Fergus, who is perhaps our best mathematician, p.r.i.c.ked up his ears when he heard there was a treatise on the relation of the Circle to the Square. Like most of the intelligent boys who will read this book, Fergus had tried his hand on the fascinating problem which deals with that proportion. Younger readers will remember that it is treated in "Swiss Family." Jack--or is it perhaps Ernest?--remembers there, that for the ribbon which was to go round a hat the hat-maker allowed three times the diameter of the hat, and a little more. This "little more" is the delicate fraction over which Archimedes studied; and Fergus, after him. Fergus knew the proportion as far as thirty-three figures in decimals. These are 3.141,592,653,589,793,238,462,643,383,279,502. When Uncle Fritz asked Fergus to repeat these, the boy did it promptly, somewhat to the astonishment of the others. He had committed it to memory by one of Mr. Gouraud"s "a.n.a.logies," which are always convenient for persons who have mathematical formulas to remember.
When those of the young people who were interested in mathematics looked at Archimedes"s solution of the problem, they found it was the same as that they had themselves tried at school. But he carried it so far as to inscribe a circle between two polygons, each of ninety-six sides; and his calculation is based on the relation between the two.
Taking the "Swiss Family Robinson" statement again, Archimedes shows that the circ.u.mference of a circle exceeds three times its diameter by a small fraction, which is less than 10/70 and greater than 10/71 and that a circle is to its circ.u.mscribing square nearly as 11 to 14. Those who wish to carry his calculations farther may be pleased to know that he found the figures 7 to 22 expressed the relation more correctly than 1 to 3 does. Metius, another ancient mathematician, used the proportion 113 to 355. If you reduce that to decimals, you will find it correct to the sixth decimal. Remember that Archimedes and Metius had not the convenience of the Arabic or decimal notation. Imagine yourselves doing Metius"s sum in division when you have to divide CCCLV by CXIII.
Archimedes, in fact, used the Greek notation,--which was a little better than the Roman, but had none of the facility of ours. For every _ten_, from 20 to 90, they had a separate character, and for every _hundred_, and for every _thousand_. The _thousands_ were the units with a mark underneath. Thus [Greek: a] meant 1, and [Greek: ,a] meant 1,000. To express 113, Archimedes would have written [Greek: rig]. To express 355, he would have written [Greek: tne]; and the place which these signs had in the order would not have affected their value, as they do with us.
We cannot tell how the greater part of Archimedes"s life was spent. But whether he were nominally in public office or not, it is clear enough that he must have given great help to Syracuse and her rulers, as an engineer, long before the war in which the Romans captured that great city. At that time Syracuse was, according to Cicero, "the largest and n.o.blest of the Greek cities." It was in Sicily; but, having been built by colonists from Greece, who still spoke the Greek language, Cicero speaks of it among Greek cities, as he would have spoken of Thurii, or Sybaris, or the cities of "Magna Graecia,"--"great Greece," as they called the Greek settlements in southern Italy. In the Second Punic War Syracuse took sides against Rome with the Carthaginians, though her old king, Hiero, had been a firm ally of the Romans. The most interesting accounts that we have of Archimedes are in Livy"s account of this war, and in Plutarch"s Life of Marcellus, who carried it on on the Roman side. Livy says of Archimedes that he was--
"A man of unrivalled skill in observing the heavens and the stars, but more deserving of admiration as the inventor and constructor of warlike engines and works, by means of which, with a very slight effort, he turned to ridicule what the enemy effected with great difficulty.
"The wall, which ran along unequal eminences, most of which were high and difficult of access, some low and open to approach along level vales, was furnished by him with every kind of warlike engine, as seemed suitable to each particular place. Marcellus attacked from the quinqueremes [his large ships] the wall of the Achradina, which was washed by the sea. From the other ships the archers and slingers and light infantry, whose weapon is difficult to be thrown back by the unskilful, allowed scarce any person to remain upon the wall unwounded.
These soldiers, as they required some range in aiming their missiles upward, kept their ships at a distance from the wall. Eight more quinqueremes joined together in pairs, the oars on their inner sides being removed, so that side might be placed to side, and which thus formed ships [of double width], and were worked by the outer oars, carried turrets built up in stories, and other battering-engines.
