CONTENTS
Introduction
Some ways of understanding the didactic use of
instruments:
The problem
The instrument
The asociated theorical knowledge
Theorem of leg
The solution to the problem
A didactic resource:
Parallactic ruler
The astrolabe
C. Huygens's clock
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THE ROLE OF
INSTRUMENTS IN THE TEACHING OF HISTORY OF SCIENCE
Introduction
In 1630 Galileo published his
"Dialogue of the two maximum systems of Ptolemaic and Copernican world". In the
third day, he says, referring to the Heliocentric Theory:
"I can not stop admiring the eminence
of ingenuity of those who have received and accepted it as true... I can not find an end
to my admiration, as I see how Aristarchus and Copernicus have been able to reason such
violence against the senses, so that, against these, it has taken over their beliefs"
In effect, Aristarchus of Samos was an
astronomer of the school of Alexandria who in the third century BC decided to measure the
distance between the Earth and the Sun. For this he took advantage of the fact that, in
the lunar quarters, the Sun, the Earth and the Moon are on the vertexes of a right angled
triangle with the right angle at the Moon, in such a way that the segment Earth-Moon is
the smaller leg, and the segment Earth-Sun is the hypotenuse. Aristarchus measured in this
position the angle Earth-Sun-Moon (with vertex on the Earth), getting a value of 87º
(this angle is really 89º 59’). Then he observed a right angled triangle with
one of its acute angles measuring 87º, similar to the previous one, proving that the
hypotenuse is twenty times longer than the smaller leg, and, therefore, as similar
triangles have proportional sides, Aristarchus concluded that the distance Earth-Sun is
twenty times longer than the distance Earth-Moon (really it is 370 times longer). He also
observed, that the apparent sizes of the Sun and the Moon are approximately the same, with
which, if the Sun is much further away than the Moon, it should be much bigger, and even
bigger than the Earth (he had got the relation between the sizes of the Earth and the Moon
by the means of other astronomical considerations; possibly observing the shadow of the
Earth over the Moon during an eclipse). In these conditions, Aristarchus thought that it
was not reasonable to suppose that such a big Sun revolves around a smaller Earth. This is
how the first heliocentric model of the universe emerges.
In the above we can distinguish, on one
hand, a change in the cosmological concept of the universe; on the other hand, geometrical
figures are resorted to, the similarity, the proportions,...; but, what is there in
between?... In between there is the measurement of an angle, only just one angle, that
together with geometry allowed Cosmology to be changed: and, evidently, this angle was
measured with a determined instrument.
Therefore, one of the possible ways of
travelling through the History of Science is to do it through the study of the scientific
instruments. In effect, we can consider Science as an "organism" whose working
order leaves, through time, a trail made up by two main "products": written
texts and the instruments. These are products quite appropriate for their
"manipulation" and whose study can be tackled from different points of view,
which gives them a high didactic applicability. The following diagram, that in a way,
could be considered trivial intends to make a summary of the working order of the
scientific "system":
In short, this diagram can be
interpreted as the working order of an organism, Science, that through History leaves a
trail of "exquisite waste", by means of which we can follow its "vital
route".
In what follows we will concentrate on one
of those products and its possible didactic applications: the instruments.
In first place we want to point out that we
are talking about instruments in a "wide sense", this is, that we consider
instrument any device capable of generating any kind of knowledge. So, a particle
accelerator is an instrument, but a simple post stuck into the ground (a gnomon) or a
numeric table, for example, are also instruments. Anyway, taking into consideration our
didactic intentions, an aspect we consider important is the simplicity; or even better,
the relation between simplicity and quantity of knowledge produced.
In second place we can see that instruments
can be considered as a refinement of our senses, which acquire through them, bigger
capacity for penetrating nature, increasing our power of observation, making induction
processes easier, the establishment of speculations, of hypothesis, etc. And mainly, the
achievement of measurements of magnitudes that can be represented in an abstract
mathematical space in which axiomatic-deductive methods proper of mathematics can be
applied, which allows us, in many cases to check, refute, or change statements established
previously. In this way the instrument becomes the key that locks (or opens?) that eternal
circle of happenings and mishaps, of doing and undoing (like Penelope), that we call
Science.
