Aug 292013

Galileo was once thought to have discovered kinetic laws important for classical mechanics by himself by means of observations and experiments in contrast to the Scholastics who confined themselves to the interpretation of Aristotle. To be sure he gave such an impression to readers, but the fact is that the Scholastics in 14th century such as Oxford Calculators and Oresme who discovered and developed the Mean Speed Theorem prepared for Galileo’s discovery of kinetic laws. They did not perform experiments except thought experiments, but since the 17th century scientific revolution is the shift from Aristotle-Thomas paradigm to Plato-Archimedes paradigm, we can consider the Oxford Calculators and Oresme to be the pioneer of the paradigm shift, because this paradigm insisted on mathematical models preceding experience.

1 : Was the Scientific Revolution a mutant from the Middle Ages?

Galileo Galilei (1564 – 1642) is recognized as one of the notable figures of the 17th century scientific revolution (usually just called the Scientific Revolution) along with Nicolaus Copernicus (1473 – 1543), Johannes Kepler (1571 – 1630) and Isaac Newton (1642 – 1727). The scientific revolution has been regarded as a mutation from the medieval stagnation of science since the Age of Enlightenment. It is generally believed that they could bring about the scientific revolution because they accepted the facts discovered by means of observations and experiments free from tradition and theoretical prejudice.

Ernst Mach (1838 – 1916), an Austrian physicist and philosopher of science, wrote in his book The Science of Mechanics: A Critical and Historical Account of Its Development in 1883 that Galileo discovered the time-squared law experimentally by rolling wooden balls down inclined planes. Mach was a positivist who asserted Galileo gave no such things as a theory of a falling body, but rather examined and stated the fact of a falling body without prejudice[1], in short, he just happened to discover the fact experimentally without any theoretical foresight.

The time-squared law (S∝T2) is a physical theorem that states the displacement of free fall (S) is proportional to the square of the elapsed time (T). Today we know that S = 0.5gT2 with g denoting gravity and that the law is right as a law of physics. Did Galileo discover it by himself? Mach thought so but Pierre Duhem (1861 – 1916), a French physicist and philosopher of science, pointed out “the Parisian precursors of Galileo” such as Nicole Oresme (ca. 1323 – 1382) in the third volume of his series, Studies on Leonardo da Vinci in 1913. According to Duhem, Oresme was not only the precursor of Galileo but also that of the 17th century scientific revolution.

Nicole Oresme not only preceded Copernicus in supporting the possibility of the diurnal movement of the earth against the Aristotelian physics; he not only preceded Descartes in making use of geometric representations by means of 2D or 3D rectangular coordinates, and in establishing the equation of the straight line; he further made a discovery that is commonly attributed to Galileo: he has recognized the law according to which the distance traveled by a mobile that continues a uniformly varied movement increases according to time. [2]

Copernican revolution and Cartesian coordinate system are taken up at the linked pages respectively. Here we take up “the law according to which the distance traveled by a mobile that continues a uniformly varied movement increases according to time” namely the Mean Speed Theorem.

The Mean Speed Theorem is also called the Merton rule because it was discovered and developed by the Merton School mathematicians such as Thomas Bradwardine (ca. 1290 – 1349), William Heytesbury (ca. 1313 – 1372/1373), John Dumbleton (ca.1310 – ca. 1349) and Richard Swineshead (fl. c. 1340 – 1354), who belonged to Merton College, Oxford in the early 14th century and were often referred to as Oxford Calculators. The Merton rule was first mentioned in Heytesbury’s magnum opus, Regulae solvendi sophismata (Rules for Solving Sophisms), written about 1335.

Oresme succeeded to the Merton tradition that classified the quality of motion into that of uniform motion (qualitas uniformis), uniformly difform motion (qualitas uniformiter difformis) and difformly difform motion (qualitas difformiter difformis). To use the current terms, these correspond to constant velocity, constant acceleration and variable acceleration respectively. The Merton rule applies to the second one, namely constant acceleration.

While the Oxford Calculators just arithmetically calculated the speed of motion, Oresme explained it geometrically. Oresme divided qualities in general into the extension (extensio) such as time and its intensity (intensio) such as the speed at a moment. He visualized the functional relation between them by diagrams (configuratio) that look like what we now call Cartesian coordinate systems, where the intensity of the quality was represented by a length (latitude) proportional to the intensity erected perpendicular to the base at a given point on the horizontal base line (longitudo). If we interpret the longitudo as time, latitudo as speed and area as distance, the interpretation is the same as that of the current physics. The figure below (Fig.OD) is a page of Oresme’s book listing a variety of diagrams.

