EXCURSE TO THE HISTORY OF WEIGHT CONCEPT: FROM ARISTOTLE TO NEWTON AND THEN TO EINSTEIN

We address teachers of high school physics course with this excurse which deals with the weight concept in physics. There are two main tasks of physics teaching: to cause learning physics knowledge and to recognize the way this knowledge is obtained and validated. In other words, this means to familiarize with ontology and epistemology of physics. Weight concept provides a unique opportunity to reveal both aspects of physics knowledge closely interwoven and mutually influencing. More precisely, following the history of weight concept, one reconstructs the way physics functions in order to understand reality in terms of conceptual pictures of the world (theories) as well as the requirements to the physical claims to be adopted as scientific truth. The story of weight concept started with physics itself (and even before), and its present understanding was obtain at the beginning of the 20 century. This progress was not fully copied by physics curriculum for which reason the history and philosophy of this concept could be elucidating and inviting implications in class instruction. The presented excurse into the history of physics reproduces how weight was understood prior to Newton, starting from the Ancient Greek science (Aristotle). From there we followed weight to the scientific revolution of the 17 century when Newton identified weight with the gravitational force and distinguished it from inertial mass. We explained why the Newtonian definition of weight had to be changed in the modern physics and weight was distinguished from the gravitational force, being defined solely through the operation of weighing. This progress followed the new understanding of the nature of gravitation attained in the Einstein's principle of equivalence within the general theory of relativity. All together, the story represents the development from the weight as a feature of material objects through being determined by the gravitational interaction between material bodies to the complementary combination of the nominal elastic definition with the operational definition of weight by means of a standard weighing. Weight implications to the practice of the modern society are addressed especially in the context of weight changes in acceleration systems, in a state of free falling (satellite of the Earth) and the rotational stations in space as dreamed by the enthusiasts of space exploration, starting by Herman Potocnik in the early 20 century.

The story of weight concept started with physics itself (and even before), and its present understanding was obtain at the beginning of the 20 th century. This progress was not fully copied by physics curriculum for which reason the history and philosophy of this concept could be elucidating and inviting implications in class instruction. The presented excurse into the history of physics reproduces how weight was understood prior to Newton, starting from the Ancient Greek science (Aristotle). From there we followed weight to the scientific revolution of the 17 th century when Newton identified weight with the gravitational force and distinguished it from inertial mass. We explained why the Newtonian definition of weight had to be changed in the modern physics and weight was distinguished from the gravitational force, being defined solely through the operation of weighing. This progress followed the new understanding of the nature of gravitation attained in the Einstein's principle of equivalence within the general theory of relativity.
All together, the story represents the development from the weight as a feature of material objects through being determined by the gravitational interaction between material bodies to the complementary combination of the nominal elastic definition with the operational definition of weight by means of a standard weighing. Weight implications to the practice of the modern society are addressed especially in the context of weight changes in acceleration systems, in a state of free falling (satellite of the Earth) and the rotational stations in space as dreamed by the enthusiasts of space exploration, starting by Herman Potocnik in the early 20 th century. * * * A deceitful balance is an abomination before the Lord: and a just weight is his will.
(The Book of Proverbs, 11:1) 1 This idea from the Old Testament (the book of proverbs, which collects the wisdom from the text of the Bible), informs us that the Lord wills correct weighing.
This obliges us to proceed in understanding what is weight, how to define it in accordance to be true. We believe that we suggest the direction that will please both teachers and students, those who want to be physicists and those who do not, but want to know about the world and the way it is organized and works. So it will be definitely the wish of the Lord that we will make sense of weight. Bible cannot help us in this, only physics can. * * *

I. Understanding of weight before Newton
The evolution of the weight concept in science started very early from the notions of heaviness (weight) and lightness (levity). Both appeared in the Greek philosophy of nature as fundamental intrinsic properties of objects.
The concept of levity lost its independence only in the Renaissance physics (Galilei 1638). Galileo argued: are two light objects create a lighter one when we combine them? The negative answer was sufficient to abandon levity and think only about heaviness of objects -their weight.
As to the weight, two theoretical conceptions prevailed in Greek science. The first was attributed to Plato. His weight was the tendency or inclination of bodies towards their kin 2 . A different approach was suggested by Aristotle 3 . His weight was a part of his cosmology. Weight manifested the tendency of objects to restore the violated order in which fundamental elements (earth, water, air and fire) were spatially organized, along the line from the centre of the Universe outwards, to the heavens. He stated that the permanent seeking of the state of rest at the appropriate location constituted the teleological cause of natural motion of any object, while its weight designated the efficient cause of such motion. 1 http://www.tldm.org/bible/old%20testament/proverbs.htm 2 Plato (1952). The Dialogues of Plato. Timaeus. Chicago: Encyclopedia Britannica, 63, p. 463. 3 Aristotle (1952). On the Heavens Chicago: Encyclopaedia Britannica. Book II,Ch. 13,295a,b,296a. Aristotle 4 ascribed absolute weight to the earth (an element) and absolute levity to fire, while the weight of other elements was relative. A compound object possessed weight in accordance with the ratio of its heavy components to the light ones.
In the natural motion of objects weight served the cause of motion: the more weight -the greater motion, whereas in violent, unnatural motion, weight resisted the mover: the greater weight is, the less quickly the object moves.
Here v -the speed of motion, F-the intensity of the mover, W -weight of the body, R -resistance of the medium.
Two manifestations of weight were recognized: weight causes the falling of nonsupported objects, and weight causes the downward pressure exerted by the object on its support, when available. The heaven bodies were not supported and did not fall; therefore, they were inferred by Aristotle to be weightless.
An alternative approach to weight appeared soon after Aristotle, in the Hellenistic science. Archimedes, saw weight as the quality opposing to the buoyant force that pushed objects immersed in water 5 resulting either its floating or sinking. Euclid took the pressure of a body on the support as measured by balance, to be its weight. This was the first operational definition of weight 6 : Weight is a measure of the heaviness and lightness of one thing, compared to another by means of a balance.
In fact, balance scale, served as an instrument to measure weight, weighing, much before any theoretical idea regarding weight was established; that is, from the very early civilizations.
Medieval science preserved the Aristotelian interpretation of weight as an inclination of the body (not as a force). Thomas Aquinas, a devoted follower of Aristotle, elaborated this distinction 7 : A thing moved by another is forced if moved against its own inclination; but if it is moved by another giving to it its own inclination, it is not forced. For example, when a heavy body is made to move downwards by that which produced it, it is not forced. In like manner God, while moving the will, does not force it, because He gives the will its own inclination.
When the medieval scholars discovered that objects accelerate while falling, the original weight had to be modified. They split it into two components, the natural Nicole Oresmea distinguished scholar of the 14 th century natural fall. It was imagined that similar attraction exists in the areas of each planet, instead of the tendency to seek the centre of the universe in Aristotle's world.
Galileo, in the 17 th century, followed the same path. He started from the medieval conception. In 1608 he suggested a way to measure the difference between 'dead weight' the weight at rest (pondus), and the weight in motion (gravitas) 11 . Galileo preserved the idea of weight as a quality causing heaviness to a body and used it somewhat similar to Archimedes 12 . Later, however, Galileo regarded weight as proportional to the amount of matter in the object (akin Newton's mass). His statements as 13 : . . . as has been often remarked, the medium diminishes the weight of any substance immersed in it . . . testify for the cumbersome concept since the amount of matter was apparently the same after the body was immersed into water. In addition, his weight concept had clear operational connotation -it is indeed easier to support the body immersed in water.
At that time, it was common to use the terms 'pondus-gravity-weight' as very close synonyms. As such they were used by Galileo, all conveying the same idea of burden, or heaviness measured by weighing 14 .
To complete the picture, we  Descartes, R. (1647Descartes, R. ( /1983. Principles of Philosophy. D. Reidel,Dordrecht, objects, being in a constant very fast whirl experience centrifugal tendency. Their radial move outwards, however, created the effective centripetal (inward) push on the bodies, making them heavy and compelling their falling to the ground. Descartes illustrated that by a thought experiment: A big bowl of gun balls had few pieces of light cork among the balls. During the rotation of the bowl, the pieces of cork moved to the center of rotation because metal balls move outwards.
Needless to say that the situation of the experiment was not even approach the reality of the bodies next to the ground, since real bodies are surrounded by air, much less dense material, but Descartes sought for the mechanism of centripetal push on the first place and kept with it regardless any other factors 16 .

