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The Works Of Archimedes - Edited In Modern Notation With Introductory Chapters
Many of the earliest books, particularly those dating back to the 1900s and before, are now extremely scarce and increasingly expensive. We are republishing these classic works in affordable, high quality, modern editions, using the original text and artwork.
514 pages, Paperback
First published January 1, 213
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April 26, 2022
The Works of Archimedes is a collection of the surviving writings of Archimedes that he composed during the 3rd century BCE. Archimedes is hands down the most notable fusion of a mathematician, physicist, and engineer of Classical Antiquity. He devised mathematical formulas and proofs that are still used today, he discovered laws on various physical phenomena that are even now relevant, and invented machines and devices never seen before his time (these inventions were not discussed in the book). Reading the book, one gets to observe Archimedes' obsession with finding relations and echoing symmetrical proportions between various geometrical shapes and figures, both in two and three dimensions, while discovering ways to apply this theoretical knowledge to real-life matters.
This English translation is excellent, although the 200-page introduction was wearisome and superfluous. I believe that the translator-author had the best of intentions in providing the reader with a thorough introduction, but something should be left for the reader to explore in the principal text.
Legend has it that the last few moments before Archimedes' death he was studying geometry and drawing shapes in the dusty ground. At this time he was approached by a Roman soldier during the the siege of Syracuse. Archimedes disobeyed the soldier's orders to come with him which resulted in the soldier drawing out his sword. Archimedes reacted and plunged to protect the dust with his hands while begging the soldier and said: "Do not disturb my circles." Nonetheless, he was killed by that soldier despite the orders of the Roman Republic that he should not be harmed.
This English translation is excellent, although the 200-page introduction was wearisome and superfluous. I believe that the translator-author had the best of intentions in providing the reader with a thorough introduction, but something should be left for the reader to explore in the principal text.
Legend has it that the last few moments before Archimedes' death he was studying geometry and drawing shapes in the dusty ground. At this time he was approached by a Roman soldier during the the siege of Syracuse. Archimedes disobeyed the soldier's orders to come with him which resulted in the soldier drawing out his sword. Archimedes reacted and plunged to protect the dust with his hands while begging the soldier and said: "Do not disturb my circles." Nonetheless, he was killed by that soldier despite the orders of the Roman Republic that he should not be harmed.
September 10, 2018
Archimedes, perhaps the greatest mathematician who ever lived (at least T.L. Heath says so in this book) is obviously considered to be one of the two towering ancient Greek mathematicians, the other being Euclid.
The book reviewed is a companion to Heath's Greek Mathematics. Sir Thomas Little Heath (1861-1940) is probably himself a greatest - the greatest historian of Greek mathematics. This book is a true classic, and is still in print (basically). I add the qualification because the Dover edition I have not only seems to be out of print, but it appears that most editions of Archimedes' "Works" and Archimedes' "Method" are nowadays printed separately.
I acquired the 1953 Dover Edition of this book twenty years after that, when I was a student in the History and Philosophy of Science Department at the University of Melbourne. One of the first year courses was on the History of Math, or Greek Mathematics … something like that. Thus, what I've actually read of the book consists of relatively small parts of it, some assigned, others perused both then and in later years. We did go into quite some detail on the way in which Archimedes proved many of his theorems.
What's in the combined volume? First, there is a Preface written by Heath in 1897, then, following the Contents, an
Introduction
which runs from page xv to page clxxxvi. Unless I'm mistaken, that's page 15 to page 186 – some 170 pages(!), divided into eight chapters(!).
I. Archimedes.
II. Manuscripts and principal editions – order of composition – dialect – lost works.
III. Relation of Archimedes to his predecessors.
IV. Arithmetic in Archimedes.
V. On the problems known as NEY∑EI∑.
VI. Cubic equations.
VII. Anticipations by Archimedes of the infinitesimal calculus.
VIII. The terminology of Archimedes.
The first section of IV deals with the Greek numeral system, and describes how the Greeks wrote and spoke numbers. It covers about everything that would be known, I'm sure, in three succinct pages.