Well now, how can we materialise these
considerations in the frame of teaching the History of Science in secondary education? To
answer this question we resort again to Galileo’s words. In the first day of the
book "Considerations and mathematical demonstrations of two new sciences",
published in 1638, Salviati (spokesman for the Academic) says to Sagredo and Simplicio:
SALVIATI: I think that the frequent
activity in your famous arsenal, Venetian Gentleman, offers a great field for
philosophising to the intellects that speculate, specially, in that part called mechanics,
where continually all kinds of instruments and machinery are built by a great number of
artisans, some of which have to be very knowledgeable and with a very acute talent due to
both the observations made by their predecessors as by what they discover themselves
without interruptions.
The meaning of this text can be interpreted
as the recognition, by Galileo, that the scientific activity that emerges in the
Renaissance is related with the abandoning of the elitist Greek prejudices against
practical arts, as well as the increasing interest for mechanical problems related with
artisan production, which makes scientific theories appear from the empirical discoveries
accumulated by the tradition of the artisan workshops. In the same way, if we want our
secondary pupils to acquire knowledge of the History of Science in a significant way, we
also should abandon our "elitist Greek prejudices" (and we could add academical)
and try to emulate the artisans, building even in a simple version, some instrument from
which we could obtain empirical knowledge.
Thus, we propose to try to build the
scientific knowledge of the pupils running over that trail of instruments left by science
through history, stopping specially at some selected points; that is to say stopping at
some instruments whose relation between simplicity and quantity of knowledge produced,
mentioned before, is satisfactory and, also, distributed along the historical period to be
studied. Each of these selected instruments will be the epistemological connection between
a determined practical question, inserted in a determined era and with all the political,
economical, religious, cosmological conditionings, etc, that are inherent to it on one
hand and, on the other some kind of theoretic knowledge such as Geometry, Mechanics,
Chemistry, etc. In this way the instrument becomes the nucleus around which the didactic
activity concentrates, which can be tackled from different disciplines, that is to say, in
a interdisciplinary way and aimed at a diversity of pupils because we use different
activities adaptable to diverse intellectual capacities: use, construction, design,
explanation of working order, theoretical fundaments etc. The above can be summarised in
the following diagram:
Here we have some examples of the
didactical use of instruments with the intention of clarifying what was exposed before.
2. SOME WAYS OF UNDERSTANDING THE
DIDACTIC USE OF INSTRUMENTS:
In the didactic application instruments can
be considered from many points of view. Below we have some examples of different ways of
understanding their use in the classroom that we consider especially interesting.
An instrument can be considered as: A
generator of new knowledge of nature's behaviour with which we can solve problems.
Let's suppose we are studying the Science
of the XVII century. One of the most significant themes of this era is the study of bodies
in motion. In effect, during the XVI and XVII centuries the Aristotelian conceptions about
motion are questioned and new ideas on the subject emerge, as for example the
crystallisation of the Law of Inertia; which lead to new approaches, that together with
the irruption of Mathematics as the language of Nature, sets the foundations of Modern
Science. We can find all this, amongst others, in the work of Galileo, who in the third
day of his book, mentioned before "Mathematical considerations and demonstrations of
two new sciences", establishes definition s of constant motion and constantly
accelerated motion that correspond to the essence of these two kinds of motion. This
correspondence, says Galileo, "... we think we have accomplished it at last, after
long reflections specially if we take into account that the properties that we have
successively demonstrated, (from our definition) seem to correspond and coincide exactly
with what natural experiments put in front of our senses. In short the study of naturally
accelerated motion has led us, as if we were taken by the hand, to the observation of
customs and rules that are followed by nature in all its remaining works, that for their
performance the most immediate, simple and easiest means are usually used..."
So, in this context we will situate the
following example, with the aim of illustrating the use of SCIENTIFIC INSTRUMENTS in the
construction of knowledge and the solving of problems, from the point of view of the
History of Science.
THE PROBLEM
If from the
points P1, P2 and P3 we drop a body to the point A, following inclined planes, which of
the three routes will be covered in less time?
In order to solve this problem, we need a
method to measure or compare the time that a body takes in sliding down each of the
inclined planes. This, in the era that we have situated ourselves, represents a
complicated technical problem, taking into consideration that the construction of precise
clocks is not very advanced.