Questions to reflect
1. Why we can talk about heaviness of the bodies as their physical characteristics and we cannot do the same for levity? 2. Aristotle did not considered weight to be a force but a tendency of the body. What was the difference?
3. The scholars of medieval science were not satisfied with one a single concept of weight (gravity) and distinguished between still-weight (or pondus) always remained unchanged, and actual gravity or accidental weight (gravitas). What was the rationale of this conception? 4. What was the idea of Descartes to explain weight? Was it reasonable to believe to such an idea? Explain

II. Weight in the classical mechanics of Newton
After Galileo, the search for the cause of gravity left the terrestrial realm. The context became astrophysical which in a sense (the number of factors that are considered) presented a simpler physical situation, at least for an initial explanation.
The logical trend of Newton is important to mention.
First, Newton introduced force-paradigm of the universe's organization, establishing the core of his theory -the laws of motion. Then, in the search for the cause for planets revolving around the sun, the system Sun-planets, Newton elicited 16 Aiton, E. J. (1959). The Cartesian Theories of Gravity, Annals of Science,15(1), the centripetal force to be in the inverse proportion to the distance between the objects, such as the Moon revolving the Earth 17 : 2 12 1 r F cp ∝ F cp stands for the centripetal force and r 12 -for the distance between two material points.
This was the attraction central force between the heavenly objects, the first step towards the Law of Universal Gravitation. Then, to relate the established force with gravity he performed a thought experiment 18 : If now the moon is imagined to be deprived of all its motion and to be let fall so that it will descend to the earth with all that force urging it by which (by Cor. Prop. III) it is [normally] kept in its orb… that force by which the moon is kept in its orbit in descending from the moon's orbit to the surface of the earth comes out equal to the force of gravity here on earth, and so (by rules I and II) is that very force which we generally call gravity.
How could Newton infer regarding the force acting on the Moon or other celestial body? In effect, he drew on the second law of motion (the axiom of his theory) implying that the net force on the body is proportional to the "change of motion": In our terms, the change of motion become the change of momentum mV. Furthermore, considering the very short time interval, Newton arrived to the inference that the net force acting on a body is proportional to the observed acceleration of its motion: a F net ∝ . Acceleration was already the quantity that he could calculate from the observed motion of the Moon.
In his thought experiment with the Moon, Newton applied his already established result regarding the centripetal force on the planets: And this centripetal force would cause this little moon if it were deprived of all the motion with which it proceeds in its orbit, to descend to the earth … and to do so with the same velocity which heavy bodies fall on the top of those mountains, because the forces with which they descend are equal.

