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LIBRARY OF WELLESLEY ‘COLLECE
PURCHASED FROM LIBRARY FUNDS
THe WORKS
OF
ARCHIME Dis.
London: C. J. CLAY anv SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, AVE MARIA LANE.
Glasgow: 263, ARGYLE STREET.
Leipsig: F. A. BROCKHAUS. fet Work: THE MACMILLAN COMPANY.
THE WORKS
OF
ΠΟ LH lM BEDE 5
2
EDITED IN MODERN NOTATION
WITH INTRODUCTORY CHAPTERS
BY
rE 1. HEATH sesh,
SOMETIME FELLOW OF TRINITY COLLEGE, CAMBRIDGE.
CAMBRIDGE: AT THE UNIVERSITY PRESS.
1897
[All Rights reserved.]
LO |e
Cambridge : PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS.
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PREFACE.
HIS book is intended to form a companion volume to my edition of the treatise of Apollonius on Conic Sections lately published. If it was worth while to attempt to make the work of “the great geometer” accessible to the mathematician of to-day who might not be able, in consequence of its length and of its form, either to read it in the original Greek or in a Latin translation, or, having read it, to master it and grasp the whole scheme of the treatise, I feel that I owe even less of an apology for offering to the public a reproduction, on the same lines, of the extant works of perhaps the greatest mathematical genius that the world has ever seen.
Michel Chasles has drawn an instructive distinction between the predominant features of the geometry of Archimedes and of the geometry which we find so highly developed in Apollo- nius. Their works may be regarded, says Chasles, as the origin and basis of two great inquiries which seem to share between them the domain of geometry. Apollonius is concerned with the Geometry of Forms and Situations, while in Archimedes we find the Geometry of Measurements dealing with the quad- rature of curvilinear plane figures and with the quadrature and cubature of curved surfaces, investigations which “gave birth to the calculus of the infinite conceived and brought to perfection successively by Kepler, Cavalieri, Fermat, Leibniz, and Newton.” But whether Archimedes is viewed as the man who, with the limited means at his disposal, nevertheless succeeded in performing what are really integrations for the purpose of finding the area of a parabolic segment and a
Vi PREFACE.
spiral, the surface and volume of a sphere and a segment of a sphere, and the volume of any segments of the solids of revolution of the second degree, whether he is seen finding the centre of gravity of a parabolic segment, calculating arithmetical approximations to the value of 7, inventing a system for expressing in words any number up to that which we should write down with 1 followed by 80,000 billion ciphers, or inventing the whole science of hydrostatics and at the same time carrying it so far as to give a most complete investigation of the positions of rest and stability of a right segment of a paraboloid of revolution floating in a fluid, the intelligent reader cannot fail to be struck by the remarkable range of subjects and the mastery of treatment. And if these are such as to create genuine enthusiasm in the student of Archimedes, the style and method are no less irresistibly attractive. One feature which will probably most impress the mathematician accustomed to the rapidity and directness secured by the generality of modern methods is the deliberation with which Archimedes approaches the solution of any one of his main problems. Yet this very characteristic, with its incidental effects, is calculated to excite the more admiration because the method suggests the tactics of some great strategist who foresees everything, eliminates everything not immediately conducive to the execution of his plan, masters every position in its order, and then suddenly (when the very elaboration of the scheme has almost obscured, in the mind of the spectator, its ultimate object) strikes the final blow. Thus we read in Archimedes proposition after proposition the bearing of which is not immediately obvious but which we find infallibly used later on; and we are led on by such easy stages that the difficulty of the original problem, as presented at the outset, is scarcely appreciated. As Plutarch says, “it is not possible to find in geometry more difficult and troublesome questions, or more simple and lucid explanations.” But it is decidedly a rhetorical exaggeration when Plutarch goes on to say that we are deceived
PREFACE. Vil
by the easiness of the successive steps into the belief that anyone could have discovered them for himself. On the contrary, the studied simplicity and the perfect finish of the treatises involve at the same time an element of mystery. Though each step depends upon the preceding ones, we are left in the dark as to how they were suggested to Archimedes. There is, in fact, much truth in a remark of Wallis to the effect that he seems “as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results.” Wallis adds with equal reason that not only Archimedes but nearly all the ancients so hid away from posterity their method of Analysis (though it is certain that they had one) that more modern mathematicians found it easier to invent a new Analysis than to seek out the old. This is no doubt the reason why Archimedes and other Greek geometers have received so little attention during the present century and why Archimedes is for the most part only vaguely remembered as the inventor of a screw, while even mathematicians scarcely know him except as the discoverer of the principle in hydro- statics which bears his name. It is only of recent years that we have had a satisfactory edition of the Greek text, that of Heiberg brought out in 1880-1, and I know of no complete translation since the German one of Nizze, published in 1824, which is now out of print and so rare that I had some difficulty in procuring a copy.
