punti as observed, in cm

253 23.75 30 24.0 337 31.64 50 31.1 394 36.99 70 36.7 451 42.34 90 42.3 495 46.48 110 46.0 534 50.14 130 50.4 574 53.89 150 53.9

"true" because it alone was capable of accounting for the experimentally determined form of the trajectory.13

111. GALILEO'S STUDY OF THE VELOCITY OF THE PROJECTILE

Having once established the relationship between the conservation of momentum and the form of the trajectory, Galileo was then faced with the difficult task of relating both ideas to the basic principle of the 1604 theory: the idea that velocity increased with distance in free fall. He pursued two lines of investigation. One, found on folio 114r, involved experiments similar to those on folio 107v. Another analysis, on folio 117r, shows that he investigated the way in which the average velocity of the projectile changed along its trajectory.

For the experiment recorded on folio 114v the inclined plane previously used in the folio 107v experiment was placed on a table and the sphere released at a series of points on the groove. (That Galileo used this equipment on a table in this way can be seen from folio 116v, the later experiment, where the height of the table is referred to as being 828 punti.) The points on the groove were selected to provide a regular series of increases in the total vertical drop from the point of release to the point of projection. According to Galileo's 1604 principle of motion (v x d), the speeds of projection in the horizontal should be proportional to these vertical drops. The principle of conservation of motion indicates that the projections (D) will be proportional to the velocities of projection.

The evidence of this experimentation is found on folio 114r, where an inclined line is joined to a series of crudely drawn curves terminating at the same level. At the end of each curve is found a single figure. The series of figures is 253, 337, 394, 451, 495, 534, and 574. The line and curves resemble the diagram on folio 116v. Here on folio 114r the larger curves in particular indicate that a change in direction occurs at the end of the inclined line. If these figures are assumed to represent horizontal projections in punti, they can be duplicated when the inclined plane is placed on a table 828 punti high and the sphere dropped from heights (H) of 30, 50, 70, 90, 110, 130, and 150 punti (see Table 6). This provides a series of vertical drops in the ratios 3:5:7:9:11:13:15. The obviously missing initial 1 from this series of odd numbers corresponds to an

13Opere, Vol. VIII, pp. 268-269; Crew and de Salvio, Dialogues, pp. 234-235.


562 R. H. NAYLOR

initial drop of 10 punti. The point that would give a vertical drop of 10 punti lies on the curved section of the inclined plane and makes the location of the point of release rather awkward. (It has to be borne in mind that an error of 1 or 2 punti would represent an overall error of 10 to 20% in a measurement of 10 punti. In fact the variation in the data from the theoretical ideal is very small-only about 5%.)

Both this experimentation and the work on folios 107 and 175 make it clear that since horizontal motion is conserved in projectile motion, then Galileo's principle that velocity increased with distance cannot be correct: the projections are not proportional to the vertical falls.

That this experimentation is related to the folio 107v investigation is suggested by some details on the far right-hand edge of the latter manuscript. Folio 107v was probably used to record two phases of the experiment, since turning it through 900 shows a series of regularly increasing vertical lines, marked a, b, c, d, e, f, representing a series of heights in the ratios 1:2:3:4:5:6. Two series of numbers appear nearby: 1, 3, 5, 7, 9, 11, 13, 15 and 1, 5, 9, 13, 15, 17, 21. The second series has been crossed out. The deletion suggests that Galileo saw the undeleted series to be more relevant or more suitable for consideration. In addition there is a triangle, one side of which is a vertical line while the base is roughly at right angles to it. There are a number of equally spaced lines drawn parallel to the base, so that the triangle resembles the diagrams Galileo used in the letter to Paolo Sarpi in 1604 and on folio 128 to represent an increase of velocity with distance according to his principle v a d.

The folio 114r experiment was evidently concerned with the increase of projected distance (D) with an increase in the initial vertical fall (H). The possible connection between velocity and projected distance is obvious. My examination of Volume 72 has not revealed another series like those found on folio 107v or folio 114v. There is of course the possibility that the relationship between these series is merely coincidental, but that seems remote.