Newton inferred:
And therefore (by Rule I and II) the force by which the moon is retained in its orbit is that very same force which we commonly call gravity; Newton reasoned by his Rules of Reasoning I and II: 20 Rule I: We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.
Rule II: Therefore to the same natural effects we must, as far as possible, assign the same causes He explained also using the rule of contraries (contradiction under opposite assumption): 21 For if gravity were different from this force, then bodies making for the earth by both forces acting together would descend twice as fast.
The final claim appears in the Scholium 22 : The force which retains the celestial bodies in their orbits has been hitherto called centripetal force; but it being now made plain that it can be no other than a gravitating force… cp grav F F ≡ It was the great moment of the great discovery indeed. What hitherto was an obscure concept of gravity, weight, etc. became from now on the force of gravity, the gravitating force, or as we call it now -the gravitational force. Weight previously equated to gravity was now married to the gravitational force.
Newton proceeded and accomplished the campaign by stating 23 : …That all bodies gravitate towards every planet; and that the weights of bodies towards any the same planet, at equal distances from the centre of the planet, are proportional to the quantities of matter which they severally contain.
This fact -the proportionality of gravity to the quantity of matter -was demonstrated by Newton by his experiments with pendulums of equal geometry and shape but different in material. He wrote: 24 I tried the thing in gold, silver, lead, glass, sand, common salt, wood, water, and wheat. I provided two wooden boxes, round and equal: I filled the one with wood, and suspended an equal weight of gold (as exactly as I could) in the centre of oscillation of the other. The boxes hanging by equal threads of 11 feet made a couple of pendulums perfectly equal in weight and figure, and equally receiving the resistance of the air. And, placing the one by the other, I observed them to play together forward and backward, for a long time, with equal vibrations.
Thus, by showing that gravitating force does not depend on the kind of the matter (all pendulums oscillated exactly the same way), Newton arrived to the gravitational force to be proportional to what he called: the quantity of matter, and we prefer today -inertial mass: m F g ∝ At the same time, here Newton made another giant step. Until him, mass and gravity were confused in one concept. Thus, the medieval concept of impetus was defined as a product of weight and speed, whereas momentum (the quantity of motion) for Newton was a product of mass and velocity. The revolutionary step was the split between the gravity and mass. Thereafter, the gravitational force was proportional to mass, not gravity.
And finally, for the symmetry of force interaction (Law III), one should add another mass into the dependence of the attraction force: g Newton could not accomplish the law beyond the proportionality for he could not make a laboratory measurement of the gravitational force between two known masses.
Cavendish, in the same university of Cambridge, performed this measurement a hundred years later in the experiment he called "weighing the Earth".
The identity between cosmic attraction and the weight of objects on the Earth seemed natural to Gilbert, Descartes, Huygens, and of course, Newton. Only after more than two centuries, this same identity of the cause (the gravitational force) and its effect (the weight of the object) was recognized as peculiar and a subject of further inquiry.
Following Newton's discovery regarding the nature of weight, as interactive force of gravitation, weight ceased to be a characteristic of objects, while mass (the quantity of matter) remained such. The often forgotten feature of the Newtonian weight was however, that it always came as a pair of forces of interaction. Newton wrote: 25 …the weights of the planets towards the sun must be as their quantities of matter… (emphasis in the original).
This meant that the weight of the Earth towards the Sun was equal to the weight of the Sun towards the Earth, and the weight of the Earth towards the Moon was equal to the weight of the Moon towards the Earth, and the weight of the Earth towards the Sun is different from the weight of the Earth towards the Moon. The Newtonian weight was not a characteristic of a body but of the pair of bodies. Such weight could not survive in the everyday life where the only practical meaning was the weight of things towards the Earth.
Newton did not forget to define weight operationally, by weighing 26 : Thus the weight is greater in a greater body, less in a less body; and, in the same body, it is greater near to the earth, and less at remoter distances. This sort of quantity is the centripetency, or propension of the whole body towards the centre, or, as I may say, its weight; and it is always known by the quantity of an equal and contrary force just sufficient to hinder, the descent of the body. (emphasis added) Here, however, problems began. 25 ibid. Book III, Proposition 6, pp. 806-810. 26 ibid. Definition VIII, pp. 407-408.
On the one hand, weight was defined by Newton as the gravitational force. On the other -weight is measured by weighing. It was known that the same body weighs differently in the locations of different latitude on the surface of the Earth. How could one explain that the same body weighs differently, is it attracted differently to the Earth? The correctness of the equation: 'weighing results = gravitational force' was questioned.
Despite of this discrepancy, which was resolved in trade and commerce by careful indication of the place where the weights were calibrated and the necessary corrections made, Newtonian equating of weight and gravitation preserved, waiting for a better account. It is this problem of weighing results, which pointed to the fact that Newton's laws are valid only in certain frames of reference -inertial frames. The rotating Earth was not such a frame. One needs to imagine himself outside the Earth, at rest relative the Sun, in order to apply Newton's laws. In any case, weighing does not reliably indicate gravitational force.
The comprehensive understanding of the situation was reached in the twentieth century within the new approach: different accounts of the world by different types of observers. Newton's concern was rather different. He did not care about any other Hitherto we have explained the phenomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of this power. This is certain, that it must proceed from a cause that penetrates to the very centres of the sun and planets, without suffering the least diminution of its force; that operates not according to the quantity of the surfaces of the particles upon which it acts (as mechanical causes use to do), but according to the quantity, of the solid matter which they contain, and propagates its virtue on all sides to immense distances, decreasing always in the duplicate proportion of the distances. … But hitherto I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses; for whatever is not deduced from the phenomena is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phenomena, and afterwards rendered general by induction. Thus it was that the impenetrability, the mobility, and 27 Newton, I. (1687/1999 the impulsive force of bodies, and the laws of motion and of gravitation, were discovered. And to us it is enough that gravity does really exist, and act according to the laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies, and of our sea.
Newton left the problem of understanding gravitation for further exploration, which since then never stopped.

Euler
Euler 28 lived in the 18 th century just after the scientific revolution, at the time that the Newtonian mechanics became the fundamental theory. In his treatise on mechanics 29 Euler presented the ideas of Newton about weight as already adopted claim in physical science, although the scientific community still debated on its validity versus Cartesian idea of vortexes which cause weight by the pressure of surrounding medium. Euler wrote 30 :

Definition 16
179. Gravity is the force, by which all bodies near the surface of the earth are forced downwards; and the force, by which anybody is acted on by gravity, is called the weight of this body.

Corollary 1
180. Gravity is the external cause, which forces terrestrial bodies downwards; and therefore it cannot be a property assigned to certain bodies themselves.

Thus a body sent off near the surface of the earth, even if it should be at rest, is urged on in a downwards motion and meanwhile it sinks until it comes upon obstacles preventing the fall.
Corollary 3

