13. Newton’s Optics

I’ve just finished reading selections from another work by Sir Isaac Newton, Optics, guided by what Mortimer J. Adler and Peter Wolff say about it in Foundations of Science and Mathematics, Reading Plan 3 of Encyclopedia Britannica’s The Great Ideas Program (Encyclopedia Britannica, 1960). In the Reading Plan Mortimer J. Adler and Peter Wolff (I) compare the methods used by Newton in Mathematical Principles and Optics; (II) examine the law of reflection and the law of refraction; (III) consider the corpuscular and wave theories of light; and (IV) pose and discuss three questions. Here I’ll compare the methods used by Newton (summarizes I), outline the parts of Optics assigned for reading in Foundations of Science and Mathematics (includes II and III), compare the corpuscular and wave theories of light (summarizes III), and pose the questions asked in IV. For a sketch of Newton’s life see https://opentheism.wordpress.com/2019/07/12/12-newtons-mathematical-principles-of-natural-philosophy/.

The Methods Used by Newton
Both of the works of Newton considered in my readings from Great Books of the Western World, Mathematical Principles of Natural Philosophy (https://opentheism.wordpress.com/2019/07/12/12-newtons-mathematical-principles-of-natural-philosophy/) and Optics, follow the pattern of Euclid’s Elements (https://opentheism.wordpress.com/2018/04/06/1-2-euclids-elements/): they begin with definitions, axioms, and postulates and then they present a series of propositions. However they differ considerably from each other in method. Mathematical Principles is a work in mathematical physics, characterized by the mathematical development of certain tentative formulations. Optics is a work in experimental physics, characterized by the experimental development of certain general principles.
All three works employ axioms, but they are of different kinds. The axioms in Elements are self-evident truths that are universally applicable. The axioms in Mathematical Principles state the very general Laws of Motion, which form the basis of the entire science of dynamics. The axioms in Optics just state what was generally accepted in the science of optics around the year 1700.
All three works present propositions, but again they are of different kinds. The propositions in Elements and Mathematical Principles state conclusions that are to be demonstrated from general principles. Optics state principles that have been found as the result of making experiments and observations and drawing general conclusions from them by induction.
[The above is based on part I of Adler and Wolff’s guide to Optics (Foundations of Science and Mathematics, pages 181-184).]

Optics
This is an outline of the parts of Optics assigned for reading in Foundations of Science and Mathematics: Book I, Part I, Definitions, Axioms, Propositions 1-2; Book III, Part I, Queries 27-31.DEFINITIONS
Newton defines eight terms. If I use a term which you don’t know the meaning of, please ask me its meaning and if it’s a term which Newton defines, I’ll give you his definition of it.AXIOMS
Newton gives eight axioms. I’ll give here just the three which Adler and Wolff explain or refer to in Part II of their guide to Optics (Foundations of Science and Mathematics, pages 184-186).
II. The angle of reflexion is equal to the angle of incidence.
IV. Refraction out of the rarer medium into the denser is made towards the perpendicular; that is, so that the angle of refraction be less than the angle of incidence.
V. The sine of incidence is either accurately or very nearly in a given ratio to the sign of refraction.
PROPOSITIONS
Newton presents 39 propositions but only the first two are assigned for reading in Foundations of Science and Mathematics. They are:
1. Lights which differ in colour, differ also in degrees of refrangibility. [Under Definitions, Newton defines refrangibility of rays of light as “their disposition to be refracted or turned out of their way in passing out of one transparent body or medium into another” (Optics in Great Books of the Western World, Encyclopedia Britannica, 1952, volume 34, page 379).]
2. The light of the Sun consists of rays differently refrangible.
QUERIES
In conjunction with presenting the 39 propositions, Newton makes 38 observations. After making them, he planned to repeat most of them and to make more to determine how rays of light are bent in their passage by bodies. However he was interrupted and instead proposed some queries to assist others in their search. The last five of his 31 queries are assigned for reading in Foundations of Science and Mathematics. They are:
27. Are not all hypotheses erroneous which have hitherto been invented for explaining the phenomena of light, by new modifications of the rays?
28. Are not all hypotheses erroneous in which light is supposed to consist in pressure or motion, propagated through a liquid fluid?
29. Are not the rays of light very small bodies emitted from shining substances?
30. Are not gross bodies and light convertible into one another, and may not bodies receive much of their activity from the particles of light which enter their composition?
31. Have not the small particles of bodies certain powers, virtues, or forces, by which they act at a distance, not only upon the rays of light for reflecting, refracting, and inflecting them, but also upon one another for producing a great part of the phenomena of Nature?
Adler and Wolff consider these queries in their III, which I summarize below.

The Corpuscular and Wave Theories of Light
In Query 29 (see above) Newton refers to the corpuscular theory of light, adding, “For such bodies will pass through uniform mediums in rights lines without bending into the shadow, which is the nature of the rays of light.” Adler and Wolff observe that this characteristic of light is easily explained by the corpuscular theory of light but gives some difficulty to its great rival, the wave theory of light, which will be encountered in the next reading in this series.
One consequence of the corpuscular theory is that light travels more swiftly in a denser than in a rarer medium. However according to the wave theory light travels more rapidly in a rarer medium than in a denser one. In 1850 Foucault performed an experiment which showed that the speed of light is greater in air than in water, thus supporting the wave theory. However since then additional phenomena have been discovered which cannot be reconciled with the wave theory. Thus the nature of light is still in doubt.

Questions about the Reading
1. What is the method employed by Newton to prove the axioms in the Optics?
2. How does the law of refraction explain the bent appearance of a stick in water?
3. Are there any practical consequences of the different refrangibility [capability of being refracted] of light rays of different color?

