Archimedes and the Science of Physics by

A. G. DRACHMA”*

Archimedes was a mathematician second to none. This is generally accepted, and if any one should doubt it, a simple reference to his works as they have come down to us should settle the question. I t is no wonder that those who study Archimedes are apt to concentrate on his mathemat- ical achievements and to show less interest in his work in other fields. This attitude is strengthened by a well known passage in Plutarchosls, which seems to indicate that this was also the attitude of Archimedes himself. He tells us how king Hieron of Syracuse persuaded Archimedes to prepare the city against attack, and how well his inventions worked against the Romans. He adds: “And yet Archimedes possessed such a lofty spirit, so profound a soul, and such a wealth of scientific theory, that although his inventions had won for him a name and fame for superhuman sagacity, he would not consent to leave behind him any treatise on this subject, but regarding the art of the engineer and every art that ministers to the needs of life as ignoble and vulgar, he devoted his earnest efforts only to those studies the subtlety and charm of which are not affected by the claims of necessity. These studies, he thought, are not to be compared with any others; in them the subject matter vies with the demonstration, the former supplying grandeur and beauty, the latter precision and surpassing power.”

To this may be added a note by Karposl2, saying that Archimedes left behind him only works on mathematics, with the sole exception of a description of a planetarium, that is an astronomical instrument, which we may suppose was neither ignoble nor vulgar.

On the strength of the quotation from Plutarchos professor Farringtons, after praising Archimedks for his theoretical writings, remarks : “The reverse of the medal was contempt for the practical applications of science. Archimedes was the greatest engineer of antiquity, but when he was

* Svends All6 47, Kgs. Lyngby.

Centmur 1967: vol. 12. no. 1: pp. 1-11

1 CLNTAURUS. VOL. XI1

2 A . G. Drachmann

asked to write a handbook on engineering he refused.” Further: “Archi- medes had expressed his contempt for the useful applications of science.”

This is repeated by Friedrich Klemml3 like this: “The following well known passage from Plutarch concerning Archimedes (who died 212 B.C.) is significant testimony to the extreme value ascribed to pure theory as against its practical application to which recourse would be had only from dire necessity and which was widely regarded as unworthy of an author’s attention.”

It may be argued that Archimedes made his inventions for the defense of Syracuse only from dire necessity; but there is evidence that he invented the water-snail2, the endless screw3, the steel-yard17 and a winch called trispustosl4 without any dire necessity at all; also that he made physical experiments and found the “law of Archimedes” about floating bodies, and the theory of specific weight18; also that he was the first to determine the centre of gravity, and that he elaborated this theory into a text-book of statics4.

From this it would seem that Plutarchos must have made a slip some- where, and at the very least that there is no foundation for the adverse judgement of Mrss Farrington and Klemm. The actual deeds of Archi- medes surely should have more weight than the words of Plutarchos written some 300 years later.

When Plutarchos writes about “the inventions that had won for him a name and fame for superhuman sagacity”, he refers to the defense of S yracuse.Bu(when would it have been possible forArchimedes to write about these inventions? Before or during the siege? No, for they would have to be kept secret; and after the siege he was not alive to write anything.

Stronger still is the following argument: Plutarchosl6 tells us that Archimedes regarded “the art of the engineer (ten peri ta mgchaniku pragmateian) as ignoble and vulgar”, but how comes it then that Archi- medes quotes himself as the author of a work “The Mechanics” (tu m~chuniku), or “The Elements of the Mechanics” ( fa stoicheia f6n mFchanik6n)? There are three quotations, all of them from the Plan. Aequil. 1, and one which is not found in this work, but, from the context, should be sought in the Mechanics4.

That Archimedes made physical experiments is told by Vitruviusl8. King Hieron had given a goldsmith a certain weight of gold, and the goldsmith had made a wreath of the same weight, but before dedicating the wreath to the god, Hieron asked Archimedes to find out if it was

Archimedes and the Science of Physics 3

made of pure gold, or if silver had been added. Since no chemical method was known at the time, it was necessary to find out the volume of the wreath; and when Archimedes with his head full of speculations about how to do this, went to a public bath, the bath-tub was full and flowed over when he entered into it. It then struck him that the amount of water flowing out must be equal to the volume of the person entering the tub. He uttered the well-known cry “Heureka”, meaning “I’ve got it!” and was in such a hurry to get home and begin his investigation that he forgot to put on his clothes.