Moreover as long as the fall is impeded, either the body being held immobile pressing on an object or it suspended, the weight of this body exerts itself by pressing down. (emphases added)
In accordance with the weight as defined by Newton, Euler mentioned that gravity "cannot be a property assigned to certain bodies themselves" (Cor. 1), meaning that 28 Leonhard Paul Euler (1707 -1783) was outstanding Swiss mathematician and physicist who worked in Russia and Germany. Euler made important discoveries in mathematical analysis (he introduced the notion of function). He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy. 29 Euler, L. (1765). Theory of the motion of solid or rigid bodies.
Leonard Euler weight characterizes a pair of objects, not one. Almost like Newton, Euler ascribes to weight to cause the downward pressure (Cor. 2).
Here Euler only defined the force of gravity and the weight similar to Newton, that is, stating the fact, without any speculation regarding the cause. This came later, in the Scholium. There, Euler openly expressed his worry of the unclear origin, the cause, of the force of gravity. Unlike Newton, however, Euler speculates and displays his confusion. Eventually, he returns to the Cartezian idea explaining gravitation by "the action of some more subtle matter that escapes the notice of our senses" 31 : 184. Those people also, who put the cause of this as a drawing together, recognize these things, that gravity is the external force, which acts extrinsically on bodies and forces them downwards. For bodies are not urged towards the earth by a certain special instinct, but they are set up to be attracted to the earth by a force drawing them together. Clearly the matter can be understood thus, as if the earth were sending out some kind of embracing forces acting on bodies, which forces send the bodies towards the earth; now nor do they consider this to happen with the help of an intervening medium, but they wish the forces to be acting in place equally, even if all the matter between the body and the earth has been taken away. Therefore the force of gravity is not a material force acting on the body, truly thus connected with the earth, in order that with this removed, the force likewise would vanish; and likewise it is therefore as if a certain spirit should move rapidly to force bodies downwards; for how otherwise the force itself is able to propagate through great distances without the support of any kind of intermediate material, cannot in any manner be considered to be understood. …What is perhaps more likely to be true is that the force of gravity arises from the action of some more subtle matter that escapes the notice of our senses; …when the admirers of attraction say that the attractive force has been put in place by a God of the earth, they say nothing else, except that bodies are to be impelled immediately by this god himself. (emphases added) Euler went further in establishing the working framework of physics. He demonstrated possibility to use weight for measuring forces and masses 32 :

191.
We may express the forces acting consistently through the equal weights from these.

192.
This expression of the forces by the weights gives no difficulty; for since the weight of each body is a force, by which that is acted on downwards, the forces acting and the weights are quantities homogeneous between each other; and whatever body may be acted on by some force, a body can always be taken to be acted on by an equal force acting downwards placed on the surface of the earth, so that just the weight of this body will show the measure of that force. And when the question concerns so great a force, that nobody near the surface of the earth is able to be present, that has an equal weight, it is sufficient to know how many times greater that force shall be than the weight of the little amounts of bodies present on the surface of the earth; if hence indeed, the magnitude of this force is surely able to be defined.
Keeping with this tradition, we in modern school laboratory calibrate force-meter by suspending different weights on a spring to calibrate it and use the calibrated spring to measure other forces.

Huygens
The influence of Newton on the framework of physical thought was enormous.
However, there was another brilliant mind -Christian Huygens 33 , a prominent Dutch physicist, who worked almost in parallel with Newton and produced alternative ideas regarding the fundamental issues in both mechanics and optics. 34 In this excurse, we touch on his fundamental idea with regard to the nature of the centrifugal force -its similarity with the force of gravity.
Huygens' view was highly intelligent and original. In 1659, before the great invention of Newtonian picture of gravitation, Huygens introduced the concept of centrifugal force 35 and determined its variables. Today we express this force by means of modern symbolism, as a formula: Huygens never wrote this formula. He could only describe the dependence of the force he termed centrifugal on different parameters: weight (not mass, as we today), speed and radius of rotation. Similar to Newton in regarding to the gravitational force, Huygens used to describe the magnitude of the centrifugal force through comparison between two cases of bodies in a similar motion.
Furthermore, the concept of force itself was ambiguous and not well defined, waiting for Newton's touch. And despite of all these, Huygens was the first who tried 33 Christian Huygens (1629 -1695) was a renowned Dutch physicist who invented the first pendulum clock, which greatly increased the accuracy of time measurement. He was a pioneer researcher in mechanics, astronomy and probability. 34 In optics, Huygens confronted the idea of particle (light rays) paradigm with the wave theory of light (elastic distortions in the ether medium Observer A, considered later by Newton, mentions the tension of the rope T and the gravitational force mg acting on the mass m. He fully describes the rotation by means of the second Newton's law. A presents an inertial observer. Observer B, considered by Huygens, does not observe the ball (mass m) in rotation but being at rest, in the state of equilibrium. Tension T and the gravitational force mg cannot provide equilibrium (they are not parallel). B needs additional force -ma to nullify the net force on the ball. This additional force (-ma), not existed for Newton, but required by Huygens, is termed today inertial force. Observer B presents a non-inertial observer (the one who needs inertial forces to account of the situation), and the rotating wheel presents a non-inertial frame of reference.
We, then, are evident to the important fact that the same situation of motion can   This implies that observer B might be misled regarding the gravitational force, given that he identifies the gravitational force with weighing result -the tension in the thread (heaviness of the suspended body) at the state of rest (mg* in Fig. 1).
At the same time, observer A clearly discerns the gravitational weight (mg), but he must depart from the identity between weight (as the tension in the thread) and gravitation. He may say that the centrifugal force increases the body weight. In effect, this could be the point to split between the concepts of "gravitation force" and "weight force", but Huygens, was still not there.
Lacking this knowledge, but being familiar with the works of Galileo and Descartes, Huygens utilized their conceptions.
Unlike Newton, he defined gravity using Descartes' notion of tendency, or conatus ( Fig. 2)   by the space FD. But these distances EC, FD, etc. increase as the series of the squares from unity, 1, 4, 9, 16, etc. Now they agree with this series ever more, exactly as the particles BE, EF are taken to be smaller, and hence at the very outset they may be considered as if they differed nothing.
Thus this tendency will clearly be similar to that which is felt when the ball is held suspended on a string, since then too it tends to recede along the line of the string with a similarly accelerated motion, i.e. such that in a first certain period of time it will traverse 1 interval, in two parts of time 4 intervals, in three 9, etc.
Huygens' work on the centrifugal force is one of the most remarkable in mechanics of the 17 th century. It was highly appreciated by his contemporaries. Newton, who rarely praised his colleagues, wrote about this study 39 : And by such propositions, Mr. Huygens, in his excellent book De Horologio Oscillatorio, has compared the force of gravity with the centrifugal forces of revolving bodies.
Huygens preceded Newton's treatment of mechanics, but his approach to force description (centrifugal force) corresponded non-inertial observer nobody considered at that time. Newton and all other scholars saw the world as in the theatre: on the stage in front of them. No other observers ever rose in mind as something important and perhaps different. Single static universe was subject to be described in absolute space and time by means of natural philosophy. Therefore, although praised, Huygens' work was not be truly evaluated by anybody, including Huygens himself.
The time was not ripe for that in the sense of conceptual worldview.