The other manuscript concerned with Galileo's study of velocity, folio 117r, like folio 107 provides only the barest indication of what Galileo was investigating. Two things are clear, however. First, Galileo was analyzing the distances traversed by a projectile in successive intervals of time; second, he was interested in the relationship between the velocity at a point on the trajectory and the average velocity of the projectile along the trajectory. Galileo's objective was to establish the way in which the average velocity of the projectile was changing in relation to time.

There is a rough sketch of a parabola on the upper half of the folio (Fig. 6). It represents a projectile with a horizontal velocity of 40 units accelerating in the vertical such that it falls through distances of 10, 30, 50, and 70 units in successive intervals of time. Galileo endeavored to obtain an approximate value for the average velocity along the successive sections of the trajectory. These values were obtained as (10 + 40)1/2 = 41.2, (30 + 40)1/2 - 50, (50 + 40)1/2 = 64 and (70 + 40)1/2 - 80.6. This series of values, though indicating in a rough way the manner in which the average velocity of the projectile was increasing, did not make clear the pattern in the sequence of values itself. Galileo therefore converted the first number of the series, 41.2, to a figure of 100 and multiplied the remaining three values, 50, 64, and 80.6, by the factor of 100/41.2, which yielded the series 100,



40 40 40

_ 160

-40

-90

Figure 6. Transcription from folio 117r.

121, 155, and 196. These figures can be seen written alongside a line drawn on folio 117r to represent those distances; they represent the values of the average velocity in succeeding intervals of time. The differences between these values could only be the result of increases in the velocity of the projectile due to an increasing vertical momentum, since horizontal momentum was constant.

The rate of increase is not steady, however: the increases are 21, 34, and 41 (i.e., 121 - 100, 155 - 121, 197 - 155). Galileo was interested in these rates of increase, for the three values 41, 34, and 21 are written on folio 117r just above two sketches of parabolic sections. These figures are significant because they reveal that the rate of increase tends toward a steady value. (Evidently as the sequence of increases is 21 + 13, 21 + 13 + 7, they form a series the terms of which converge towards a limiting value.) As has already been pointed out, these increases in the average velocity of the projectile could only be due to the successive increases in its vertical momentum.

This evidence conflicts with the principle Galileo had adopted in 1604, that velocity in fall increases with distance fallen. According to the theory developed in 1604, the average velocity over the spaces traversed in successive equal intervals of time increases at an ever greater rate. This results in a diverging series for the successive increases in average velocity in succeeding equal intervals of time: the series of average velocities is 1, 15, 65, 175, 369, 671, . . .; the series of increases in average velocity is 1, 14, 40, 120, 194, 302,.

Figure 7, which appears on folios 85v and 128v, depicts Galileo's earlier demonstration of his law s a t2 from the principle that instantaneous velocity increased with distance. AB, BC, CD, and DE in the figure are equal intervals of distance in the path of fall of the body from A to F. The instantaneous velocity at B is represented by the horizontal line BM and at C by CN and at D by DO. As the velocity is increasing at every point on the line AF, these points are infinite and the infinite number of lines from these points constitute the triangle AEP. Therefore, Galileo argues, the velocities at all points on the line


564 R. H. NAYLOR

A A

B M

C N

D 0 X

E P

F 0

Figure 7. Transcription from Figure 8. Second transcription folios 85v and 128 v. from folio 117r.

AB to those at all points on the line AC are as the area of triangle ABM to the area of the triangle ACN, that is, as the square of line AB to the square of line AC. The crucial link between instantaneous velocity and average velocity is provided on folio 128 as follows: the velocitd with which the line AD is traversed to the velocita with which the line AC is traversed is as the square AD to the square of A C.