"Against this naval armament Archimedes placed, on different parts of the walls, engines of various dimensions. Against the ships which were at a distance he discharged stones of immense weight; those which were nearer he a.s.sailed with lighter and more numerous missiles. Lastly, in order that his own men might heap their weapons upon the enemy without receiving any wounds themselves, he perforated the wall from the top to the bottom with a great number of loop-holes, about a cubit in diameter, through which some with arrows, others with scorpions of moderate size, a.s.sailed the enemies without being seen. He threw upon their sterns some of the ships which came nearer to the walls, in order to get inside the range of the engines, raising up their prows by means of an iron grapple attached to a strong chain, by means of a _tolleno_ [or derrick], which projected from the wall and overhung them, having a heavy counterpoise of lead which forced the line to the ground. Then, the grapple being suddenly disengaged, the ship, falling from the wall, was by these means, to the utter consternation of the seamen, so dashed against the water that even if it came back to its true position it took in a great quant.i.ty of water."
"Fancy," cried Bedford, "one of their double quinqueremes, when she had run bravely in under the shelter of the wall. Just as the men think they can begin to work, up goes the prow, and they all are tumbled down into the steerage. Up she goes, and fifty rowers are on each other in a pile; when the old pile-driver claw lets go again, and down she comes, splash into the sea. And then Archimedes pokes his head out through one of the holes, and says in Greek, "How do you like that, my friends?" I do not wonder they were discouraged."
The bold cliff of the water front of Syracuse gave Archimedes a particular advantage for defensive operations of this sort. They are described in more detail in Plutarch"s Life of Marcellus, who was the Roman general employed against Syracuse, and who was held at bay by Archimedes for three years.
Here is Plutarch"s account:--
Marcellus, with sixty galleys, each with five rows of oars, furnished with all sorts of arms and missiles, and a huge bridge of planks laid upon eight ships chained together,[1] upon which was carried the engine to cast stones and darts, a.s.saulted the walls. He relied on the abundance and magnificence of his preparations, and on his own previous glory; all which, however, were, it would seem, but trifles for Archimedes and his machines.
These machines he had designed and contrived, not as matters of any importance, but as mere amus.e.m.e.nts in geometry,--in compliance with King Hiero"s desire and request, some little time before, that he should reduce to practice some part of his admirable speculations in science, and by accommodating the theoretic truth to sensation and ordinary use, bring it more within the appreciation of people in general. Eudoxus and Archytas had been the first originators of this far-famed and highly prized art of mechanics, which they employed as an elegant ill.u.s.tration of geometrical truths, and as a means of sustaining experimentally, to the satisfaction of the senses, conclusions too intricate for proof by words and diagrams. As, for example, to solve the problem so often required in constructing geometrical figures, "Given the two extremes to find the two mean lines of a proportion," both these mathematicians had recourse to the aid of instruments, adapting to their purpose certain curves and sections of lines. But what with Plato"s indignation at it, and his invectives against it as the mere corruption and annihilation of the one good of geometry, which was thus shamefully turning its back upon the unembodied objects of pure intelligence, to recur to sensation, and to ask help (not to be obtained without base subservience and depravation) from matter; so it was that mechanics came to be separated from geometry, and when repudiated and neglected by philosophers, took its place as a military art.
Archimedes, however, in writing to King Hiero, whose friend and near relative he was, had stated that, given the force, any given weight might be moved; and even boasted, we are told, relying on the strength of demonstration, that if there were another earth, by going into it he could move this.
Hiero being struck with amazement at this, and entreating him to make good this a.s.sertion by actual experiment, and show some great weight moved by a small engine, he fixed upon a ship of burden out of the king"s a.r.s.enal, which could not be drawn out of the dock without great labor by many men. Loading her with many pa.s.sengers and a full freight, sitting himself the while far off, with no great endeavor, but only holding the head of the pulley in his hand and drawing the cord by degrees, he drew the ship in a straight line, as smoothly and evenly as if she had been in the sea.
The king, astonished at this, and convinced of the power of the art, prevailed upon Archimedes to make him engines accommodated to all the purposes, offensive and defensive, of a siege. These the king himself never made use of, because he spent almost all his life in a profound quiet and the highest affluence. But the apparatus was, in a most opportune time, ready at hand for the Syracusans, and with it also the engineer himself.