The plan of this work is then the same as that which I followed in editing the Conics of Apollonius. In this case, however, there has been less need as well as less opportunity for compression, and it has been possible to retain the numbering of the propositions and to enunciate them in a manner more nearly approaching the original without thereby making the enunciations obscure. Moreover, the subject matter is not so complicated as to necessitate absolute uniformity in the notation used (which is the only means whereby Apollonius can be made
Vill PREFACE.
even tolerably readable), though I have tried to secure as much uniformity as was fairly possible. My main object has been to present a perfectly faithful reproduction of the treatises as they have come down to us, neither adding anything nor leaving out anything essential or important. The notes are for the most part intended to throw light on particular points in the text or to supply proofs of propositions assumed by Archimedes as known; sometimes I have thought it right to insert within square brackets after certain propositions, and in the same type, notes designed to bring out the exact significance of those propositions, in cases where to place such notes in the Intro- duction or at the bottom of the page might lead to their being overlooked.
Much of the Introduction is, as will be seen, historical; the rest is devoted partly to giving a more general view of certain methods employed by Archimedes and of their mathematical significance than would be possible in notes to separate propo- sitions, and partly to the discussion of certain questions arising out of the subject matter upon which we have no positive historical data to guide us. In these latter cases, where it is necessary to put forward hypotheses for the purpose of explaining obscure points, I have been careful to call attention to their speculative character, though I have given the historical evidence where such can be quoted in support of a particular hypothesis, my object being to place side by side the authentic information which we possess and the inferences which have been or may be drawn from it, in order that the reader may be in a position to judge for himself how far he can accept the latter as probable. Perhaps I may be thought to owe an apology for the length of one chapter on the so-called νεύσεις, or inclinationes, which goes somewhat beyond what is necessary for the elucidation of Archimedes; but the subject is interesting, and I thought it well to make my account of it as complete as possible in order to round off, as it were, my studies in Apollonius and Archimedes.
PREFACE. 1x
I have had one disappointment in preparing this book for the press. I was particularly anxious to place on or opposite the title-page a portrait of Archimedes, and I was encouraged in this idea by the fact that the title-page of Torelli’s edition bears a representation in medallion form on which are endorsed the words ‘Archimedis effigies marmorea in vetert anaglypho Romae asservato. Caution was however suggested when I found two more portraits wholly unlike this but still claiming to represent Archimedes, one of them appearing at the beginning of Peyrard’s French translation of 1807, and the other in Gronovius’ Thesaurus Graecarum Antiquitatum ; and I thought it well to inquire further into the matter. I am now informed by Dr A. S. Murray of the British Museum that there does not appear to be any authority for any one of the three, and that writers on iconography apparently do not recognise an Archimedes among existing portraits. I was, therefore, re- luctantly obliged to give up my idea.
The proof sheets have, as on the former occasion, been read over by my brother, Dr R. 5. Heath, Principal of Mason College, Birmingham ; and I desire to take this opportunity of thanking him for undertaking what might well have seemed, to any one less genuinely interested in Greek geometry, a thankless task.
Τ. 1. BATH.
March, 1897.
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LIST OF THE PRINCIPAL WORKS CONSULTED.
JOSEPH TORELLI, Archimedis quae supersunt omnia cum LHutocii Asca- lonitae commentariis. (Oxford, 1792.)
Ernst ΝΊΖΖΕ, Archimedes von Syrakus vorhandene Werke aus dem griechischen tibersetzt und mit erliuternden und kritischen Anmerk- ungen begleitet. (Stralsund, 1824.)
J. L. Herpere, Archimedis opera omnia cum commentariis Eutociv. (Leipzig, 1880-1.)
J. Τὸ HerBereG, Quaestiones Archimedeae. (Copenhagen, 1879.)
F. Huutscn, Article Archimedes in Pauly-Wissowa’s Real-Encyclopidie der classischen Altertumswissenschaften. (Edition of 1895, τι. 1, pp. 507-539.)
C. A. BRETSCHNEIDER, Die Geometrie und die Geometer vor Euklides. (Leipzig, 1870.)
M. Cantor, Vorlesungen iiber Geschichte der Mathematik, Band 1, zweite Auflage. (Leipzig, 1894.)
G. Frrep.etn, Procli Diadochi in primum Euclidis elementorum librum commentarii. (Leipzig, 1873.)
JAMES Gow, A short history of Greek Mathematics. (Cambridge, 1884.)
StzemunD GUtnrHerR, Abriss der Geschichte der Mathematik und der Naturwissenschaften im Altertum in Iwan von Miiller’s Handbuch der klassischen Altertumswissenschaft, v. 1.
HERMANN HANKEL, Zur Geschichte der Mathematik in Alterthum und Mittelalter. (Leipzig, 1874.)