According to the theory explicating Figure 7 that Galileo expounds on folio 128, the average velocities through AB, AC, AD, and AE in Figure 6 will be as 12, 22, 32, and 42-that is, 1, 4, 9, 16. The average velocities through AB, BC, CD, DE will be as the areas ABC, BCNM, CDON, DEPO. These are in the ratio 12, 22 - 1, 32 - 22, 42 - 32-that is, 1, 3, 5, 7, 9. But Galileo does not refer to these distances being traversed in equal time intervals-nor could he, as this would be paradoxical. In accelerated motion successive equal distances must be traversed in ever smaller time intervals, as he appreciated. On folio 117r Galileo investigates the issue of the increase of average velocity in relation to time beginning with the series of distances actually traversed in successive equal time intervals, that is, 1, 3, 5, 7. According to the theory developed on folio 128, the average velocities over the distances traversed in these successive equal intervals of time should be as 12, 42 - 12, 92 - 42, 162- 92_that is, as 1, 15, 65, and 175.

When we examine the series of average velocities along the parabolic trajectory as Galileo did, we find the velocities 100, 121, 155, 196 and the series of differences 21, 34, 41 . . ., both of which indicate that the rate of change decreases, giving rise to a converging series. This shows that the rate of increase in the average velocity of the projectile along its trajectory cannot be that required by the folio 128 theory. On the contrary, the convergence of the series on folio 117r indicates that the rate of increase in average velocity with time tends to be a steady value, that is, the rate of increase of the increase in average velocity tends to zero as convergence occurs. The earlier theory required that instantaneous velocity increased with distance, but this in turn required that

"4Opere, Vol. II, pp. 181-183; Drabkin and Drake, On Motion, pp. 173-174.



average velocity increased ever more rapidly with time-a requirement impossible to reconcile with the parabolic trajectory.

Galileo had an important insight about the direction of motion of a body moving along or tending to move along a curved path. In De motu and Le meccaniche he had argued that the tendency to motion of a body on the point of moving in a circular path was along the tangent at that point. 14 The same idea proved invaluable when he analyzed the projectile trajectory. That Galileo recognized that the direction of the instantaneous velocity of the projectile at a point on the trajectory was along the tangent is indicated by the sketches on folio 117r (see Fig. 8). The tangent property of the parabola gave its direction at 0 as OA, where AB =,BC, AC being the axis and B the apex of the parabola.

The instantaneous velocity V has two components, one due to the horizontal motion H, the other due to the vertical motion U. Thus V2 = H2 + U2. From this simple relationship and from the tangent property of the parabola Galileo could establish the vertical component of the projectile's motion after the first, second, third, and fourth intervals of time. The parabola gave the tangent after those time intervals, and since the tangent indicated the ratio of the vertical velocity to the steady horizontal velocity of 40 units, it followed directly that the vertical velocities were 20, 40, 60, and 80 units-that is, there was increase with time.

In 1604 Galileo had not regarded the relationship between instantaneous velocity and average velocity in uniformly accelerated motion as a simple one: average velocity was proportional to the square of the instantaneous velocity. His rejection of the older 1604 view relating to average velocity in accelerated motion would have opened the way to a reappraisal of the relationship between instantaneous velocity and average velocity.

The tangent property indicated a steady increase in instantaneous velocity with time, but not with distance. Indeed, the instantaneous velocity appeared to increase with the square root of the distance fallen. That is, the instantaneous velocity doubled, from 20 to 40 and from 40 to 80, as the total vertical fall quadrupled, from 10 to 40 and from 40 to 160 respectively.

Again, the average velocity along the projectile path indicated that the average vertical velocity increased at a steady rate with time. By calculating the average vertical velocity from the ratio of distance to time, as he had calculated the average velocities along the trajectory on folio 117r, Galileo could obtain the values for the average velocity in one, two, three, and four of the intervals of time. The values of these average velocities would be 10, (10 + 30)/2, (10 + 30 + 50)/3, (10 + 30 + 50 + 70)/4-that is, 10, 20, 30, 40. This shows a constant rate of increase in average velocity, a feature Galileo did expect in uniform acceleration, though originally in 1604 he had regarded it as a steady increase with regard to distance. The steady increase in average vertical velocity indicated by the parabolic trajectory was in relation to time and thus in agreement with the values for the changes in average velocity of the projectile along its trajectory. Thus all four features of the projectile's motion could be brought into agreement. The uniform horizontal motion could be related to instantaneous vertical velocity by the tangent property, average vertical velocity could be related to the uniform horizontal velocity by consideration of time and distance, the average velocity along the trajectory could be related to average horizontal velocity and the average vertical velocity, and finally instantaneous vertical velocity could be related to average vertical velocity.


