From the President:
Dear Reader,
As the new year approaches, those of us involved in the Arts of Liberty Project have much to be grateful for. January 6 will mark the first birthday of its new parent organization, the Boethius Institute for the Advancement of Liberal Education. Read More
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Creativity in STEM and Bill McLean
by Liam Collins
I am blessed to have received a classical liberal arts education. I was homeschooled through high school, and then graduated with a Bachelor’s in liberal arts from the Great Books program of Thomas Aquinas College. After that I found myself at a bit of a loss. Read More
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The Path Less Traveled: Early Education in the Liberal Arts
by Forest Barnette
I fell in love with liberal education during the pandemic. I was teaching first grade at a poor school that had only recently decided to renew its curriculum and embrace the liberal arts. But through all of the training sessions, retreats, and curriculum writing, I continually encountered the same frustration: All of this would be so useful if only my students could read! Read More
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Einstein's Imagination
Part of Einstein’s genius was his ability to think things through using just his imagination. In this excerpt, Einstein shows how imagining an elevator accelerating in empty space led him to posit that gravity can be understood as a relative phenomenon. Read More
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Further Enrichment
The Revolutionary Influence of the First English Children’s Novel — Children's novels are an indispensable tool for elementary school teachers, but few know the origin of the genre. Read this brief account of the genre-defining book and its lasting impact.
Rediscover the Magic of Mathematics — As an antidote to the dull practicality Liam Collins describes in his article, mathematician Satyan Devadoss highlights the beauty of mathematics found in exploring the unknown.
Aquinas' The Office of the Wise Man — On our The Examining Life podcast, Dr. Andrew Seeley dives into Aquinas' account of the wise man with Dr. John Boyle. They also discuss St. Thomas More as an an example of the wise man.
The Hopeful Future of Catholic Education — Chris Weir, the executive director of the Camino Schools discusses his journey and his hope for the future of Catholic schools on this episode of Anchored, a podcast by the Classical Learning Test
Uncovering the Hidden Blessings of the Melancholic — Understanding the four temperaments can help us know ourselves. Melancholics sometimes seem to have the worst temperament. This article reveals the temperament's beauty.
Floriani — Floriani is a men’s vocal ensemble dedicated to serving the Church and saving the culture through the beauty of sacred music. They also have a podcast that teaches listeners to sing ancient Christian chant.
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Dear Reader,
As the new year approaches, those of us involved in the Arts of Liberty Project have much to be grateful for. January 6 will mark the first birthday of its new parent organization, the Boethius Institute for the Advancement of Liberal Education. We have had a solid first year and are ready to expand our efforts in 2024, including publishing our Introductory Geometry and Arithmetic text by Michael Augros, author of Who Designed the Designer? Dr. Augros does an excellent job re-presenting key content from all of Euclid’s Elements in a clear, accessible way for high school students and life-long learners. This volume has been downloaded for years by many around the world from the Arts of Liberty website; its new existence in a durable, easy-to-read printed volume will introduce the wonders of classic geometry to thousands more learners.
Returning the Quadrivium to its rightful place in liberal arts studies is high on the Boethius Institute priority list. Properly presented, mathematics introduces the young mind to the life of knowledge by arousing wonder through careful reasoning. As Dr. Augros writes,
Geometry is full of wonders. At every level of this science, from the most elementary to the most advanced, we are confronted with the unexpected. Often the seemingly possible turns out to be impossible, and conversely what at first seemed impossible turns out to be possible.
Our Quadrivium students at the Pascal Instituut experienced this in a recent discussion of Euclid’s treatment of incommensurable magnitudes. Said one, “It is not possible that two lines can be incommensurable! You can divide them into parts as small as you want. You must be able to find a common measure!” And yet, Euclid shows it is not so.
Classical mathematics is so formative because it occupies the sweet spot for human knowing by relying on both imagination and argumentation. For this reason among others, encouraging the development of the imagination in the young is crucial for learning ordered to knowledge, as elementary teacher Forest Barnette points out in her article “On Early Education in the Liberal Arts:”
Imagination is not an alternative to reality – it’s the key to reality. Imagination helps us to understand the most fundamental truths around us: it helps us to explore what is beyond the physical limitations of the moment; it helps us to explore what isn’t by showing us what is preventing it from being so; and it helps us to explore what could be by going beyond the is and isn’ts and into the unknown.