J. Τὸ Herpere, Litterargeschichtliche Studien iiber Euklid. (Leipzig, 1882.)
J. L. Herpere, Luclidis elementa. (Leipzig, 1883-8.)
F. Huxtscn, Article Arithmetica in Pauly-Wissowa’s Real-Encyclopiidie, i. 1, pp. 1066-1116.
ΧΙ LIST OF PRINCIPAL WORKS CONSULTED.
F. Houurscw, Heronis Alexandrini geometricorum et stereometricorum reliquiae. (Berlin, 1864.)
F. Huurscw, Pappi Alexandrini collectionis quae supersunt. (Berlin, 1876-8.)
Grvo Loria, Jl periodo aureo della geometria greca. (Modena, 1895.)
Maximitien Mariz, Histoire des sciences mathématiques et physiques, Tome I. (Paris, 1883.)
J. H. T. Miuuer, Bertriige zur Terminologie der griechischen Mathematiker. (Leipzig, 1860.)
G. H. F. Nessetmann, Die Algebra der Griechen. (Berlin, 1842.)
F. SusEMIHL, Geschichte der griechischen Litteratur in der Alexandrinerzett, Band 1. (Leipzig, 1891.)
P. Tannery, La Géométrie grecque, Premiére partie, Histoire générale de la Géometrie élémentaire. (Paris, 1887.)
H. G. ZeutHen, Die Lehre von den Kegelschnitten im Altertum. (Copen- hagen, 1886.)
H. G. ZeurHen, Geschichte der Mathematik im Altertum und Mittelalter. (Copenhagen, 1896.)
CHAPTER I.
CHAPTER II.
CuHapter ITI.
ΕΠ § 2.
CHAPTER IV.
§ 1. ὃ 2. ὅτι § 4. ὃ 5. ὃ 6.
§ 7. § 8.
CONTENTS. INTRODUCTION.
ARCHIMEDES
MANUSCRIPTS AND PRINCIPAL EDITIONS—ORDER OF COMPOSITION—DIALECT—LOST WORKS
RELATION OF ARCHIMEDES TO HIS PREDECESSORS
Use of traditional geometrical methods
Earlier discoveries affecting quadrature and cubature .
Conic Sections
Surfaces of the second Slee :
Two mean proportionals in continued Sonor: tion .
ARITHMETIC IN ARCHIMEDES
Greek numeral system
Addition and subtraction .
Multiplication
Division :
Extraction of fib square root :
Early investigations of surds or incommensu- rables
Archimedes’ sprronmations to V3
Archimedes’ approximations to the square roots of large numbers which are not complete squares
Note on alternative πον Ati spent © the approximations to ./3
PAGE XV
XXill
XXX1X xl
xlvii li
liv lxvii xviii xix ΙΧΧῚ ΙΧΧῚΪ lxxili |xxiv
Ixxvil Ixxx
lxxxiv
ΧΟ
X1V
CHAPTER V.
CHapter VI.
CuHapter VII.
CONTENTS.
ON THE PROBLEMS KNOWN AS NEYSEIS
Nevoes referred to by Archimedes
Mechanical constructions: the conchoid of Nico- medes :
Pappus’ solution of the νεῦσις reed Ὧ in Props. 8, 9 On Spirals .
The problem of the two mean pear anipeate
The trisection of an angle
On certain plane vevoes
CUBIC EQUATIONS
ANTICIPATIONS BY ARCHIMEDES OF THE INTE- GRAL CALCULUS
CHapteR VIII. THE TERMINOLOGY OF ARCHIMEDES .
THE WORKS OF ARCHIMEDES.
ON THE SPHERE AND CYLINDER, BOOK I. .
”
MEASUREME ON CONOIDS ON SPIRALS
” ” ” BOOK II. NT OF A CIRCLE AND SPHEROIDS
ON THE EQUILIBRIUM OF PLANES, BOOK I.
9?
BOOK II.
” 37 ”
THE SAND-RECKONER QUADRATURE OF THE PARABOLA ON FLOATING BODIES, BOOK I.
” ”
5: BOOK II.
BOOK OF LEMMAS. THE CATTLE-PROBLEM
PAGE
CV
151 189 203 221 233 253 263 301 319
INTRODUCTION.
CHAPTER I. ARCHIMEDES,
A ΠΡῈ of Archimedes was written by one Heracleides*, but this biography has not survived, and such particulars as are known have to be collected from many various sources}. According to Tzetzest he died at the age of 75, and, as he perished in the sack of Syracuse (B.c. 212), it follows that he was probably born about 287 B.c. He was the son of Pheidias the astronomer§, and was on intimate terms with, if not related to, king Hieron and his
* Kutocius mentions this work in his commentary on Archimedes’ Measwre- ment of the circle, ὥς φησιν ‘Hpaxdeldns ἐν τῷ ᾿Αρχιμήδους βίῳ. He alludes te it again in his commentary on Apollonius’ Conics (ed. Heiberg, Vol. τι. p. 168), where, however, the name is wrongly given as Ἡράκλειος. This Heracleides is perhaps the same as the Heracleides mentioned by Archimedes himself in the preface to his book On Spirals.