When, therefore, the Romans a.s.saulted the walls in two places at once, fear and consternation stupefied the Syracusans, believing that nothing was able to resist that violence and those forces. But when Archimedes began to ply his engines, he at once shot against the land forces all sorts of missile weapons, with immense ma.s.ses of stone that came down with incredible noise and violence, against which no man could stand; for they knocked down those upon whom they fell in heaps, breaking all their ranks and files. In the mean time huge poles thrust out from the walls over the ships [these were the derricks, or _tollenos_, of Livy]
sunk some by the great weights which they let down from on high upon them; others they lifted up into the air by an iron hand or beak like a crane"s beak, and when they had drawn them up by the prow, and set them on end upon the p.o.o.p, they plunged them to the bottom of the sea. Or else the ships, drawn by engines within, and whirled about, were dashed against the steep rocks that stood jutting out under the walls, with great destruction of the soldiers that were aboard them. A ship was frequently lifted up to a great height in the air (a dreadful thing to behold), and was rolled to and fro and kept swinging, until the mariners were all thrown out, when at length it was dashed against the rocks, or let fall.
At the engine that Marcellus brought upon the bridge of ships,--which was called _Sambuca_ from some resemblance it had to an instrument of music of that name,--while it was as yet approaching the wall, there was discharged a piece of a rock of ten talents" weight,[2] then a second and a third, which, striking upon it with immense force and with a noise like thunder, broke all its foundation to pieces, shook out all its fastenings, and completely dislodged it from the bridge. So Marcellus, doubtful what counsel to pursue, drew off his ships to a safer distance, and sounded a retreat to his forces on land. They then took a resolution of coming up under the walls, if it were possible, in the night; thinking that as Archimedes used ropes stretched at length in playing his engines, the soldiers would now be under the shot, and the darts would, for want of sufficient distance to throw them, fly over their heads without effect. But he, it appeared, had long before framed for such occasion engines accommodated to any distance, and shorter weapons; and had made numerous small openings in the walls, through which, with engines of a shorter range, unexpected blows were inflicted on the a.s.sailants. Thus, when they, who thought to deceive the defenders, came close up to the walls, instantly a shower of darts and other missile weapons was again cast upon them. And when stones came tumbling down perpendicularly upon their heads, and, as it were, the whole wall shot out arrows against them, they retired.
And now, again, as they were going off, arrows and darts of a longer range inflicted a great slaughter among them, and their ships were driven one against another, while they themselves were not able to retaliate in any way. For Archimedes had provided and fixed most of his engines immediately under the wall; whence the Romans, seeing that infinite mischiefs overwhelmed them from no visible means, began to think they were fighting with the G.o.ds.
Yet Marcellus escaped unhurt, and, deriding his own artificers and engineers, "What," said he, "must we give up fighting with this geometrical Briareus, who plays pitch and toss with our ships, and with the mult.i.tude of darts which he showers at a single moment upon us, really outdoes the hundred-handed giants of mythology?" And doubtless the rest of the Syracusans were but the body of Archimedes"s designs, one soul moving and governing all; for, laying aside all other arms, with his alone they infested the Romans and protected themselves. In fine, when such terror had seized upon the Romans that if they did but see a little rope or a piece of wood from the wall, instantly crying out that there it was again, that Archimedes was about to let fly some engine at them, they turned their backs and fled, Marcellus desisted from conflicts and a.s.saults, putting all his hope in a long siege. Yet Archimedes possessed so high a spirit, so profound a soul, and such treasures of scientific knowledge, that though these inventions had now obtained him the renown of more than human sagacity, he yet would not deign to leave behind him any commentary or writing on such subjects; but, repudiating as sordid and ign.o.ble the whole trade of engineering, and every sort of art that lends itself to mere use and profit, he placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life,--studies the superiority of which to all others is unquestioned, and in which the only doubt can be whether the beauty and grandeur of the subjects examined or the precision and cogency of the methods and means of proof most deserve our admiration.
It is not possible to find in all geometry more difficult and intricate questions, or more simple and lucid explanations. Some ascribe this to his natural genius; while others think that incredible toil produced these, to all appearance, easy and unlabored results. No amount of investigation of yours would succeed in attaining the proof; and yet, once seen, you immediately believe you would have discovered it,--by so smooth and so rapid a path he leads you to the conclusion required. And thus it ceases to be incredible that (as is commonly told of him) the charm of his familiar and domestic science made him forget his food and neglect his person to that degree that when he was occasionally carried by absolute violence to bathe, or have his body anointed, he used to trace geometrical figures in the ashes of the fire, and diagrams in the oil on his body, being in a state of entire preoccupation, and, in the truest sense, divine possession, with his love and delight in science.