12. Newton’s Mathematical Principles of Natural Philosophy

“Occasionally in the history of thought there occurs a moment when some man or some book shatters preceding tradition by a great leap. Such a moment certainly occurred with the publication of Newton’s Mathematical Principles” (Mortimer J. Adler and Peter Wolff, Foundations of Science and Mathematics, Encyclopedia Britannica, 1960, page 161). I’ve just finished reading the passages in that work assigned for reading in Foundations of Science and Mathematics, Reading Plan 3 of Encyclopedia Britannica’s The Great Ideas Program.
In the Reading Plan Mortimer J. Adler and Peter Wolff (I) quote tributes paid to Sir Isaac Newton by the astronomer Edmund Halley and the poet Alexander Pope; (II) summarize Newton’s accomplishments; (III) discuss the characteristics of Newton’s method; (IV) discuss some of the characteristic concepts of Newton’s physics; and (V) pose and discuss three questions. Here I’ll sketch Newton’s life, outline the passages in Mathematical Principles of Natural Philosophy assigned for reading in Foundations of Science and Mathematics, and pose the questions asked in V.

Sir Isaac Newton

Isaac Newton was born on Christmas Day, 1642, in the English town of Woolsthorpe. His father, a farmer, died a few months before his birth. In 1645 his mother remarried and left him with his maternal grandmother at Woolsthorpe. In 1656 his stepfather died and his mother returned to Woolsthrope to take care of the farm. She took Isaac out of school and brought him home so that he could prepare himself to manage the farm. However before long she realized that he wasn’t suited for farm life and sent him back to school. In 1661 he entered Trinity College in Cambridge University, and he graduated in 1665.
Newton wanted to stay on at the university to continue his studies but it was closed because of the Black Plague and he returned to Woolsthorne. In the eighteen months that he was there, he conducted experiments in optics and chemistry and continued his mathematical speculations. During the time he hit upon a new mathematical tool, now called calculus; began working out the law of attraction between all objects in the universe, the law of gravitation; and experimented with light, succeeding in showing that a beam of sunlight is made up of bands of colour from red to violet which he called the spectrum.
After the plague ended Newton returned to Cambridge and continued working on light and colour. This work led to the discovery of the reflecting telescope. In recognition of his work in mathematics and optics (the science of light), he was appointed professor of mathematics at Trinity College in 1669. Although he experimented mainly with optics, his mind always returned to the problem of gravitation. Finally he completed the mathematics of the laws of gravitation Using this law, in 1682 he proved mathematically a law of planetary motion that had been figured out by the astronomer Johannes Kepler in the early 1660’s. Encouraged by friends, in 1685 he plunged into the task of writing a book explaining his work on planetary motion, gravitation, and other matters. The book, The Mathematical Principles of Natural Philosophy, appeared in 1687 and “is not only Newton’s masterpiece but also the fundamental work for the whole of modern science” (“Newton, Sir Isaac,” Encyclopedia Britannica, Encyclopedia Britannica, 1974, volume 13, page 19).
In 1696 Newton was appointed Warden of the Mint and, although he didn’t resign his Cambridge appointments until 1701, he moved to London and from then on centred his life there. He was made Master of the Mint in 1699, became president of the Royal Society in 1703, and was knighted in 1705. When he died in 1727 (March 20), he was buried in Westminster Abbey, among the great men of England. A statue of him stands today in the hall of Trinity College, Cambridge University.

Mathematical Principles of Natural Philosophy

DEFINITIONS
[Newton defines the quantity of matter, the quantity of motion, the vis insita or inertia of matter, an impressed force, a centripetal force, the absolute quantity of a centripetal force, and the accelerative quantity of a centripetal force; and he distinguishes between absolute and relative time, absolute and relative space, absolute and relative place, and absolute and relative motion.]

AXIOMS OR LAWS OF MOTION
LAW I. Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed on it.
LAW II. The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.
LAW III. To every action there is always an equal reaction: or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.
[Newton also gives six corollaries to the three laws of motion and a scholium, but I didn’t work through them.]

RULES OF REASONING IN PHILOSOPHY
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.
RULE III. The qualities of bodies, which admit neither intensification nor remission of degrees, and which are found to belong to all bodies within the reach of our experiences, are tio be esteemed the universal qualities of all bodies whatsoever.
RULE IV. In experimental philosophy we are to look upon propositions inferred by general induction from phenomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phenomena occur, by which they may either may be made more accurate, or liable to exceptions.
[Adler and Wolff call these the rules of simplicity, consilience, empiricism, and induction. They observe that Rules 1 and 2 are so closely related that they might almost have been combined into one rule and that Rules 3 and 4 also belong closely together with Rule 4 just reaffirming Rule 3.]

GENERAL SCHOLIUM
[Newton summarizes the motions of the sun, planets, moons, and comets.] This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being.… This Being governs all things, not as the soul of the world, but as Lord over all; and on account of his dominion he is wont to be called Lord God. [Newton considers the attributes of this God.]
Hitherto we have explained the phenomena of the heavens and of our sea by the power of gravity…but hitherto I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypothesis. [Adler and Wolff maintain that Newton does make hypotheses in the sense of tentative formulations to be explored mathematically and verified or disproved by experiment but agree that he doesn’t make them in the sense of fictional explanations such as nature’s abhorring a vacuum.]

Questions about the Reading

1. How does Newton describe his method of reasoning? [see RULES OF REASONING IN PHILOSOPHY above.]
2. Does Newton continue the “Keplerian revolution”?
3. How does Newton distinguish between absolute and relative motion?

10. Francis Bacon’s “Novum Organon”

I’ve finally read another Great Books of the Western World selection, parts of Francis Bacon’s Novum Organum, the tenth reading for which guidance is given in Foundations of Science and Mathematics of Encyclopedia Britannica’s The Great Ideas Program. In introducing the reading Mortimer J. Adler and Peter Wolff emphasize that Bacon was a pioneer in a new scientific attitude—that the aim of science should be to make nature useful to us rather than to accumulate facts about the world. They say that only recently has his view of science been challenged, concluding, “Today we witness the hot debate between the advocates of ‘basic research’ and the advocates of useful knowledge. Bacon is the earliest and perhaps the greatest exponent of the latter view” (Mortimer J. Adler and Peter Wolff, Foundations of Science and Mathematics, volume 3 of The Great Ideas Program, Encyclopedia Britannica, 1960, page 135).