After the Introduction comes the section called
The Works of Archimedes.
This has thirteen sections in it:
On the Sphere and Cylinder, Books I & II
Measurement of a Circle
On Conoids and Spheroids
On Spirals
On the Equilibrium of Planes, Books I & II
The Sand-Reckoner*
Quadrature of the Parabola
On Floating Bodies, Books I & II
Book of Lemmas
The Cattle-Problem*
The two sections with asterisks are ones I have found especially astounding, because they illustrate in a fairly simple way the breadth of Archimedes' mathematical brilliance.
The Sand Reckoner finds Archimedes disagreeing with an expressed view that, were the earth entirely filled with grains of sand, the number of such grains would be greater than any actual numeric quantity expressible by man.
In the treatise, addressed to King Gelon, Archimedes goes such doubters one better by first stating how the size of the universe itself can be estimated (based on various assumptions); then introduces a methodology and numeric concepts allowing the writing and naming of extremely large numbers (involving Orders and Periods of numbers). The Greeks had a simple name for the largest number they commonly spoke of, the "myriad". This was the quantity 10,000. Thus a myriad myriads can be expressed (equal in modern terms to 100,000,000 – 100 million). Over the next couple pages Archimedes builds a system of expressing larger and larger numbers. Heath explains that the last number in the first Period would be a 1 followed by 800,000,000 zeros; and the last number of the tenth Period would require 100,000,000 times as many zeros.
In the few remaining pages, Archimedes brings together the estimates of the size of the universe, assumptions about the size of a grain of sand, and the scheme of Orders and Periods of immense numbers, to conclude that the number of grains of sand which could be contained in a sphere the size of our 'universe' is less than 1,000 units of the seventh order of numbers [10 to the 51st power, though the Greeks had not this method of stating powers of ten]
The Cattle Problem is one of Diophantine analysis. The problem requires solving for the smallest number of bulls and cows in each of four different-colored herds of the Sun God which satisfy stated conditions. There are seven equations given involving these eight unknowns. Archimedes states that if the student can solve for these eight unknowns, "thou art no novice in numbers, yet can not be regarded as of high skill." To achieve the "high skill" level, the student must find a solution which fulfills two additional conditions:
White bulls + black bulls = a square number,
Dappled bulls + yellow bulls = a triangular number.
There's a good description of the problem at https://en.wikipedia.org/wiki/Archime.... In fact the additional conditions to the problem make it so difficult that it was not solved until 1880, and the numbers involved were never printed until modern computers accomplished the task - the numbers involved have 100,000 digits.
Following all of this is a Supplement,
The Method of Archimedes.
The Method is a "supplement" (with pagination starting over at 1) because the treatise now called The Method of Archimedes was only introduced to the modern world in 1906, nine years after publication of Heath's original work; it was discovered to be part of a Greek MS. (known as the Archimedes Palimpsest) by J.L. Heiberg in that year. As Wiki states, "Heiberg inspected the vellum manuscript in Constantinople in 1906, and realized that it contained mathematical works by Archimedes that were unknown to scholars at the time."
This amazing discovery excited historians of ancient mathematics no end, Heath himself as much as any of them. Thus in the next edition of his work on Archimedes he added this Supplement, almost fifty pages in length, complete with an Introductory Note, a translation of the complete text, modern representations of the various equations therein, and diagrams illustrating the geometric figures.
The significance of this discovery, as Heath states, is that for the first time it allowed modern scholars to see how Archimedes came to believe that certain theorems were true, and that the effort of finding a proof was a task to be pursued. "Nothing is more characteristic of the classical works of the great geometers of Greece, or more tantalising, than the absence of any indication of the steps by which they worked their way to the discovery of their great theorems."
This "Method" is so-called because it illustrates how Archimedes came to believe that certain theorems were true, without having yet found a satisfactory proof. Heath continues,
. . . . . . . . . . . . . . . . . . . .