Fig.OD. Oresme’s diagrams drawn in his book. Source:Tractatus de latitudinibus formarum

Oresme formulated the mean speed theorem in his treatise written about 1350: Every quality, if it is uniformly difform, is of the same quantity as would be the quality of the same or equal subject that is uniform according to the degree of the middle point of the same subject[3]. This formula is too abstract to understand. So, let me show it by the following Oresme-style diagram (Fig.MR).

Fig.MR. Geometrical explanation for the mean speed theorem.

Suppose the horizontal axis to be time and the vertical axis to be speed (positive velocity). The line OB represents the speed of constant acceleration motion. The distance at the moment A is equal to the area of a triangle OAB and also to that of the blue square OACD. That is to say, the distance a body in constant acceleration motion travels in the interval OA equals the distance a body travels at half speed MN in constant velocity motion. As MN is the speed at the moment M, the median point of OA, it can be “mean” speed in two senses, average speed and speed in the middle.

We can easily deduce the time-squared law from the fact that the speed is proportional to time. Oresme also demonstrated that the quality of uniformly difform motion follows the law of odd numbers[4] according to which the first n consecutive odd numbers amount to n2. Galileo rediscovered the time-squared law and the law of odd numbers. Did Oresme contribute to Galileo’s rediscovery?

2 : Did Oresme’s theory influence Galileo’s physics?

According to the popular legend Galileo dropped two balls of the different masses from the Leaning Tower of Pisa to show that they reached the ground at the same time. But there is no proof that Galileo actually conducted such an experiment against the Aristotelian idea that heavy objects fall faster than lighter ones. The Aristotelian idea had already been denied in 1st century BC[5] and its first demonstration by the experiment was given by Simon Stevin (1548 – 1620) in 1586[6]. Even if Galileo had carried out the legendary experiment, he had no priority over it.

Generally speaking, achievements tend to be attributed to a big-name researcher. A researcher’s work is less likely to be cited, the less famous he or she is. Readers usually read only famous works written by famous researchers and tend to assume everything new for readers to be the researcher’s original views. As a result famous researchers are apt to monopolize the credit of originality, if we neglect careful investigations. The current concern is whether a similar tendency can be found in the mean speed theorem or not.

Galileo stated a theorem similar to the mean speed theorem in his book Two New Sciences published in 1638.

The time in which any space is traversed by a uniformly accelerated body starting from rest is equal to the time in which that same space would be traversed by the same body moving at a uniform speed whose value is the mean of the highest and last speed of the prior uniformly accelerated motion.[7]

There is a small difference between this formula and the Merton rule. The Merton school and Oresme used the speed at the middle moment, while Galileo used half of the highest and last speed of the constant acceleration motion to apply the mean speed theorem. To explain Galileo’s rule by reference to the Fig.MR, the area of the triangle OAB is equal to that of the square OACD whose height is half of OA. Galileo’s rule is true of a special case when the initial velocity is zero and the acceleration is positive, while the Merton rule holds true even when the initial velocity is not zero and the acceleration is negative. Galileo’s rule, however, has its own merit of being easy to verify experimentally. Measuring the meantime speed of constant acceleration motion was difficult at that time, but the last speed was easily measured by converting the constant acceleration motion into constant velocity motion.

Galileo proposed the double distance rule in Dialogue Concerning the Two Chief World Systems: Ptolemaic and Copernican published in 1632. This rule is even easier to demonstrate experimentally. Galileo formulated the rule as follows: when a ball rolled down an inclined plane and continued to move with the same degree uniformly, i.e. without acceleration or retardation, it would pass a space along double the inclined plane in as much time as it rolled down the inclined plane.[8] To explain Galileo’s rule by reference to the Fig.MR, the area of the square OABE is double that of the triangle OAB. Galileo had no stopwatches but he could utilize a water clock. If the amount of water flowed while a ball is rolling down an inclined plane is the same as that while it travels double the distance of the inclined plane at a constant velocity, the double distance rule is demonstrated.

