Constructing an educational message from a comparison between
scientists: Leonardo and Galileo compared
Abstract. Physics textbooks usually present linearly scientists and their contributions isolated
from other contributions as well as approaches. Discussing prominent scientists through
comparing and contrasting them could attain a richer picture of scientific enterprise. The
tradition of this approach, started by Plutarch in history and developed by Koyré in the history
and philosophy of science, can also be beneficial in science education. Comparing scientists is
supported within the discipline-culture structure of the subject matter (2005) and scientific
methodology. Two prominent scientists of the Italian Renaissance, Leonardo and Galileo, were
contrasted to illustrate the educational potential of this approach. Although both figures
contributed much to the developing paradigm of modern science, their contributions were
essentially different. We have illustrated this difference in a number of aspects relevant to
physics curricula of high school and introductory university courses.
Introduction
Among the recognized benefits of including history and philosophy of science into
teaching is providing students with knowledge about scientific methodology and the
nature of scientific activity. The situation is different with respect to the parallel question
of teaching the subject matter of a science discipline. This is because the subject matter
is often identified with some normative knowledge and any deviation from this (provided
by the history of science) is suspected of confusing the learner and impeding the progress
of learning1. In our experience, an oversimplified, inaccurate and colorless image of
science is what is commonly presented as science in education.
Usually textbooks make reference to the great names with regard to their most
famous contributions to the disciplinary knowledge (Galileo – falling and ballistic
motion, Newton – laws of motion and gravitation, Coulomb – electrostatic force and so
on). Often nothing, but general statements appraising the empirical approach, is n said
about the scientific method, the ways in which the knowledge was attained. The great
methodological variety, which in fact discriminates between different scientists, makes
them very different individuals, and thus humanizes science, often remains out of the
scope of science teaching. The result is a line-up of scientists who differ mainly due to
the fashion of their clothes, and common in their success of advancing physics because
they were smart. Knowledge creation is presented as a linear accretion in the "correct"
1
This stance is opposite to our own one which is presented in several publications (e.g. Author 2005). In
our personal experience we have often met severe resistance to the integration of the history and
philosophy of science as an unnecessary and confusing addition.
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Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
direction ("Whig history"), celebrating the triumph of rationality and objectivity, the
inductive-empiricist view of unfolding truth in presenting a rising line of successive
contributions, each improving on the previous one (e.g. Jacoby & Spargo 1989).
This particular way of using history of science suffers from a variety of pedagogical
shortcomings. For example, citations from the great names are made in iconic style one
can be easily skipped. Even when presented, the original sketches of scientists and their
apparatus are isolated from the main text, lack any explanation of their meaning or
bridging with the parallel modern counterparts.
The challenge of constructing an alternative curriculum in our days of prevailing
pragmatic values of science is nontrivial. Today it is not sufficient to quote Mach and
promise that “Historical investigation not only promotes the understanding of that which
now is, but also brings new possibilities before us” (1883/1960: 316). Even if many
would agree that presenting history might be beneficial, they also know that the “real”
history presents an ocean of facts, events and stories. Deliberate selection following a
convincing rationale is badly needed.
Such a rationale should draw on a certain
conceptual framework, a guiding paradigm.
Here we will elaborate one possible view of historical materials for educational
presentation that compares between the approaches of renowned scientists to research in
physics. This step introduces a non-linear representation of curricular items, revealing
the difference in the compared philosophical, sociological, methodological and
disciplinary views and the social behavior of the scientists.
This approach, being
culturally appropriate to our times, might be more appealing to the contemporary
generation of physics learners.
In our study we chose to consider several facets of the activities of Galileo Galilei,
who symbolizes for many the start of modern science (Koyré 1943a), and Leonardo da
Vinci, who is famous as a great engineer, artist, inventor, sculptor, architect, but is much
less frequently credited as being a great scientist too (White 2000).
We will start by presenting the theoretical rationale of our approach, which is of
central importance to us. We will proceed by elaborating the benefits of comparison
between contributions to particular disciplinary subjects.
In the discussion we will
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Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
compare between the two scientists in the general aspects of scientific methodology and
sociology.
Theoretical background
The major theoretical framework of our vision is provided by the structure of disciplineculture of the scientific knowledge and its curricular counterpart (Author 2005). This
structure develops the concept of scientific research program by Lakatos (1970), which
could represent common disciplinary knowledge of a certain fundamental theory: nucleus
("hard core"), containing the paradigm: principles and fundamental laws and concepts,
and normal area ("protecting belts"), which includes application of the nucleus to
problem solving and explanations of phenomena, devices, and effects (Fig. 1a).
The major change introduced to this structure by discipline-culture is the area of
periphery that includes a variety of alternative approaches, views contradicting the
nucleus and effects and phenomena that remained unexplained drawing on the given
nucleus contents (Fig. 1b).
Educational constructivism would add to the periphery
students' misconceptions of the subject.
a
b
1
2
1
2
3
Figure 1: (a) the structure of scientific research program (a discipline), (2) the structure
of discipline-culture. 1- nucleus, 2 – normal area, 3- periphery.
It is the periphery area that distinguishes a regular discipline from a disciplineculture. The educational rational of this step is that it is not sufficient to present the
"correct" knowledge results (nucleus and normal area) for the genuine understanding of
their contents, their meaning, can be attained mainly through comparison with their
alternatives (periphery).
This structure was discussed mainly with regard to presentation of the subject matter.
However, this rationale can be extended to the domain of metaphysical knowledge, even
if this domain is not conceived as having a singular nucleus. With regard to the nature of
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Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
scientific knowledge one may conceive as a more appropriate the structure which
included several nuclei representing competitive approaches (empiricist, rationalistic,
logical-postivistic, etc.) each legitimate in science practice (Fig. 2). Non-scientific ideas
about the method (guessing, revelation, mystical, magic, traditional, etc.) may remain
within the horizon of interest and be included in the periphery.