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The book reviewed is a companion to Heath's Greek Mathematics. Sir Thomas Little Heath (1861-1940) is probably himself a greatest - the greatest historian of Greek mathematics. This book is a true classic, and is still in print (basically). I add the qualification because the Dover edition I have not only seems to be out of print, but it appears that most editions of Archimedes' "Works" and Archimedes' "Method" are nowadays printed separately.
I acquired the 1953 Dover Edition of this book twenty years after that, when I was a student in the History and Philosophy of Science Department at the University of Melbourne. One of the first year courses was on the History of Math, or Greek Mathematics … something like that. Thus, what I've actually read of the book consists of relatively small parts of it, some assigned, others perused both then and in later years. We did go into quite some detail on the way in which Archimedes proved many of his theorems.
What's in the combined volume? First, there is a Preface written by Heath in 1897, then, following the Contents, an
Introduction
which runs from page xv to page clxxxvi. Unless I'm mistaken, that's page 15 to page 186 – some 170 pages(!), divided into eight chapters(!).
I. Archimedes.
II. Manuscripts and principal editions – order of composition – dialect – lost works.
III. Relation of Archimedes to his predecessors.
IV. Arithmetic in Archimedes.
V. On the problems known as NEY∑EI∑.
VI. Cubic equations.
VII. Anticipations by Archimedes of the infinitesimal calculus.
VIII. The terminology of Archimedes.
The first section of IV deals with the Greek numeral system, and describes how the Greeks wrote and spoke numbers. It covers about everything that would be known, I'm sure, in three succinct pages.
After the Introduction comes the section called
The Works of Archimedes.
This has thirteen sections in it:
On the Sphere and Cylinder, Books I & II
Measurement of a Circle
On Conoids and Spheroids
On Spirals
On the Equilibrium of Planes, Books I & II
The Sand-Reckoner*
Quadrature of the Parabola
On Floating Bodies, Books I & II
Book of Lemmas
The Cattle-Problem*
The two sections with asterisks are ones I have found especially astounding, because they illustrate in a fairly simple way the breadth of Archimedes' mathematical brilliance.
The Sand Reckoner finds Archimedes disagreeing with an expressed view that, were the earth entirely filled with grains of sand, the number of such grains would be greater than any actual numeric quantity expressible by man.
In the treatise, addressed to King Gelon, Archimedes goes such doubters one better by first stating how the size of the universe itself can be estimated (based on various assumptions); then introduces a methodology and numeric concepts allowing the writing and naming of extremely large numbers (involving Orders and Periods of numbers). The Greeks had a simple name for the largest number they commonly spoke of, the "myriad". This was the quantity 10,000. Thus a myriad myriads can be expressed (equal in modern terms to 100,000,000 – 100 million). Over the next couple pages Archimedes builds a system of expressing larger and larger numbers. Heath explains that the last number in the first Period would be a 1 followed by 800,000,000 zeros; and the last number of the tenth Period would require 100,000,000 times as many zeros.
In the few remaining pages, Archimedes brings together the estimates of the size of the universe, assumptions about the size of a grain of sand, and the scheme of Orders and Periods of immense numbers, to conclude that the number of grains of sand which could be contained in a sphere the size of our 'universe' is less than 1,000 units of the seventh order of numbers [10 to the 51st power, though the Greeks had not this method of stating powers of ten]
The Cattle Problem is one of Diophantine analysis. The problem requires solving for the smallest number of bulls and cows in each of four different-colored herds of the Sun God which satisfy stated conditions. There are seven equations given involving these eight unknowns. Archimedes states that if the student can solve for these eight unknowns, "thou art no novice in numbers, yet can not be regarded as of high skill." To achieve the "high skill" level, the student must find a solution which fulfills two additional conditions:
White bulls + black bulls = a square number,
Dappled bulls + yellow bulls = a triangular number.
There's a good description of the problem at https://en.wikipedia.org/wiki/Archime.... In fact the additional conditions to the problem make it so difficult that it was not solved until 1880, and the numbers involved were never printed until modern computers accomplished the task - the numbers involved have 100,000 digits.
Following all of this is a Supplement,
The Method of Archimedes.