Did Galileo discover the mean speed theorem or the double distance rule independent of the Oxford Calculators or Oresme? This is a point in dispute among historians of science. Takahashi Kenichi, a Japanese historian of science, denies the influence, taking up a letter from Galileo to Paolo Sarpi dated 16 October 1604 as a counterevidence. The letter reads as follows:

Thinking again about the matters of motion, in which, to demonstrate the phenomena observed by me, I lacked a completely indubitable principle which I could pose as an axiom, I am reduced to a proposition which has much of the natural and the evident: and with this assumed, I then demonstrate the rest; i.e., that the spaces passed by natural motion are in double proportion to the times, and consequently the spaces passed in equal times are as the odd numbers from one, and the other things. And the principle is this: that the natural moveable goes increasing in velocity with that proportion with which it departs from the beginning of its motion.[9]

To translate into the current terms, the expression, “in double proportion to the times (proporzione doppia dei tempi)”, is equivalent to “proportional to the squares of the elapsed time (S∝T2)”. Galileo tried to derive the time-squared law from the proposition that the velocity is proportional to the traversed distance (v∝S). If the proposition is true, the velocity must be proportional to the squares of the elapsed time (v∝S∝T2). It is apparently contradictory to Oresme’s theory that the velocity of constant acceleration motion is proportional to the elapsed time. Based on the letter, Takahashi claims that Galileo cannot have known the Merton rule or Oresme’s theory.

Since Oresme stated that v∝T is true to “uniformly difform motion”, Galileo could have recognized S∝T2, if he had applied the “Merton rule” to it. Then it would be completely needless and incomprehensible that he adhered to the erroneous principle (v∝S) in the letter to Sarpi and later had much difficulty escaping from the error.[10]

There is a logical leap here. A scientist does not necessarily accept a theory, even if it is true. Galileo knew Kepler’s theory that the orbit of every planet is an ellipse but refused to accept it nor did he accept criticism of his erroneous theory that the tides were caused by the Earth’s rotation on its axis and revolution around the Sun. It is highly probable that in spite of knowing the Merton rule and Oresme’s theory Galileo refused them to persist in his original theory.

There is evidence of Galileo knowing the Merton rule and Oresme’s theory. We can find Heytesbury, Oxford Calculators, the mean speed theorem, “uniformly difform (uniformiter difformis)” quality and “Parisian doctors (Doctores Parisienses)” in manuscripts that Favaro, chief editor of the National Edition of the works of Galileo Galilei, called Youthful Writings[11]. Judging from the contents of the manuscript, the “Parisian doctors” must include not only Jean Buridan (1295 – 1358), who developed the concept of impetus close to the modern concept of inertia, but also his disciple, Oresme. Their theories were made so popular by Albert of Saxony (Albertus de Saxonia; ca. 1320 – 1390)and Domingo de Soto (1494 – 1560) that it is natural that Galileo could hear of them.

Although Galileo wrote the manuscripts by himself, they were supposed to be mere copies of the lecture delivered by Francesco Buonamici at the University of Pisa in 1584 and not necessarily what convinced him of their truth. At that time he was tired of the Scholastic lecture at the university. Outside the University he was devoted to reading the writings by Euclid (Εὐκλείδης, Eukleidēs; FL. 300 BC) and Archimedes (Ἀρχιμήδης; c. 287 BC – c. 212 BC) that Niccolò Tartaglia (1500‐57) translated into Italian. He finally left the University in 1585. His later works show his contempt for and hostility toward the Scholastics and the Aristotelians. It is no wonder that he did not accept all of the theories by the “Parisian doctors” that he considered Scholastic and Aristotelian. His contempt for and hostility toward the Scholastics and the Aristotelians is the key to the riddle of Galileo’s theory of falling bodies.

Read the letter to Sarpi again. The time-squared law, “the spaces passed by natural motion are in double proportion to the times,” and the law of odd numbers, “the spaces passed in equal times are as the odd numbers from one,” were presupposed as established facts. He wrote that the phenomena were observed by him and his note 107v[12] indicates that he actually conducted an experiment to verify the laws. Though he wrote the note before the letter to Sarpi, he did not write a letter or a paper to report the result of the experiment to acquire a priority. This means that he was aware that he could not show originality in the discovery of the laws.

As is evident from the letter to Sarpi, what he tried to display originality is the axiom to deduce the laws: “that the natural moveable goes increasing in velocity with that proportion with which it departs from the beginning of its motion (v∝S).” Presuming the time-squared law (S∝T2), Galileo’s axiom (v∝S) is incompatible with Oresme’s law (v∝T ⇔ v2∝S). If Galileo’s axiom had turned out to be true, his original kinematics could have overcome Oresme’s theory. That is why he concentrated on the study in kinematics after 1604.