1
1
1
2
3
Figure 2: The cultural structure of the metaphysical views regarding scientific method.
Both structures Fig. 1b and Fig. 2 draw on the same fundamental assumptions:
1. The structure of scientific knowledge is hierarchical;
2. The meaning of things is understood in comparison;
3. All fundamental concepts of science were historically consolidated in discourse
and therefore their meaning is interdependent,
4. Neither ontological nor epistemological knowledge of science reside in one
person. They are social, cumulative and complementary (rather than replacing
each other in the course of history).
It is not surprising therefore that in their psychological implications these statements
perfectly match the claims of educational psychology. Marton and Tsui (2004) claimed
the effectiveness of learning in variation and the necessity to provide learners with a
space of learning. These assertions additionally support the discipline-culture approach
in the sense of directing the learning beyond a single and strictly correct type of
knowledge as often assumed.
Comparative approach
Human cognition is comparative and hence the comparative method has been used since
ancient times in humanities: education, history and literature. The most famous example
is Plutarch (46 - 127), who in his Parallel Lives (1989) depicted in contrasting pairs the
renowned Grecians and Romans, such as Alexander the Great and Julius Caesar. In
philosophy, in general, and in the philosophy of science, in particular, comparison is
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Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
fundamental to almost any presentation, especially educational. Thus, the regular course
in philosophy or philosophy of science usually starts by comparing Parmenides and
Democritus and the introduction culminates with the contrast between Plato (the
rationalist trend of thought) and Aristotle (the empiricist trend of thought).
Alexander Koyré (1892-1964) essentially strengthens the comparison approach in his
ingenious historical studies that showed equally mature knowledge of history, physics
and philosophy: Galileo versus Kepler (1943a), Aristotle versus Plato (1943b), Newton
versus Descartes (1968).
These studies inspired generations of historians by his
elucidating comparative analysis of the leading figures of intellectual history. Thus,
unlike common textbooks in physics, Galileo, Kepler, Descartes, Newton did not appear
in a dull ascending in time ladder, but as a group of bright minds, essentially different in
their worldviews, maintaining the eternal dialogue about the nature of the World and the
way humans can explore and comprehend it. Koyré, who coined the term scientific
revolution, demonstrated, among other points, the fundamental differences between
Galileo's and Newton's approaches to physics study. Those are of high importance for
the broad span of students' interests including understanding the subject matter and the
nature of this knowledge, regardless of whether or not they intend to be physics
practitioners or people literate in science.
Why, if so, do physics textbooks normally ignore the approach of comparison which
has demonstrated itself as being so powerful in the humanities? The answer seemingly
lays in the fact that humanities are used to support pluralistic presentation without
censorship of the ‘right’ views. At the same time, physics teaching, as well as physics
itself, is commonly conceived by the practitioners in normative terms, introducing a clear
demarcation between "right" and "wrong". It appeared, however, that physics contents
allow cultural interpretation (Tseitlin & Galili 2005). It was stated that the fundamental
theories of physics present cultural (pluralistic) knowledge and as such should be
presented in teaching, to reach genuine comprehension of the disciplinary contents. This
should be a fortiori correct if one considers the interdisciplinary topics of the nature of
physics, its methodology and sociology. This assumption motivated us to check what
could be learned by students who would compare the activities of great minds which are
learned in the secondary school physics.
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Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
Leonardo and Galileo, compared in science
Following the above-mentioned trend of thought we considered two prominent figures in
the history of science, Leonardo and Galileo. To match the required format of a short
article we intend to bring out only several illustrative points, relevant for teaching physics
at school, out of a long list of possible elaborations.
Both scientists were pioneers who criticized and rejected much of the knowledge that
was contemporaneous with them. Their contributions were unique, coinciding in some
points and different in others, complementary and contradictive in various aspects.
1. THE FIRST PRINCIPLE OF MECHANICS
The first principle of mechanics, the principle of inertial motion, can serve a probe of the
conceptual system held by the researcher, his/her theoretical affiliation in physics. It is
instructive to check the views of Leonardo and Galileo on this subject.
Leonardo wrote:
The straightness of the transverse movement continues in the movable thing as long as
the whole of the power given to it by its mover continues. The straightness fails in the
transverse movement because the power which the movable thing acquires from its
mover becomes less. (MacCurdy 1955: 587)
The expressions emphasized by us do not leave much space for interpretation:
Leonardo refuted Aristotelian theory of (violent) motion (provided by a continuous
external mover) and followed the trend of thought originated by Hyparchus and
Philoponus of Hellenistic science, and expressed in the theory of impetus – the charge of
motion by Buridan and Oresme in 13-14 c. (e.g. Pederson & Pihl 1974: 237-241).
Galileo addressed the subject of inertial movement right from his early works. In his
Letter on Sunspots he wrote:
…all external impediments removed, a heavy body on a spherical surface concentric
with the earth will be indifferent to rest and to movements toward any part of the
horizon. And it will maintain itself in that state in which it has been once placed; that is
if placed in a state of rest, it will conserve that: and if placed in a movement toward the
west (for example), it will maintain itself in that movement. (Galilei 1613/1957, 113)
Skipping over the possible erroneous interpretation of ‘circular inertia’, Galileo’s –
‘maintains itself’, regarding inertial motion remained dubious (maintain might mean
either a presence of an active agent, such as impetus, or being physically indifferent to
rectilinear uniform motion, as understood by us). The problem remained in Galileo’s
mind. Later, in his Dialogue, Sagredo (the chairman of the discussion) said:
6
Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
I see in a movable body is the natural inclination and tendency it has to an opposite
motion. . . . I said ‘internal resistance’, because I believe that this is what you meant,
and not external resistances, which are many and accidental. (Galilei 1632/1953, p.
213) (emphasis added)
Salviati (Galileo’s representative) readily confirmed and refined:
I wonder whether there is not in the movable body, besides a natural tendency in the
opposite direction, another intrinsic and natural property which makes it resist motion
(ibid.) (emphasis added).