He then filled a bowl with water, put in the wreath and took it out again, and filled up the bowl from a measuring jug; so he knew the volume of the wreath. Next he took a lump of gold of the same weight as the wreath and found its volume in the same way; it proved to be less than that of the wreath. Then he determined the volume of a lump of silver, again of the same weight, and this was found to be greater than that of the wreath. These three figures enabled him to calculate the amount of silver contained in the wreath. In this way he laid the foundation of the theory of the specific weight of the elements, and it is very probable that he also found out the law of the loss of weight of a submerged body while he was working with lumps of gold and silver, a balance and a bowl of water. There is not the slightest evidence that he should have looked down upon this experimental work; on the contrary, he used the results for his Floating Bodies.

Turning now to the work of his known to himself as “The Mechanics” or “The Elements of Mechanics”, we find that we do not possess all that Archimedes has written on this subject. Starting with what we have, we find that Archimedes quotes his Plan. Aequil. 1 five times, giving the title twice as “The Mechanics”, once as “The Elements of Mechanics”, once as “The Equilibrics” and once as “The Equilibrilia”, and that there is a sixth quotation, evidently from the “Mechanics”, which is not found in the Plan. Aequil. 1.5 This indicates that the Plan. Aequil. 1 is to be regarded as part of a greater work called “The Mechanics”, and there are some. other indications that this is right. J. L. Heiberg7 has called attention to the fact that the two books generally called Plan. Aequil. 1-2, do not belong together, since the titles are not the same, Book 1 being On the Equilibrium of Plane Figures, while Book 2 is merely: On Equilibria; more important is that the book on the Squaring of the Parabola comes in between, since it quotes Plan. Aequil. 1 and is quoted by Plan. Aequil.2.

4 A. G. Drachmann

A close study of Plan. Aequil. 1 shows that it is very far from being a complete presentation of the centre of gravity of plane figures; it gives only proof of the centre of gravity in a rectangle, a triangle, and a trapeze, and these three propositions together with the proof that two magnitudes are in equilibrium if the distances from their centers of gravity to the point of suspension are in inverse proportion to their weight are the propositions needed for the squaring of the parabola, neither more nor less. This suggests that the Plan. Aequil. 1 as we have it is a selection made from a larger work just to serve as a stepping-stone for the squaring of the parabola.

In Heron’s Mechanics, which have come down to us in an Arabic translation only, there is evidence both of some works by Archimedes on the centre of gravity, to come before the Plan. Aequil. 1, and of some works by him on statics and the centre of gravity in plane figures, which supplement the Plan. Aequil. 1. In all we have traces of four different works: I) On the centre of gravity in bodies. 2) On the centre of gravity in plane figures. 3) On uprights, that is the statics of beams carried by posts. 4) On balances. I have written at length upon these fragments in the Centaurus4, so here a short summary will do.

In the first part, on the centre of gravity in solid bodies, Archimedes shows, by means of two “axioms” (definitions) and two “postulates” (stipulations), followed by five propositions, that every body has a center of gravity, and that the body will be in equilibrium if supported in this point.10

Heron writes9 “That it is in truth not possible to speak of inclination and declination (that is, of gravity) except in bodies, that is what no one will deny. But if we speak of geometrical figures, corporeal and flat, as having a center of inclination or a center of gravity in a certain point, Archimedes has given a sufficient explanation about it. So you have to understand it in the way in which we present it.”

From this it appears that Heron had a work by Archimedes in which he explained how it is possible to reckon with plane figures having a center of gravity and a weight corresponding to their surfaces. How Archimedes explained this I do not venture to guess, but though Heron is right in saying that such an idea “in truth” is impossible, it cannot be denied that it has proved very fruitful.

If we next study the stipulations (“postulates”) at the beginning of Plan. Aequil. 1, we shall find:

Archimedes and the Science of Plysics 5

4. If equal and similar plane figures are fitted upon each other, the centres of gravity will coincide.

5. In unequal, but similar plane figures the centres of gravity will be at corresponding places. We say that points are in corresponding places, when straight lines drawn from them to the same angle points form the same angles with the corresponding sides.

7. In every figure whose perimeter is hollow to one side only the centre of gravity must be inside the figure.

These stipulations we must regard as proven in the preceding work and placed here as an introduction to the investigations into the Equilibrium of Plane Figures. This work is preliminary to the main problem, the squaring of the parabola, since it contains only the propositions necessary for this work; from Heron’s Mechanics we know that Archimedes had made many more propositions on the equilibrium of plane figures, so Plan. Aequil. 1 is a selection made on purpose.