+ An exhaustive collection of the materials is given in Heiberg’s Quaestiones Archimedeae (1879). The preface to Torelli’s edition also gives the main points, and the same work (pp. 363—370) quotes at length most of the original references to the mechanical inventions of Archimedes. Further, the article Archimedes (by Hultsch) in Pauly-Wissowa’s Real-Encyclopidie der classischen Altertumswissenschaften gives an entirely admirable summary of all the available information. See also Susemihl’s Geschichte der griechischen Litteratur in der Alexandrinerzeit, τ. pp. 723—733.
+ Tzetzes, Chiliad., 11. 35, 105.
§ Pheidias is mentioned in the Sand-reckoner of Archimedes, τῶν προτέρων ἀστρολόγων Hvddeov...Pecdla δὲ τοῦ ἁμοῦ πατρὸς (the last words being the correction of Blass for τοῦ ᾿Ακούπατρος, the reading of the text). Cf. Schol. Clark. in Gregor. Nazianz. Or. 34, p. 355a Morel. Φειδίας τὸ μὲν γένος ἢν Συρακόσιος ἀστρολόγος ὁ ᾿Αρχιμήδους πατήρ.
Xvl INTRODUCTION.
son Gelon. It appears from a passage of Diodorus* that he spent a considerable time at Alexandria, where it may be inferred that he studied with the successors of Euclid. It may have been at Alexandria that he made the acquaintance of Conon of Samos (for whom he had the highest regard both as a mathematician and as a personal friend) and of Eratosthenes. To the former he was in the habit of communicating his discoveries before their publication, and it is to the latter that the famous Cattle-problem purports to have been sent. Another friend, to whom he dedicated several of his works, was Dositheus of Pelusium, a pupil of Conon, presumably at Alexandria though at a date subsequent to Archi- medes’ sojourn there.
After his return to Syracuse he lived a life entirely devoted to mathematical research. Incidentally he made himself famous by a variety of ingenious mechanical inventions. These things were however merely the ‘diversions of geometry at play t,” and he attached no importance to them. In the words of Plutarch, “he possessed so high a spirit, so profound a soul, and such treasures of scientific knowledge that, though these inventions had obtained for him the renown of more than human sagacity, he yet would not deign to leave behind him any written work on such subjects, but, regarding as ignoble and sordid the business of mechanics and every sort of art which is directed to use and profit, he placed his whole ambition in those speculations in whose beauty and subtlety there is no admixture of the common needs of life{.” In fact he wrote only one such mechanical book, On Sphere-making§, to which allusion will be made later. :
Some of his mechanical inventions were used with great effect against the Romans during the siege of Syracuse. Thus he contrived
* Diodorus v. 37, 3, ods [τοὺς κοχλίας] ᾿Αρχιμήδης ὁ Συρακόσιος εὗρεν, ὅτε παρέβαλεν εἰς Αἴγυπτον.
+ Plutarch, Marcellus, 14.
ac ἡδιτα. 17:
8 Pappus vu. p. 1026 (ed. Hultsch). Κάρπος δὲ πού φησιν ὁ ᾿Αντιοχεὺς ᾿Αρχιμήδη τὸν Συρακόσιον ἕν μόνον βιβλίον συντεταχέναι μηχανικὸν τὸ κατὰ τὴν σφαιροποιΐαν, τῶν δὲ ἄλλων οὐδὲν ἠξιωκέναι συντάξαι. καίτοι παρὰ τοῖς πολλοῖς ἐπὶ μηχανικῇ δοξασθεὶς καὶ μεγαλοφυής τις γενόμενος ὁ θαυμαστὸς ἐκεῖνος, ὥστε διαμεῖναι παρὰ πᾶσιν ἀνθρώποις ὑπερβαλλόντως ὑμνούμενος, τῶν τε προηγουμένων γεωμετρικῆς καὶ ἀριθμητικῆς ἐχομένων θεωρίας τὰ βραχύτατα δοκοῦντα εἶναι σπουδαίως συνέγραφεν" ὃς φαίνεται τὰς εἰρημένας ἐπιστήμας οὕτως ἀγαπήσας ὡς μηδὲν ἔξωθεν ὑπομένειν
αὐταῖς ἐπεισάγειν.
ARCHIMEDES. xvil
catapults so ingeniously constructed as to be equally serviceable at long or short ranges, machines for discharging showers of missiles through holes made in the walls, and others consisting of long moveable poles projecting beyond the walls which either dropped heavy weights upon the enemy’s ships, or grappled the prows by means of an iron hand or a beak like that of a crane, then lifted them into the air and let them fall again*. Marcellus is said to have derided his own engineers and artificers with the words, “Shall we not make an end of fighting against this geo- metrical Briareus who, sitting at ease by the sea, plays pitch and toss with our ships to our confusion, and by the multitude of missiles that he hurls at us outdoes the hundred-handed giants of mythology!+”; but the exhortation had no effect, the Romans being in such abject terror that “if they did but see a piece of rope or wood projecting above the wall, they would cry ‘there it is again,’ declaring that Archimedes was setting some engine in motion against them, and would turn their backs and run away, insomuch that Marcellus desisted from all conflicts and assaults, putting all his hope in a long sieget.”