That was Galileo's comprehension of inertia: an inherent resistance to motion
(perhaps, in the meaning of the change of motion) and of inertial movement: the dubious
maintaining of motion.
A contemporary learner studies Newton’s first law with a rather different meaning
from the meaning that both Leonardo and also Galileo held (Author 2003). As such, both
conceptions present a potentially beneficial environment for introducing the
contemporary conception of inertial movement and of the Newton's first law, using the
views of Leonardo and Galileo as alternative conceptions, facilitating pedagogy by
drawing on genetic epistemology.
2. FALLING BODIES
The discourse on falling objects played a central role in the transition from the medieval
to modern science (e.g. Dijksterhius 1986). Therefore, Galileo's law of falling bodies is
commonplace in every regular introductory physics course. In his Discourse (1638/1914:
107) Galileo refuted the Aristotelian claim regarding falling bodies through the use of the
thought experiment suggested by Benedetti (e.g. Dijksterhius 1986).
Galileo
demonstrated the logical inconsistency of the claim that the heavier body falls faster. The
latter was shown to be erroneous, but not more than that. That is, no new theory was
suggested. Unlike what is often mentioned to students, Galileo did not claim that his
revealing of the Aristotelian fallacy presented, at the same time, a proof that all bodies
fall in the same way (Author 2007).2 Despite the readiness of Simplicio – a peripatetic
2
And indeed they do not. In accordance with Newtonian theory, the acceleration with which two masses
approach each other directly depends on their masses. However, close to the Earth and for masses much
smaller than that of the Earth, in a very good approximation, all bodies fall with the same acceleration
(Galileo's law). Ignoring the approximate nature of Galileo's law presents conceptually deficient
instruction.
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Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
philosopher – to accept the logical refutation, Galileo proceeded turning to the empirical
realm:
It is clear that Aristotle could not have made the experiment, yet he wishes to give us the
impression of his having performed it when he speaks of such an effort as one which we
see. (ibid.: 110) (italics added)
Lacking an alternative theory Galileo had to turn to experimentation.3
A long
discussion and detailed elaboration of empirical considerations of bodies falling in a
space filled with matter of decreasing density lead Galileo to the conclusion:
Having observed this I came to the conclusion that in a medium totally devoid of
resistance all bodies would fall with the same speed. (ibid.: 116) (italics added)
This empirical law stating the independence of falling from a body's weight is known
as Galileo's law, which later, within the Newtonian theory, appeared to be approximately
correct for regular bodies falling to the ground.
Was Galileo the first to state this regularity? Looking at Leonardo's notebooks we
read:
The heavy thing descending freely gains a degree of speed with every stage of
movement and the part of the movement which is made in each degree of time is
always longer successively, the new one than that which preceded it. (MacCurdy
1955: 573)
This is exactly the statement of the fact that all bodies ("the heavy thing") fall in a
motion of constant acceleration! Our amazing grows even higher when we read there the
following:
If two bodies of equal weight and the same shape fall one after the other from the
same height in each degree of time the one will be a degree more distant than the
other. (ibid.)
This is already a very subtle effect stating, in modern terms, that thrown one after
another two bodies linearly increase the separation between them. Using the modern
notions one may write for the distances S1 and S2 between such a pair of bodies dropped
within the time interval t0 (g stands for the free fall acceleration):
1 2
1
gt and S 2 ! g (t " t0 ) 2
2
2
This implies for the distance between the bodies:
S1 !
3
This was a remarkable recourse. In other cases Galileo showed quite an opposite attitude to the
experiment, preferring convincing reasoning free from the factors which may deform the
appearance and mask the true phenomenon (McAllister 1996).
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Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
1 2
gt0
2
The meaning of this result is that the separation between the bodies increases linearly
S1 " S 2 ! gtt0 "
with time:
S1-S2 # t + const
or in terms of Leonardo: "in each degree of time the one will be a degree more distant
than the other". Assuming that Leonardo meant exactly what he stated a question arises
of how possibly a human may see this effect by a naked eye? It is just an inconceivably
acute perception. Since Leonardo often did not prove his statements (we will return to
this point) this leaves the question without an answer.
The topic of falling is directly related to the notion of weight as the cause of falling.
This subject was touched by both scientists in the especially interesting context: a thought
experiment of throwing a body into a tunnel through the Earth’s globe. Seemingly
Leonardo read about this imaginary experiment suggested originally by Albert of Saxony
(the birth of pendulum) (Clagett 1959) and wrote:
And if one should make a hole which was with its diameter or indeed its center the
diameter of the world, and there were thrown there a weight, the more it were to
move, the greater would its weight become. So having arrived at the center of the
earth which has only the name and it being itself equal to nothing, the weight thrown
would not find any resistance at this center but would rather pass and then return.
(MacCurdy 1955: 592)
In his Dialogue Galileo's representative Salviati said:
If a tunnel were made that went past the center, a clod would not pass beyond this
center except in so far as it was carried by an impetus pushing it further, to return
there afterwards and finally come to rest there. (Galilei 1632/1953: 134)
We observe that in this point both scientists shared similar views: rejecting
Aristotelian physics (according to which the body should stop at the center of the world)
and keeping with the medieval theory of impetus (implying oscillations relative to the
center). Also, Leonardo following other medieval scholars, talked about weight increase
during the falling. This was the famous accidental gravity (e.g. Clagett 1959):
Gravity is a certain accidental power which is created by movement and infused into
one element which is either drawn or pushed by another… (MacCurdy 1955: 630)
For Galileo weight of a body remained constant and served as the cause for falling.
He, however, refrained from speculation regarding the cause for acceleration (Galilei
1638: 202). If one equates weight with the gravitational force, one sees that Galileo
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Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
stopped only a little short of the Newton's second law in its form G=mg (G- gravitational
force, m – mass, g – free fall acceleration).
So why present this material in a physics class?