So we find a series of four books: On the centre of gravity in solid bodies, On the centre of gravity in plane figures, On the equilibrium of plane figures, On the squaring of the parabola, forming together a single work: The Squaring of the Parabola. The first part may be reconstructed with some degree of probability from Heron’s text; of the second one we know only what it contained, but nothing of how the results were found; the third part, Plan. Aequil. 1 , we know to be a selection from a greater work. These three works together seem to be quoted by Archimedes himself as ‘‘The Mechanics”.

But now a curious problem presents itself: how did all this come about? Was Archimedes investigating the centre of gravity, and did he then see that here was a way towards the squaring of the parabola? It does not seem possible.

Or was he trying to square the parabola, and did he then realize that he had to discover the centre of gravity in order to do it‘? This does not look so very probable either.

But I think that we can find an explanation by studying his book On the Method. Introd. (p. 428 sq.). He writes that while a mechanical perception, the&iu, does not offer a proof, apodeixis, it will often be a help towards finding the proof to have a clear perception of what is to be proven. So although Eudoxos was the first to prove that the cone and the pyramid have one third of the volume of a cylinder, respectively a prism, of the same base and height, honour is also due to Democritos for having first announced this, even without a proof.

6 A. G. Drachmann

Archimedes then gives first a mechanical perception, based upon the use of equilibria and centres of gravity, of the squaring of the parabola, next a mathematical proof, without any “mechanics”. In the Quad. Parab. he does the same thing, his perception and his proof being different from those of the Method.

From this it would at first seem a reasonable conclusion that Demokritos had reached his results by a similar mechanical perception; but this is impossible, since the centre of gravity was discovered by Archimedes, and so was not available to Demokritos, who died long before Archimedes was born.

A very likely explanation is that Demokritos used a mechanical experi- ment, turning a cylinder and a cone out of wood or clay and comparing their weights. And then it seems only reasonable to suppose that Archi- medes did the same thing: he made a parabolic section and a triangle out of thin plate and compared their weights. Then he probably got the surprise of his life when he found that they were commensurable.

But if he did this, why does he not say so? The answer is plain, if we read Plutarchos in the right way: Because,

whatever he wrote, he wanted to state it in the language of mathematics. So he went through his physical, mechanical experiment and translated it into mathematical terms.

First he studied what is meant by weighing. You place on one scale pan a body of unknown weight and on the other pan known weights, until the balance beam is horizontal and will go back to this horizontal position, if you depress either end of it. Then there is equilibrium, and the weight is the same in both pans, provided that the arms of the balance are of equal length. These lengths are measured from the point of sus- pension of the balance to the points of suspension of the pans; but what if the bodies are placed directly on the beam? If the process of weighing is to be unambiguous, there must be just a single point in each body from which we measure its distance from the suspension point of the balance beam.

So Archimedes undertook to define this point, which he called first the centre of inclination, kentron ropps, later the centre of gravity, kentron bareos. We know from Papposls that he wrote a book On Balances, peri zyg&z, and this will have been the result of these studies.

For this very first investigation into the centre of gravity Archimedes had to use a mechanical definition of equilibrium: “By equilibrium we

Archimedes and the Science of Physics 7

understand the state when two bodies are suspended in a balance, and the beam regains its horizontal position no matter how it is moved.”g But in the further studies his definition of equilibrium is purely mathematical : “ln a magnitude consisting of two magnitudes the centre of gravity will be found on the straight line connecting the centres of gravity of the two magnitudes, in such a place that the distances to the two centres of gravity is inversely proportional to the weights of the magnitudes.” It is C. Juel who first has called attention to this factll. In the Plan. Aequil 1 : 6-7 Archimedes proposes to prove that magnitudes are in equilibrium at distances in inverse proportion to their weights; but what he does prove is that the magnitude consisting of the two magnitudes in question has its centre of gravity in this point.

The first book of the Mechanics: On the Centre of Gravity in Bodies, deals with bodies only, but what is given by Heron in his Mechanics must be regarded as no more than a selection of this work; we might refer to this book some of the lemmata found in The Method: 8. The centre of gravity of a cylinder is the middle point of its axis. 9. The centre of gravity of a prism is the middle point of its axis.

10. The centre of gravity of a cone is found in its axis in the point whose distance to the vertex is three times that of the rest.