Weight is the gravitational force
In a way, Huygens' vision of the centrifugal force as a force on the whiling body remained on the margins of theoretical mechanics of those days. He tried to explain gravitational force by means of centrifugal force: 40 The mechanism envisaged by Huygens involved a fluid vortex rotating at such a high speed that all bodies on the earth are pushed toward its center because their centrifugal force is smaller than that of an equivalent volume of the vortex. Thus gravity results from a difference of centrifugal forces, and in this sense centrifugal force does produce motion, i.e., every time a heavy body falls. the way it fit his paradigm of force interaction. For him, the centrifugal force was the pair companion of the centripetal force, and acted on the constraint (the sling).
Discussing the motion of a body rotating inside a hollow cylinder or circle he wrote: 42 This is the centrifugal force with which the body urges the circle; and the opposite force, with which the circle continually repels the body toward the center, is equal to this centrifugal force.
In practice, however, it was the Huygens' meaning of the centrifugal force that was adopted. Indeed, one may regard our Earth as a giant revolving wheel.  questions did not change anything regarding the conception of weight. As long as the old framework of thought of the true observer preserved, all deviations in weighing, including falling, could be explained within the Newtonian frame of thought and adjusted to the practical needs by means of using the notions of true (the gravitational force) and apparent (the result of weighing) weights. It all changed when physics entered into the period of conceptual reconstruction in the new scientific revolution.

Questions to reflect
1. How did Newton show that gravitational force is proportional to the quantity of matter (inertial mass)?
2. In what way did Newton change the medieval concept of weight?
3. Characterize Newton's concept of weight. What was in Newton's weight concept that was abandoned in the following use of his definition of weight? 4. Why Newton was not satisfied with his understanding of gravitation? 5. What was, in Euler's view, the condition of weight to manifest itself? 6. What mechanism for gravity did Euler imagine to himself? 7. Huygens demonstrated that gravity force is similar to the centrifugal force. What strategy he used for this purpose? 8. What was the interpretation of the centrifugal force by Newton? 9. Huygens was the first to ask about the description of situation in view of rotating observer. Why do you think this approach was abandoned in physics until the 20 th century? * * *

Einstein: the principle of equivalence
The great change took place at the beginning of the 20 th century.
Physicists revealed the special role of observer in physics. Albert Einstein 45 was the first who in 1905 put inertial observers in the center of physical account of the world with his demand for physics laws to be indistinguishable for any inertial observer. The idea of the special theory of relativity was very nice, but the special 45 Albert Einstein (1879 -1955), the outstanding physicist and physics philosopher who shaped the modern physics in the 20 th century. There is another way to express the meaning of Einstein's result. As we learned from Newton, each two bodies gravitate to each other with forces proportional to their inertial masses. In principle one could expect that they could attract each other in proportion to their gravitational masses. Therefore, one may see at this Newton's result the demonstration of the fact that inertial and gravitational masses are equal. It means that regardless of the mass of bodies in the field of gravity, they fall with the same acceleration (which is commonly labeled as g). This is the explanation of the empirical law of falling which was established by Galileo in the beginning of the 17 th century but only as an empirical law. Galileo left it without theoretical account.

Reichenbach wrote 50 :
Although the equality of the inert mass and the heavy [gravitational] mass was long known, nevertheless Einstein was the first man to recognize the basic significance of this fact. He realized that here lies the reason why the distinction between accelerated motion and gravitation cannot be made and why the physicist in the box cannot, therefore, determine whether he is moving upward in an accelerated motion or gravitational field interferes from. Hence, Einstein calls both conceptions equivalent, and maintains that it is meaningless to look for a truth-distinction between them.

New definition of weight
As we see, the rediscovered identity of inertial and gravitational masses implies uncertainty in the interpretation of weighing in the sense that that weighing results cannot testify for the action of the gravitational force. In this situation, there is no other way but to change the weight concept definition.
In 1928, rather soon after the introduction of the principle of equivalence,

Reichenbach wrote 51 :
What is the basis of this indistinguishability? According to Einstein, its empirical basis is the equality of gravitational and inertial mass. This new distinction must be added to the usual distinction between mass and weight. There are therefore three concepts: inertial mass, gravitational mass and weight.
Newton's distinction between mass and gravitational force became insufficient. Now there was a need for further refinement -to distinguish between gravitational force and weight force. After the alliance of more than two hundreds of years, the gravitational force was conceptually divorced from weight (Fig. 5). Weight of the body was defined as following (theoretical definition): Weight is the force that body exerts on its support at the state of rest as claimed by certain observer (in the correspondent system of reference). In accordance with this definition, weight is the force that is measured by the calibrated spring, exactly as the observer in Einsten's thought experiment did. One may also provide the operational definition of weight:

Weight is the result of standard weighing
This definition actually repeats the described above inability of the internal observer to know whether the weight is due to gravitation of accelerated motion. For him the stationary state is interpreted as following: W = F elastic = mg* The external observer may interpret the weighing differently: Two observers interpret the same reality differently, unless a = 0. The situation is especially interesting in the special case, when the laboratory is freely falling: a = g. This is the state of weightlessness inside the laboratory. It may be explained by the external observer A (Fig. 4) as cancellation of inertial and gravitational forces. The internal observer B remains ignorant of the origin of the weight or its lacking, reflecting the incapability to distinguish between inertia and gravitation.
This approach allows physicists to generalize the concept of gravity in its old sense, in the meaning of objects being heavy whether due to the gravitational force (mg) or due to the inertial force (-ma). The origin of weight may be unknown as long as the definition of weight relate it to the results of standard weighing (or the force acting on the support at the state rest for the certain observer). Yet, the important comment may clarify more the modern weight definition.
Weight force is spread along the whole body and grows in the direction up-to-down.
This can be understood as pressing force of each layer of the body on the subsequent one below, supporting it. Thus, tension gradient within the body is what accompanies weight. Therefore, no magnetic boots, which may stick the body to some surface, can replace weight in the state of weightlessness -the whole body remains weightless.