These episodes reflect the consolidation of knowledge of falling, weight and
pendulum motion. By using them in the class the teacher may render an often boring
topic of physics curricula, kinematics, into a vivid discussion on the major contents of
physics. The activities of Galileo and Leonardo may show students the naïveté of the
polar separation between the Rationalism and Empiricism in the epistemology of physics.
The advantage of the mathematical account of motion will become conspicuous,
motivating the student to overcome the initially repelling dry mathematical treatment.
3. PROJECTILE MOTION
Both scientists tried to account for the phenomenon of projectile motion –also an item in
a standard curriculum. Both were dissatisfied with the current ideas on the subject which
was based on the Aristotelian theory of violent motion and its medieval successor: a
primitive combination of rectilinear and circular motions (e.g. Dibner 1974: 181).
Leonardo, the best ever observer of the Nature, made the first step when he
understood that the motion of projectiles is reproduced by the stream of water pushed
from an open pipe. Besides careful drawing, demonstrating his super acute vision, he
suggested a special device which produced a water jet launching from a container at
different angles with the horizon (Fig. 3a). This drawing suggested controlled variation
of the relevant parameter, the core of the scientific method. This approach could bring
Leonardo close to the recognition of the actual trajectory, parabola. However, the full
understanding was not attained, since in reality, due to air resistance, the trajectory is
slightly asymmetrical, the parabola smashed a little at its second half (Figs. 3b, c).
Leonardo caught it even in the case of heavy cannon shells (less influenced by the air),
but without theoretical account. Such results could promote practical benefits, providing
manuals for artillery, but not physics knowledge. The latter was made by Galileo.
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Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
a
b
c
Figure 3: (a) Leonardo's apparatus to investigate trajectory dependence on the angle of launch
(Marcolongo 1956: 490). (b, c) Leonardo's sketches of ballistic motion (Calvi 1956:
284-285).
In contrast, Galileo's approach to the same problem is purely theoretical. His major
discovery was to see the projectile motion as a composition of horizontal uniform
movement and vertical uniformly accelerated motion. Galileo constructed a parabola and
this theoretical progress put him in a position of the full control: ability to make exact
predictions of maximal distance, to demonstrate the feature of angle duality (existence of
two inclinations causing the same distance of the projectile) and others. Galileo totally
neglected air resistance. His approach was to detect the phenomenon as an idealized
effect, something like a result in a thought experiment (McAllister 1996). The advantage
of such a rationalistic method is a clear conceptual understanding of the considered
phenomenon and ability to further investigate it theoretically. Two empirically supported
principles (of the uniform inertial motion horizontally and the uniformly accelerated free
falling vertically) allowed Galileo to manage without the knowledge of dynamics, only
later provided by Newton.
The comparison between Leonardo's and Galileo's treatment of projectiles is highly
beneficial pedagogically. By such a comparison contemporary students may learn that no
precise observation, whatever the accuracy is, and no matter how much of a genius the
observer is, can achieve either understanding, or control of natural phenomena. The full
power of physics knowledge is achieved only in terms of rigorous mathematical
statements and applied physical principles. The empiricist approach is restricted to a
limited success in practical use and is forgotten when a theory is invented. Theory is the
queen of physics allowing universal knowledge of natural phenomena.
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Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
4. ASTRONOMY AND ILLUMINATION
Leonardo knew that the Moon reflects the light of the Sun and therefore is observed in
phases. One could believe that only illuminated part of the Moon is seen. In fact one can
see also the not illuminated by the Sun area, although faintly.
Leonardo made a
qualitative guess and explained the visibility of the non-illuminated part of the Moon by
the reflected light from the Earth (Fig. 4) (Helden 1989: 22). This guess was correct but
lacked any quantitative account. Only much later, scientists could furnish this guess with
a solid support within the laws of photometry. The first was Kepler in his Astronomia
pars Optica of 1604, the author of the fundamental 1/r2 law for the decrease of light
intensity with the increase in distance.4
Figure 4: Leonardo's study of light reflection from the Moon (Emanuelli 1956: 207).
a
b
c
Figure 5: (a) Galileo's sketch of the Moon (Galilei 1610/1989: 44). (b) Leonardo's sketch of light
reflection from the Moon. (c) Leonardo's sketch of a shaft of water pouring into a pool
and producing turbulent motion of clear structure (Zimmattio 1974: 193).
4
This law was rediscovered by Bouguer, the founder of photometry, in the 18th century (Wolf 1961:167).
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Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
Galileo, a pioneer of telescopic observation of the Moon, discovered the irregular line
separating illuminating between non-illuminated and illuminated by the Sun areas of the
Moon (Fig. 5a, Galilei 1610/1989: 44). He interpreted this fact as an evidence for the
profile of the Moon's surface. In fact, Galileo described gradual change of illumination
of the mountain peaks on the Moon. Moreover, Galileo proceeded and by applying
geometrical reasoning and principle of rectilinear light expansion, succeeded to figure out
the height of the Moon's mountains (Galilei 1610/1989: 52).
Unlike Leonardo, Galileo correctly identified lunar dark spots as a smoother surface
of water. Leonardo made an opposite inference and thought the water surface to be
brighter (Drake 1957: 34)5.
Galileo reasoned by light reflection from the flat and
irregular surfaces and interpreted dark areas on the Moon surface as an evidence of
smooth, mirror like surface of water – Moon's "seas". Although literally incorrect this
statement remained valid in its evidence of more or less flat areas of the Moon ground,
plains, which till now are named as seas and oceans.