Adler and Wolff go on to: (I) introduce The Great Instauration, which Novum Organon is part of; (II) describe Bacon’s attempt to seek preferment and position at the Court and, after his seeming to have done so, his rapid decline in fortune; (III) consider Bacon’s insistence that the arts and sciences be rebuilt by induction as described in Novum Organon; (IV) summarize the example of induction given by Bacon in Book II of Novum Organon; and (V) pose and discuss three questions about the reading. Here I’ll sketch Bacon’s life, describe his theory of induction, and pose the questions asked in V.

Francis Bacon

Francis Bacon was born January 22, 1561, in London, the youngest son of Sir Nicholas Bacon, Lord Keeper of the Seal under Queen Elizabeth I. At the age of twelve he entered Cambridge University where he resided with his older brother, and in 1576 when he finished his studies there his father sent him to Paris with the English ambassador to complete his political education. Recalled to England by the death of his father in 1579, he found that as the youngest son he was virtually penniless. He applied himself to the study of the law at Gray’s Inn (one of four inns of court that served as institutions for legal instruction), to which he had been admitted before leaving for Paris, and in 1582 was admitted to the bar.

Bacon obtained a seat in Parliament in 1584 and immediately began advocating for political advancement. However Queen Elizabeth took offence at his taking a stand in 1593 against the government’s demand for subsidies to help meet the expenses of the war against Spain and, despite the support of the Earl of Essex, a favourite of the Queen, he lost to his constant rival, Sir Edward Coke, in his bid for the vacant position of Attorney-General in 1594. His fortunes changed when he participated in the trial of the Earl of Essex (his patron) in 1600-01 and even more with the accession of James I in 1603. Writing in defence of the royal policies, he became one of the leaders of the King’s cause against Parliament. Knighted in 1603, he became Solicitor General in 1607, Attorney-General in 1613, Lord Keeper of the Seal in 1617 (following the dismissal of Coke—see above), and Lord Chancellor in 1618. In 1618 he was made Baron Verulam and in 1621 Viscount St. Albans.

However, also in 1621, he was accused of receiving bribes as a judge. He admitted receiving “gifts” but denied that his judgment had been affected by them. He was found guilty by the high court of Parliament, which decided that he should pay a fine, be imprisoned in the Tower of London at the King’s pleasure, and not hold any public office. Although he was released from the Tower after a few days and the king remitted his fine, he never again held public office. In her Encyclopedia Britannica article on Bacon, Kathleen Marguerite Lea comments: “Certainly, adversity discovered in Bacon the virtues of patience, unimpaired intellectual vigour, fortitude, and a good sense of acceptance. Physical deprivation distressed him for himself and his wife [he’d married Alice Barnham, the daughter of a London alderman in 1606] but what hurt most was the loss of favour; it was not until January 20, 1622/23, that he was admitted to kiss the king’s hand; the full pardon that would have cleared his name never came.” (“Bacon, Francis”; Encyclopedia Britannica; Chicago, etc: Encyclopedia Britannica, 1974; volume 2, page 563; see also https://www.britannica.com/biography/Francis-Bacon-Viscount-Saint-Alban.)

Even under Elizabeth Bacon was engaged in scientific and literary work and in 1597 he published the first edition of his Essays, his best-known writings, noted for their style and observations about life. In 1605 he published Advancement of Learning, which revised and extended later became the first part of his Instauratio Magna (Great Instauration), a comprehensive plan to reorganize the sciences and to restore to man the command of nature to the glory of God and the relief of man’s estate. After his political fall, he concentrated on the Instauratio Magna but his efforts were limited almost entirely to his own writings. The Novum Organon (1620) was to serve as the second of its six parts. When he died April 9, 1626, he was still engaged with the proposed work.

Bacon’s Theory of Induction

Induction derives from observation and experimentation conclusions that are more general than the evidence on which they are based. Bacon describes it as “construct[ing] its axioms from the senses and particulars, by ascending continually and gradually, till it finally arrives at the most general axioms” (Novum Organum in Great Books of the Western World, Encyclopedia Britannica, 1952, volume 30, page 108). He illustrates his method of induction by describing its use in investigating what he terms the “form of heat.”

Bacon starts by making a series of three tables. Table I lists known cases of heat, Table II lists cases where we would expect from the first table that we’d find heat but don’t find it, and Table III notes the extent to which various things exhibit different degrees of heat. For example, he includes the rays of the sun in Table I and the rays of the moon and stars in Table II and notes in Table III that the sun provides more heat as it approaches the perpendicular or zenith and provides more heat at its perigee (closest point) than at its apogee (farthest point) from the earth.

Then Bacon constructs a fourth table, an “Exclusive Table,” consisting of negative inferences drawn from the other three tables. For example, because the sun’s rays exhibit heat, he infers that heat cannot be something that is only terrestrial in nature. On the other hand, because fire exhibits heat, he also infers that heat cannot be something that is only celestial in nature. He goes on to reject twelve more natures, concluding that none of them is of the form of heat.

From this Bacon finally derives the “form or true definition of heat”: “Heat is an expansive motion restrained, and striving to exert itself in the smaller particles. The expansion is modified by its tendency to rise, though expanding towards the exterior; and the effort is modified by its not being sluggish, but active and somewhat violent” (Novum Organum, pages 152-153). (I adopted the word “finally” from Adler and Wolff’s account of Bacon’s investigation of the form of heat; Adler and Wolff, page 141).

Adler and Wolff observe that Bacon’s language being old-fashioned gives his remarks about heat a quaint quality. However they go on to note that when over two hundred years later John Stuart Mill considered induction in his A System of Logic, he followed in Bacon’s footsteps. His first canon corresponds to Bacon’s Table I, his second canon corresponds to Bacon’s Table II, and his fifth canon corresponds to Bacon’s Table III. (Adler and Wolff, pages 141-142)

In her Encyclopedia Britannica article on Bacon, Kathleen Marguerite Lea identifies two serious objections to Bacon’s method. The first is that “the laboriousness of his method requires the aid of hypotheses, and Bacon gave no help toward the formulation of such concepts,” and the second is that “the method breaks down when it is related to the end proposed, that is the discovery of ‘forms.’ Bacon himself had great difficulty in giving an adequate and exact definition of what he meant by a form” (“Bacon, Francis,” page 566).