Early in 1609 Galileo noticed that his axiom (v∝S) might be wrong. The note 116v (Fig.GN) about the experiment to decide which is true, v∝S or v2∝S, suggests his bewilderment.

Fig.GN. The note 116v written early in 1609 about the experiment to decide which is true, v∝S or v2∝S (part) Source: Electronic representation of Galilei’s notes on motion, folio 116v (media) Max-Planck-Institut für Wissenschaftsgeschichte

He changed the velocity of a ball flying out of the table by changing the height of the slope and measured how far it reached. He then found that the prediction based on v2∝S was closer to the result of the experiment than that based on v∝S. It was humiliating for him to admit that the hypothesis advanced by a Scholastic more than 200 years ago was superior to his. He seems to have been reluctant to accept the result. He tried modifying measured values to justify v∝S, only to fail.

His kinematic study reached a deadlock. Around 1609 he interrupted it and started an astronomical study, making a telescope with high magnification. The result was fruitful. He made crucial discoveries of astronomical facts such as Jupiter’s satellites and the achievement was published in Sidereus Nuncius in 1610. This book made a big stir and he was appointed Chief Mathematician of the University of Pisa and Philosopher and Mathematician to the Grand Duke of Tuscany. Still he did not give up his original kinematics based on the axiom (v∝S). He arranged a plan to write three books on an entirely new kinematics founded on a new principle[13] along with many research plans in the letter to promote himself Chief Philosopher and Mathematician to the Grand Duke of Tuscany.

Then his astronomical study reached a deadlock, though this time it is not a scientific but a religious one. He was accused of heresy for his support of the Copernican theory in 1616 and was ordered to cease to defend it in any way whatsoever. He interrupted his astronomical study and returned to the kinematic study for a while. In 1616 he made his disciples, Niccolò Arrighetti and Mario Guiducci, arrange his manuscripts and copy them out fair in 1618. Just at that time three new comets appeared, He was involved in the debate over their nature and returned to the astronomical study.

This is the situation in which Galileo published Dialogue Concerning the Two Chief World Systems: Ptolemaic and Copernican in 1632. As is known from the title, the main topic of this book is astronomical, but kinetic topics were also taken up. It was in this book that Galileo first proved the double distance rule from the right axiom (v∝T) of Oresme. It is not clear why he abandoned his original axiom (v∝S), but my hypothesis is that at that time it became no longer necessary for him to show originality in this axiom, because the main topic of this book was not kinetics and he could show off originality in other discoveries such as the so-called “Galilean transformation”.

He had more originality in reserve, when he published Two New Sciences. In this book he formulated the mean speed theorem in a more similar way to Oresme’s and proved the law of odd numbers in such a geometrical way as reminds us of Oresme. The law prescribes that the distance traveled by a freely falling body during equal intervals of time stands to one another in the same ratio as the odd numbers beginning from unity. Today we can easily deduce the law algebraically from the time-squared law (S∝T2).


Galileo, however, demonstrated it geometrically, drawing the following figure (Fig.ON).

Fig.ON. The demonstration of the law of odd numbers Source: Two New Sciences

The figure (1) was the one inserted in Two New Sciences. As it is too crammed, I divided it into (2) and (3). The vertical line of figure (2) represents time flowing from A to O and its horizontal line stands for the velocity at its moment. Owing to the proportion v∝T, the relation of time and velocity forms the triangle APO. The congruent triangles ABC, BFG and FPQ can be transformed into quadrilaterals in Oresme style, with their area unchanged, that is to say, figure (2) can be transformed into (3), whose ratio of area produces a progression of odd numbers (1:3:5…).

Here we can find Oresme’s influence on Galileo, but he never mentioned his name, although he often heaped the highest praise on the works of Archimedes. Maybe he did not want to credit the achievement to the Aristotelian Scholastics. Oresme arranged the axis of time rightward and the axis of velocity upward, while Galileo arranged the former downward and the latter leftward. Oresme’s arrangement is so natural that it is adopted today. It would be probably because Galileo wanted to display the difference from Oresme that he invented the unnatural diagram. Takahashi asserts that the difference of the diagram denies Oresme’s influence on Galileo[14]. He seems to be dancing to Galileo’s tune. Anyway the difference is not so essential as to deny the influence.

Oresme is usually regarded as an Aristotelian Scholastic. To be sure, he was engaged in the interpretation of Aristotle and not as critical of the Scholastic tradition as Galileo, but, as Marshall Clagett pointed out, Oresme was subject to the influence of Archimedes whose original Greek texts were translated into Latin by Willem van Moerbeke in 1269[15]. We can find Archimedes’ influence on Oresme in his geometrical and quantitative analysis and his usage of an infinite series for mensuration. In this sense Oresme was the precursor of Galileo or the 17th century scientific revolution.