As for the lemmata 1 and 3 the matter is a little more doubtful, for what, after all, is a magnitude? We shall get to that a little later.

The second book of the Mechanics, the book On the Centre of Gravity in Planes Figures, is known to us only through the remark by Heron that Archimedes has shown that it does give sense to speak of such a thing. In the sequence of the physical experiment to gain a “mechanical perception” it represents the fact that Archimedes used a thin plate of homogenous material and equal thickness for cutting out his two figures. His mathematical treatment of this I do not venture to try to reproduce, but we may refer to this work two lemmata from The Method: 4. The centre of gravity of a straight line is its middle point. 7. The centre of gravity of a circle is the point that is its centre. To these should be added the three stipulations from the Plan. Aequil. 1 :

4, 5, 7, quoted above. But if we go on, we shall find that quite a number of propositions in

Heron’s Mechanics and by him referred to “Archimedes and others” may be placed in this book.

2 : 35 “To find the centre of gravity in a triangle” must have been

8 A . G. Drachmann

taken from “others”, for not only it is different from the two propositions given by Archimedes in the Plan. Aequil 1 , but it is incorrect, as shown by Dijksterhuisl. Pappos has the same false proof, and it may be due to Poseidonios the Stoic. But the following are correct, and so may be Archimedean :

2 : 36 To find the centre of gravity in a quadrangle. 2 : 37 To find the centre of gravity in a pentagon. The next chapter, 2 : 38 “We want to explain, if there is a triangle

ABG of equal thickness and weight and there are uprights under the points ABG in equal position, how we can find the amount of the weight of the triangle ABG that every one of them will have to carry”, leads us back to I : 25-31, the Book on Uprights, and so do the ch. 2 : 3941, which deal with variations of the same theme.

But though the Book on Uprights speaks of beams and walls and pillars and columns, mathematically it is just a question of plane figures, for the thickness of the beams and posts is nowhere mentioned. So I think that the Book on Uprights may have been part of the Book o n , the Centre of Gravity of Plane Figures. Its connexion with the experiment of Archimedes is evident, since Heron’s Mechanics 2 : 38, 39 speak of “a triangle of equal weight and thickness”, though in the propositions 40 and 41 only a triangle and a pentagon are mentioned.

From all this we get the impression that Archimedes’s way of working was such that when he took up a subject he tried to go as far as he could.

He wanted to find a mathematical expression for the act of weighing, and this not only made him discover the centre of gravity, which was what he needed for the squaring of the parabola, but also caused him to write the work “On Balances” quoted by Papposls. Of this work we have only that one quotation “For it has been proved by Archimedes in his book “On Balances” that the greater circles overpower the smaller circles, when their turning takes place on the same centre”.

To this we may add the remark in Simpliciusl7, that Archimedes invented the steel-yard, charistion.

“When Archimedes made the instrument for weighmg called charistion by the proportion between that which is moving, that which is moved, and the way travelled, then, as the proportion went on as far as it could go, he made the well-known boast “Somewhere to stand, and f shall move the earth”.”

Aristoteles’s Mechanical Problems ch. 20 speaks of another sort of

Archimedes and the Science of Physics 9

steel-yard, phalanx, for weighing meat, where the pan is hung from one end of the beam, a weight is fixed on the other end, and a number of loops are sitting on the beam between them. The weight of the meat is determined by the loop that makes the beam stay horizontal. The charis- tion, which is a steel-yard with a sliding weight, was at once more handy and more accurate.

As I have pointed out already58, there is a possibility that Heron’s Mechanics I : 34 comes from On Balances. It shows a pulley-wheel, to which two strings are fastened at the ends of a horizontal diameter. As long as they carry equal weights, the wheel will be at rest, but if a certain weight is added to the burden on the left, it will sink down, while the burden on the right will rise as the string is wound round the wheel.

To find out where the wheel will stop, we produce the left-hand string upwards till it intersects the horizontal diameter. The distance froin the centre to this point will be to the radius in inverse proportion as the two burdens. “And so it is possible that any weight can balance a weight smalIer than itself in this way.”

The connexion between this figure and the quotation from Simplicius is obvious.

Next, according to Heron, Archimedes goes on to show that the theory of the centre of gravity may be used in plane figures, so that the area of the plane figure is used as a mathematical expression for its weight. This corresponds to the fact that he used for his experiment a thin plate of equal thickness and weight. A trace of this may be seen in Heron’s Mechanics 2 : 38, 39, where a “triangle of equal thickness and weight” is used. Now Archimedes could deal with bodies, plane figures and even straight lines in the same way; so he uses the word “magnitude” to indicate that the propositions are valid for all three; when necessary he uses only “plane figures”, e.g. in the stipulations 4,5 and 7 to the Plan. Aequil. 1 .