Questions about the Reading

1. What is Bacon’s aim in Novum Organum?

2. What are the four idols which make the advancement of knowledge difficult?

3. Was Bacon an accurate prophet of the future course of science?

9. Galileo’s Dialogues Concerning the Two New Sciences

In many respects, Galileo typifies the man of science. His devotion to the pursuit of learning and to the truth was so great that he let nothing stand in its way. All the stories about him–and they are many–indicate that he had but one interest throughout his life, namely, the investigation of nature. In the cathedral at Pisa, instead of listening to the service, he watched the chandeliers swing regularly and so discovered the isochronism [the characteristic of having a uniform period of vibration] of pendulums. (Mortimer J. Adler and Peter Wolff, Foundations of Science and Mathematics, volume 3 of The Great Ideas Program, Encyclopedia Britannica, 1960, page 117)

Thus Mortimer J. Adler and Peter Wolff introduce their guide to Galileo’s Dialogues Concerning the Two New Sciences, which I’ve now reached in my rereading of selections from Great Books of the Western World guided by The Great Ideas Program. They go on to: (I) introduce Dialogues Concerning the Two New Sciences itself; (II) identify the subject of the Third Day of the four days into which Dialogues Concerning the Two New Sciences is divided; (III) describe Galileo’s method; and (IV) pose and discuss four questions about the reading. Here I’ll sketch Galileo’s life, summarize (I) and (II) together and (III) separately, and pose the four questions asked in IV.

Galileo Galilei

Galileo Galilei was born in Pisa, Italy, on February 15, 1564. When he was a young boy, the family moved to Florence, where he obtained his early education in a nearby monastery. In 1581 he was sent to the University of Pisa to study medicine.

It was while he was there that the incident occurred that is referred to in the quotation above from Adler and Wolff. Not only did Galileo work out the laws that govern a pendulum, but also he applied those findings to inventing a timing device. “What is most important, Galileo had set the pattern of his life: a simple observation led to questions and experiments, which led to new knowledge and applications” (Patricia G. Lauber, “Galileo,” The New Book of Knowledge, Grolier, 1976, volume 7, page 5).

Lacking money Galileo left medical school in 1585 and returned to Florence but continued his investigations. His writings on physics made him known, and he became professor of mathematics at the University of Pisa in 1589. During the next two years he conducted experiments on the motion of falling bodies. Aristotle had said that heavy objects fall faster than light ones, but when Galileo dropped balls of various weights from the same height they reached the ground together. His questioning of Aristotle’s ideas made Galileo unpopular and he left Pisa.

Shortly afterwards Galileo became professor of mathematics at the University of Padua, where he taught from 1592 to 1610, his fame as an experimental physicist attracting students from all parts of Europe. Hearing about the invention of the telescope, which had appeared in Europe by 1608, Galileo figured out how to make one and night after night viewed the heavens, making many discoveries. Although he had personally accepted the teaching of Copernicus as early as 1597, he continued to teach the Ptolemaic system while at Padua. However his discovery that Jupiter had moons and that Venus had phases convinced him that he should try to establish the new theory. His doing so brought was opposed by the Roman Catholic Church, which still believed that everything revolved around the earth. In 1616 it condemned the Copernican system and the pope ordered Galileo to stop supporting it. Galileo promised to obey.

Leaving Padua for Florence, Galileo continued his other work. However finally he turned back to the Copernican system, and in 1632 he published Dialogues on the Two Chief Systems of the World, in which the winner of the argument among three men was clearly the one who supported the Copernican system. Galileo was called to Rome to appear before the Inquisition (a tribunal set up by the Church to suppress heresy), which required him to deny his belief and sentenced him to imprisonment (later changed to house arrest).

Galileo spent the rest of his life almost cut off from the world. But he was allowed to teach and to carry on his research. One of the things that he did in these years was to write the book that this article is about, Dialogues Concerning the Two New Sciences, which summed up Galileo’s work on motion, acceleration, and gravity. Above all, Galileo continued searching for the truth, which he was still doing when he died on January 8, 1642.

Dialogues Concerning the Two New Sciences

Dialogues Concerning the Two New Sciences is divided into four days instead of into books or chapters. Each day three men talk about a different subject. Two of the men, Salviati and Sagredo, represent two real friends of Galileo. The third man, Simplicio, is apparently a fictitious person meant to represent the Aristotelian school of thought. Most of the Third Day, which the selections assigned in the reading are from, consists of propositions written by “our author,” Galileo, and read by Salviati to the other two men.

The subjects of the Third Day are uniform motion and natural non-uniform motion. The natural non-uniform motion Galileo considers is the downward, falling motion of heavy bodies; he doesn’t consider the natural upward motion of fire and similar objects. On the Fourth Day he considers the violent (non-natural) motion of heavy bodies, the motion of projectiles.

Galileo’s Method

A mathematical physicist, Galileo’s method is mathematical and he gives an appropriate mathematical formula as the solution to any problem. The Third Day illustrates his method. In the section on uniform motion, he begins with a definition. From it and four axioms, he derives a number of propositions, the last of which is equivalent to the standard formula for uniform motion: distance traversed = speed x elapsed time. The section on falling motion also begins with a definition, one equivalent to the formula: speed = acceleration x time. As he did in the section on uniform motion, Galileo follows the definition with a series of propositions derived from it. Most of the Third Day consists of such propositions and their proofs.

Thus the first two steps in Galileo’s method are the proposal of a definition or hypothetical formulation and the mathematical development of it. However a third step must be taken for the method to be complete: the experimental verification of the hypothesis and its mathematically developed consequences. Adler and Wolff point out that the method does not include an investigation of causes, quoting a speech by Salviati in which he says, “At present it is the purpose of our Author [Galileo] merely to investigate and to demonstrate some of the properties of accelerated motion (whatever the cause of this motion may be” (Adler and Wolff, page 125; quoting from Galileo, Dialogues Concerning the Two New Sciences, Great Books of the Western World, Encyclopedia Britannica, 1952, volume 28, page 202).