If we are rightly informed, Archimedes died, as he had lived, absorbed in mathematical contemplation. The accounts of the exact circumstances of his death differ in some details. Thus Livy says simply that, amid the scenes of confusion that followed the capture of Syracuse, he was found intent on some figures which he had drawn in the dust, and was killed by a soldier who did not know who he was§. Plutarch gives more than one version in the following passage. ‘‘ Marcellus was most of all afflicted at the death of Archimedes ; for, as fate would have it, he was intent on working out some problem with a diagram and, having fixed his mind and his eyes alike on his investigation, he never noticed the incursion of the Romans nor the capture of the city. And when a soldier came up to him suddenly and bade him follow to
* Polybius, Hist. vin. 7—8; Livy xxiv. 34; Plutarch, Marcellus, 15—17.
+ Plutarch, Marcellus, 17.
+ ibid.
§ Livy xxv. 31. Cum multa irae, multa auaritiae foeda exempla ederentur, Archimedem memoriae proditum est in tanto tumultu, quantum pauor captae urbis in discursu diripientium militum ciere poterat, intentum formis, quas in puluere descripserat, ab ignaro milite quis esset interfectum ; aegre id Marcellum tulisse sepulturaeque curam habitam, et propinquis etiam inquisitis honori praesidioque nomen ac memoriam eius fuisse.
H. A. δ
XVlll INTRODUCTION.
Marcellus, he refused to do so until he had worked out his problem to a demonstration; whereat the soldier was so enraged that he drew his sword and slew him, Others say that the Roman ran up to him with a drawn sword offering to kill him; and, when Archimedes saw him, he begged him earnestly to wait a short time in order that he might not leave his problem incomplete and unsolved, but the other took no notice and killed him. Again there is a third account to the effect that, as he was carrying to Marcellus some of his mathematical instruments, sundials, spheres, and angles adjusted to the apparent size of the sun to the sight, some soldiers met him and, being under the impression that he carried gold in the vessel, slew him*.” The most picturesque version of the story is perhaps that which represents him as saying to a Roman soldier who came too close, “Stand away, fellow, from my diagram,” whereat the man was so enraged that he killed himy+. The addition made to this story by Zonaras, representing him as saying παρὰ κεφαλὰν καὶ μὴ παρὰ γραμμάν, while it no doubt recalls the second version given by Plutarch, is perhaps the most far-fetched of the touches put to the picture by later hands.
Archimedes is said to have requested his friends and relatives to place upon 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}; from which we may infer that he himself regarded the discovery of this ratio [On the Sphere and Cylinder, 1. 33, 34] as his greatest achievement. Cicero, when quaestor in Sicily, found the tomb in a neglected state and restored itS.
Beyond the above particulars of the life of Archimedes, we have nothing left except a number of stories, which, though perhaps not literally accurate, yet help us to a conception of the personality of the most original mathematician of antiquity which we would not willingly have altered. Thus, in illustration of his entire preoccupation by his abstract studies, we are told that he would forget all about his food and such necessities of life, and would be drawing geometrical figures in the ashes of the fire, or, when
Plutarch, Marcellus, 19.
Tzetzes, Chil. 11. 85, 1385; Zonaras Ix. 5. Plutarch, Marcellus, 17 ad jin.
Cicero, Tusc. vy. 64 sq,
ὧν tt + *
ARCHIMEDES. ΧΙΧ
anointing himself, in the oil on his body*. Of the same kind is the well-known story that, when he discovered in a bath the solution of the question referred to him by Hieron as to whether a certain crown supposed to have been made of gold did not in reality contain a certain proportion of silver, he ran naked through the street to his home shouting εὕρηκα, εὕρηκα .
According to Pappust it was in connexion with his discovery of the solution of the problem Zo move a gwen weight by a given force that Archimedes uttered the famous saying, “Give me a place to stand on, and I can move the earth (δός μοι ποῦ στῶ καὶ κινῶ τὴν γῆν). Plutarch represents him as declaring to Hieron that any given weight could be moved by a given force, and boasting, in reliance on the cogency of his demonstration, that, if he were given another earth, he would cross over to it and move this one. ‘And when Hieron was struck with amazement and asked him to reduce the problem to practice and to give an illustration of some great weight moved by a small force, he fixed upon a ship of burden with three masts from the king’s arsenal which had only been drawn up with great labour and many men; and loading her with many passengers and a full freight, sitting himself the while far off, with no great endeavour but only holding the end of a compound pulley (πολύσπαστος) quietly in his hand and pulling at it, he drew the ship along smoothly and safely as if she were moving through the sea$.” According to Proclus the ship was one which Hieron had had made to send to king Ptolemy, and, when all the Syracusans with their combined strength were unable to launch it, Archimedes contrived a mechanical device which enabled Hieron to move it by himself, insomuch that the latter declared that “from that day forth Archimedes was to be believed in every- thing that he might say|.” While however it is thus established that Archimedes invented some mechanical contrivance for moving a large ship and thus gave a practical illustration of his thesis, it is not certain whether the machine used was simply a compound
* Plutarch, Marcellus, 17.
