Standing on the Shoulders of Giants: The Case of Archimedes

by CJ Uberroth

It is said that during the Roman siege of Syracuse (212-214 BC), the Romans were stalemated by the Syracusans’ complex methods of defense. Marcus Claudius Marcellus, a renowned Roman General, found himself at a loss during the siege. Apparently, the mathematician, Archimedes, had used his knowledge to invent a claw capable of toppling siege weapons and sinking ships, as well as a mirror designed to focus light and heat on the sails of advancing ships, in order to set them ablaze. The defense was so potent that upon the final push into Syracuse, Marcus ordered that Archimedes be spared so that he might be honored for his genius. Though this order was ignored and Archimedes was killed in his home, it is said that Marcus arranged special funeral for his adversary in honor of the battle. With Archimedes’ death, we would wait centuries for another equally brilliant mind to emerge.

There are any number of achievements in modern mathematics that demonstrate the depth and scope of Archimedes’ intelligence, but three iconic moments in mathematical and scientific history exemplify it.

The trial of Galileo by the Catholic Church for his assertion that the earth revolves around the sun along with the other planets is well-known, but what is much less known or appreciated is the fact neither Galileo’s nor even Copernicus’s advancement of the theory of heliocentrism was the first time that this picture of the universe had been proposed.

Archimedes wrote a letter commonly known as The Sand Reckoner in which he set out to debunk the “sand-problem.” The idea was the notion that the sands of the world were either infinite or uncountable due to the lack of a number system large enough to count them. While The Sand Reckoner is considered one of Archimedes’ most accessible works, it is not the beauty of his proof that is relevant in this case.  Archimedes’ method of proof was exhaustive: he showed that if you imagine a quantity of sand greater than that of the earth and prove that it is countable and finite, then this would imply that the sands of the world were countable and finite. In his attempt to demonstrate this, Archimedes applies the concept of heliocentrism:

Now you are aware that ‘universe’ is the name given by astronomers to the sphere whose center is the center of the earth and whose radius is equal to the straight line between the center of the sun and the center of the earth. This is the common account (τὰ γραφόμενα), as you have heard from astronomers. But Aristarchus of Samos brought out a book consisting of some hypotheses, in which the premises lead to the result that the universe is many times greater than that now so called. His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun in the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of the fixed stars, situated about the same center as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the center of the sphere bears to its surface.

As it turns out, then, Galileo and Archimedes had more in common than simply dying in their homes. Indeed, one can only wonder how vigorously Archimedes must have been turning in his grave, when the Inquisition sentenced Galileo, or how human cultures around the world could have held fast to false ideas for so long.

In the 17th and 18th centuries, mathematics really began to take off, with the work of Gauss, Euler, and L’Hopital, and was the period which saw the development of infinitesimal calculus. Some may know the story of Leibniz and Newton’s battle to create calculus as we know it today, and they are credited with developing the Fundamental Theorem of Calculus, which connects differential calculus and integral calculus. Essentially, what they did was give prove of and provide a notation for the idea that you can imagine rectangles below a curve that estimates the value of the space between said curve and the x-axis. If you then  shrink these rectangles to an infinitely small width, you can obtain the exact area of the space below the curve.

graph 1

This theorem and its subsequent use have led to countless advancements in mathematics and science.  And yet, again, Leibniz and Newton were not the first to do it, but rather, Archimedes, who had made these comparisons of figures in his letters On the Sphere and the Cylinder and On the Measurement of a Circle. Most notably, he used inscribed polygons within circles to demonstrate formulas for areas of conic sections and then “rotated” them into three dimensional figures to extrapolate the formulas for volumes. These are all things we teach in modern calculus, but the history of these formulas and their creation largely goes unnoticed.

graph 2

Finally, it is important to mention Leonhard Euler. From his development of a modern notation for algebra and calculus to the discovery of his namesake value, e, Euler really informed just about every facet of the mathematics of his day. Given what we have already seen, however, I’m sure that readers will not be surprised to discover that like Galileo, Leibniz, and Newton, Euler was standing on the shoulder of a much older mathematical giant.  For whether it is his creation of common notations for summations and pi, his analysis of arithmetic, geometric, and recursive series, or his incredible work in physics and mechanics with Bernoulli, what we find is that Euler’s achievements arose on top of Archimedes’ work.