Questions about the Reading

1. After defining uniform motion, Galileo adds a “Caution.” What is the purpose of it? Is it really necessary?

2. How valid is the method of proof Galileo employs in Proposition I?

3. What does the experiment with the inclined plane accomplish?

4. For his experiment on the inclined plane, Galileo employs a water clock. As he describes it, is it suitable for his purpose?

8. Kepler’s Epitome of Copernican Astronomy

Johannes Kepler is the first modern scientist whom we encounter in this Reading Plan. Although the Copernican revolution upset the ancient views of astronomy, Copernicus cannot be truly called a modern; he still clung to many of the concepts and prejudices of antiquity. Kepler, too, wrought a revolution, and it propelled astronomy headlong into modernity. (Mortimer J. Adler and Peter Wolff, Foundations of Science and Mathematics, volume 3 of The Great Ideas Program, Encyclopedia Britannica, 1960, page 105)

Thus Mortimer J. Adler and Peter Wolff introduce their guide to Kepler’s Epitome of Copernican Astronomy, which I’ve now reached in my rereading of selections from Great Books of the Western World guided by The Great Ideas Program. Further on in their introduction to Epitome of Copernican Astronomy, they attribute Kelper’s genius to his imagination, to be “able to free himself from the mass of detailed observations and see things in the heavens that no one before him had noticed” (Adler and Wolff, pages 105-06).

Adler and Wolff go on to: (I) consider the works of Kepler and purpose of Epitome of Copernican Astronomy; (II) identify and explain Kepler’s three laws of planetary motion; (III) argue that Kepler did more than substitute ellipses for circles; (IV) pose and discuss two questions about Kepler’s work. Here I’ll sketch Kepler’s life, comment on I, outline II and III, and pose the two questions asked in IV.

Johannes Kepler

Kepler was born on December 27, 1571, in the German town of Weil of poor parents. Recognized as being of superior intelligence, he was able to attend the University of Tübingen as a charity student in 1587 and trained to become a Lutheran minister. While there he studied astronomy under Michael Mästlin, who introduced him to the work of Copernicus. In his last year of study at Tübingen he was persuaded to accept the vacant position of teacher of mathematics in the Lutheran high school at Graz in Austria in 1594.

While filling that position, he began to speculate on the order and distances of the planets. He speculated that they were based on the five regular bodies of geometry and embodied his theory on this in Cosmographic Mystery in 1596, The book brought him to the attention of Tycho Brahe, the imperial mathematician of the Holy Roman Empire, who in 1560 invited him to join his research staff at the observatory near Prague. When Tycho Brahe died the next year, Kepler was appointed his successor as imperial mathematician.

Using the records of Tyccho Brahe’s observations and the results of his own observations, Kepler published a series of works which gained him a European reputation. After the death of his wife and (of smallpox) his three children, in 1611 he accepted an offer to become mathematician to Upper Austria and moved to Linz, where he re-married and resumed his astronomical investigations. His twelve years there saw the publication of his most important astronomical works, including Epitome of Copernican Astronomy (1618-21).

Unfortunately Kepler’s salary as imperial mathematician was continually in arrears. In 1628 the debt was transferred to Duke Wallenstein of Friedland and Kepler moved with his family to Silesia. On November 15, 1630, he died of a fever. His epitaph, composed by him, reads: “I had measured the heavens; now I measure earth’s shadows. Mind came from the heavens, Body’s shadows has fallen” (Great Books of the Western World, Encyclopedia Britannica, 1952, volume 16, page 842).

Epitome of Copernican Astronomy

Adler and Wolff observe that Epitome of Copernican Astronomy was intended as a popular work giving the highlights of Kepler’s system. However, as with the works of Ptolemy and Copernicus, I found it difficult. I noted in “7. Copernicus’ On the Revolutions of the Heavenly Spheres” how grateful I was to Adler and Wolff for helping me understand (somewhat) the readings from Ptolemy and Copernicus. I feel the same way about the help that they gave me in trying to understand Kepler’s Epitome of Copernican Astronomy.

“Kepler is famous for his three laws of planetary motion. The first law states that the planets move in ellipses (not circles around the sun … The second law tells us how long the planet takes to traverse any portion of its orbit … Kepler’s third law [demonstrates] that a definite relationship exists between the periodic times of the planets (the length of time for one revolution around the sun) and their distances from the sun.” (Adler and Wolff, pages 107-08; 108; 109)

“Kepler’s achievement involves much more than the substitution of ellipses for circles.… [He] deliberately and self-consciously rejects the principles of traditional astronomy. [Although] he agrees with that there must be regularity in the planetary movements … he rejects as illegitimate the conclusion that therefore the planets’ movements must be circular and uniform” (Adler and Wolff, pages 110-11). Adler and Wolff support their claim by observing that Kepler lists four arguments given by the ancients to support their conclusion and showing how he rejects each as either false or inconclusive.

Questions about Kepler’s Work

1. Was there a Keplerian revolution?

2. What is the character of Kepler’s reasoning?

7. Copernicus’ On the Revolutions of the Heavenly Spheres

A major contribution to Western thought was the publication in 1543 of … On the Revolutions of the Celestial Spheres … by Nicolaus Copernicus, Polish astronomer, who is noted for the Copernican theory of the heavens. By attributing to the Earth a daily motion around its own axis and a yearly motion around the stationary Sun, Copernicus developed an idea that had far-reaching implications for the rise of modern science. Henceforth, the Earth could no longer be considered the center of the cosmos; rather, as one celestial body among many, it became subject to mathematical description.

Thus opens the article “Copernicus” in The New Encyclopedia Britannica (Encyclopedia Britannica, 1974, volume 5, page 145). I’ve now reached Copernicus’ On the Revolutions of the Heavenly Spheres in my rereading of selections from Great Books of the Western World guided by Encyclopedia Britannica’s The Great Ideas Program. Mortimer J. Adler and Peter Wolff present a guide to its Introduction and Book I, Chapters 1-11, in volume 3 of The Great Ideas Program, Foundations of Science and Mathematics (Encyclopedia Britannica, 1960, pages 91-103).