But in the stipulations of the Plan. Aequil. 1 and in the work itself every trace of the physical experiment has vanished, and we have a mathematical treatment of the “mechanical perception” we are seeking. But Heron’s work shows plainly that the Plan. Aequil. 1 is merely a selection ad hoc for the Squaring of the Parabola. Archimedes has really found out the centre of gravity of all sorts of rectilinear plane figures. And he goes further. The “Book on Uprights” shows how to use the idea of a centre of gravity in straight lines for the purpose of finding the

10 A . G. Drachmann

distribution of the weight of a beam or a wall supported by two or more pillars placed anyhow, or of a balk carried by any number of men, all of it a work on statics of the greatest use for practical purposes. After treating of the beam by itself, he goes on and hangs weights on it; then he takes up plane figures, shows the distribution of weight in their angles, and then proceeds to place weights on them also. AU of it most useful.

The picture given by Plutarchos of the unworldly mathematician, who scorned to write of anything practical or banausic, is found to be some- what lopsided as a portrait of Archimedes. In stead we find a mathematical and mechanical genius, who whenever he found something in his mathe- matical studies that could be of any practical use, followed it up as far as it would take him, and then for the benefit of those who could use it, expressed the result in a clear mathematical language. He distinguished sternly between a “mechanical perception” and a mathematical proof, but he wanted to express the “mechanical perception” in mathematical terms, because that was the best way to present it.

The mistake of Plutarchos is quite easy to understand. He found that Archimedes was writing in a mathematical language, and so he concluded that he was writing about mathematics. But such was not the case; he was writing about physics. Only it took some 1200 years before other people began to write about physics in this way; but then Archimedes was Archimedes.

My best thanks are due to the Carlsberg Foundation for a grant to- wards the writing of this paper.

R E F E R E N C E S

1. Dijksterhuis, E. J. : Archimedes. Copenhagen 1956. (Acta historica scientiarum naturalium et medicinalium 12) p. 301 sq.

2. Drachmann, A. G.: The Screw of Archimedes. Actes du VIIIe Congrbs Internat. d’Histoire des Sciences. Florence 1956, p. 940-943.

3. The same: How Archimedes Expected to Move the Earth. Centaurus 1958: 5: 278-282. 4. The same: Fragments from Archimedes in Heron’s Mechanics. Centaurus 1963: 8:

5. Same work: p. 96. 5a. Same work: p. 140 sqq 6. Famngton, Benjamin: Greek Science. Its Meaning for Us. Penguin Books 1961

(Pelican Books A 142) p. 216, 309. 7. Heiberg, I. L. : Geschichte der Mathematik und Naturwissenschaften im Altertum.

Munchen 1925. (Handbuch der Altertumswissenschaft. 5. Bd., 1. Abt., 2. HAlfte) p. 67 sq.

91-146.

8.

9. 10. 11.

12. 13. 14. 15.

16. 17.

18.

Archimedes and the Science of Physics I 1

Heron: Mechanik. Hrsg. von L. Nix. Leipzig 1900. (Heronis Alexandrini opera qvae supersunt, Vol. 2, fasc. 1) 1 :24 (b) (The small letters from my translation, ref. 4). Same work, 1:24 (d, e, f, h). Same work, 1:24 (i, f, h, d, e, k, p, 9). Juel, C.: Note om Archimedes’ Tyngdepunktslere. Oversigt over det Kgl. Danske Videnskabernes Selskabs Forhandlinger 1914, p. 421-441. Karpos. Pappos (ref. 15) p. 1026. Klemm, Friedrich: A History of Western Technology. London 1959, p. 20. Oreibasios: Coliectiones medicae 49:23. (Ed. H. Rieder, Vol. 4, Leipzig 1933, p. 33). Pappos. Pappi Alexandrini Collectionis quae supersunt. Instr. Fridericus Hultsch. Vol. 3, Tom. 1. Berlin 1878. p. 1068. Plutarchos: Marcellus ch. 17. Simplicius p. 1 1 10. (In Aristotelis Physicorum libros commentaria ed. Diels. Berlin 1845). Vitruvius: De architectura 9: introd. 9-12.