Archimedes demonstrated the constancy of the ratio between a circle’s circumference and area (pi) and also developed his own arguments for using arithmetic to solve for the value of a series (a series is a long chain of adding values, created by some mathematical rule — an example is adding the first one-hundred integers). Indeed, Archimedes even dabbled in mechanics. In a letter to Eratosthenes, Archimedes laid out proofs involving centers of gravity and the change in these centers under different circumstances, and advancement that we really would not fully grasp until the modern scientific and industrial revolutions. What is particularly amazing is the fact that this letter had no real practical purpose behind it.  Archimedes simply wanted to share his knowledge with a respected colleague, who had shown interest in the relationship between mathematics and the principles of mechanics.

The truth is that all of Archimedes’ letters have this quality, their ultimate aim being to create a commonwealth of knowledge that ultimately would know no borders, in time or space. It would seem that he never kept anything for himself or for his people, and in that sense, he imagined the international, apolitical community of mathematicians and scientists that exists today and has proven to be resilient even in the most politically contentious of times.  The only shame is that the long history of human contentiousness, not to mention an increasing and appalling level of historical ignorance, has contributed to Archimedes going largely un-credited for the intellectual civilization to which, in many ways, he gave life.

21 Comments »

  1. Okay, I have an ignoramus question:

    With regard to the area between the curve and the bars in the graph, if you have to imagine rectangles of an “infinitely small width” within the space, then how can you calculate *exactly* the area of the space? Rectangles of “infinitely small width” don’t have a specific area.

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  2. This is a fascinating essay on so many levels. How do we determine who are the greats? It is an opinion, but whose opinion matters? Charles Murray(Human Accomplishment) approached this question in a careful, methodological way by surveying the writings of experts in the field. In the field of technology he concluded the 10 greatest people were, [normalized score], and (when they flourished):

    1. [100] Watt (1776)
    1. [100] Edison (1187)
    3. [60] Leonardo (1492)
    4. [51] Huygens (1669)
    5. [51] Archimedes (-247)
    6. [50] Marconi (1914)
    7. [43] Vitruvius (-40)
    8. [37] Smeaton (1764)
    9. [34] Bessemer (1853)
    10.[33] Newcomen (1703)

    I do have some problems with your concluding claim:
    … history of human contentiousness, not to mention an increasing and appalling level of historical ignorance, has contributed to Archimedes going largely un-credited for the intellectual civilization to which, in many ways, he gave life

    His position at #5 in the list shows that he has a high degree of recognition for his contributions to technology. But in an equivalent list for mathematics he only comes in at #20, which evidently accounts for your judgement. However, even at #20 he is recognised as a preeminent figure, one of the greats.

    It is only when one looks at the date they flourished that the true greatness of Archimedes becomes apparent. He flourished in -247 while the remaining figures, with the exception of Vitruvius, flourished from 1492 to 1914. He was an astonishing 1700 years ahead of his time. That is true greatness. Vitruvius(architect, civil and military engineer) is the other notable figure, far ahead of his time.

    Mind you, I agree with your statement, “an increasing and appalling level of historical ignorance“, except I think it is far worse than that. For today’s generation history began when Facebook invented its Timeline.

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  3. True. Since the width of the rectangles would be indefinite, it wouldn’t be possible to calculate a specific area. This is however a key concept for calculus to solve. As the width of the rectangles approaches zero, we needed a new method of mathematics to determine the values. Hence, the integral. Essentially what the integral does is identify the function for which the rate of change is equal to the value we are analyzing under our curve. Therefore we can evaluate how much it changed which will represent the value under the original curve (the second fundamental theorem of calculus)

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  4. The person that fascinates me is Vitruvius, commonly known as the father of architecture and author of De architectura, dedicated to his patron, Caesar Augustus.

    He is known for his assertion that buildings should have the properties of utilitas, firmitas and venustas (utility, strength and beauty).