Weight in rotating systems
Weight in continuously revolving or spinning systems is an especially important case.
Such are a revolving container, connected by a beam to the axis of rotation, and the Earth itself, of course. Huygens was the first who considered the phenomena in the rotating systems. In his study, he addressed the observer inside such a system -a revolving wheel.
Detailed analysis led Huygens to the conclusion that the observer on the wheel will account for the reality of the rotating objects without essential distinguishing between the gravity and centrifugal force in radial outward direction. Today we know what Huygens did not: that both forces are proportional to body's mass.
Consequently, both forces may contribute to the weight (gravity) of the objects being in a circular motion around certain axis.
The fact that the centrifugal force can be regulated by the rate of rotation allows regulation weight magnitude of objects in a rotating system. This is especially important in cases where one intend to reproduce regular weight in the case the gravitational force cannot provide it. This is the case in a simple satellite, the space station which is in the state of weightlessness. Humans cannot survive this state for a long time (more than a year). Irreversible biological changes eventually cause serious damage to the functioning of organism at the level of biochemical processes resulting in the deterioration of health and ultimately the death of the organism. As living organisms, we essentially need weight. Rotation can help us in the state of a free gravitational movement (free falling) to avoid weightlessness and create weight. This idea is crucial for future space projects.

Questions to reflect
1. What was the idea of splitting the weight concept used after Newton as suggested by Reichenbah? What was the rationale of this split? 2. What is essential for functioning of the human organism gravitation or weight?
Why do you think so?

Summary
We may summarize our excurse to the history of weight through thousands of years.
From the beginning, the idea of weight reflected the perception of heaviness and quantity of matter. Until Newton, weight was explained as an inherent feature of the body. Aristotle related it also to the inherent intention of a body to move and take its natural place, in accordance with the kind of matter comprising this body.
Newton identified the gravitational attraction between any material objects as the cause for their weights. Newton suggested that the supporting hand senses and evaluates weight force of the object. Euler, however, stressed that the pressure upon the support only informs about weight, the weight itself is not the pressure, but is the force of gravity causing the pressure.
Weight concept of Newton became a characteristic of particular pair of objects.
The amount of matter was characterized by a different quantity -inertial mass.
Newton discovered that inertial mass determines the gravitational attraction as well.
For this very fact, in a regular everyday practice, one may often ignore the difference between mass and weight.
During the 17-19 centuries, weight concept was used mainly in the terrestrial environment where the only important gravitational interaction is between the Earth and each object on its surface, not between the objects. In this situation, weight lost again its meaning as a characteristic for a pair attraction. Weight of a body was identified with the force of the gravitational attraction to the Earth exerted on the body. As such, weight returned to be a characteristic of any particular object, rather similar to its pre-Newtonian use.
After the introduction of modern physics, the physics descriptions of reality expanded to any observer (inertial and non-inertial) and weight lost its univocal correspondence to the gravitational force. Inertial forces are legitimate contributors to weighing results. Therefore, weight was defined as equal to weighing results or, equally, as the force the object exerts on its support being at rest for certain observer (reference frame).
The important point to discover is that the definition of weight, as defined by us operationally and theoretically appears to be observer independent: the weighing results remain the same regardless the forces that the particular observer introduces to explain it. The answer to the question "What is the cause of weight?" is, however, observer dependent: the deformation of the spring may indicate centrifugal force for the rotating non-inertial observer, or centripetal elastic force -for the outside inertial observer.
We may summarize the history of weight in the following conceptual diagram ( Fig. 7):

Conceptual aspects
Weight concept initially entered to the Hellenic science as an intrinsic characteristic of any object characterizing its heaviness. As regular in ancient Greek philosophy, it was accompanied with a counter-concept -levity, the lightness of objects. Levity was removed only by Galileo through the argument: two light objects together never were lighter but always heavier than separately. In the world-picture of Plato, weight represented attraction of alike whereas for Aristotle, weight manifested an inclination of the object to get its natural place, specific for each object. As it seemed to the scholars of that time concepts reflected the reality as it is, rather as a direct description the essence.
Newtonian weight was formulated through an abstract concept -force -that described interaction. Weight appeared in pairs of equal forces between each two objects. Newton introduced weight between celestial objects gravitating towards each other. 52 In the terrestrial context, due to the unobserved gravitational interaction between regular bodies, weight remained a characteristic of a single body, the force pulling it towards the ground.
Descartes ascribed a special value to revealing the mechanism of natural phenomena. He suggested the mechanism of weight of bodies as caused by a deficiency of centrifugal force produced by vortices of fine matter surrounding each body. This mechanism, which Descartes tried to illustrate in his thought experiment with a bowl of gun balls and few pieces of cork among the balls. Today a similar principle bases the function of centrifugal separator. The efforts of Huygens, Leibnitz, young Newton and later Euler could not prevent the total failure of this program. The demands to the medium whiling around objects were contradictive and the reproduction of the features of gravity failed.
Facing this failure, Newton decided that it is upon the Natural Philosophy to describe the gravitation and eschew any speculation of its origin not based on the 52 e.g. Newton, Op.cit., Book III, Proposition 6, page 806.

and afterwards rendered general by induction
In accordance to this view, the program of Newton was to provide the mathematical description of the reality, as accurate as possible in the quantitative sense. Indeed, the mathematical formalism produced by Newton, allowed his numerous successes: the account of celestial mechanics, tidal phenomena, flatness of the Earth globe, the account of projectiles and satellites motions and many others, all these at a high accuracy.
In the 20 th century, Schrödinger expressed a similar idea when he said that he always felt that his equation was smarter than he was. Although nice and surprisingly powerful, this idea, however, never prevented people from curiousity and inquiry about how and what takes place in reality. They continued to seek the unknown mechanism behind the laws and principles, often applying hypothesis and speculations (abduction). Left without explanation, the gravitational interaction-at-adistance expanded to electricity and magnetism in the 18 th and 19 th centuries.
Eventually, the ontological awkwardness of the interaction-at-a-distance caused the invention of the field theory -reviving the Aristotelian and Cartesian idea of plenum (medium filled space) by Faraday and Maxwell. Interaction-at-a-contact replaced interaction-at-a-distance in the field theory of electromagnetism. By analogy, it expanded to gravitation. This step prepared a completely different mechanism of gravitation which was suggested by Einstein in 1916, in his theory of general relativity. In the modern physics of the 20 th century, gravitation became a manifestation of the curved space-time, caused by matter. And the concept of weight, divorced from gravitation, it was identified with the pressing force that the object exerts on its support left at rest without any intrusion. 53 Newton, Op.cit., General Scholium, page 943.
One may represent the steps in the development of understanding of weight in the following