The Method is a "supplement" (with pagination starting over at 1) because the treatise now called The Method of Archimedes was only introduced to the modern world in 1906, nine years after publication of Heath's original work; it was discovered to be part of a Greek MS. (known as the Archimedes Palimpsest) by J.L. Heiberg in that year. As Wiki states, "Heiberg inspected the vellum manuscript in Constantinople in 1906, and realized that it contained mathematical works by Archimedes that were unknown to scholars at the time."
This amazing discovery excited historians of ancient mathematics no end, Heath himself as much as any of them. Thus in the next edition of his work on Archimedes he added this Supplement, almost fifty pages in length, complete with an Introductory Note, a translation of the complete text, modern representations of the various equations therein, and diagrams illustrating the geometric figures.
The significance of this discovery, as Heath states, is that for the first time it allowed modern scholars to see how Archimedes came to believe that certain theorems were true, and that the effort of finding a proof was a task to be pursued. "Nothing is more characteristic of the classical works of the great geometers of Greece, or more tantalising, than the absence of any indication of the steps by which they worked their way to the discovery of their great theorems."
This "Method" is so-called because it illustrates how Archimedes came to believe that certain theorems were true, without having yet found a satisfactory proof. Heath continues,
… we have here a sort of lifting of the veil, a glimpse of the interior of Archimedes' workshop as it were. He tells us how he discovered certain theorems in quadrature and cubature, and he is at the same time careful to insist on the difference between (1) the means which may be sufficient to suggest the truth of theorems, although not furnishing scientific proofs of them, and (2) the rigorous demonstrations of them by irrefragable geometrical methods which must follow before they can be finally accepted as established.To fully explain what's going on would require quoting most of the next four pages of Heath's Introduction. However, there is a detailed description at https://en.wikipedia.org/wiki/The_Met....
. . . . . . . . . . . . . . . . . . . .
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September 21, 2020
In fact, how many theorems in geometry which have seemed at first impracticable are in time successfully worked out!
Many of the most influential and ingenious books ever written possess the strange quality of being simultaneously exhilarating and quite boring. Unless you are among that rare class of people who enjoy a mathematical demonstration more than a symphony, this book will likely possess this odd duality. This was the case for me. Reading this book was a constant exercise in fighting the tendency for my eyes to glaze over. But I am happy to report that it is worth the trouble.
Archimedes lived in the 3rd century BCE, somewhat after Euclid, in Syracuse on the island of Sicily. Apart from this, not much else can be said with certainty about the man. But he is the subject of many memorable stories. Everybody knows, for example, the story of his taking a bath and then running through the streets naked, shouting “Eureka!” We also hear of Archimedes using levers to move massive boats, and claiming that he could move the whole earth if he just had a place to stand on. Even his death is the subject of legend. After keeping the invading Romans at bay using ingenious weapons—catapults, cranes, and even mirrors to set ships afire—Archimedes was killed by a Roman soldier, too preoccupied with a mathematical problem to care for his own well-being.
True or not, good stories tend to accumulate around figures who are worthy of our attention. And Archimedes is certainly worthy. Archimedes did not leave us any extended works, but instead a collection of treatises on several topics. The central concern in these different works—the keystone to Archimedes’s method—is measurement. Archimedes set his brilliant mind to measuring things that many have concerned impossible to reckon. His work, then, is an almost literal demonstration of the human mind’s ability to scan, delimit, and calculate things far outside the scope of our experience.
As a simple example of this, Archimedes established the ratios between the surface areas and volumes of spheres and cylinders—an accomplishment the mathematician was so proud of that he apparently asked for it to be inscribed on his tombstone. (Cicero describes coming across this tombstone in a dilapidated state, so perhaps this story is true.) Archimedes also set to work on giving an accurate estimation of the value of pi, which he accomplished by inscribing and circumscribing 96-sided polygons around a circle, and calculating their perimeters. If this sounds relatively simple to you, keep in mind that Archimedes was operating without variables or equations, in the wholly-geometrical style of the Greeks.