Nurturing the mathematical imagination is not only delightful in itself, but also immensely helpful in making science both practical and liberating, as aeronautics engineer Liam Collins witnesses in his article, “Dr. William McLean and Imaginative Creativity.” Albert Einstein shows us his creative imagination at work in using a magical space elevator to provide the fundamental insight for his theory of general relativity.
As Socrates and Plato experienced, classical mathematics can set ablaze the love of wisdom in a budding philosopher. We taste truth and yet cannot help but question existence. Working through the Elements was my first experience of learning indubitable truth. At the same time, doubts about the reality of points without parts and breadthless lengths were also present from the very beginning. And, as far as we can tell, regular 15-sided polygons inscribed in a circle exist nowhere in the natural world, much less inscribed dodecahedrons; probably we can't really make tangents to circles. But the delight in geometry and arithmetic does not depend upon being able to find their objects in physical reality; in some way it is enhanced because we are easily convinced that we cannot. We find them in our imagination. And yet they are true and objective -- the imagination is fed by and determined by our experience of sensible reality, as it is empowered by the intellect.
I am very grateful to announce that last month we received IRS approval of our tax-exempt status. We hope the publication of Dr. Augros’s volume will just be the beginning of our Library of Liberal Arts series. Several more volumes are ready to be edited, and we need to commission our volumes on grammar, rhetoric, and music. If you are able to make a financial contribution, we will be deeply grateful. Small gifts go a long way in a new organization like ours, and also help us to show potential major benefactors that we are serving a widely felt need.
We hope your hearts rejoice in the peace of this season.
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Creativity in STEM and Bill McLean
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by Liam Collins
Co-Founder, Human-Centered Tech
I am blessed to have received a classical liberal arts education. I was homeschooled through high school, and then graduated with a Bachelor’s in liberal arts from the Great Books program of Thomas Aquinas College. After that I found myself at a bit of a loss. My education did what it promised: it ignited wonder, vivified my imagination, and engaged my heart and emotions while at the same time developing my calculating mind. But, though I liked the philosophy I’d studied, I didn’t think that I was capable of doing it for the rest of my life while staying attached to reality. Moreover, I didn’t really like the idea of relying on the charity of others for my livelihood, as I would likely have to do at least indirectly in becoming a professor and taking a job at any school whose existence rests on the beneficence of its donors. Not that there’s anything the slightest bit wrong with doing so; we need great teachers, which is why successfully wealthy people are willing to donate to the institutions that foster them! But we need the donors too, and part of me wanted the challenge of seeing if I could be a provider in that way.
So academia was out. But then, how was I to bring the goods of my education into the rest of my life? This talk of joy and wonder and imagination sounds great, in theory, but the modern world is hard, scientific, competitive, complex, and process driven. Did a traditional education in the ways of wonder and imagination really prepare me as a young graduate to thrive in the 21st century world?
In hindsight, this question of how to integrate what’s wonderful with what’s practicable has been one of the central themes of the 33 years I’ve lived thus far. The liberal arts tradition of education might itself be to blame here. The “liberal arts” are often defined sharply in opposition to “servile arts” as those that befit free men versus the tasks given to slaves. Was I choosing mental servitude for the sake of material thriving? This left me wondering: could I pursue the so-called servile arts in a way that exercised my wonder and imagination, in a way which led towards freedom?
To express my dilemma more generally, does the student trained in the ways of wonder and imagination have the wherewithal to bountifully provide food, shelter, and security for himself, his family, and his countrymen, to be as free physically as he is intellectually? Ideally, should we not only be able to be free both practically and intellectually, but able to do so in a well-integrated way, without having to painfully wait it out through a boring workday while hoping for a precious little time afterwards with which to dwell on things which actually feel worth freely pursuing?
I decided to pursue a so-called STEM career, having some hope that it would not only allow me to support a family and be in a position to be generous, but also would involve interesting work. I had always liked math and science and airplanes, so I decided on aerospace engineering, this time starting with a second bachelor’s at a state school.