3 : Was Oresme’s theory just imaginary and not scientific?

Acknowledging the formula of Oresme is equivalent to Galileo’s, some do not acknowledge him as a physicist. For example, Yamamoto Yoshitaka, a Japanese historian of science, lays great emphasis on the difference between Oresme and Galileo.

The definite difference of Galileo from the preceding Parisian Scholastics consists not in the formula itself or in its deduction but in the status they gave to the formula. That is to say, while Galileo thought of the formula as the representation of the motion of an actual body that could and should be verified and measured experimentally, the Scholastics were engaged in mere fiction or intellectual exercise. Oresme himself declared that the line representing the intensity of quality was imaginary and the Scholastics took no measurement, let alone carry out any experiment.[16]

It is true that Oresme argued “according to the imagination (secundum imaginationem).” In his Le Livre du ciel et du monde Oresme discussed a range of evidence for the daily rotation of the Earth on its axis, but the bottom line of the book is that not the Earth but the heavens move. Oresme said that he discussed it only for amusement[17]. That is why Yamamoto says Oresme’s argument about physics is mere “intellectual exercise.”

Taking the religious environment of Paris at that time into consideration, however, you will notice that you should not interpret his words literally. In 1270 Etienne Tempier issued a formal condemnation of thirteen heretical doctrines that could violate the omnipotence of God. In 1277 Pope John XXI (Ioannes XXI; 1215 – 1277), former professor of theology at the University of Paris, encouraged Tempier to expand the number of condemned doctrines to 219. The condemnation had exerted a wide influence on Parisian intellectuals until the 14th century. Even Thomas Aquinas (1225 – 1274), who was later honored as a saint by the Catholic Church, was suspected of heresy after his death. In the end, Albertus Magnus (ca. 1193 – 1280) journeyed to Paris to defend the memory of his disciple and Thomas Aquinas escaped excommunication, but this episode indicates how intolerant the religious environment was at that time.

Judging from Oresme’s contention in Le Livre du ciel et du monde, he would have believed that the Earth actually rotates on its axis, but it was too dangerous to advocate such a heretic doctrine against the Bible. Therefore he concluded the negation of the rotation and said he argued for it just “for amusement”. So was his commitment to the mean speed theorem. If he had insisted on the reality of his theory, the risk of being suspected of heresy would have increased. To reduce the risk he had to say that his theory was imaginary. Galileo also disguised his heliocentric theory as a mere mathematically possible hypothesis to evade the attack from the Church. Galileo was not greatly different from Oresme at this point as well.

Some might argue that the Oxford Calculators and Oresme do not deserve the precursor of the 17th century scientific revolution, because unlike Galileo they did not perform experiments except thought experiments. Such an empiricist argument is based on a misunderstanding of the essence of the 17th century scientific revolution. Although the interpretation by Alexandre Koyré (1892 – 1964), a French historian of science, in his book Studies of Glileo in 1939 went too far, it is still true that Galileo was a kind of Platonist. The Copernican theory was also based on Neoplatonism. While Aristotle was close to an empiricist, Plato and Archimedes were rationalists who gave priority to the idea. If you recognize the 17th century scientific revolution to be the shift from Aristotle-Thomas paradigm that was dominant in the late Middle Ages to Plato-Archimedes paradigm, you cannot say the Oxford Calculators and Oresme were not pioneers of the 17th century scientific revolution because they made only thought experiments.

Galileo made Salviati, the spokesman for Galileo in Two New Sciences, say, the knowledge of one single effect acquired through its causes opens the mind to the understanding and certainty of other effects without need of recourse to experiments[18]. For example, those who know why the projectile range of a shot becomes maximum when the angle of elevation is half a right angle can also demonstrate that of the other shots whose angle exceed or lack an equal degree is the same without need of recourse to experiments. Suppose the initial velocity is v0, deviation from half a right angle is θ. To apply the current notation, the projectile range S is

S=frac{v_{0}^{2}}{g} cos2theta.

Those who know the projectile range S becomes maximum when θ=0 can also demonstrate that S is the same whether θ is positive or negative, so long as its absolute value is identical. Even if you try to verify it experimentally, air resistance and other disturbing factors hinder the exact measurement. Moreover, you cannot perform experiments unlimitedly, putting unlimited values into θ. The most fundamental defect of empiricism lies, as Galileo pointed out, in that those who engage only in observations and experiments and do not know its theoretical necessity cannot apply the result of the discovery to the prediction of other discoveries.