    Venustas is now translated as ‘beauty’, though it was originally translated as ‘delight’ by Sir Henry Wotton. I much prefer Wotton’s translation, believing that good architecture should be a source of delight. I relate to this because my former boss at VW was forever urging us to strive for ‘customer delightment‘ 🙂 But achieving this turned out to be a Sisyphean task.

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  5. Thanks labnut, I appreciate the enthusiasm. I do agree that he is recognized within the math community but everyone knows of Newton and Leibniz. Unfortunately most people’s knowledge of Archimedes refers back to the owl in Sword in the Stone. But I do think his recognition is growing now that we are finding more and more of his writings.
    As for your generational comment, I do have to agree it’s rather frightening.

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  6. Archimedes, at least as transcribed by scholars, is careful to say that he is not using Aristarchos’ heliocentric system because he believes it is correct, but because its larger circumference of the universe gives him an opportunity to play with larger numbers in his estimations. His father was apparently an astronomer, so he would have known the subject well.

    Eratosthenes at the time was the head librarian at Alexandria, and the letters from Archimedes addressed to him would have also served as a form of ancient peer-review publishing, at a time when everything had to be copied by hand. The sand-reckoner was not addressed to Eratosthenes, but to a young king of Syracuse. Its accessibility may have been due to its educational nature: he was sort of myth-busting the ancient saying about grains of sand on shore being without measure. The maths in it is very much applied, he describes how to estimate the size of smallest sands using poppy seeds, and how to construct an apparatus to measure the parallax of the Sun, and so on.

    In his more mathematical works he is interested in the exact measurements of different geometrical objects, and some of his proofs do resemble integration as we understand it today. He of course did not have cartesian coordinates, but in some ways he was also not limited in his thinking by them. His proofs are very creative.

    There was a general-audience book published a few years ago which I liked: Reviel Netz, William Noel: “The Archimedes Codex”. The name and the cover are probably meant to resemble Da Vinci Code, unfortunately, but the actual contents of the book is very readable.

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  7. Hi CJ, this was quite interesting. I want to agree with Labnut that Archimedes might have some popularity outside of math. I knew of him and at least one of the stories you mentioned. But it is certainly worth discussing and getting an idea how knowledge we think belongs to a more recent scientist/mathematician/philosopher might really be traced further back.

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  8. Very neat reminder of the fact that developments in math and physics have a real history, and not just a series of inspired ‘discoveries.’

    I think this a good place to hype a blog I follow, by a scholar of math and science history, obsessed with making sure that Galileo is recognized for his actual accomplishments – neither more nor less: The Renaissance Mathematicus.. From his most recent post, replying to a commentator:

    “Phillip then asks, ‘So what was the shock of the Copernican Revolution (how many even get that pun?)? Was it demoting humanity from the centre of the universe, or promoting the Earth to be on par with the other heavenly bodies?’ (…) The answer is as simple as it is surprising, there wasn’t one. The acknowledgement and acceptance of the heliocentric hypothesis was so gradual and spread out over such a long period of time that it caused almost no waves at all. First up, there was nothing very new in Copernicus suggesting a heliocentric cosmos. As should be well known it had already been proposed by Aristarchus of Samos in the third century BCE and Ptolemaeus’ Syntaxis Mathematiké (Almagest) contains a long section detailing the counter arguments to it, which were well known to all renaissance and medieval astronomers. Also in the centuries prior to Copernicus various scholars such as Nicholas of Cusa had extensively discussed both geocentric models with diurnal rotation and full heliocentric ones. All that was new with Copernicus was an extensive mathematical model for a heliocentric cosmos.” https://thonyc.wordpress.com/2017/05/18/perpetuating-the-myths-addendum-the-copernican-shock/

    Despite the title of this interesting article, the truth is that we do not stand on the shoulders of giants. Before us, some build sand castles; sometimes these are kicked aside, leaving only ruins. But from these ruins others build great cathedrals of knowledge, belief, and understanding; in which we now dwell.

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  9. CJ,
    Unfortunately most people’s knowledge of Archimedes refers back to the owl in Sword in the Stone. But I do think his recognition is growing now that we are finding more and more of his writings.