Unfortunately, many of the things which I’d loved most about science and engineering up to that point were significantly lacking in the engineering schooling I experienced, things like exercising imaginative creativity, or the joy of seeing the incarnation of abstract theories in real physical devices. Much of what I actually found seemed to be a sort of advanced box-checking exercise. I hoped that this was an anomaly, perhaps due to the field of aerospace engineering being past its prime or to my having chosen a lackluster engineering program, and kept doggedly on, ultimately getting a job as an aerospace engineer at China Lake Naval Base, the U.S. Navy’s last remaining live fire test range, whose vast expanse stretches out at the southeast base of the beautiful Sierra Nevada mountains. But there too I experienced a certain deadness, an acedia whose sources I couldn’t completely pin down. There were clearly embers of what had once been a fire of inspiration at China Lake; I could sense them in the glow in an old engineer’s eyes, or the cool artifacts around the base, or here or there in the pages of a dusty book in the library. But despite the many ostensibly cool projects and the billions of dollars of annual budget on the base, the original fire was clearly long since gone, burned out in a sea of red tape and wasted time and money. I was tempted to give up on engineering altogether. However, there was one particular ember which really stood out, giving me confidence that engineering had, in fact, existed at least at one time in something like the way I had always idealized it, and in turn giving me something to continue to strive for in the engineering world.
That glowing remnant from a past age was found in the collected speeches of Dr. William B. McLean. The son of a Presbyterian minister, William Burdette McLean (1914–1976) was a civilian physicist at what was then known as the Naval Ordnance Test Station (NOTS) in the desert of China Lake, California, during the early Cold War. He led the development of the Sidewinder air-to-air missile, a brilliant and innovative piece of engineering, which was carried by aircraft defending the west from encroaching communism around the globe in greater numbers than any other missile before or since. After leading the Sidewinder team in the late '40s and early '50s, McLean was promoted to Technical Director at NOTS during which time he led what is remembered by those involved as a golden era of engineering and innovation on the station in the service of America's freedom, with the engineering output to prove it. In other words, he was neither a slave nor a mere dreamer; he was both a capable and inspiring leader and one of the true practical geniuses of American history.
While McLean didn't write books, he was often asked to speak, particularly after the success of the Sidewinder program, and we are fortunate enough to have the transcripts of many speeches. I found these typewritten transcripts fascinating; ultimately they renewed my belief that imagination and creativity should be an integral part of my career field.
One of the things which comes through most clearly in reading Bill McLean's speeches is the centrality of his regard for creativity.
“I believe if the United States is to be successful in either its military or economic competition, we will in the future need to learn to appreciate and to foster creative design capabilities.”
But he also saw that the typical formation of the young squelched creativity.
The number of people who start life with a high degree of creative ability and creative drive is unknown because the forces of society begin so rapidly to act to repress and restrain the curiosity and experimental operations of the young child.
He believed the central effort of the creative scientist is to see a good solution in his imagination.
The designer… needs to outline as many ways of accomplishing the design as he can imagine… Industrial laboratories are handicapped by a natural desire to improve on what exists, by military specifications that are unimaginative.
This means that managers must encourage the creative freedom of those on their teams.
As a man responsible to others for the function of managing research… I need to be in a position to understand and accept new ideas and eventually to judge the ability of people to carry out the work which they are interested in doing. In this type of judgment I would place first priority on the interest and enthusiasm which a man shows in the work which he is doing and, second, on his skill in visualizing and planning the crucial experiments which must be carried out in order to check new theories or hypotheses.
This visualization is so critical for effective and elegant design that McLean is willing to recommend a radically unconventional design methodology, namely design residing in the imagination of a single designer, along the lines of a wall mural.
It seems to me that the creation of a missile system would progress more effectively if it were recognized to have many of the same problems as the creation of a large mural painting. Many useful analogies might then result. The creation of a mural is obviously too large a job for one man and yet, at the same time, it must represent an integrated whole, rather than a collection of parts. In the case of the mural, we have adopted the practice of selecting a master artist whose responsibility is to conceive a picture in accord with the general message which is to be conveyed. He then uses his imagination, his understanding of the materials and tools available, and his knowledge of the abilities of his assistants to lay out an overall design. Committees can review his work and make suggestions, but they cannot take over his responsibility for it. Once the general concept has been sketched out, many people can begin to work using their own specific abilities to fill in the various parts of the picture. As a result, we have an integrated creation that reflects primarily the skill, ability, and experience of the master artist, but which also uses the individual skills of his assistants to a maximum.