Physics teachers may use the opposite interpretations of lunar spots by Galileo and
Leonardo to promote better understanding of diffusive and mirror light reflections (a
matter of common misconception for students learning optics). It was this understanding
that both Leonardo and Galileo used to refute the Aristotelian view of celestial bodies as
perfect spheres:
… why the Moon is not smooth, I reply that and all the other planets are inherently
dark and shine by light from the Sun. Hence they must have rough surfaces, for if
they were smooth as mirrors no reflection would reach us from them and they would
be quite invisible to us…(Galilei 1623/1957: 263)
Furthermore, addressing Galileo's sketches of the Moon’s surface physics teachers
can compare them to the much more accurate drawings produced by Leonardo, for
example, when depicting fluids in motion (Fig. 5c); lacking a telescope Leonardo6 could
not draw in detail the Moon's surface. One may discuss the aspect of accuracy in
5
We could not find the explicit reasoning of Leonardo for this inference. Perhaps his assumption was that
water surface on the Moon was turbulent and so reflected light diffusively, as usually the case on the
Earth, and therefore stronger than the lunar soil. We saw such reflection shown by Leonardo in his
sketch (Fig. 5b) which might also confirm Leonardo's believe that the Moon possesses its own
atmosphere.
6
Although there is evidence that Leonardo developed the idea of a telescope (Argentieri 1956: 416-426),
he seemingly never realized this idea in a model and so no never had a chance to see the Moon through a
telescope. His note regarding the telescope remained: "…here only one star is seen, but it will be large.
And so the Moon will be seen larger and its spots in a more defined form." (ibid.: 422)
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Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
scientific drawings. On the one hand, Leonardo's eyes seemingly penetrated to the
essence of the phenomenon: the details of the fluid stream profile (Fig. 5c), which can be
identified on close inspection as displaying the pattern structure of turbulent motion
recognized by researchers centuries after Leonardo.
On the other hand, Galileo's
sketches, despite their schematic character and lack of many details (Fig. 5a), were useful
and helped him in his calculations of the Moon’s mountains’ height.
The lesson for students could be the existence of various standards of drawings in
science. Drawings may catch the essential features of the depicted phenomenon (usually
missed in a superficial viewing) helping its further analysis and serving as a source of
information (e.g. Leonardo). But there could be also drawings of another kind, rather
representing the idea, highlighting and isolating the feature that is the focus of the
investigation (e.g. Galileo). The use and interpretation of both types of drawings are
crucial and indispensable for physics research into the nature of the world.
A precise account for the illumination of bodies was of special interest to Leonardo.
How could it be performed without basic laws of photometry7 and calculus? Despite all
these, Leonardo developed a qualitative procedure to account for the nature of partial
shadow created behind an object in case of extended light source (Fig. 6). This drawing
shows Leonardo's construction of the shadow area behind a disc caused by a spherical
light source. Leonardo applied successive summing of shadow areas created by light
cones emanating from different points of the source. The drawing is pedagogical too.
Leonardo started with overlaying only two areas (aside from the major drawing), then
proceeded to three and four areas, and only after that presented the general case of many
areas.
The procedure resulted in the partial shadow (penumbra) with a gradually
decreasing intensity (astronomy textbooks usually inaccurately present the penumbra area
as homogeneously illuminated).
Although such a structure of penumbra correctly
represents reality behind any planet illuminated by the Sun, the method remains only
qualitative.
7
The fundamental laws of photometry relate the illumination at a point on a lit surface in inverse square
dependence with the distance to the light source (Bouger law) and in direct proportion to the cosine of
the angle of incidence of the light ray (Lambert law).
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Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
Figure 6: Leonardo's study of creation of complex shadow by a circular screen and spherical
light source (Argentieri 1956: 413).
In contrast, Galileo's approach to the problem of illumination was different.
Discussing illumination and light reflection from celestial bodies, which by their
appearance refute the idea of perfect spheres in favor of an Earth like planet, Galileo
asserted:
You must know, then, that a given surface receives more or less illumination from the
same light according as the rays of light fall upon it less or more obliquely; the greatest
illumination occurs where the rays are perpendicular. (Galilei 1632/1953: 80)
Such an account, however, could be sufficient for Leonardo, but not for Galileo.
Sagredo asked for explanation and Salviati (Galileo) provided a semi-quantitative
demonstration of the statement (Fig. 7). Galileo represents a homogeneous light flux by
means of parallel rays and pointed to the fact that the amount of rays, which collide with
the tablet immersed in the flux, decreases with the angle between the plane of the tablet
and the rays. In fact, this statement foresees Lamberts' cosine law for illumination
intensity as was suggested about a hundred years later, in 1729 (Wolf 1961: 168)8.
Figure 7: Galileo's reasoning regarding the angular dependence of light illumination (Galilei
1632/1953: 80).
8
The amount of rays crossing the tablet is in the direct proportion to the cosine of the angle between the
normal to the plane and the direction of the light rays.
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Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
5. LEONARDO'S CONSTRUCTION AND GALILEO'S STATICS
Our next example deals with applied mechanics, or theory of construction. In this field,
which facilitates engineering, Leonardo was a pioneer and his contribution was
tremendous (Uccelli 1956, Zammattio 1974).
Galileo entered the area as a mature
researcher, and presented the results of his study in his Dialogue Concerning Two New
Sciences (1638/1914: Day Two) of which one was in fact Statics.
Quite in the style of his days, Leonardo often registered his finding in the form of
questions and answers (a pedagogically beneficial style for the students). Thus treating
the case of a rod fastened into a wall at one end and supporting a weight at the other end,
Leonardo performs experiments and makes inferences regarding the load sustained and
the produced deformation (Fig. 8a):
If a rod projected by one hundred thickness [100 times its diameter] from a wall
carries ten pounds, how much will be borne by a hundred similar rods similarly
projecting and bound together into a unit? - I say that if [a rod of] hundred thickness
will carry ten pounds, the [bundle of] five thickness will carry ten time as much as
the [rod] of 100 [thickness], and if ab is the [bundle] of five thickness, the hundred
rods will carry 20,000 pounds. (ibid.)
a
b
Figure 8: (a) Leonardo's drawing of the experiment of the loaded rod (Uccelli 1956: 271). (b)
Galileo's drawing of the model for the loaded rod (Galilei 1638/1914, 143).
The similar context was treated by Galileo in his Discourse (Galilei 1638/1914).