566 R. H. NAYLOR

IV. THE DOUBLE-VELOCITY RULE

It is clear from studies of the kind found on folio 117r that average vertical velocity to a point on the trajectory is always one-half the instantaneous vertical velocity at that point. This form of analysis provided Galileo with the double- velocity rule. The rule is found on folio 163v as well, where its expression suggests that Galileo regarded it as a basic rule before he had fully grasped the implications of the rule itself.

On folio 163v a body is first considered to move from A to B with naturally accelerated motion beginning from rest A (see Fig. 9). The body is then considered to move from A to B with a uniform motion equal to the maximum speed attained by the accelerated motion. In this case, it is said, the body would complete the space twice as quickly as in the first case. Galileo argues that this conclusion follows from the ratio of the instantaneous velocities in accelerated motion and in uniform motion. The total number of points on AB and hence of instantaneous velocities is the same in uniform motion and accelerated motion.'5 The ratio of the to- tals of these instantaneous velocities is said to be the ratio of the areas of ACB to ADCB.

Consideration of folio 163v suggests that it represents Galileo's ideas during the stage when he was beginning

A _ _ _C

X ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i

\\I

B D Figure 9. Transcription from folio 163v.

to see the implications of the idea that average velocity increased steadily during natural motion.

The diagram used to demonstrate the double-velocity rule was similar to that which Galileo was accustomed to use in discussing the increase of instantaneous velocity with distance. But examination of the consequences of the new approach very soon shows that the double-velocity rule cannot be validly related to the old principle of motion. Galileo's adoption of the view that a comparison could be made directly between a uniform motion and the average velocity of a uniformly accelerated motion was of fundamental importance. The key to Gali- leo's method on folio 163v was that he compared motions made in the same time. It is just such a comparison that the parabolic trajectory allows.

Consider the case of accelerated motion over a distance AE equal to 2AB. Then if v oc d the final instantaneous velocity will be double that at the point B and the overall average velocity will be double that through AB. Thus the time required to complete AB will be the same as that required to complete AE. That Galileo came to regard this to be a disproof of the v oc d principle is clear from the Two New Sciences.16 More than this could be seen, however. Accor- ding to the folio 128 theory, the average velocities through AB and AE are as AB2 and AE2, that is, 1:4. Combining this with the new double-velocity rule is clearly impossible. The average velocities cannot be in this ratio. A ratio of 1:2, which the double-velocity rule leads to when combined with v ac d, does as we have seen require the completion of any given distances in the same time; a greater ratio would suggest that the greater distance is completed in less timet

15Opere, Vol. VIII, pp. 383-384; Wisan, "New Science of Motion," pp. 203-204.

'6Opere, Vol. VIII, pp. 203-204; Crew and de Salvio, Dialogues, pp. 160-162.



We know that Galileo examined this problem, which arose directly from the adoption of the new principle of the steady increase in average velocity with time. His consideration of the relationship between average velocity and instantaneous velocity is found on folio 152r, which dates from the same period as folio 163v. On folio 152r Galileo examined the consequence of the new relationships between average velocities, instantaneous velocities, and distances.17 He also examined the average velocities for the case in point as predicted by his old theory of motion.

Galileo considered the case of a body accelerated through a distance of 4 miles for 4 hours with an average velocity of 10 degrees. Beneath this is asked the question: how long will it take the same body to move 9 miles with 15 degrees of speed? That Galileo was still in the process of considering the implications of his newly developed double-velocity rule is suggested by his questioning approach. On the manuscript appears a diagram representing the case under consideration. The distances traversed are represented by a vertical line abc, the instantaneous velocity at b by a horizontal line be, and the instantaneous velocity at c by a horizontal line cf. The consequence of Galileo's new theory that velocity increases with the square root of distance fallen is represented by a parabolic line joining a, e, and f. On the left-hand side are written corresponding values of distance, time, and instantaneous velocity:

Distances Instantaneous velocities Times through ab 4 ab 4 ad 6 be 20 ac 6 ac 9 cf30

The values 20 and 30 are those determined by the double-velocity rule from the average values of 10 and 15. The average values are written out again.