McLean proposed that management strategy that aims to maximize imaginative creativity and enjoyment is the necessary way to both practice and preserve the freedom we so deeply treasure.
I hope that we as a Nation can choose in the management of our business and our military programs the type of management which maximizes enjoyment, participation, and the contributions of individual creativity, rather than the type of management whose goals and objectives are set from the top and which is budgeted, planned, and integrated to achieve objectives on schedule without consideration of possible creative inputs. One type of management will strengthen what we have variously called ‘The Free Competitive System,’ ‘The American Way of Life,’ or ‘Life Liberty, and the Pursuit of Happiness.’ The other type of management by overinsistence on the importance of budget and schedule, comes perilously close to conditioning us to the type of organization which believes that man's highest goal is to achieve and surpass through successive five and ten year plans.
What will it profit a man if he gains the whole world, yet forfeits his soul?
Modern culture stands in awe of the pragmatic fruits of empirical science. However, many are content (as the central planners were) to extort these fruits by any means, leaving the wonder and gratitude which ought to accompany their uncovering (and which originally gave rise to scientific inquiry itself) as a relic of the past. Especially in large corporations and in government, technological development is often seen as the product of a vast machine, the result of a method, in which individual people are merely cogs; taken to its extreme, this view sees imagination, inspiration, and even freedom as no longer necessary. The classically educated liber, on the other hand, sees the wonder in the world, the necessity of a rightful ordering of technology, and the value of knowledge for its own sake as well as for its fruits, but may not always have a ready answer to the often earnestly asked question, “but what are you going to do with that education if not teach or become a priest?” After the initial shock of the encounter with this widening gulf between practicality and wonder, however, one discovers that it is not only possible but necessary that we bridge the gulf, both for the sustenance of wonder (and wonderers) and also for the fullest attainment of the pragmatic. And with the transformation offered by the Christian understanding of the redemptive power of suffering and the Cross, classical thought becomes capable of seeing the full truth, that man is called to imitate his Creator with smaller creations of his own, taking joyful hope not only in the fruits of his labors but also in loving acts of labor itself. But fully carrying this spirit of wonder filled creativity into the pragmatic world of modern technology is a difficult task, undertaken by few and done well by fewer. Those rare few who have really done so well are examples worth treasuring and learning from. Bill McLean is one such treasure.
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The Path Less Traveled: Early Education in the Liberal Arts
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by Forest Barnette
Academic Renewal Specialist, Archdiocese of Denver's Office of Catholic Schools
I fell in love with liberal education during the pandemic. I was teaching first grade at a poor school that had only recently decided to renew its curriculum and embrace the liberal arts. But through all of the training sessions, retreats, and curriculum writing, I continually encountered the same frustration: All of this would be so useful if only my students could read!
Our school primarily served immigrant families, so a variety of factors – particularly competing spoken languages between home and school – delayed literacy. I knew these children deserved the freedom provided by the liberal arts and, in fact, that the very philosophical underpinnings of liberal education all but demand that preliterate learners be included in this fully human way of engaging reality. But so few of our resources could accommodate these sweet, eager minds.
There’s much work to be done in exploring best practices of modern-day early liberal education. While I’m convicted of that, I’m not qualified to provide much of the necessary scholastic momentum. Instead, I would like to humbly highlight three qualities that, it seems to me, set early liberal education apart from other pedagogies. Perhaps these can be a starting point for deeper consideration by those wiser and more experienced than me.
First: imaginative.
Modern pedagogy often uses children’s natural propensity for imagination as an engagement tool for otherwise sterile lessons. This use is improper. Imagination is not an alternative to reality – it’s the key to reality. Imagination helps us to understand the most fundamental truths around us: it helps us to explore what is beyond the physical limitations of the moment; it helps us to explore what isn’t by showing us what is preventing it from being so; and it helps us to explore what could be by going beyond the is and isn’ts and into the unknown.