Adler and Wolff divide their guide into five sections: I. an introduction to the reading; II. an explanation of what Copernicus adopted unchanged from Ptolemy’s theory (Ptolemy’s theory was the subject of my last reading from Foundations of Science and Mathematics; see https://opentheism.wordpress.com/2018/06/15/6-ptolemys-the-almagest/); III. an explanation of how much simpler Copernicus’ heliocentric (sun-centred) system is than Ptolemy’s geocentric (earth-centred) system; IV. an explanation of how Copernicus’ hypothesis showed that the universe is much bigger than it had been conceived to be by Ptolemy’s hypothesis; and V. a discussion of three questions about the two hypotheses.

Here I’ll just sketch Copernicus’ life and present the three questions which Adler and Wolff pose and summaries of their answers to the questions. But first I’ll note how impressed I am by the work of both Ptolemy and Copernicus and how grateful I am to Adler and Wolff for helping me understand (somewhat) the readings from Ptolemy and Copernicus.

Copernicus

Nicolaus Copernicus was born in Torun, Poland, on February 19, 1473. His father (a prosperous merchant) died when he was ten, and an uncle (a priest who became a bishop in 1489) adopted him and saw that he got a good education. In 1491 he entered the University of Cracow (in Poland), where he became interested in the study of astronomy. In 1496 he went to Italy to study canon law in preparation for administrative work in the Church. However astronomy (and mathematics) remained his main interest. In 1506 he returned to Poland, where he served as advisor and physician to his uncle until his uncle’s death in 1512. He then became a canon at the Cathedral of Frauenburg–he had been appointed to it in 1497 but took a leave of absence from it to do further studies and then to work under his uncle–and remained there until his death.

All the time that he was a canon, Copernicus carried on his work in astronomy. Although making observations with the unaided eye, using mathematics and logic he concluded that Ptolemy’s system was wrong and worked out his system. Not wanting to revolt against the authority of the Church, he didn’t announce his findings. However he did share his findings with friends and at their urging finally allowed his work to be published. He saw an advance copy of On the Revolutions of the Heavenly Spheres on the day of his death, May 24, 1543.

Questions

– What are the similarities and differences between Ptolemy’s and Copernicus’ methods?
Adler and Wolff identify these similarities: both emphasize the role of hypothesis in astronomy; both insist on uniform circular motion as the appropriate motion of heavenly bodies; and both are non-mechanical, not paying nay attention to any forces, attractions, or repulsions between the heavenly bodies.

– Can the Copernican hypothesis be confirmed by experimental or observational evidence?
Copernicus presented and defended his theory without the aid of such evidence, but Adler and Wolff demonstrate that today there is ample evidence for the validity of his theory. That the earth has a daily revolution is confirmed by the Foucault pendulum; see https://en.wikipedia.org/wiki/Foucault_pendulum. That the earth makes an annual revolution around the sun is conformed by Venus’s having phases as seen from the earth.

– Without the aid of the Foucault pendulum or the telescope (neither of them possessed by Copernicus), is there any way of judging between Copernican and Ptolmaic hypotheses? Both hypotheses explain the appearances, but Copernicus’ theory is much simpler.

6. Ptolemy’s The Almagest

The motions of the heavenly bodies have played a prominent role in the thoughts and lives of men since the earliest known times, The regularities of the stars in their courses have guided men in the planning and harvesting of food, in hunting and fishing, and other vital activities. The heavenly order and harmony has also played a role in religious life and in thought about the basic nature of the universe. This intense practical and spiritual interest in the heavens accounts for astronomy’s becoming the most highly developed natural science of the ancient world. The work of Ptolemy of Alexandria represents the peak of this development. ( Mortimer J. Adler and Peter Wolff, Foundations of Science and Mathematics, volume 3 of The Great Ideas Program, Encyclopedia Britannica, 1960, page 75)

Thus Mortimer J. Adler and Peter Wolff open their guide to Ptolemy’s The Almagest, which I’ve now reached in my rereading of selections from Great Books of the Western World guided by The Great Ideas Program. They go on to observe that although we no longer accept the picture of the world which Ptolemy drew–that the earth is the centre of the planetary system and that the heavens have perfect regular movement, we do accept many of his purposes and methods, pursuing astronomy as a science. They conclude their introduction to their guide to The Almagest thus: “In short, the heritage which Ptolemy has left us is that the entire physical universe must and can be explored by the mind of man” (Adler and Wolff, page 76).

In the guide itself, Adler and Wolff consider what is known about Ptolemy, summarize Chapters 1-8 of Book I of The Almagest, investigate Ptolemy’s purpose in The Almagest, examine the apparent irregularity in the motion of the sun, and discuss these questions:
– What do you think of the principle that all heavenly motions must be reduced to uniform circular motion?
– Is Ptolemy’s The Almagest a work of “celestial mechanics”?
– Are Ptolemy’s explanations of planetary motion true or false?
Here I’ll just sketch what is known about Ptolemy’s life and list the titles of the chapters of The Almagest assigned for reading in The Great Ideas Program.

Ptolemy
Ptolemy’s full name was Claudius Ptolemaeus. His birth and death dates aren’t known, but Adler and Wolff suggest that they were about 100 and 178 A.D. He worked in or near Alexandria in Egypt. Greek astronomy culminated in Ptolemy. His work drew heavily from his predecessors, especially Hipparchus (about 130 BC) but he had no successors. His fame rests chiefly on The Almagest (originally known as “The Mathematical Collection”), but he also composed many shorter astronomical and mathematical works. After The Almagest, his most important work is his Guide to Geography, which was for geography what The Almagest was for astronomy until well into the Renaissance. (The Renaissance was the great revival of art, literature, and learning in Europe in the 14th, 15th, and 16th centuries.)