According to the proof of s x t2 on folio 128 (see above) the average velocities are given as follows:

velocity through ab area of abc 20 velocity through ac area of acf 671/2

If as stated the velocity through ab is 4, then the velocity through ac would be

671/2 X 4 = 131/2 20 These would be the values that would follow if instantaneous velocity did increase with distance to the value of 30 degrees at c and 20 degrees at b. The statement

through ab velocity as 4 through ac velocity as 13 1/2

appears on the top right-hand corner of the manuscript. But these values cannot be reconciled with the analysis of the parabolic trajectory on folio 117r, and therefore cannot be correct.

'7Published by Stillman Drake in "Galileo's Discovery of the Law of Free Fall," Scientific American, May 1973, 228(5):84-92. Discussed by R. H. Naylor in "Galileo's Theory of Motion: Processes of Conceptual Change in the Period 1604-1610," Ann. Sci., 1977, 34:365-392.


568 R. H. NAYLOR

On the central right-hand section of the manuscript Galileo provides a sliort demonstration showing that if instantaneous velocity (vi) is proportional to the square root of the distance traveled (s) then v12 a s; he concludes that the line representing diagrammatically the change of velocity with distance is a parabola.

Thus folio 152r reveals that Galileo had reassessed the relationship between average velocity and instantaneous velocity and therefore changed his view of the relationship between instantaneous velocity and distance. Both changes can be seen to relate to the analyses of the projectile trajectory found on folios 117r and 175v. Those analyses also provided another rule of naturally accelerated motion-Galileo's so-called double-distance rule.

V. THE DOUBLE-DISTANCE RULE

Since a projectile's trajectory is parabolic, its horizontal motion continues unaffected by its vertical acceleration; this relationship is clear in folio 175v. What is more, the distance traversed vertically during fall can be directly related to the final instantaneous velocity. Thus on folio 117r the consideration of the vertical falls over one, two, three, and four intervals of time shows that average vertical velocity equals distance divided by time and that it increases steadily with time. As the average velocity is clearly also equal to half the final instantaneous velocity, from the tangent property, it follows directly that the instantaneous velocity equals double the distance traversed divided by time.

A different and more general way of expressing this relationship is to consider a vertical fall followed by a deflection into a horizontal direction. In such a case the average velocity in the horizontal will be double the average velocity of the preceding fall. In equal intervals of time the distance traveled in the horizontal will be double the intitial free fall. This is Galileo's double-distance rule.

The double-distance rule makes its first apearance in the manuscript on folio 163v with the double-velocity rule. They were evidently developed at the same time, and the brief statement of the velocity rule leads into the statement of the double-distance rule.

The double-distance rule, however, assumes a direct proportionality between instantaneous velocity and average velocity considered in relation to time. All this was a radical break with the older view where no such direct relationship existed. On folio 128 the average velocity of an accelerated motion had been shown to be proportional to the square of the final instantaneous velocity. The double-distance rule explicitly assumes that average velocity in accelerated motion is equal to half the final instantaneous velocity for any time interval considered. Thus instantaneous velocity and average velocity both increase with time in the same manner.

The combination of the double-distance rule with the law of fall relating distance and time (s a t2) shows that the instantaneous velocity in free fall is directly proportional to time. 18 Galileo deduced this on folio 91v and that manuscript provides the earliest surviving demonstration of the principle. The handwriting, the language, and watermark of the manuscript all indicate a date in the late Paduan period. That the demonstration itself was developed during Galileo's work on the projectile trajectory is clear from the context. It appears below a draft of what was to become Proposition II on projectile motion and

x'Wisan, "New Science of Motion," pp. 227-228.



above an outline draft for Proposition IV. The demonstration itself was later adapted to become Proposition III on projectile motion. Evidently Galileo's draft on folio 91 was developed as a direct consequence of his discovery of the parabolic trajectory recorded on folios 81r and 107v.