Although both researchers observe similar deformation of the loaded rod they depict and
treat the case very differently. Instead of appealing to experiment and manipulation with
data as Leonardo did, Galileo provided a detailed theoretical treatment in general terms,
built a model, scrutinized it, employing more fundamental laws and inferred deformation
of the rod ascribing to it the shape of a parabola, with no specific numbers (Fig. 8b). In
his concluding statements Galileo seeks algebraic generality, for instance we read in the
same chapter:
16
Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
In the case of two cylinders, one hollow the other solid but having equal lengths, their
resistances [bending strengths] are to each other in the ration of their diameters.
(ibid.: 150)
The difference between the compared treatments by the two scientists is striking and
very important for science students. It is more than a transfer between the arithmetic to
algebra9; rather it was a transfer from concrete descriptive to abstract theoretical science.
6. PENDULUM AS A DEVICE AND AS A PHENOMENON
Both Leonardo and Galileo addressed the pendulum in their studies, but the difference
between their approaches employed by the scientists to study the topic was remarkable
and therefore deserves the attention of physics teachers. Leonardo's pendulum was an
item in the ocean of technical devices he imagined and elaborated, all in his style:
impressive in their novelty and creativity, but lacking depth in the accompanying
theoretical explanation (Bedini & Reti 1974). For Galileo, the pendulum was one of the
central theoretical topics, which he used as a stage for demonstrating the fundamentals of
physics and the principles of exploration of nature, as he conceived this activity of the
human intellect (Matthews 2000).
Leonardo's image of pendulum motion was of photographic accuracy. His drawing of
a swinging pendulum presents a unique artistic presentation of natural oscillations (Fig.
9a). The sketch anticipates a series of movie stills for making a dynamic picture.10
Leonardo sees the swinging as it is: a non tautochronic process of oscillations. If this is
the nature's reality, Leonardo "corrects" it and suggests a contrivance which makes the
oscillations exactly isochronal (Fig. 9b).
This step, which in fact replaced free
oscillations with the motion of the pendulum driven by an external mechanical agent,
would blow up with Galileo who saw in the pendulum the motion of natural falling
congenial to a free falling, the free fall under constrain. Leonardo "solved the problem"
of isochronicity, at the expense of the total elimination of the context of natural falling, so
important for Galileo's analysis of the theory of motion (Matthews 2000: Ch. 5).
9
This is despite the fact that Galileo never wrote formulas as we do. Instead he expressed himself in long
verbal descriptions as we have exemplified.
10
The picture is in opposition to stroboscopic pictures usually used in physics textbooks to represent
motion. Thus stroboscopic picture would show more crowded images at the extremes of a swing
deviation, where the velocity of the bob is low. Leonardo makes denser images next to the lowest point
of the swing, nadir, where the velocity is the highest. Leonardo's approach corresponds to the
appearance of the images in an observer's eye.
17
Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
a
b
Figure 9: (a) Leonardo's drawing of swinging pendulum (Bedini & Reti 1974: 262). (b)
Leonardo's model of regulated pendulum (Bedini & Reti 1974: 258).
Unlike Leonardo, Galileo approached the pendulum as a physical phenomenon and
treated it exclusively in a theoretical manner. He started with the establishing of an
elucidating connection between the pendulum motion and the motion along the inclined
plane and vertical fall (Galilei 1590/1960: 65).
He preceded this treatment in his
Dialogue (Galilei 1632/1953: the Fourth day) with the unproved assertion of
isochronicity of the falling of a body along a quadrant (a quarter of a circle), regardless of
the initial point (this claim is equivalent to the independence of pendulum motion from
the amplitude). This result initiated the immature account of celestial movement of
planets and lead to the mistaken theory of tides. Eventually, the proof of isochronicity
was delivered, but only in his Discourse (Galilei 1638/1914: the Third day). It was,
however, given for the bodies descending along the chords of a circle ("the law of
chords") (Fig. 10). Not only was the geometric demonstration rather cumbersome and
complex (it had to draw on the empirical principles11), but the extension of the validity of
the law of chords to the motion along the corresponding arcs was just mistaken, as well
as the statement of the minimum of the time of descent between two points along the arc
of a circle (brachistochrone)12. Galileo was passionately driven by the wrong idea: the
law of chords was too seductive as if suggesting applicability to arcs although it had been
proved for chords.
11
Not to forget that Newton's laws which make the proof simple and straightforward for the modern
student were not available. They were replaced with empirical principles of inclined plane developed in
the medieval physics.
12
The problem of isochronicity was solved in 1673 by Huygens: the trajectory should be along cycloid.
This curve appeared to be also brachistochrone between two given points, as was proved in 1697 by
Leibniz, Newton and Bernoulli brothers (e.g. Matthews 2000: 124-133).
18
Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
Figure 10: Galileo's drawing illustrating the law of chords (Galilei 1638/1914: 237).
Didn't Galileo see that the statement of amplitude being independent of swing time
did not correspond to the reality (experiment), as Leonardo saw before him? Why, then,
did he write:
It must be remarked that one pendulum passes through it arcs of 180$, 160$ etc. in the
same time as the other swings through its 10$, 8$, etc. ... if two persons start to count
the vibrations, the one the large, the other the small, they will discover that after
counting tens and even hundreds they will not differ by a single vibration, not even
by a fraction of one. (Galilei 1638/1914: 254-255)
Here is an important pedagogical point. We may understand Galileo's mistake as
following. The laws of physics are stated for the idealized phenomenon, free of all
foreign impediments, which may mask and change the appearance. This principle was
known since Aristotle and in fact was traditionally used in the medieval physics to
explain the apparent deviation of reality from the Aristotelian tenets.
Thus, both
Aristotelian and Galilean predictions regarding falling "heavy" and "light" bodies did not
correspond to many observations in a regular environment. While interpreting empirical
data physicists always face the choice: either the used theoretical framework is correct,
but the appearance was impeded by foreign factors, or the adopted theoretical framework
is wrong. To resolve the dilemma, careful physicists usually postpone the decision and
go for additional experimenting. People in the throes of passion, however, often do not
wait and perform abduction, making decisive claims without additional testing (which
might be of course not available or even possible at a particular time or situation).