Archimedes’s works on conoids, spheroids, and spirals show a similar preoccupation with measurement. What all of these figures have in common is, of course, that they are composed of curved lines. How to calculate the areas contained by such figures is not at all obvious. To do so, Archimedes had to invent a procedure that was essentially equivalent to the modern integral calculus. That is, Archimedes used a method of exhaustion, inscribing and circumscribing ever-more figures composed of straight lines, until an arbitrarily small gap remained between his approximations and what he was attempting to measure. To employ such a method in an age before analytic geometry had even been invented is, I think, an accomplishment difficult to fully appreciate. When the calculus was finally invented, about two thousand years later, it was by men who were “standing on the shoulders of giants.” In his time, Archimedes had few shoulders to stand on.
The most literal example of Archimedes’s concern with measurement is his short work, The Sand Reckoner. In this, he attempts to calculate the number of grains of sand that would be needed to fill up the whole universe. We owe to this bizarre little exercise our knowledge of Aristarchus of Samos, the ancient astronomer who argued that the sun is positioned at the center of the universe. Archimedes mentions Aristarchus because a heliocentric universe would have to be considerably bigger than a geocentric one (since there is no parallax observed of the stars); and Archimedes wanted to calculate the biggest universe possible. He arrives at a number that is quite literally astronomical. The point of the exercise, however, is not in the specific number arrived at, but in formulating a way of writing very large numbers. (This was not easy in the ancient Greek numeral system.) Thus, we partly owe to Archimedes our concept of orders of magnitude.
Archimedes’s contributions to natural science are just as significant as his work in pure mathematics. Indeed, one can make the case that Archimedes is the originator of our entire approach to the natural sciences, since it was he who most convincingly demonstrated that physical relationships could be described in purely mathematical form. In his work on levers, for example, Archimedes shows how the center of gravity can be found, and how simple principles can explain the mechanical operation of counterbalancing weights. Contrast this with the approach taken by Aristotle in his Physics, who uses wholly qualitative descriptions and categories to give a causal explanation of physical motion. Archimedes, meanwhile, pays no attention to cause whatever, but describes the physical relationship in quantitative terms. This is the exact approach taken by Galileo and Newton.
Arguably, the greatest masterpiece in this collection is On Floating Bodies. Here, Archimedes describes a physical relationship that still bears his name: the relationship of density and shape to buoyancy. While everyone knows the story of Archimedes and the crown, it is possible that Archimedes’s attention was turned to this problem while working on the design for an enormous ship, the Syracusia, built to be given as a present to Ptolemy III of Egypt. This would explain Book II, which is devoted to finding the resting position of several different parabolas (more or less the shape of a ship’s hull) in a fluid. The mathematical analysis is truly stunning—so very far beyond what any of his contemporaries were capable of that it can seem almost eerie in its sophistication. Even today, it would take a skilled physicist to calculate how a given parabola would rest when placed in a fluid. To do so in ancient times was simply extraordinary.
Typical of ancient Greek mathematics, the results in Archimedes’s works are given in such a way that it is difficult to tell how he originally arrived at these conclusions. Surely, he did not follow the steps of the final proof as it is presented. But then how did he do it? This question was answered quite unexpectedly, with the discovery of the Archimedes Palimpsest in the early 1900s. This was a medieval prayer book that contained the remains of two previously unknown works of Archimedes. (Parchment was so expensive that scribes often scraped old books off to write new ones; but the faded impression of the original work is still visible on the manuscript.) One of these works was the Ostomachion, a collection of different shapes that can be recombined to form a square in thousands of different ways (and it was the task of the mathematician to determine how many).
The other was the Method, which is Archimedes’s account of how he made his geometrical discoveries. Apparently, he did so by clever use of weights and balances, imagining how different shapes could be made to balance one another. His method of exhaustion was also a crucial component, since it allowed Archimedes to calculate the areas of irregular shapes. A proper Greek, Archimedes considered mechanical means to be intellectually unsatisfactory, and so re-cast the results obtained using this method into pure geometrical form for his other treatises. If it were not for the serendipitous discovery of this manuscript, and the dedicated work of many scholars, this insight into his method would have been forever lost to history.