+ Vitruvius, Architect. 1x. 3. For an explanation of the manner in which Archimedes probably solved this problem, see the note following On jloating bodies, τ. 7 (p. 259 sq.).
+ Pappus vir. p. 1060.
§ Plutarch, Marcellus, 14.
|| Proclus, Comm. on Eucl. 1., Ὁ. 63 (ed. Friedlein).
62
XX INTRODUCTION.
pulley (πολύσπαστος) as stated by Plutarch; for Athenaeus*, in describing the same incident, says that a heliw was used. This term must be supposed to refer to a machine similar to the κοχλίας described by Pappus, in which a cog-wheel with oblique teeth moves on a cylindrical helix turned by a handley. Pappus, how- ever, describes it in connexion with the βαρουλκός of Heron, and, while he distinctly refers to Heron as his authority, he gives no hint that Archimedes invented either the βαρουλκός or the par- ticular κοχλίας ; on the other hand, the πολύσπαστος is mentioned by Galen 1, and the τρίσπαστος (triple pulley) by Oribasius§, as one of the inventions of Archimedes, the τρίσπαστος being so called either from its having three wheels (Vitruvius) or three ropes (Oribasius). Nevertheless, it may well be that though the ship could easily be kept in motion, when once started, by the τρί: σπαστος or πολύσπαστος, Archimedes was obliged to use an appliance similar to the κοχλίας to give the first impulse.
The name of yet another instrument appears in connexion with the phrase about moving the earth. Tzetzes’ version is, ‘‘ Give me a place to stand on (za Bw), and I will move the whole earth with a χαριστίων ||”; but,.as in another passage] he uses the word τρίσπαστος, it may be assumed that the two words represented one and the same thing**.
It will be convenient to mention in this place the other mechanical inventions of Archimedes. The best known is the
* Athenaeus v. 207 a—b, κατασκευάσας yap ἕλικα TO τηλικοῦτον σκάφος εἰς THY θάλασσαν κατήγαγε" πρῶτος δ᾽ ᾿Αρχιμήδης εὗρε τὴν τῆς ἕλικος κατασκευήν. ΤῸ the same effect is the statement of Eustathius ad Il. 111. p. 114 (ed. Stallb.) λέγεται δὲ ἕλιξ Kal τι μηχανῆς εἶδος, ὃ πρῶτος εὑρὼν ὁ ᾿Αρχιμήδης εὐδοκίμησέ, φασι, dc αὐτοῦ,
+ Pappus virt. pp. 1066, 1108 sq.
+ Galen, in Hippocr. De artic., 1v. 47 (=xvut. p. 747, ed. Kiihn).
§ Oribasius, Coll. med., xu1x. 22 (iv. p. 407, ed. Bussemaker), ᾿Απελλίδους ἢ ᾿Αρχιμήδους τρίσπαστον, described in the same passage as having been invented πρὸς Tas τῶν πλοίων καθολκάς.
|| Tzetzes, Chil. τι. 180.
q Ibid., ut. 61, ὁ γῆν ἀνασπῶν μηχανῇ τῇ τρισπάστῳ βοῶν' ὅπα Bw καὶ σαλεύσω τὴν χθόνα.
** Heiberg compares Simplicius, Comm. in Aristot. Phys. (ed. Diels, p. 1110, 1. 2), ταύτῃ δὲ τῇ ἀναλογίᾳ τοῦ κινοῦντος καὶ τοῦ κινουμένου Kal τοῦ διαστήματος τὸ σταθμιστικὸν ὄργανον τὸν καλούμενον χαριστίωνα συστήσας ὁ ᾿Αρχιμήδης ὡς μέχρι παντὸς τῆς ἀναλογίας προχωρούσης ἐκόμπασεν ἐκεῖνο τὸ πᾶ BQ καὶ κινώ τὰν γᾶν.
ARCHIMEDES. ΧΧῚ
water-screw* (also called κοχλίας) which was apparently invented by him in Egypt, for the purpose of irrigating fields. It was also used for pumping water out of mines or from the hold of ships.