THE INSTRUMENT
Let's consider a circumference situated
on a vertical plane, where various inclined planes have been built (in this case two),
forming strings that meet at the lowest point of a vertical diameter. These have different
inclinations, and on them we can release simultaneously two equal objects; for example,
two crystal balls (to avoid, as far as possible, friction). In the following figure we can
see a diagram of the instrument:
With this instrument we can prove
"empirically" that the falling time along all the strings considered, is the
same. This means, that the instrument PRODUCES new knowledge: "the falling times
along the strings that meet at the lowest point of a vertical diameter are all the
same", or said in a different way, the mentioned strings are TAUTOCRONAS (from the
Greek: equal time).
THE ASOCIATED THEORICAL KNOWLEDGE
But, as we have pointed out on the
diagram, each instrument has some theoretical fundaments associated of a mechanical,
geometrical or arithmetical nature, etc; accumulated through history, until the era in
which we are situated. In the case of our instrument we can PROVE the property obtained
empirically using the II and III theorems on naturally accelerated motion included in the
third day of "Considerations and mathematical demonstrations of two new
sciences" and the "Theorem of Leg" that we can find enunciated and
demonstrated in the "Elements" of Euclid:
THEOREM II: If an object in motion falls,
starting from rest with a constantly accelerated motion, the spaces covered by it in
whatever time, are between each other like the quadrate of the times.
THEOREM III: If one and the same object in
motion moves, starting from rest, upon an inclined plane and along a vertical one both
having the same height, the times of motion between each other will be like the length of
the plane and the vertical.
THEOREM OF LEG:
The leg is the proportional media
between the hypotenuse and its projection upon it. Using the results above, we can do the
following demonstration:
To conclude, we will use the knowledge
acquired through the instrument to solve the problem that we had set out:
THE SOLUTION TO THE PROBLEM
If we take into consideration the
theoretical results acquired by means of the instrument, we can observe that the times
that the moving object in sliding from the points Q, P2, R towards A, are the same. So the
time from P2 to A is the shortest, because in the other inclined planes the moving object
has to cover a supplementary space (the space outside the circumference), so the times
will be longer. The following drawing shows the situation:
2. A didactic resource:
It is to say, an object around which
different activities can be organised aimed at diverse types of pupils. Around an
instrument activities can be designed that range from handcraft (using tools for its
construction) to the most abstract formalisation proper of geometry that explains its
working order, passing through empirical activities (use of the instrument to make
observations and to obtain information). This way we can try to solve one of the biggest
problems product of the new educational legislation, that assures the access of education
to all the society, independent of their social, economical, cultural conditions, etc,
that means, on the one hand, an advance in social justice, but, on the other hand, results
in teachers finding themselves in the classroom with some pupils who want to pursue
universities studies, together with others that do not know what to do with their future,
and others that do not even want to be in the classroom; However, something must be
offered to each of them. So, we can start by proposing to those who do not want to be in
the classroom, the construction of an instrument like the one in the following drawing
(called
PARALLACTIC RULER):
The instrument consists in a ruler AB
length 1 with a longitudinal slot in which the extreme of the segment Pk, ½ long, slides,
which is articulated at P with a segment AP of the same length as PK, and that can turn
around point A. This way, to each angle with a vertex at A corresponds a segment of length
AK, which can be used to indirectly measure the said angle.
Once built, it is very possible that we
could convince one of the pupils unsure of his / her future, to try to use this instrument
to measure angles, after those who are interested in achieving a certain level of
knowledge have investigated the geometrical relation needed to be able to associate to
each angle, a length on the ruler. In this way, the building up of knowledge, covers a
process that goes from the use of hands (handcraft) up to certain levels of abstraction,
passing through the acquisition of data about what surrounds us, using instruments
(empirical knowledge).
3. A storeroom of knowledge, or what is the
same, a book where the History of Science is written. To study Science means to study its
history, its genesis; for this we must read these books, and so we must understand the
language in which they are written : GEOMETRY.