By the time he had reassessed the relationship between velocity, time, and distance, Galileo would have been able to return to folio 114r and to confirm that the horizontal projections (D) were, as folio 117r suggested, proportional to the square root of the corresponding vertical drops (H). At this point the evidence was brought into agreement with the theory.

The last stage in the process was thus very probably the experimentation related to folio 116v. Here Galileo appears to have examined the relationship between his theory of projectile motion and his theory for motion in the inclined plane.

Galileo's study of the parabolic trajectory had thus led him ultimately to formulate the double-velocity and double-distance rules as applicable to natural motions. The parabolic form of the trajectory demonstrated the validity of both of these rules beyond doubt. At the same time the unraveling of the explanation of the form of the trajectory provided Galileo with a means of conducting a further investigation into motion on the inclined plane. The experiment recorded on folio 114r examined the relationship between velocity and distance. It could also be used to examine the validity of the double-distance rule and Galileo's postulate. According to the postulate the velocity gained by a sphere in rolling down any incline depends only on the vertical distance fallen. The experiment on folio 116v quite obviously aimed to check the validity of the double-distance rule and presumably also involved a cross check on the postulate.

The measurements on folio 116v were obtained using the same experimental arrangements as those used earlier in the folio 114r experiment. In one instance the sphere rolled down the incline and in so doing fell through a vertical distance of 828 punti. After being projected horizontally, it fell through a further vertical distance of 828 punti before striking the floor. According to the double- distance rule it should have moved 2 x 828 punti horizontally in this time, that is, 1656 punti. However, as folio 116v shows, the sphere only traveled 1340 punti horizontally. The reason for this is that rolling down an incline cannot be linked to free fall in the simple way Galileo assumed. Galileo did not solve the puzzle presented by this result. One obvious source of the discrepancy might have been that the curve used to deflect the sphere was slowing the sphere down. How- ever, the use of different inclines and more gentle curves would have shown that this was not the case. At the same time conducting the experiment would have provided evidence in favor of the postulate, for it shows that over a considerable range of inclinations the projections were dependent only on the vertical drop (H).

Galileo would have been able to recognize from his 1602 experiments on semicircular inclined planes that the sphere met surprisingly little resistance while moving in smooth curves of large radius of curvature. The use of a semicircular groove in the folio 116v experiment makes no difference to the projections obtained. This remained an unsolved problem for Galileo. His inability to confirm the double-distance rule, though probably puzzling, could not bring the physical validity of the rule into doubt. The truth of the rule was clearly established by the parabolic form of the projectile trajectory itself.


570 R. H. NAYLOR

Galileo was left with the problem of providing a suitable demonstration of his double-distance rule. He recognized, after he had reconsidered the proof technique on folio 152r, that the proof technique was not a very satisfactory means of establishing physical relationships. To avoid using it in establshing the foundations of his theory of motion he apparently expended some effort in the search for an independent demonstration of the validity of the double- distance rule, and indeed it seems he may even have hoped to base the theory of motion on this fundamental rule. As in other of his mathematical investigations his search proved unsuccessful. He reached a position in which the only demonstration available to him required the subtraction of two infinite magnitudes. Nevertheless, though he evidently never felt he had obtained a completely satisfactory demonstration, he included what he had in Two New Sciences. What is more, to emphasize its importance for him, he went so far as to place it before the demonstration employing the proof technique and the definition that velocity increased with time in natural motion.

VI. CONCLUSION

The clear impression created by the manuscript evidence of Galileo's work on the projectile trajectory is that it was not, indeed could not have been, either a purely "rational" or a simply "empirical" investigation. Beginning with an initial hypothesis, Galileo used experiment to identify the shape of the trajectory. This established, he proceeded to an extended analysis of the relationship between its parabolic form and the two basic features of his theory-the principle of inertia and the law of fall. In this Galileo inevitably used analysis, as it was in any case impossible to devise independent empirical tests, because the parabolic trajectory was only produced by the combined effects of inertia and fall. Only by means of a repeated process of comparison of theory and experiment and the analysis of theory was Galileo able to bring this work to a successful conclusion.