His discoveries were numerous and admirable; but he is said to have requested his friends and relations that when he was dead they would place over his tomb a sphere containing a cylinder, inscribing it with the ratio which the containing solid bears to the contained.
The boys were highly edified by this statement of the difficulty which Archimedes"s friends found in making him take a bath, and chaffed Jack, who had asked if he were not the inventor of bath-tubs.
When the reading from Plutarch was over, Fergus asked if that were all, and was disappointed that there was nothing about the setting of ships on fire by mirrors. It is one of the old stories of the siege of Syracuse, that he set fire to the Roman ships by concentrating on them the heat of the sun from a number of mirrors. But this story is not in Livy, nor is it in Plutarch, though, as has been seen, they were well disposed to tell what they knew which was marvellous in his achievements. It is told at length and in detail by Zonaras and Tzetzes, two Greek writers of the twelfth century, who must have found it in some ancient writers whose works we do not now have.
"Archimedes," says Zonaras,[3] "having received the rays of the sun on a mirror, by the thickness and polish of which they were reflected and united, kindled a flame in the air, and darted it with full violence upon the ships, which were anch.o.r.ed within a certain distance, in such a manner that they were burned to ashes."
The same writer says that Proclus, a celebrated "mathematician" of Constantinople, in the sixth century, at the siege of Constantinople set fire to the Thracian fleet by means of bra.s.s mirrors. Tzetzes is yet more particular. He says that when the Roman galleys were within a bow-shot of the city walls, Archimedes brought together hexagonal specula (mirrors) with other smaller ones of twenty-four facets, and caused them to be placed each at a proper distance; that he moved these by means of hinges and plates of metal; that the hexagon was bisected by the meridian of summer and winter; that it was placed opposite the sun; and that a great fire was thus kindled, which consumed the ships.
Now, it is to be remembered that these are the accounts of writers who were not so good mechanics as Archimedes. It should be remembered, also, that in the conditions of war then, the distance at which ships would be anch.o.r.ed in a little harbor like that of Syracuse was not great. By "bow-shot" would be meant the distance at which a bow would do serious damage. Doubtful as the story of Zonaras and Tzetzes seems, it received unexpected confirmation in the year 1747 from a celebrated experiment tried by the naturalist Buffon.
After encountering many difficulties, which he had foreseen with great acuteness, and obviated with equal ingenuity, Buffon at length succeeded in repeating Archimedes"s performance. In the spring of 1747 he laid before the French Academy a memoir which, in his collected works, extends over upwards of eighty pages. In this paper he described himself as in possession of an apparatus by means of which he could set fire to planks at the distance of 200 and even 210 feet, and melt metals and metallic minerals at distances varying from 25 to 40 feet. This apparatus he describes as composed of 168 plain gla.s.ses, silvered on the back, each six inches broad by eight inches long. These, he says, were ranged in a large wooden frame, at intervals not exceeding the third of an inch, so that, by means of an adjustment behind, each should be movable in all directions independent of the rest; the s.p.a.ces between the gla.s.ses being further of use in allowing the operator to see from behind the point on which it behooved the various disks to be converged.
In this last statement there is a parallel with that of Tzetzes, who speaks of the division of Archimedes"s mirrors.
At the present moment naturalists are paying great attention to plans for the using of the heat of the sun. It is said that on any county in the United States, twenty by thirty miles square, there is wasted as much heat of the sun as would drive, if we knew how to use it, all the steam-engines in the world.
Fergus asked Uncle Fritz if he believed that Archimedes threw seven hundred pounds of stone from one of his machines. The largest modern guns throw shot of one thousand pounds, and it is only quite recently that any such shot have been used.
Uncle Fritz told him that in the museum at St. Germain-en-Laye he would one day see a modern catapult, made by Colonel de Reffye from the design of a Roman catapult on Trajan"s Column. This is supposed to be of the same pattern which is called an "Onager" in the Latin books. This catapult throws, when it is tested, a shot of twenty-four pounds, or it throws a sheaf of short arrows. In one catapult the power is gained by twisting ox-hide very tightly, and suddenly releasing it. Another is a very stout bow, worked with a small windla.s.s. Of course this will give a great power.