A good example to illustrate this could be:
THE ASTROLABE
It is very possible that astronomy appeared
when someone combined the observation of the heavens with memory. If we observe the
heavens on a clear night we can see an infinity of stars. If we observe it a while later
we see that the position of the stars has changed, this is, we prove that there is motion
in the heaven. Repeating the observation and memorising the previous positions of the
different stars, the idea of trajectory appears; geometry of the cosmos appears, that
later would become simply Geometry.
But something that moves spontaneously is
not perfect, because if it moves, it is not occupying its place in an orderly cosmos. Does
this mean that the cosmos is not perfect? To answer this question, let us observe, let us
memorise, that is, let us do geometry.
In this way a trajectory appears that will
forever determine the geometry of the cosmos: the circumference (the wheel?). In any other
geometrical figure, if we make it rotate a certain angle, its points will abandon the
initial figure; however, if we make a circumference rotate around its centre, its points
will always remain within it, that is to say, it is inmutable, or in other words, it is
closed in respect to the movement of rotation around its centre. In consequence, the only
natural motion possible in an orderly and inmutable cosmos, is that which follows a
circular trajectory, which made the circle and its extension to a tridimentional space,
the sphere, arise as the basic geometrical figures for the construction of explicative
models of the cosmos.
This is how the theories in the antiquity
appear, ranging from Euxodus of Cnido ( IV Century BC), creator of the first model known
that used concentric spheres to explain the motion of stars, to Ptolomy( II Century AD),
who managed to project the celestial sphere upon a circle, which allowed him to build an
instrument whose circular geometry reproduces "the working order" of the cosmos:
the astrolabe, the noctolabe, etc.
These theories developed in the Hellenic
world, together with the contributions of eastern cultures, as well as the techniques of
construction of instruments, were assimilated and improved by the Arabs, and spread to
western Europe during the Middle Ages.
Lastly, we can observe that the different
"signs" which appear over the circles that constitute an astrolabe, are the
traces left by the contributions made to astronomy over the centuries, so to understand
its meaning (understand its working order), means to understand the history of astronomy,
or what is the same, understand the genesis of the astronomical knowledge. And it is
obvious that these signs are written in a geometrical language on circular books called
astrolabe, noctolabe,...
4. The connection between a practical or
theoretical problem set in the society at a determined era under economical, political,
cultural religious conditions, etc; and a scientific, mathematical theory, more or less
abstract. The study of this aspect of the instruments allows us to make interdisciplinary
approaches, which involve matters related to all branches of knowledge.
C.HUYGEN'S CLOCK
One of the main problems that
navigation encountered during the modern ages, was the exact determination of the position
of ships at sea. In the XVI century the problem of the determination of latitude had been
solved by the observation of the apparent height of the stars in the different points of
the globe, this was not the case of longitude. The importance of this problem was such,
due to its economical, political, and geographical implications, that for three centuries
it turned into a permanent object of investigation and it was a real challenge for the
science of that era.
From the XVI century the European
monarchies were interested in the solution to this problem and international contests were
held, with important prizes for a definite resolution; as well, official boards were
created in order to discuss the available methods.
The first attempts at a solution came from
Astronomy. In effect, the observation of astronomical happenings done simultaneously in
different places allowed the position to be established easily. In this way the methods
based upon the observation of sun and moon eclipses appeared, the one of Saturn's
satellites, and especially, the method of the lunar distances.
The scarce results of these methods,
together with the difficulty of calculus for the elaboration of tables, drew the attention
towards the use of clocks that allowed the comparison of the time at the dock of departure
with the time on the boat, deducing in this way, the difference of longitude.
One of the attempts to find a solution is
due to Christian Huygens, who in the book "HORULOGIUM OSCILLATORUM", published
in 1673, describes, among other things, the construction of a pendulum based on the
properties of the cycloid curve that would work on a ship, independently of the movement
of this due to the waves. It turns out to be a proposition of great "geometrical
beauty" due to the surprising properties of the mentioned curve.
The theoretical knowledge associated to
these instruments are mainly its mechanical properties, the BRACHISTOCRONY and the
TAUTOCHRONY, and their geometrical properties; that can be found in the mentioned work of
Huygens: "Horulogium Oscillatorum". These properties can be proved empirically
with the instruments mentioned in the previous section, PROVED using the Euclidean
geometry and the results of Galileo's study of motion of falling bodies.
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