    I don’t think it is quite as bad as that. (when one excludes the Facebook generation) 🙂

    For example, one of the most widely known Greek words in English is Eureka and we owe that to Archimedes. Google finds 72 million references to it. To put that into context, psyche has 33 million references and pathos 12 million. But none of the other common Greek words that survived unchanged in English can be so strongly attributed to one person. Or another way of looking at it, Einstein has 129 million references in Google. Archimedes has had quite a profound influence on the modern consciousness even if many are not aware of the origin.

    I still remember clearly our physics lesson in secondary school when we were told the story of Archimedes in the public baths exclaiming eureka, eureka! Perhaps that story is no longer taught?

    I’m just quibbling over details and I don’t want that to detract from your overall thesis about Archimedes, which I completely agree with, except that perhaps you may have understated his achievements.

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  10. EJ,
    Despite the title of this interesting article, the truth is that we do not stand on the shoulders of giants

    Please expand on this and tell us what you mean. I was going to dash off a strong refutation but decided to wait for a more complete motivation from you. Perhaps you are over-interpreting a metaphor? When Newton used the phrase “If I have seen further it is by standing on the shoulders of Giants” he was both expressing humility and giving recognition to others whose work made his possible. He borrowed the phrase from Bernard of Chartres, which in itself is an amusing example of what he was saying.

    Dan-K
    The title is my fault, not CJ’s!

    I wouldn’t call it a fault. That is a jolly good title.

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  11. labnut,
    the development of any knowledge is a collective endeavor over many years, as we grope through the dark for some certainty concerning the world we live in. Individual names are remembered only when they signify a convergence of previous effort into a new understanding of reality. We think of these as naming ‘giants’ because we can’t imagine how knowledge would have developed without them; but it fact such convergence of effort is probably inevitable. It wasn’t just Newton, but Leibniz as well. Had there been no Newton, would Leibniz have applied his particular form of calculus to the problem of gravity? Possibly not; but probably someone would have, since the problem was irritating a number of thinkers at the time.

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  12. Dan,

    A way to do this, is as follows (not saying Archimedes used this method – I studied his proof, but it’s a very long time ago).

    Take rectangles at the inside. Take whichever set of rectangles you want. Fine, less fine …
    Take the surface areas of these sets of rectangles at the inside.
    You get a new set ( S1, S2, S3 … ) of areas that approximate the area of the circle.
    Now such a set had the property (in the real numbers) that it has a “supremum” X.
    In a certain sense, the supremum is the smallest number that “bounds” S1, S2, S3, …
    The supremum is bigger than all these S’s (or equal to some of them), but it is the smallest number with this property.
    It’s not self-evident that the supremum exists, but accept for a moment that it does.

    Do the same thing from the outside (fit the circle *in* sets of rectangles).
    You get again a set of areas, and this set has an infimum. The infimum is the biggest number that’s smaller (or equal to) the areas in this set (it’s the mirror image of the supremum).

    Sometimes you can prove that the infimum = supremum.
    In that case you get the area.

    (More follows)

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  13. EJ,
    We think of these as naming ‘giants’ because we can’t imagine how knowledge would have developed without them; but it fact such convergence of effort is probably inevitable.

    First, I think it is wrong to claim that we simply “can’t imagine how knowledge would have developed without them“. You have wrongly attributed a belief to the science world that simply won’t stand up to examination.

    Second, if that is your definition of ‘giants’ it is excessively narrow and then there are none because you have defined them out of existence. But that is hardly useful and it simply does not accord with the experience and intuitions of the scientific world.

    Our experience and our intuitions tell us that greatness does really exist in the many fields of intellectual endeavour. But what is this ‘greatness’? It is readily recognised in retrospect but is a slippery concept difficult to define. Your definition simply won’t do. You have treated it as if it were a monochromatic thing with only one dimension and compounded the problem by choosing the wrong dimension. In fact greatness has several dimensions.

    I will return to the discussion tomorrow morning 🙂

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  14. Now the absolutely fabulous thing …

    The Ancient Greeks didn’t know what the reals are. They could compare fractions, and say things like A/B > C/D.
    But the reals?