One of my favorite and oft-repeated lessons comes to mind. Each year, my own five- and six-year-old students spent our much-anticipated Dinosaur Day studying fossils, biological adaptation, deductive reasoning, and earning “doctorates” in paleontology. The crowning moment of the event was when, donning their handmade T-Rex hats, they “became” dinosaurs. With elbows tucked to their sides and secured with soft, oversized yarn, the young T-Rexes were simply asked to extend two fingers from their closed fists and then go about the rest of the day. It wasn’t a particularly interesting itinerary for human students – eating a snack, putting on a backpack, opening the door, drawing a picture, free play with friends – but the dinosaurs alternated between laughter, frustration, and exhaustion as they discovered the evolutionary disadvantages of a T-Rex’s short arms and few, non-opposable digits. Some students resorted to holding pencils in their mouths. A pair of boys playing Tic-Tac-Toe with sidewalk chalk repeatedly lost balance as their truncated arms failed to reach the ground, even from a kneeling position. Catching a fall was hard, getting up was even worse. Duck-Duck-Goose had to be adapted. Cretaceous chaos reigned.
No adult merely informing them about evolutionary adaptation would have seared the reality into their minds the way that imaginative play did all on its own.
Such an example confirms that when learning, young children ascend a ladder which is equal parts imagination and reality–often done through nature’s own pedagogy: play. Children need to begin in imagination, measure it against reality, and then return to imagination to process what they’ve learned. This means finding the virtuous middle between sterile lessons which employ imagination as an afterthought and abandoning children to their own devices in largely unstructured “play education.”
Second: nurturing of true schola.
To explain this, please pardon a brief departure from the topic at hand.
Schola refers to leisure devoted to learning. For the ancients the ability to study was leisure – that is, time away from the physical demands of survival. But education as leisurely seems contradictory to modern sensibilities. For myself, when I think of school, I think “restriction” and “stress.” When I think of leisure, I think “engaging” and “freedom.”
Freedom is the intention of the liberal arts; that is, freedom to see the Truth of things. The ability to teach oneself well is a freedom that opens up greater access to Truth, and knowing that Truth allows us to work with things as they are, rather than being restrained by assumptions, projections, and guesses.
Beyond even that, discovery is what happens when we see what is (that is: the Truth), embrace it, and make new connections. More properly, we make connections that are new to us. This discovery deepens our delight in the complex nexus of truths made by Him who is Truth so that we may further delight in Him. True understanding, then, is that which allows us to more deeply delight in Truth. If the liberal arts free us to see the Truth of things, the understanding gained therein frees us to delight in that Truth.
Here we return more directly to the topic at hand: the ideal of the liberal arts – to connect schola to the modern understanding of leisure and further, to the freedom to delight in God – is perhaps most easily achievable in early childhood.
These things should, of course, be intuitively connected at any age, but we live in a fallen world with a further fallen education system which has masterfully divorced leisure from discovery and discovery from delight for many of its students.
Early education is the ideal time to bring ancient and modern understandings of leisure together by making learning truly delightful. Examples are truly endless. Preschoolers may encounter evaporation firsthand as they “paint” with water on a hot sidewalk and watches their art disappear before their very eyes; kindergarteners may dissolve into fits of giggles as they learn to manipulate words by changing the first letter of “cat” to an “f”; first graders may be confronted with the difficulty of making a teepee stand on its own as they explore the difficult implications of a nomadic lifestyle; second graders alternating between laughter, frustration, and gratitude for the human form as they go about their day with their elbows tucked to their sides and only two fingers extended from their closed fists, emulating the evolutionary disadvantages of a T-Rex’s short arms and few, non-opposable digits.
Liberal arts education for preliterate learners must be marked by fostering the natural eagerness and delight of children while buttressing that posture against the empty cynicism of modern education.
It is a disservice to our children to strengthen them by building walls and obstacles against the rest of the world. Instead, we must foster a love of schola that is strong within itself by nurturing, to borrow Tolkien’s poetry, deep roots that are not reached by the frost. This creates a difficulty, addressed by the next mark of liberal arts early education.
Third: patient.
Deep roots are formed in secret, sometimes without measurable changes above the soil. One cannot measure wonder nor grade the gradual integration of numeracy, literacy, and reasoning into the bedrock of a child’s mind. This lack of qualitative measurement is a difficulty when assuring parents of the value of a slow and steady approach in these critical years. Parents–particularly young ones–almost unconsciously measure their children’s progress against the perceived progress of the offspring of their peers. While young students of more popular pedagogy may be able to spout off math facts or identify words memorized by sight, young liberal arts students may not necessarily display such superficial knowledge at the outset.