The work revealed by the manuscripts establishes the continuity of Galileo's work in mechanics in that it shows a progressive development of his theory as he attempted to account for what was undoubtedly a problem of great topical interest in the sixteenth and seventeenth centuries. There is no evidence that the work on the trajectory was part of a general program to establish the Copernican theory. And if instead Galileo had been attracted to the problem as a means of demonstrating the principle of inertia, he would surely have deduced the form of the trajectory directly. Such views of both Galileo's metaphysical commitment and his method have in the past led to just such claims. His notes reveal a much more tentative approach. Logically Galileo would have been unable to claim that he had explained the trajectory using his theory until he had resolved all the theoretical problems it posed. Eventually he was able to do this-up to a point. The nature of the relationship between the principle of horizontal inertia and the principle of circular inertia as it applied to the Copernican theory remained an unsolved problem for Galileo. Thus Galileo's work on the trajectory, while it destroyed once and for all the Aristotelian view of such motions, never really provided strong evidence for Copernicanism-as has often been implied.


Article Contentsp.550p.551p.552p.553p.554p.555p.556p.557p.558p.559p.560p.561p.562p.563p.564p.565p.566p.567p.568p.569p.570

Issue Table of ContentsIsis, Vol. 71, No. 4, Dec., 1980Volume Information [pp.695-710]Front Matter [pp.525-529]The Industrial Relations of Science: Chemical Engineering at MIT, 1900-1939 [pp.531-549]Galileo's Theory of Projectile Motion [pp.550-570]Graham Island, Charles Lyell, and the Craters of Elevation Controversy [pp.571-588]Critiques & ContentionsPublic Science in Britain, 1880-1919 [pp.589-608]

Notes & CorrespondenceVico and the Continuity of Science: The Relation of His Epistemology to Bacon and Hobbes [pp.609-620]The Students of Ira Remsen and Roger Adams [pp.620-626]

News of the ProfessionConference Reports [pp.627-629]Teaching Programs [pp.629-631]Appointments, Archives, Associations [pp.631-632]

Essay ReviewsThe DSB: A Review Symposium [pp.633-652]

Book ReviewsHistory of Scienceuntitled [pp.653-654]untitled [p.654]untitled [pp.654-655]

Philosophy of Scienceuntitled [pp.655-656]untitled [pp.656-657]

Social Relations of Scienceuntitled [pp.657-658]untitled [pp.658-659]untitled [pp.660-661]untitled [pp.661-662]

Women and Scienceuntitled [pp.662-664]untitled [pp.664-665]

Humanistic Relations of Scienceuntitled [p.665]untitled [pp.665-666]

Mathematicsuntitled [pp.666-667]

Physical Sciencesuntitled [p.667]untitled [p.668]untitled [p.668]

Earth Sciencesuntitled [p.669]

Biological Sciencesuntitled [pp.669-670]

Sciences of Manuntitled [p.670]untitled [pp.671-672]untitled [p.672]untitled [pp.672-673]

Medicineuntitled [pp.673-674]untitled [p.674]untitled [p.675]untitled [pp.675-676]

Technologyuntitled [p.677]untitled [pp.677-678]untitled [p.678]

Classical Antiquityuntitled [pp.678-679]

Middle Agesuntitled [pp.679-680]untitled [pp.680-681]untitled [pp.681-682]

Islamic Culturesuntitled [pp.682-683]

Far Eastuntitled [pp.684-685]

Renaissance & Reformationuntitled [p.685]

Seventeenth & Eighteenth Centuriesuntitled [p.686]untitled [p.686]

Nineteenth & Twentieth Centuriesuntitled [p.687]untitled [p.687]untitled [pp.687-688]untitled [pp.688-689]untitled [p.689]

Contemporaryuntitled [pp.689-690]

Back Matter [pp.691-694]

galileo theory proyectil

Documents