As I hope you can see, Archimedes was a genius among geniuses, a thinker of the rarest caliber. His works are exhilarating demonstrations of the power of the human mind. And yet, they are also—let us admit it—not the most exciting things to read, at least for most of us mere mortals. Speaking for myself, I would need a patient expert as a guide if I wanted to understand any of these works in detail. Even then, it would be hard work. Indeed, I have to admit that, on the whole, I find mathematicians to be a strange group. For the life of me I cannot get excited about the ratio of a sphere to a cylinder—something that Archimedes saw as the culmination of his entire life.
Archimedes is the very embodiment of the man absorbed in impractical pursuits—so obsessed with the world of spirals and curves that he could not even avoid a real sword thrust his way. And yet, if subsequent history has shown anything, it is that these apparently impractical, frigid, and abstract pursuits can reveal deep truths about the universe we live in—much deeper than the high-flown speculations of our philosophers. I think this lesson is worth suffering through a little boredom.
September 1, 2019
I mean, it’s math.
I didn’t understand most of it, but I can appreciate that the man was a genius. And I believe there are gospel truths found in circles and spheres. Math is black and white, right and wrong. TRUTH. Order. It’s pretty impressive.
I didn’t understand most of it, but I can appreciate that the man was a genius. And I believe there are gospel truths found in circles and spheres. Math is black and white, right and wrong. TRUTH. Order. It’s pretty impressive.
reference
April 26, 2021“...Archimedes had at least as much imagination as Homer.” -Voltaire
March 29, 2021
We know a person by what they want on their tombstone. Where we end is where we begin, for the tombstone is merely a sign, one among many, of where we stand and what we value. It is with the deepest respect that I relate the importance of Archimedes' tombstone.
According to Heath, a diligent historical researcher if there ever was one, "Archimedes is said to his friend that he wanted on his tomb a representation of a cylinder circumscribing a sphere within it, together with an inscription giving the ratio which the cylinder bears to the sphere..." (page xviii) Heath draws from this that Archimedes was very pleased with the discovery of the ratio in Book One, Proposition 33, 34 of his work On the Sphere and Cylinder.
Proposition 33: The surface of any sphere is equal to four times the greatest circle in it.
Proposition 34: Any sphere is equal to four times the cone which has its base equal to the greatest circle in the sphere and its height equal to the radius of the sphere.
Your turn: What do you want on your rectangle, slab, sphere, or tombstone?
According to Heath, a diligent historical researcher if there ever was one, "Archimedes is said to his friend that he wanted on his tomb a representation of a cylinder circumscribing a sphere within it, together with an inscription giving the ratio which the cylinder bears to the sphere..." (page xviii) Heath draws from this that Archimedes was very pleased with the discovery of the ratio in Book One, Proposition 33, 34 of his work On the Sphere and Cylinder.
Proposition 33: The surface of any sphere is equal to four times the greatest circle in it.
Proposition 34: Any sphere is equal to four times the cone which has its base equal to the greatest circle in the sphere and its height equal to the radius of the sphere.
Your turn: What do you want on your rectangle, slab, sphere, or tombstone?
Read
February 23, 2009Syracuse, c287 - 212 BCE in sack of Syracuse
(spent bunch of time in Alexandria)
The Works of
T.L. Heath, ed.
too much like a high school math textbook; not to be read
If we are rightly informed, Archimedes died, as he had lived, absorbed in mathematical contemplation.
(spent bunch of time in Alexandria)
The Works of
T.L. Heath, ed.
too much like a high school math textbook; not to be read
If we are rightly informed, Archimedes died, as he had lived, absorbed in mathematical contemplation.
Want to read
May 10, 2008Started on it. Ended up reading 50 pages on the history of the various manuscripts. I regret that.
December 12, 2024
A good English-translation of the surviving works of the great Archimedes of Syracuse for modern readers interested in the history of mathematics.
January 22, 2008
Eureka! Not as studied as Euclid, interesting for some alternate solutions.
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