One student in particular comes to mind, who could barely associate letters with their sounds through kindergarten and most of first grade while her peers steadily progressed beyond her. Barring seasons of discouragement, she was engaged by the pedagogy. She paid attention, let herself be enamored by wonder at the content, but displayed little or no measurable growth. Discussions about retention were in progress. Suddenly, two weeks before the end of the academic year, she successfully sounded out a two-syllable spelling word in front of the class. Given another, she nailed it. And another. And one even more complex. Just in the nick of time, she had her breakthrough. Her determined mind had finally – and seemingly all at once – synthesized and integrated two years of patient, steady work and burst forth, shining with pride.
We as educators know the wait is well worth it, but will the parents? Will they fear the early years wasted if not immediately able to impress with shallow appearances? Will they maintain hope in the time that the seed is drinking, germinating and diving deep into darkness before it pushes through the soil to stretch in the sun? The inexperienced gardener may believe the sowing in vain and scoop up the kernels from their rich soil for fear they will never bear fruit. Therefore, without patience, the liberal arts model is unsustainable for early education. Schools will wither and die from dropping enrollment if impatient parents quickly transplant their children to shallow germination plates for sake of meeting arbitrary expectations set by trending pedagogy.
Whether metrics such as grades are appropriate at this age is a topic beyond the purview of this summary. What is clear is this: any method used to gauge the learning of preliterate students finds its proper place secondary to the pursuit of wonder and establishment of a deep love of learning Truth.
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Beyond the practical need to adapt curriculum and the reality that young learners deserve freedom, it makes sense to establish the foundations of human learning from the outset of each student’s academic adventure, rather than to try and patch in human pedagogy later.
Again, these waters are somewhat uncharted, so I’ll leave it to those wiser and more experienced than myself to more fully plot them out. But consider this the official call to action.
May we indiscriminately echo the invitation given by Truth Himself to all the children: come to me.
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Excerpt from Relativity: The Special and General Theories by Albert Einstein.
Part of Einstein’s genius was his ability to think things through using just his imagination. In this excerpt, Einstein shows how imagining an elevator accelerating in empty space led him to posit that gravity can be understood as a relative phenomenon.
In contrast to electric and magnetic fields, the gravitational field exhibits a most remarkable property, which is of fundamental importance for what follows. Bodies which are moving under the sole influence of a gravitational field receive an acceleration, which does not in the least depend either on the material or on the physical state of the body. For instance, a piece of lead and a piece of wood fall in exactly the same manner in a gravitational field (in vacuo), when they start off from rest or with the same initial velocity…We then have the following law: The gravitational mass of a body is equal to its inertial mass.
It is true that this important law had hitherto been recorded in mechanics, but it had not been interpreted. A satisfactory interpretation can be obtained only if we recognize the following fact: The same quality of a body manifests itself according to circumstances as “inertia” or as “weight” (lit. “heaviness”). In the following section we shall show to what extent this is actually the case, and how this question is connected with the general postulate of relativity.
WE imagine a large portion of empty space, so far removed from stars and other appreciable masses that we have before us approximately the conditions required by the fundamental law of Galilei. It is then possible to choose a Galileian reference-body for this part of space (world), relative to which points at rest remain at rest and points in motion continue permanently in uniform rectilinear motion. As reference-body let us imagine a spacious chest resembling a room with an observer inside who is equipped with apparatus. Gravitation naturally does not exist for this observer. He must fasten himself with strings to the floor, otherwise the slightest impact against the floor will cause him to rise slowly towards the ceiling of the room.