The Almagest
The Almagest contains 13 books (indicated below by Roman numerals), each with several chapters (indicated below by Arabic numerals).
I, 1. Preface
I, 2. On the Order of the Theorems
I, 3. That the Heavens Move Spherically
I, 4. That the Earth, Taken as a Whole, is Sensibly Spherical
I, 5. That the Earth is in the Middle of the Heavens
I, 6. That the Earth has the Ratio of a Point to the Heavens
I, 7. That the Earth Dies Not in any way Move Locally
I, 8. That There Are Two Different Prime Movements in the Heavens
(In I, 8, the two movements are the general daily motion from east to west and the slower movement of the planets, including the sun and moon, from west to east.)
II, 3. How, with the Same Things Given, the Height of the Pole is Given, and Conversely
II, 4. How One is to Calculated at What Places, When, and How Many Times, the Sun Comes to the Zenith
(In II, 3 and 4, Ptolemy explains the apparent irregularity in the motion of the sun. Adler and Wolff devote almost half of their guide to commenting on the two chapters.)

5. Archimedes’s On Floating Bodies

In my rereading of selections from Great Books of the Western World guided by The Great Ideas Program, I’ve reached the fifth reading in the third volume of The Great Ideas Program, Foundations of Science and Mathematics by Mortimer J. Adler and Peter Wolff (Encyclopedia Britannica, 1960), Archimedes’s On Floating Bodies. I considered his On the Equilibrium of Planes in an earlier post. See it for some information about Archimedes.

In their guide to On Floating Bodies, Adler and Wolff describe it as “a masterpiece of simplicity.” They continue:

All he asks us to grant him is a single postulate stating the characteristics of water and other fluids. The rest is a matter of geometric reasoning. This is all we need for the study of floating bodies, with Archimedes as our teacher.
The odd and wonderful thing about this is that you do not even need water or objects to put in it. It may be fun to verify some of the theoretical demonstrations in the washbowl or bathtub. But Archimedes gives the scientific essentials about floating bodies without experimentation. As you follow him along, you, too, may want to cry out “Eureka.”
( Mortimer J. Adler and Peter Wolff, Foundations of Science and Mathematics, volume 3 of The Great Ideas Program, Encyclopedia Britannica, 1960, pages 61-62)

“Eureka” refers to a story told about Archimedes. Hiero, the king of Syracuse (a Greek city on the island of Sicily), had ordered a new crown. The crown was to be made of pure gold, but the king suspected that the jeweller had mixed in some less costly silver. He asked Archimedes to determine whether the crown was made of pure gold or was a mix of gold and silver. The solution came to Archimedes while he was in the bathtub, and he was so excited that he jumped out of the tub and ran naked through the streets shouting “Eureka [I have found it]!”) Archimedes’s solution was based on equal weights of gold and silver having different volumes and thus that if a crown made of gold and silver were lowered into water it would displace more volume than a crown made of pure gold would.

The postulate which Archimedes asks us to grant him is:

Let it be supposed that a fluid is of such character that, its parts lying evenly and being continuous, that part which is thrust the less is driven along by that which is struck the more; and that each of its parts is thrust by the fluid which is above in a perpendicular direction if the fluid be sunk in anything and compressed by anything else. (Archimedes, On Floating Bodies, Great Books of the Western World [Encyclopedia Britannica, 1952], volume 11, page 538)

The postulate is followed by seven propositions. The first four are preliminary and are summarized thus by Adler and Wolff:

The cross-section of a sphere, through the center, is always a circle (Prop. 1), If a body equal in specific gravity with a certain fluid is submerged in that fluid, it will neither sink to the bottom nor stick out of the fluid, but will rest just below the surface (Prop. 3). A body lighter than the fluid in which it is submerged will partially submerge and partly project out of the fluid (Prop. 4). (Adler and Wolff, page 66).

The other propositions are:

Proposition 5. Any solid lighter than a fluid will, if placed in the fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced.
Proposition 6. If a solid lighter than a fluid be forcibly immersed in it, the solid will be driven upwards by a force equal to the difference between its weight and the weight of the fluid displaced.
Proposition 7. A solid heavier than a fluid will, if placed in it, descend to the bottom of the fluid, and the solid will, when weighed in the fluid, be lighter than its true weight by the weight of the fluid displaced.
(Archimedes, pages 540-41)

Archimedes bases his proof for each of the propositions on a mathematical diagram, which he explains. I also found Adler and Wolff’s explanations of them (Adler and Wolff, pages 66-67) helpful.

Besides commenting on the propositions in On Floating Bodies, Adler and Wolff summarize Archimedes’s accomplishments in mathematics, recount two of the “fabulous” stories told about Archimedes, and discuss the following questions:
– How might the “crown problem” been solved?
– How would you measure the amount of weight a body loses by being immersed in water?
– What is the empirical evidence in which On Floating Bodies is based?

4. Nicomachus’s Introduction to Arithmetic

In my rereading of selections from Great Books of the Western World guided by The Great Ideas Program, I’ve reached the fourth reading in the third volume of The Great Ideas Program, Foundations of Science and Mathematics by Mortimer J. Adler and Peter Wolff (Encyclopedia Britannica, 1960), Nicomachus’s Introduction to Arithmetic.

Nicomachus of Gerasa (probably a Greek city in Palestine) flourished around the end of the first century. His Introduction to Arithmetic was the first work to treat arithmetic as a discipline independent from geometry. Setting out the elementary theory and properties of numbers, it was used as a textbook for a thousand years.

Adler and Wolff consider the first sixteen chapters in Book One of Introduction to Mathematics and divide their study of them into these sections:
I ‒ a comparison of the concerns of Nicomachus’s and today’s arithmetic.
II ‒ an explanation of Nicomachus’s classification of numbers.
III – a presentation and discussion of three questions on the reading.
Here I’ll summarize briefly I and II, present the three questions, and indicate how Adler and Wolff answer them.

Comparison of Nicomachus’s and Today’s Arithmetic
Today we expect an arithmetic textbook to show us how to perform arithmetical operations such as addition, subtraction, multiplication, division. However Introduction to Arithmetic studies numbers themselves and their properties, relations, and classification. Adler and Wolff attribute this to Nicomachus’s belonging to the school of Pythagorus, a group of mathematicians and philosophers who thought that the principles of mathematics were the principles of all things.