Today scientists do not employ the inductive method of randomly repeating observations or experiments and thereafter abstracting the general law. They first construct an ideal theory and thereafter verify it by means of observations and experiments. So, the order of Oresme first making a thought experiment and then Galileo verifying it by a real experiment is not unnatural.

Scientists allow a margin of error for the result of an experiment. They do not think it negates the completeness of a mathematical model, because they are convinced that the mathematical order should rule the nature. We can find such a conviction in the following famous statement in Galileo’s book The Assayer .

Philosophy is written in that great book which ever lies before our eyes (I mean the universe) but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth. [19]

“Philosophy (la filosofia)” here is equivalent to today’s natural science. The quoted text is often referred to as the declaration of the spirit of modern science, but it is not a new idea in the history of the Western thought. The idea dates back to the Pythagorean idea that the world is mathematical. The Pythagorean philosophy influenced Plato, Euclid and Archimedes. After the 12th century Renaissance, treatises of Archimedes were introduced to Europe and made an impact on Oresme and Galileo, who attempted to apply Archimedes’ approach to statics to kinetics. Their attempt was succeeded to and completed by Isaac Newton’s Principia, the terminal point of the 17th century scientific revolution. In fact, Archimedes influenced Newton’s geometrical approach to the calculus. Now that the 17th century scientific revolution is the shift from Aristotle-Thomas paradigm to Plato-Archimedes paradigm, we can consider the Oxford Calculators and Oresme to be the pioneer of the paradigm shift, though they were forgotten after the Age of Enlightenment.

4 : References

  1. Die Mechanik in ihrer Entwickelung: historisch-kritisch dargestellt (author) Ernst Mach (page) 128
  2. Les précurseurs parisiens de Galilée, Études sur Léonard de Vinci, Troisième série (author) Pierre Duhem (page) 388-389
  3. Tractatus de configurationibus qualitatum et motuum (author) Nicole Oresme (media) Nicole Oresme and the Medieval Geometry of Qualities and Motions (editor) Marshall Clagett (page) 408
  4. Quaestiones super geometriam Euclidis (author) Nicole Oresme (media) Nicole Oresme and the Medieval Geometry of Qualities and Motions (editor) Marshall Clagett (page) 560
  5. De Rerum Natura, Liber Secundus, line 225 – 229 (author) Titus Lucretius Carus
  6. De Beghinselen des Waterwichts, Anhang (author) Simon Stevin
  7. Discorsi e dimostrazioni matematiche intorno a due nuove scienze attinenti alla mecanica ed i movimenti locali (media) Le opere di Galileo Galilei Vol. 8 (page) 208
  8. Dialogo sopra i due massimi sistemi del mondo tolemaico e copernicano (media) Le opere di Galileo Galilei Vol. 7 (page) 52
  9. La lettera di Galileo a Paolo Sarpi del 16 ottobre 1604 (media) Le opere di Galileo Galilei Vol. 10 (page) 115
  10. ガリレオの迷宮 (page) 86 (author) 高橋憲一
  11. Iuvenilia (media) Le opere di Galileo Galilei Vol. 1 (page) 7-177
  12. Electronic representation of Galilei’s notes on motion, folio 107v (media) Max-Planck-Institut für Wissenschaftsgeschichte
  13. La lettera di Galileo a Belisario Vinta in Firenze del 7 maggio 1610 (media) Le opere di Galileo Galilei Vol. 10 (page) 350-351
  14. ガリレオの迷宮 (page) 312 (author) 高橋憲一
  15. Archimedes in the Middle Ages: The Fate of the Medieval Archimedes (part) 3 (author) Marshall Clagett
  16. 一六世紀文化革命 2 (page) 700-701 (author) 山本義隆
  17. The Compatibility of the Earth’s Diurnal Rotation with Astronomical Phenomena and Terrestrial Physics (author) Nicole Oresme (media) A Sourcebook in Medieval Science (page) 504, 510 (editor) Edward Grant
  18. Discorsi e dimostrazioni matematiche intorno a due nuove scienze attinenti alla mecanica ed i movimenti locali (media) Le opere di Galileo Galilei Vol. 8 (page) 296
  19. Il Saggiatore (section) 6 (author) Galileo Galilei
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