To the middle of the lid of the chest is fixed externally a hook with rope attached, and now a “being” (what kind of a being is immaterial to us) begins pulling at this with a constant force. The chest together with the observer then begin to move “upwards” with a uniformly accelerated motion. In course of time their velocity will reach unheard-of values—provided that we are viewing all this from another reference-body which is not being pulled with a rope. But how does the man in the chest regard the process? The acceleration of the chest will be transmitted to him by the reaction of the floor of the chest. He must therefore take up this pressure by means of his legs if he does not wish to be laid out full length on the floor. He is then standing in the chest in exactly the same way as anyone stands in a room of a house on our earth. If he release a body which he previously had in his hand, the acceleration of the chest will no longer be transmitted to this body, and for this reason the body will approach the floor of the chest with an accelerated relative motion. The observer will further convince himself that the acceleration of the body towards the floor of the chest is always of the same magnitude, whatever kind of body he may happen to use for the experiment.
Relying on his knowledge of the gravitational field (as it was discussed in the preceding section), the man in the chest will thus come to the conclusion that he and the chest are in a gravitational field which is constant with regard to time. Of course he will be puzzled for a moment as to why the chest does not fall in this gravitational field. Just then, however, he discovers the hook in the middle of the lid of the chest and the rope which is attached to it, and he consequently comes to the conclusion that the chest is suspended at rest in the gravitational field.
Ought we to smile at the man and say that he errs in his conclusion? I do not believe we ought if we wish to remain consistent; we must rather admit that his mode of grasping the situation violates neither reason nor known mechanical laws. Even though it is being accelerated with respect to the “Galileian space” first considered, we can nevertheless regard the chest as being at rest. We have thus good grounds for extending the principle of relativity to include bodies of reference which are accelerated with respect to each other, and as a result we have gained a powerful argument for a generalised postulate of relativity.
We must note carefully that the possibility of this mode of interpretation rests on the fundamental property of the gravitational field of giving all bodies the same acceleration, or, what comes to the same thing, on the law of the equality of inertial and gravitational mass. If this natural law did not exist, the man in the accelerated chest would not be able to interpret the behavior of the bodies around him on the supposition of a gravitational field, and he would not be justified on the grounds of experience in supposing his reference-body to be “at rest."
Suppose that the man in the chest fixes a rope to the inner side of the lid, and that he attaches a body to the free end of the rope. The result of his will be to stretch the rope so that it will hang “vertically” downwards. If we ask for an opinion of the cause of tension in the rope, the man in the chest will say: “The suspended body experiences a downward force in the gravitational field, and this is neutralized by the tension of the rope; what determines the magnitude of the tension of the rope is the gravitational mass of the suspended body.” On the other hand, an observer who is poised freely in space will interpret the condition of things thus: “The rope must perforce take part in the accelerated motion of the chest, and it transmits this motion to the body attached to it. The tension of the rope is just large enough to effect the acceleration of the body. That which determines the magnitude of the tension of the rope is the inertial mass of the body.” Guided by this example, we see that our extension of the principle of relativity implies the necessity of the law of the equality of inertial and gravitational mass. Thus we have obtained a physical interpretation of this law.
From our consideration of the accelerated chest we see that a general theory of relativity must yield important results on the laws of gravitation. In point of fact, the systematic pursuit of the general idea of relativity has supplied the laws satisfied by the gravitational field. Before proceeding farther, however, I must warn the reader against a misconception suggested by these considerations. A gravitational field exists for the man in the chest, despite the fact that there was no such field for the co-ordinate system first chosen.
Now we might easily suppose that the existence of a gravitational field is always only an apparent one. We might also think that, regardless of the kind of gravitational field which may be present, we could always choose another reference-body such that no gravitational field exists with reference to it. This is by no means true for all gravitational fields, but only for those of quite special form. It is, for instance, impossible to choose a body of reference such that, as judged from it, the gravitational field of the earth (in its entirety) vanishes.
We can now appreciate why that argument is not convincing, which we brought forward against the general principle of relativity at the end of the general principle of relativity at the end of Section XVIII. It is certainly true that the observer in the railway carriage experiences a jerk forwards as a result of the application of the brake, and that he recognises in this the nonuniformity of motion (retardation) of the carriage. But he is compelled by nobody to refer this jerk to a “real” acceleration (retardation) of the carriage. He might also interpret his experience thus: “My body of reference (the carriage) remains permanently at rest. With reference to it, however, there exists (during the period of application of the brakes) a gravitational field which is directed forwards and which is variable with respect to time. Under the influence of this field, the embankment together with the earth moves non-uniformly in such a manner that their original velocity in the backwards direction is continuously reduced.
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