Classification of Numbers
Like Euclid Nicomachus classifies numbers as even, odd, even-times even, even-times odd, odd-times even, and odd-times odd numbers. For Nicomachus even-times even numbers are numbers that have only even factors; even-times odd numbers are even numbers that are the product of 2 and an odd number; odd-times even numbers are numbers that can be divided by 2 several times eventually arriving at an odd number; and odd-times odd numbers are numbers that are the product of two odd numbers. Nicomachus also talks about perfect numbers, which are considered below in the first question.

Questions
– What is a perfect number?
A perfect number is a number which is the sum of its factors. Examples are 6 and 28, 6 because it is the sum of 1, 2, and 3 (its factors) and 28 because it is the sum of 1, 2, 4, 7, 14 (its factors)..
– What is a prime number?
Nicomachus says that a prime number “is found whenever an odd number admits of no other factor save the one with the number itself as denominator, which is unity, for example, 3, 5, 7, …” (Nicomachus, Introduction to Arithmetic, Great Books of the Western World [Encyclopedia Britannica, 1952], volume 11, page 817). However we generally consider 2 to be a prime number too because it has no other factors but 1 and itself.
– Is the Pythagorean concern with numbers foolish and superstitious or is there some point to it?
Adler and Wolff answer, “Modern mathematicians are still concerned with numbers, and with the properties of them as primeness, evenness, perfectness, etc. All these properties are treated in the Theory of Numbers.” However they go on to concede that “perhaps the Pythagoreans went to extremes when they made number a cosmological principle and considered numbers as the elements or principles of things.” (Adler and Wolff, Foundations of Science and Mathematics, Encyclopedia Britannica, 1960, pages 58-59).

3. Archimedes’s Equilibrium of Planes

The names of many ancient mathematicians and scientists are quite unfamiliar to most of us. But there is one shining exception‒Archimedes of Syracuse. His name usually brings to mind a man sitting in a bathtub with the water running over, then running out into the streets unclad, crying “Eureka (I have found it),” the “it” being the answer ro a scientific problem. We like to think of him as typical of the man of science who is so immersed in his studies that he pays no attention to his surroundings. We also remember Archimedes’ famous boast: “Give me a fulcrum (a place to rest a lever on), and I will move the earth.” (Mortimer J. Adler and Peter Wolff, Foundations of Science and Mathematics, volume 3 of The Great Ideas Program, Encyclopedia Britannica, 1960, page 33)

In my rereading of selections from Great Books of the Western World guided by The Great Ideas Program, I’ve reached the third reading in the third volume of The Great Ideas Program, Foundations of Science and Mathematics by Mortimer J. Adler and Peter Wolff (Encyclopedia Britannica, 1960), Archimedes’s On the Equilibrium of Planes. Adler and Wolff consider the first seven propositions in Book One of On the Equilibrium of Planes and divide their study of them into these sections:
I ‒ a consideration of what the subject of the book is; after observing that it is neither a purely mathematics treatise nor an introduction to experimental natural science, they describe as an example of mathematical physics.
II ‒ a discussion of how a book about physics can get along without any experiments; they conclude that they aren’t necessary because “The Equilibrium of Planes takes existing experimental and observational knowledge and, with the help of mathematics tools, refines it and recasts it in different language” (Adler and Wolff, page 37).
III ‒ a close look at Propositions 6 and 7, which together constitute the Law of the Lever. They show how Propositions 1-5 prepare for Propositions 6-7; then they assign numerical values to the weights and distances involved in Proposition 6 and go through Archimedes’s proof of it (I got stuck going through Archimedes’s proof but was able to follow Adler and Wolff’s explanation of it); and finally they use Proposition 6 to prove Proposition 7.
IV – a presentation and discussion of three questions on the reading; see below.

Here I’ll identify Archimedes, present the questions which Adler and Wolff present on On the Equilibrium of Planes, and indicate how Adler and Wolff answer the questions.

Archimedes

Archimedes was born in the Greek city-state of Syracuse around 287 B.C. As a young man he spent some time in Egypt, where he may have studied with the pupils of Euclid in Alexandria and where he invented the water-screw as a means of drawing water out of the Nile for irrigating the fields. His mechanical inventions won great fame for him and figure largely in the traditions about him, such as those referred to in the passage from Adler and Wolff with which I introduced this article. However, except for a lost work On Sphere-Making, he wrote only on strictly mathematics subjects. His absorption in his mathematical investigations that he forgot his food and neglected his person. It may even have caused his death. “In the general massacre which followed the capture of Syracuse by Marcellus in 212 B.C., Archimedes was so intent upon a mathematical diagram that he took no notice, and when ordered by a soldier to attend the victorious general, he refused until he should have solved his problem, whereupon he was slain by the enraged soldier” (Great Books of the Western World, Encyclopedia Britannica, 1952, volume 11, page 400).

Questions

– What are the assumptions underlying Archimedes’ proof of Proposition 6?
Archimedes doesn’t define “centre of gravity” but seems to use two different meanings for it. “The first meaning, used in Propositions 4 and 5, is apparently that of a point of balance, or the point at which a fulcrum should be located in order to balance a system of weights.… The second meaning [which is used in Proposition 6] is a point at which we imagine the entire weight of a body or system of bodies concentrated” (Adler and Wolff, page 41). In Proposition 6 it has to be assumed that both meanings are the same.
– Why must there be two proofs of the Law of the Lever, one for commensurable and one for incommensurable magnitudes?
Adler and Wolff give a lengthy explanation, which includes: “Archimedes’ proof of Proposition 6 depends on finding a common measure of weights A and B and dividing each of them into parts equal to the common measure. These equal parts are then strung out along the line LK. It is obvious that this method of proof will not work for incommensurable magnitudes, since by definition they have no common measure.” (Adler and Wolff, pages 43-44)
– What is the Law of the Lever for magnitudes that do not balance?
Adler and Wolff show that Archimedes seems to know it in proving Proposition 7 and express it in modern terms as “The side with the smaller moment will rise, while the side with the larger moment will be depressed,” where the moment (of force) is “the product of the weight and the distance from the fulcrum at which that weight is applied.” (Adler and Wolff, page 45)