Galileo's choice in observing falling bodies was correct, but the decision made with
regard to pendulum motion was wrong.
Still in the context of the pendulum, learners may compare the views of Leonardo and
Galileo regarding motion in general. Leonardo conceived separately of that part of the
descending motion of a bob (identified as "natural" motion) and the other part of its
19
Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
ascending motion ("accidental" motion). He even stated the rule: the accidental motion is
always weaker (seemingly reflecting the fact of natural damping) (Bedini & Reti 1974:
262). This view was much closer to the Aristotelian dichotomy of natural and violent
motions. Galileo considered pendulum motion holistically, as one motion, and in this he
clearly surpassed his great predecessor.
Discussion
In a series of examples we have displayed several concrete points of subject matter taught
in school physics, in which comparison between the knowledge of Leonardo and that of
Galileo could benefit the contemporary learners of physics. This pedagogy may support
students' construction of the image of physics as possessing the structure of disciplineculture (Fig. 1b), that is, to include peripheral knowledge. The merits of this pedagogy,
however, surpass better understanding of the disciplinary contents and introduce learners
to the scientific method and the culture of science.
Too often science is presented in classes as if it is based solely on experimental
evidence and straightly emerges directly from it. This understanding corresponds most of
all to the Aristotelian philosophy – the inductive-deductive cycle of scientific exploration
of the Nature (e.g. Losee 1993: 5-15) – and no longer convinces philosophers of science
who have surpassed this naïve version of empiricism. However, this conception often
prevails in many designers of science curricula. Since it is not realistic to oblige them all,
together with the community of science teachers, to take a course in the philosophy of
science, one may suggest an alternative way: to attain an adequate understanding through
discussing purposely chosen representative examples which by contrast would deliver a
more realistic image of science. Although no activity of a real scientist can represent
pure empiricism, episodes from Leonardo's activity could provide a sufficiently close
evidence of the limited scientific power of the approach that mainly draws on the sense
experiences.
Indeed, one could start with mentioning that in Leonardo's view Art, specifically
painting, was a science, moreover, the "Queen of all sciences", which "provided not only
the means of obtaining knowledge, but also communicating it to all the generations of the
20
Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
world".13
This method of knowledge construction, as Leonardo conceived it, was
elaborated by him in the following stages:
1) Close observation;
2) Repeated testing of the observation from various viewpoints;
3) Drawing the object or phenomenon so skillfully that it would become a "fact" which all the
world could see, or could grasp with the aid of brief explanatory notes.14
Needless to say that Leonardo himself could not fully keep with these highly intuitive
principles when he tried to make statements about the world beyond its mere description.
Lacking systemic university education, he educated himself through his own reading,
inevitably limited and not well organized.15
However, a strong common sense, an
extremely acute sense perception, a smart and creative mind, an extraordinary skill of
reflective representation by exquisite drawings, all these brought Leonardo to many
discoveries.
Driven by a permanent thirst for innovation, Leonardo produced an
unbelievable variety of innovative technical contrivances.
What did not happen,
however, was that Leonardo made any significant progress in theoretical knowledge, a
new fundamental theory of reality or a method of knowledge manipulation (mathematical
description and analysis). Isn't this an invitation to learn theory even when students are
extremely successful in puzzle resolving and pass problem solving tests?
There is another important and interesting feature teachers may discuss in physics
class: Leonardo's scientific activity was always covered by secrecy. He did not maintain
any open scientific discourse, did not participate in public debates or discussions of
scientific issues with colleagues; he did not teach and had no pupils in the domain of
science. Not only his writings were not published, they were coded by him in a way
which should prevent access to them by others by using mirror writing. His notebooks
served him as a private journal, and they were published much after his time (MacCurdy
1955). Is this the way to make science? Is this a way to know science?
13
Leonardo quoted by Wallace (1966: 104).
Ibid.
15
Scientific texts in natural philosophy were rare and were written in Latin, which Leonardo learned by
himself limitedly. Language restricted him in science. Leonardo was an autodidact and mostly learned
from experience and from conversations with educated people around. His mathematics knowledge was
due to Luca Pacioli with whom he worked for short period. Leonardo mastered only rudimentary
geometry and was often inaccurate with his arithmetic. He never excelled in mathematics. Alberti of
Renaissance Florence (among the founders of perspective method in painting) seemingly influenced
Leonardo's theoretical knowledge (White 2000).
14
21
Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
In the perspectives of science education and scientific discourse, Galileo presented a
contrasting case. He was well schooled, taught at universities (the University of Padua
was among the leading universities of his time). His activity included intensive discourse
with various academics, correspondence and meetings with distinguished colleagues, and
pupils (e.g. Drake 1978).
In his publications Galileo, in contrast to Leonardo's
codification, chose the style of dialogue with opponents and for the first time in scientific
publishing wrote in vernacular, colloquial Italian.
For this reason, some academic
readers found the free style of the Dialogue "shapeless and undisciplined" (Sharratt 1994:
153). But just in this way, Galileo, a brilliant teacher and entertaining debater, extended
his highly eloquent debate beyond the limits of scientific community, making it public,
that is, involving many contemporary intellectuals: writers, artists, enlightened amateurs
(e.g. de Santillana 1955; Segre 1991: 15-32). The ideas of Galileo, his approach to
science, were thus widely disseminated and influenced generations of scientists in Europe
with unprecedented speed. Galileo became the leader of the new science and a symbol of
the scientific revolution.