Another invention was that of a sphere constructed so as to imitate the motions of the sun, the moon, and the five planets in the heavens. Cicero actually saw this contrivance and gives a description of itt, stating that it represented the periods of the moon and the apparent motion of the sun with such accuracy that it would even (over a short period) show the eclipses of the sun and moon. Hultsch conjectures that it was moved by water {. We know, as above stated, from Pappus that Archimedes wrote a book on the construction of such a sphere (περὶ σφαιροποιἴας), and Pappus speaks in one place of “those who understand the making of spheres and produce a model of the heavens by means of the regular circular motion of water.” In any case it is certain that Archimedes was much occupied with astronomy. Livy calls him “unicus spectator caeli siderumque.” Hipparchus says§, “From these observations it is clear that the differences in the years are altogether small, but, as to the solstices, I almost think (οὐκ ἀπελπίζω) that both I and Archimedes have erred to the extent of a quarter of a day both in the observation and in the deduction therefrom.” It appears therefore that Archimedes had considered the question of the length of the year, as Ammianus also states|). Macrobius says that he discovered the distances of the planets{/. Archimedes himself describes in the Sand-reckoner the apparatus by which he measured the apparent diameter of the sun, or the angle subtended by it at the eye.
The story that he set the Roman ships on fire by an arrange- ment of burning-glasses or concave mirrors is not found in any
* Diodorus 1. 34, v. 837; Vitruvius x. 16 (11); Philo m1. p. 330 (ed. Pfeiffer) ; Strabo xvi. p. 807; Athenaeus v. 208 f.
+ Cicero, De rep., τ. 21-22; Tusc., 1.63; De nat. deor., τι. 88. Cf. Ovid, Fasti, νι. 277; Lactantius, Instit., τι. 5, 18; Martianus Capella, 1. 212, νι. 583 sq.; Claudian, Epigr. 18; Sextus Empiricus, p. 416 (ed. Bekker).
+ Zeitschrift f. Math. wu. Physik (hist. litt. Abth.), xx. (1877), 106 sq.
8 Ptolemy, σύνταξις, τ. p. 153.
|| Ammianus Marcell., xxvr. i. 8.
4 Macrobius, in Somn. Scip., 11. 3.
XX1l INTRODUCTION.
authority earlier than Lucian*; and the so-called loculus Archi- medius, which was a sort of puzzle made of 14 pieces of ivory of different shapes cut out of a square, cannot be supposed to be his invention, the explanation of the name being perhaps that it was only a method of expressing that the puzzle was cleverly made, in the same way as the πρόβλημα ᾿Αρχιμήδειον came to be simply a proverbial expression for something very difficult Ἷ.
* The same story is told of Proclus in Zonaras xtv. 3. For the other references on the subject see Heiberg’s Quaestiones Archimedeae, pp. 39-41. + Cf. also Tzetzes, Chil. x11. 270, τῶν ᾿Αρχιμήδους μηχανών χρείαν ἔχω.
CHAPTER. 1|.
MANUSCRIPTS AND PRINCIPAL EDITIONS—ORDER OF COMPOSITION—-DIALECT—-LOST WORKS.
THE sources of the text and versions are very fully described by Heiberg in the Prolegomena to Vol. 111. of his edition of Archi- medes, where the editor supplements and to some extent amends what he had previously written on the same subject in his dis- sertation entitled Quaestiones Archimedeae (1879). It will there- fore suffice here to state briefly the main points of the discussion.
The MSS. of the best class all had a common origin in a MS. which, so far as is known, is no longer extant. It is described in one of the copies made from it (to be mentioned later and dating from some time between A.D. 1499 and 1531) as ‘most ancient’ (παλαιοτάτου), and all the evidence goes to show that it was written as early as the 9th or 10th century. At one time it was in the possession of George Valla, who taught at Venice between the years 1486 and 1499; and many important inferences with regard to its readings can be drawn from some translations of parts of Archimedes and Eutocius made by Valla himself and published in his book entitled de expetendis et fugiendis rebus (Venice, 1501). It appears to have been carefully copied from an original belonging to some one well versed in mathematics, and it contained figures drawn for the most part with great care and accuracy, but there was considerable confusion between the letters in the figures and those in the text. This MS., after the death of Valla in 1499, became the property of Albertus Pius Carpensis (Alberto Pio, prince of Carpi).. Part of his library passed through various hands and ultimately reached the Vatican; but the fate of the Valla MS. appears to have been different, for we hear of its being in the possession of Cardinal Rodolphus Pius (Rodolfo Pio), a nephew of Albertus, in 1544, after which it seems to have disappeared.
XX1V INTRODUCTION.
The three most important MSS. extant are:
F (=Codex Florentinus bibliothecae Laurentianae Mediceae plutei xxvii. 4to.).
B (=Codex Parisinus 2360, olim Mediceus). C (=Codex Parisinus 2361, Fonteblandensis).