    And then there was Eudoxus. He designed an incredibly clever way to compare reals by using fractions – and those, the Greeks understood.
    I’m modernizing Eudoxus a bit here – he used a slightly different trick – but it was incredibly clever, and what’s more, it was rigorous, even to modern standards.

    Archimedes uses this trick of Eudoxus – again in a very subtle, but very rigorous way(°) – and without even mentioning them, he uses exactly the relevant properties of the supremum and the infimum.

    I’m modernizing Archimedes here a bit too, but if you read him with modern eyes, that exactly what he does.

    The reals where put on a sure footing by mathematicians like Cauchy and Dedekind in the 19th century. But if you read Eudoxus with modern eyes, and you then look at the approach of Dedekind, you get the strange feeling that Dedekind only formalized what Eudoxus and Archimedes already knew – twenty centuries earlier!

    (°) the problem with the approach of Archimedes is that it’s not constructive. You have to know the answer beforehand to find the proof. But there are strong indications that Archimedes used less rigorous ways to find the answer (and then gave a rigorous proof).

    Actually, the proofs of Archimedes are far more rigorous than anything Newton or Leibniz ever did in this area of mathematics.

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  15. Dan,

    to clarify things a bit: you don’t need infinitely small rectangles in your circle.
    You only need all the possible sets of rectangles that fit into the circle. All these rectangles are *finite*.
    The trick then is to understand that there’s a supremum that bounds the stuff from above.

    It’s often misunderstood, but you don’t need infinitely small stuff to do calculus. Mathematicians found a way around that.

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  16. Just one last thing, and then I’ll stop bothering you all.

    The idea of a supremum may sound strange, but it’s actually simple (proving that it exists is subtle, though).

    Take the sequence 0.3 ; 0.33 ; 0.333 ; 0.33333 ; and so on.
    *All* the numbers in this sequence are smaller than 1/3.
    You *never* get there, no matter how far you go in the sequence.

    But 1/3 is the supremum of the set ( 0.3 ; 0.33 ; 0.333 ; 0.33333 ; and so on )

    1/3 is the *smallest* fraction that’s *bigger* than the elements in the set.

    If you take the sequence 1/2 ; 1/4 ; 1/8 ; 1/16 ; 1/32 ; and so on, then the elements in this sequence *never* become zero.

    But zero is the infimum of this sequence. It’s *biggest* real that’s *smaller* than all the elements in the sequence.

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  17. labnut,
    we may have to agree to disagree here, since a conversation about whether there really are ‘great’ men or women in any field, or whether such a thing as ‘genius’ exists would actually require quite a bit to unpack and argue. Certainly there are men and women of history I admire; but I confess I long ago gave up on hero-worship. People just do their best; some stumble forward, others sprint, other’s get nowhere, and oft wisdom or talent disappears unnoticed. Much of ‘greatness’ is merely luck.

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  18. EJ,
    Certainly there are men and women of history I admire; but I confess I long ago gave up on hero-worship.

    Talking about greatness is not ‘hero-worship’. Greatness, in the Charles Murray sense, is a careful judgement based on countable public facts, stripped of the emotion contained in your phrase ‘hero-worship’.

    Much of ‘greatness’ is merely luck.

    This is undoubtedly true in some cases but to ascribe it merely to ‘luck’ is a grave disservice to most of them. Certainly we live in a harsh, unfair, chance driven world. There will be many a goat herder who potentially could have been the next Einstein or Aristotle but instead languishes on our drought stricken hillsides.

    So yes, there is some truth in what you say. And yet certain individuals have risen above their origins and surroundings in ways that excite universal admiration. We call them the ‘greats’ and so we should. Their example motivates the rest of us.

    The golfing ‘great’, Gary Player, on shooting one of his many hole-in-ones, was congratulated on his lucky shot. He replied that luck was where opportunity meets preparedness(this was not originally his phrase). You might say that capable people make their own luck.

    Life is a struggle where we try to rise above desperate circumstances. It is a world imbued with suffering. But do we struggle, accept or succumb? A cow, mired in quick sands, will struggle until her dying breath. Lacking the ability to conceive of the future she cannot see her struggle is hopeless so she continues to struggle uselessly.