In contrast to Leonardo, Galileo was perfectly knowledgeable about the major results
of the Hellenic, Hellenistic and medieval science. The merit of this knowledge was
striking. For instance, while performing his own studies of falling and projectile motions
Galileo incorporated the account of the uniform and accelerated motions as it had been
developed at the Merton school in Oxford, in the 14th century. From Oresme in Paris,
also in the 14th c., Galileo learned about the introduced by him graphical representation of
time dependence of velocity in uniform and uniformly accelerated motions.16 Galileo
performed a synthesis of both methods (Mertonian and Parisian) and added to them his
own part – addressing the experiment, that is, an interpretation of the real data. From
Buridan, Oresme, Benedetti and others Galileo adopted the skill and taste for thought
experiments which significantly enhanced his power of persuasion in scientific polemics.
All together it was a new scientific method that founded the modern science, the one that
16
Many results that we use to attribute to Galileo, such as the formula for the distance: S ! 1 at 2 , the
2
&
v
v
t
theorem of the average speed: S ! 0
% t and their graphical representations were developed in
2
Oxford and Paris (e.g. Pederson & Pihl 1974: 219-225), waiting for the time to be incorporated into the
new science which crystallized within the mediaeval one.
22
Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
fused a mathematically formulated theory with experiment, both actual and also
imaginary.
Galileo did rely on the authorities from the past (especially Archimedes) and did not
hesitate to reproach others (especially Aristotle). The major revolt against Aristotelian
physics was the Galileo's introduction of a quantitative account of the experiments into
physics (Koyré 1943a).
However, in his global approach to scientific exploration,
Galileo's style was compatible with both major trends of the Greek thought. In the best
tradition of Aristotle Galileo sought for the patterns of law-like regularities in the real
environment given to our senses. At the same time, he sought for and tried to distill
idealized situations, which could represent the laws of Nature in their "pure" form, as
they were written in "the Book of Nature", a rather Platonic idea. Only such were
considered experiments by Galileo (McAllister 1996) and were used by him for
illustrating of the principles which govern the paramount order in the Nature.
While Leonardo often refrained from displaying and discussing his reasoning in
support of his claims, Galileo, when adopting the heliocentric view, opened a whole
program to provide justification for this conception. He interpreted his discovery of the
moons of Jupiter as an evidence against the unique status of the Earth, as the center for all
celestial objects which revolved around it. He identified the expected phases of Venus.
Challenged by Bellarmino, the Chief Inquisitor, Galileo sought for an empirical evidence
for the true and not just apparent Earth's movement (e.g. Sharratt 1994: 114-116). It
seemed to him that he founded the required evidence in the phenomenon of ocean tides
(Galilei 1632: The Fourth Day; Finocchiaro 1989: 119-133). Albeit erroneous, this
argument manifested the method, seeking empirical evidence which could beyond any
doubt indicate the correctness of the Copernican theory. This was a delicate interplay of
theory and experiment in the scientific exploration.
In contrast to Leonardo's method, which we quoted above, Galileo subordinated
sensory evidence and even the common sense, that provides people with naïve
explanations, to mathematical reasoning (demonstration) combined with illustrative
experiments:
Viewed as a whole, Galileo's method then can be analyzed into three steps, intuition
or resolution, demonstration, and experiment. (Burtt 1927: 70)
23
Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
For our discussion here, however, it is especially important to see the complex role
Galileo ascribed to experiments (Segre 1991: 47). We can illustrate it quoting Galileo:
Did you [Salviati] make an experiment? – No, I do not need it, as without any
experience I can affirm that it is so, because it cannot be otherwise. … I without
experiment am certain that the effect will follow as I tell you, because it is necessary
that it should. (Galilei 1632: 145).
And the same Galileo said:
…the fact that all human reasoning must be placed second to direct experience.
Hence, they will philosophize better who give assent to propositions that depend
upon manifest observations, than they who persist in opinions repugnant to the senses
and supported only by probable reasons. (Galilei 1613: 118).
On the face of it, Galileo may confuse the learner with his inconsistency and
especially those who try to classify him as wholly belonging either to the tradition of
rationalism or to that of empiricism. Koyré summarized this amazing mixture:
The new science is for him [Galileo] an experimental proof of Platonism (Koyré
1968: 43)
In fact, however, learners of Galileo observe a consolidation of the scientific method
as it is practiced by the modern science: an amalgam of the approaches, kept in balance
by numerous practitioners, sharing scientific discourse. On some occasions physics
research may look like pure empiricism (remember the continuous empirical research of
high temperature superconductivity in ceramics which proceeds and still lacks a
grounded theory), and, in other cases, physics research appears as a pure rationalism
(string theory is still lacking any empirical evidence, but continues to occupy many
brilliant minds who have developed an extensive theory in this frontier domain). By
considering contemporary physics research it is meaningless to state a principle
advantage of either of the two trends of thought over the other. Both are indispensable,
they share complementary importance. A discussion, which starts with a critique of
Leonardo's insufficient theoretical account and proceeds to Galileo's complex and
seemingly controversial empiricist-rationalist method, may serve as a good introduction
to the contemporary pluralistic method of science, which draws on the balance between
the two extremes, both equally unacceptable if one method tries to monopolize. It was on
this occasion that Cohen (1950/1993) expressed his sympathy for "the poor teacher of
elementary physics who seeks a simple picture for his students".
Conclusion
24
Constructing an educational message from a comparison between scientists: Leonardo and Galileo compared
We may conclude that a comparison between Leonardo and Galileo fits the approach of
discipline-culture to teaching and learning physics in several important aspects. Firstly,
by concrete examples of different accounts of the same phenomena, suggested by the
prominent scientists, one can contribute to understanding of the essential points of the
disciplinary material (ontological aspect). Secondly, this approach displays different and
powerful methodological approaches (epistemology, using mathematics) and thus allows
for learning aspects of the subject often ignored by regular learning.
Thirdly, the
comparative approach could be effective for demonstrating the social nature of scientific
activity, the crucial importance of maintaining an open discourse to facilitate the progress
of knowledge.
Leonardo and Galileo provide an indispensable resource for an
instructional strategy which seeks a culturally valid education.
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