Absolute and relative concepts
Newton never considered how different observers perceive reality, how they would describe the world if want to apply his theory. This was not relevant: he conceived the Universe as an object, if looking from aside, or flying above. The expression of this perspective was that the whole picture was placed in absolute space and time -as a natural container. These concepts were different from all others in physics, they were taken as self-evident, beyond any need for definition, 54 in a sensemetaphysical.
Yet, Newton did distinguish between true and relative rest; as well as true and relative movement. However, his understanding of those notions was special. By true rest and motion, he understood those in view of his unique observer, the only one, who perceived the universe exactly as it was. And by relative, or apparent, Newton meant any state or quantity as perceived by all other regular observers, humans who unavoidably make errors in their perception and measurement: 55 54 Newton, Op.cit., Definitions,Scholium, ibid. Only I must observe that the common people conceive those quantities under no other notions but from the relation they bear to sensible objects. And from these arise certain prejudices, for the removing of which it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common.
Regarding absolute and relative time, he wrote 56 : But we may distinguish rest and motion, absolute and relative, one from the other by their properties, causes and effects. It is a property of rest, that bodies really at rest do rest in respect to one another. And therefore as it is possible, that in the remote regions of the fixed stars, or perhaps far beyond them, there may be some bodies absolutely at rest; but impossible to know, from the position of bodies to one another in our regions whether any of these do keep the same position to that remote body; it follows that absolute rest cannot be determined from the position of bodies in our regions… Absolute time, in astronomy is distinguished from relative, by equation or correction of the apparent time.
In case of rotation, Newton discriminated between relative and absolute movement by means of forces that appear only in the true movement. 57 But, even then, Newton never considered anything which would remind us different frames of reference.
Koyre summarized this philosophy of Newton 58 : In the Newtonian world and in Newtonian science, it is not man, but God, who is the measurer of things.
In somewhat similar manner, the notion of true-weight was reserved for the gravitational force. As to the apparent-weight -the term fitting to the Newton's philosophy -it represents the results of weight measurement in presence of impediment factors, which might deceive a practitioner, but not a philosopher.
Newton illustrated by one of such misleading factor: the buoyant force. For him, apparent-weight, similar to relative time, may cause misunderstanding. For the case of an object immersed in liquid he wrote: 59 But those things (immersed in water) which neither by preponderating descend, nor, by yielding to the preponderating fluid, ascend, although by their true weight they do increase the weight of the whole, yet comparatively, and as commonly understood, they do not gravitate in water (emphasis added by us).
The gravity of bodies in fluids is therefore two-fold: the one true and absolute; the other apparent, common, and relative. Absolute gravity is the whole force with which a body tends downwards; relative or common gravity is the excess of gravity with which the body tends downward more than the surrounding fluid. … Those things which are in air, and do not preponderate are commonly looked upon as not heavy. Those which do preponderate are commonly considered to be heavy, inasmuch as they are not sustained by the weight of the air. The common weight is nothing but the excess of the true weight above the weight of the air (emphasis added by us).

Nominal definition
The question of relationship between the theoretical and operational knowledge (and hence definition) of weight never rose in classical science. It was not before the end of the 19 th century that the new trend in the philosophy of science -positivismbrought into physics a special sensitivity to the empirical basis for any theoretical concept. The goal was to reduce the arbitrary (metaphysical, as they were called) statements, especially in the basis of science. Physicists had to worry and eschew the claims non-supported directly by empirical procedures that could determine the objective meaning of the concept. 60 This way, a dichotomy between the theory based and empirical based definitions arose. The definition which introduces a concept basing on the pertinent theoretical knowledge was called -nominal (nomos means law in Greek). In the case of weight, Newton was the pioneer of the nominal gravitational definition of weight, according to which: Weight of an object is the gravitational force exerted on that object.
As was mentioned already, the gravitational force, and so the gravitational weight, were introduced by Newton in regarding the gravitational attraction between a couple of material bodies, in pairs. Hitherto, many physics textbooks keep with a half of this definition and consider as weight of an object the force of gravitation towards the Earth acting it.
In the 20 th century, after the introduction of equivalence principle, the gravitational nominal definition was replaced by the nominal elastic definition: Weight is the force that body exerts on its support at the state of rest as viewed by certain observer in the correspondent system of reference.

Operational definition
Alternatively, positivists insisted on the operational definitions of all physical concepts. This demand was so important that it gave rise to a special philosophical trend known as operationalism. The latter demands definition of any physical concept by an explicit and unique measuring procedure by which the considered concept is defined. Thus, in the case of weight, the operational definition is: Weight of a body is defined as the result of its weighing.
Since the operation itself (such as weighing) is not defined here, numerous procedures could be suggested. This conceptual ambiguity should be avoided and therefore, the type of apparatus and the procedure of measurement all should be clearly mentioned. Hence, in the case of weight, one should use a more accurate definition:

Weight of a body is defined as the result of standard weighing.
Such a definition matches the requirements of Bridgman, 61 and is considered as the operational definition of weight, in contrast to the above introduced nominal one.
One may refine standard in the definition and write:

Weight of a body is determined by weighing it at the state of rest by means of a calibrated spring (spring scale).
Using spring scale (Fig. 8) presents an important constraint of the standard weighing. This is because balance, using horizontal lever, compares the forces (torques, in general), rather than evaluates their magnitudes. Balance, therefore, is more convenient to infer about mass of an object in comparison with some standard. Balance is not sensitive to weight changes due to geographical latitude.    Margenau, H. (1950). The Nature of physical Reality. New York,Ch. 12, This is the situation in our country -Israel.
However, the subject of weight is highly relevant for physics and science teaching at schools. Regular physics/science curricula for schools include the concept of weight through all levels of instruction. Therefore, there is a need to adjust teaching at all levels, making it consistent, from kindergarten to high school (K-12 idea of the 2061 project in the USA).
However, the situation teaching is as following: • The curricula of many elementary schools (6-12 years of age) usually include operational definition of weight: weight of a body is obtained by its weighing.
• The curricula of many middle schools (12-15 years of age), however, usually include nominal gravitational definition of weight: weight is the gravitational force exerted on the body. The inconsistency with the previous level is especially apparent in the account for the state of weightlessness. Several levels may be distinguished in concept teaching. The first level is "the level of things". At this level, weight is associated with some known things as their inherent feature. Weight constitutes an intuitive scheme by which the learner ascribes heaviness to the familiar objects (things are "light" and "heavy") 65 . Weight within this scheme may draw on individual perception and be directly to related to weighing.
At the second level the concept of weight is related to other physical concepts. The teacher may define weight as the force exerted by the object on its support (or suspending cord). This definition explains weight through the pressure on the support and tension in the thread. The knowledge becomes theoretical and so the definitionthe nominal definition. The teacher completes by adding the operational definition: weight is the result of weighing. Thus the students have a couple: nominal and operational definitions of weight.
The gravitational force (the invention of Newton) is taught as the factor which causes falling of objects and their weight. The instruction proceeds to weight changes with geographical location (latitude) explained qualitatively, by Earth rotation.
Weight, although caused by the gravitational force, appears different from it.
At the third level, one refines the concept of weight pointing to the fact that weight could be due to either gravitational or inertial forces. This step involves introduction of inertial forces (non-inertial observer in a rotating system). The important point to emphasize is that the definition of weight (weighing results) is observer independent. Yet, the account for weight as caused by other forces is observer dependent.
The didactical benefit of this approach is natural introduction of non-inertial observers. While considering weight changes in different situations (accelerating vehicles, satellites), students usually identify themselves with the observer inside the system under acceleration. Thus, introduction of non-inertial observers make this view legitimate and matching the intuition, removing tension and misconceptions.
A special didactical benefit is reserved to the case of weight in the rotational space station. Caused by centrifugal force, this weight strengthens conceptual knowledge of students distinguishing between weight and gravitational forces as independent. 66 The suggested way to teach weight draws on the scientific diachronic debate and essential appeal to the philosophy of science. The latter is involved with regard to the 65 e.g. Piaget, J. (1972) 5. Try to calculate the radius of the station, rate of spinning, tangential and angular velocity at the rotational station taking into account that gravity intensity (g) created by the rotation will not change more than 1% along the height of a person (h=2m) 70 .
Where the dimensions of the spaceship going to Jupiter shown in the Kubrick's movie "2001: Space Odyssey" realistic?
6. In the course of the historical excurse and while presenting the historicophilosophical background we provided questions for reflections which can be discussed with the students. Here we present several additional questions that might be used to probe understanding of the concept of weight by the learners. e. Consider a tunnel crossing the Earth globe along its diameter.
A body was dropped into the tunnel (this situation was discussed by the medieval scholars in the University of Paris in the 14 th century). As the body starts to fall down what happens to its weight? Explain your considerations.
f. In the following pictures you observe four cases: (i) jumping from the plane, (ii) floating under water, (iii) descending on a parachute, (iv) jumping from a hill.
Compare these cases in terms of weight changes of the body of the jumpers (assume their equal mass).
(i) (ii) (iii) (iv) • g. In the following pictures you observe a person in a free falling elevator cabin, next to the ground, and the astronaut left the satellite for a free "walk" in space.
Compare and characterize their situations in terms of their weight and the gravitational force acting on them. Are the situations different, the same, else?
h l. American Space Agency -NASAuses water pools for training astronauts.
Does floating in water put divers into the state of weightlessness? What could be the rationale of using water pools in astronauts' training?
m. Discuss the different and common with respect to weight in the four following pictures.

* * *
Another serious difficulty might follow from the fact that many teachers were instructed within the curriculum, which adopted the gravitational definition of weight, as still prevailing in many countries.
Thus, among the textbooks in English, one may distinguish between two groups.
The authors from the first group are adherent to the old tradition of the gravitational definition of weight 73 , whereas the authors in the second group follow the new trend presented in this excurse. The new trend which started at the 60 th of the previous century defines weight as the result of weighing by a calibrated spring scale. 74 The curriculum policy is determined by the educational institutions, 75 and the debate between physics educators continues. 76 Facing this difficulty one may suggest to the teachers to learn the historical arguments for the change of the weight definition and the requirements of the contemporary philosophy of science as common in physics practice.
The operational definition of weight might help in removing another obstacle in teaching -neglecting multiple observers. It may make apparent that keeping with the restriction of school teaching solely to the inertial observers may clash with students' approach who usually prefer, albeit intuitively, the point of view of the accelerated observers -the active participants of the situation and not its passive observer.
Naturally, any change of curriculum should start from learning the alternatives.
For example, the explanation of the state weightlessness under the restriction to the inertial observers is known as very difficult to students and teachers 77 . Indeed to ascribe weight to the floating in space astronaut, being at the state weightlessness, is not a simple task. Legitimizing non-inertial observers in school curriculum may simplify the transition to the new understanding matching the modern physics.
The cultural approach to teaching physics would imply discursive teaching, the one that presents a wider perspective of more than one explanation: the one by an on the ground (on the Earth surface) observer, as well as by one inside the satellite. The operationally based definition of weight as a contact elastic force suites this teaching approach.

Pedagogical skills
Besides the numerous skills required from the teacher in regular teaching of science 78 , one may mention here that using the history and philosophy of science in physics lessons needs a special skill to teach culturally rich materials. The characteristic feature of cultural material is its dialogical character which presents the knowledge in its conceptual variation. 79 The position of Newton was shaped in a dialogue with Aristotelian ideas as well as with those of Descartes, Hooke and Huygens. Einstein struggled in the very aggressive debates with many scholars until his theory was adopted. Accordingly, the teacher may organize a dialogue in class to promote students understanding and success in learning. The validity of this strategy draws on the dialectical nature of the scientific truth, the complementary contribution of several aspects to the contemporary scientific knowledge.
Within such teaching it is a special skill to monitor incorporation of metaphysical components (knowledge about science: historical, philosophical, social) into the actual teaching of scientific disciplinary contents. A "free" discussion in the class can indeed lead the students astray of the topic to be learned. It is upon the teacher to facilitate by mediation the discussion that will lead the students to the construction of the valid knowledge of weight and gravitation from a culturally rich context. The materials of this excurse were directed to suit this strategy.