    Cursed with cognition, we can imagine the future and so we succumb. More often than not we succumb uselessly, paralysed by our own cognition. But cognition also allows us to hope, it ignites the spark of possibility and this spark of possibility drives some to climb far beyond their circumstances. We need sources of hope and possibility which is why we create the category of the ‘greats’.

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  19. labnut,
    In 1979, an English teacher at the University of South Carolina, who had previously only enjoyed a brief notoriety for a book on the aesthetics of pornography, and another arguing that Romanticism was born of a psychological urge towards ‘chaos,’ published what I consider the definitive text linking semiotics and behaviorism – Explanation and Power, by Morse Peckham. You have probably never heard of him, or of this book, although I have referred to it a number of times here.

    I have spent more than 25 years trying to think of an adequate counter argument to Peckham, or some way around it. I mean, as much as I love Umberto Eco, I successfully deconstructed his theory of mirrors as paradigmatic of iconography, which is no mean feat! But after 25 years, I can’t touch Peckham’s base theory.

    But outside of my references to him here, you’ve never heard of him. Philosophers take no stock in him, nor any psychologists, nor even most semioticians.

    It should be noted that Explanation and Power was Peckham’s ’emeritus text;’ that is, it was considered the summation of what he had always wanted to say, publishable only after his retirement from the Academy. That’s important, because there are a lot of such texts that get dismissed because the authors have already had their professional say in peer-reviewed journals. Most ’emeritus texts’ are considered summations; whatever they have to say has already been said.

    But Explanation and Power is no summation, although it relies heavily on Peckham’s previous research. It may be the most important text on the relation between semiotics and behavior written in the 20th Century.

    But without my mention of it here, you’ve almost certainly never heard of it. very few have. Likely the text will disappear into oblivion.

    But perhaps sometime in the future, someone will resurrect the ideas in that text, in a social environment that allows that thinker attention and fame. Peckham, and his text will remain forgotten, but the ideas will at last be noticed. And that author will be considered ‘great’ in his or her field. Reviewers may even declare him or her a ‘genius’ for ‘discovering’ what Peckham in fact first organized as a clear explanation of ‘explanation and power.’

    Luck and environment determines the ‘greatness’ of any thinker.

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  20. Hi EJ,
    Luck and environment determines the ‘greatness’ of any thinker.

    Replace the word “determines” with “influences” or “plays a role in” and then I will agree with you. For my taste your statement is too stark and deterministic. It neglects to consider what really makes a person great.

    The starting point of greatness is a mind far and beyond the normal. One that is curious, intelligent, possessing a penetrating vision, a remarkable clarity of insight and an extraordinary breadth of understanding This is the first primary determinant. The second primary determinant is the power of the person’s character to inquire, to persevere, to articulate, to persuade and to overcome obstacles. These two factors are the primary determinants of greatness and they reside in the mind and personality of the person.

    And then we have the secondary determinants, external to the person, native to his environment and society. They shape, limit, encourage, dissuade or enable the person to realise their innate greatness.

    Both the primary and secondary determinants must be present for greatness to occur and be realised. But to simply ascribe it to luck and environment, as you have done, is to only see a small part of the whole picture. What is remarkable is how often greatness rises above an unfavourable environment.

    When seen in the way I have outlined it makes sense to recognise and celebrate greatness. But I will go further than that and claim it is highly desirable to celebrate intellectual and artistic greatness because:
    1) It encourages society to liberate funds to support intellectual inquiry.
    2) It encourages more bright, young entrants to the field of intellectual inquiry.
    3) It assists in the spread of their contributions.
    4) It motivates others to emulate their example.
    5) It encourages the development of an inquiring spirit.
    6) It steers society, giving direction to its inquiring spirit
    7) Recognising the contributions of others is a generous, outward looking, non-solipsistic attitude to life. We need more of that.

    Most of all, we need more moral greatness, of the kind seen in Jesus, Paul, Moses, Buddha, Confucius, Laozi, etc. We need more of Pope Francis and Dalai Lama. We need towering figures of compassionate love that lead us to care for the countless suffering, that teach us to treat each other with dignity, understanding, tolerance, forgiveness, concern and love.

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