Seven hundred pounds, however, seems beyond the ability of any such machines as this; but from his higher walls Archimedes could, of course, have rolled such stones down on the decks of the ships below. And if he were throwing other stones or leaden b.a.l.l.s to a greater distance with his _Onagers_, it may well be that Plutarch or Livy did not take very accurate account of the particular engine which threw one stone or another.
Archimedes was killed by a Roman soldier, to the great grief of Marcellus, when the Romans finally took Syracuse. The city fell through drunkenness, which was, and is, the cause of more failure in the world than anything else which can be named. Marcellus, in some conversations about the exchange or redemption of a prisoner, observed a tower somewhat detached from the wall, which was, as he thought, carelessly guarded. Choosing the night of a feast of Diana, when the Syracusans were wholly given up to wine and sport, he took the tower by surprise, and from the tower seized the wall and made his way into the city. In the sack of the city by the soldiers, which followed, Archimedes was killed. The story is told in different ways. Plutarch says that he was working out some problem by a diagram, and never noticed the incursion of the Romans, nor that the city was taken. A soldier, unexpectedly coming up to him in this transport of study and meditation, commanded him to follow him to Marcellus; which he declining to do before he had worked out his problem to a demonstration, the soldier, enraged, drew his sword, and ran him through. "Others write that a Roman soldier, running upon him with a drawn sword, offered to kill him, and that Archimedes, looking back, earnestly besought him to hold his hand a little while, that he might not leave what he was then at work upon inconsequent and imperfect; but the soldier, not moved by his entreaty, instantly killed him. Others, again, relate that as Archimedes was carrying to Marcellus mathematical instruments, dials, spheres, and angles by which the magnitude of the sun might be measured to the sight, some soldiers, seeing him, and thinking that he carried gold in a vessel, slew him.
"Certain it is, that his death was very afflicting to Marcellus, and that Marcellus ever after regarded him that killed him as a murderer, and that he sought for the kindred of Archimedes and honored them with signal honors."
Archimedes, as has been said, had asked that his monument might be a cylinder bearing a sphere, in commemoration of his discovery of the proportion between a cylinder and a sphere of the same diameter. A century and a half after, when Cicero was quaestor of Sicily, he found this monument, neglected, forgotten, and covered with a rank growth of thistles and other weeds.
"It was left," he says, "for one who came from Arpinas, to show to the men of Syracuse where their greatest countryman lay buried."
III.
FRIAR BACON.
"All the world seems to have known of Columbus"s discoveries as soon as he came home, but all the world did not know at once of Archimedes"s inventions; indeed, I should think the world did not know now what all of them are."
Hester Van Brunt was saying this in the hall, as the girls laid off their waterproofs, when they next met the Colonel.
"I think that may often be said of what we call Inventions and what we call Discoveries," he said, "till quite recent times. When a man invented a new process, it was supposed that if he could keep the secret, it might be to him a very valuable secret. But when one discovered an island or a continent, it was almost impossible to keep the secret. They tried it sometimes, as you know. But there must be a whole ship"s crew who know something of the new-found land, and from some of them the secret would leak out.
"But there has been many a process in the arts lost, because the man who discovered the new quality in nature or invented the new method in manufacture kept it secret, so that he might do better work than his compet.i.tors. This went so far that boys were apprenticed to masters to learn "the secrets of their trades.""
Fergus said that in old times inventors were not always treated very kindly. If people thought they were sorcerers, or in league with the Devil, they did not care much for the invention.
Uncle Fritz said they would find plenty of instances of the persecution of inventors, even to quite a late date. It is impossible, of course, to say how many good things were lost to the world by the pig-headedness which discouraged new inventions. It is marvellous to think what progress single men made, who had to begin almost at the beginning, and learn for themselves what every intelligent boy or girl now finds ready for him in the Cyclopaedia. It is very clear that the same beginnings were made again and again by some of the early inventors. Then, what they learned had been almost forgotten. There was no careful record of their experiments, or, if any, it was in one ma.n.u.script, and that was not accessible to people trying to follow in their steps.
"I have laid out for you," said Uncle Fritz, "some of the early accounts of Friar Bacon,--Roger Bacon. He is one of the most distinguished of the early students of what we now call natural philosophy in England. It was in one of the darkest centuries of the Dark Ages.
"But see what he did.