Of these it is certain that B was copied from the Valla MS, This is proved by a note on the copy itself, which states that the archetype formerly belonged to George Valla and afterwards to Albertus Pius. From this it may also be inferred that B was written before the death of Albertus in 1531; for, if at the date of B the Valla MS. had passed to Rodolphus Pius, the name of the latter would presumably have been mentioned. The note re- ferred to also gives a list of peculiar abbreviations used in the archetype, which list is of importance for the purpose of com- parison with F and other MSS.
From a note on C it appears that that MS. was written by one Christophorus Auverus at Rome in 1544, at the expense of Georgius Armagniacus (Georges d’Armagnac), Bishop of Rodez, then on a mission from King Francis I. to Pope Paul III. Further, a certain Guilelmus Philander, in a letter to Francis I. published in an edition of Vitruvius (1552), mentions that he was allowed, by the kindness of Cardinal Rodolphus Pius, acting at the instance of Georgius Armagniacus, to see and make extracts from a volume of Archimedes which was destined to adorn the library founded by Francis at Fontainebleau. He adds that the volume had been the property of George Valla. We can therefore hardly doubt that C was the copy which Georgius Armagniacus had made in order to present it to the library at Fontainebleau.
Now F, B and C all contain the same works of Archimedes and Eutocius, and in the same order, viz. (1) two Books de sphaera et cylindro, (2) de dimensione circuli, (3) de conoidibus, (4) de lineis spiralibus, (5) de planis aeque ponderantibus, (6) arenarius, (7) quadratura parabolae, and the commentaries of Eutocius on (1) (2) and (5). At the end of the quadratura parabolae both F and B give the following lines:
εὐτυχοίης λέον γεώμετρα
πολλοὺς εἰς λυκάβαντας ἴοις πολὺ φίλτατε μούσαις. F and C also contain menswrae from Heron and two fragments περὶ σταθμῶν and περὶ μέτρων, the order being the same in both
MANUSCRIPTS. XXV
and the contents only differing in the one respect that the last fragment περὶ μέτρων is slightly longer in F than in C.
A short preface to C states that the first page of the archetype was so rubbed and worn with age that not even the name of Archimedes could be read upon it, while there was no copy at Rome by means of which the defect could be made good, and further that the last page of Heron’s de mensuris was similarly obliterated. Now in F the first page was apparently left blank at first and afterwards written in by a different hand with many gaps, while in B there are similar deficiencies and a note attached by the copyist is to the effect that the first page of the archetype was indistinct. In another place (p. 4 of Vol. πι., ed. Heiberg) all three MSS. have the same lacuna, and the scribe of B notes that one whole page or even two are missing.
Now C could not have been copied from F because the last page of the fragment περὶ μέτρων is perfectly distinct in F; and, on the other hand, the archetype of F must have been illegible at the end because there is no word τέλος at the end of F, nor any other of the signs by which copyists usually marked the completion of their task. Again, Valla’s translations show that his MS. had certain readings corresponding to correct readings in B and C instead of incorrect readings given by F. Hence F cannot have been. Valla’s MS. itself.
The positive evidence about F is as follows. Valla’s trans- lations, with the exception of the few readings just referred to, agree completely with the text of F. From a letter written at Venice in 1491 by Angelus Politianus (Angelo Poliziano) to Lau- rentius Mediceus (Lorenzo de’ Medici), it appears that the former had found a MS. at Venice containing works by Archimedes and Heron and proposed to have it copied. As G. Valla then lived at Venice, the MS. can hardly have been any other but his, and no doubt F was actually copied from it in 1491 or soon after. Confirmatory evidence for this origin of F is found in the fact that the form of most of the letters in it is older than the 15th century, and the abbreviations etc., while they all savour of an ancient archetype, agree marvellously with the description which the note to B above referred to gives of the abbreviations used in Valla’s MS. Further, it is remarkable that the corrupt passage corresponding to the illegible first page of the archetype just takes up one page of F, no more and no less.
XXV1 INTRODUCTION.
The natural inference from all the evidence is that F, B and C all had their origin in the Valla MS.; and of the three F is the most trustworthy. For (1) the extreme care with which the copyist of F kept to the original is illustrated by a number of mistakes in it which correspond to Valla’s readings but are cor- | rected in B and C, and (2) there is no doubt that the writer of B was somewhat of an expert and made many alterations on his own authority, not always with success.
Passing to other MSS., we know that Pope Nicholas V. had a MS. of Archimedes which he caused to be translated into Latin. The translation was made by Jacobus Cremonensis (Jacopo Cas- siani*), and one copy of this was written out by Joannes Regio- montanus (Johann Miiller of Koénigsberg, near Hassfurt, in Fran- conia), about 1461, who not only noted in the margin a number of corrections of the Latin but added also in many places Greek readings from another MS. This copy by Regiomontanus is pre- served at Niirnberg and was the source of the Latin translation given in the editio princeps of Thomas Gechauff Venatorius (Basel, 1544); it is called ΝΡ by Heiberg. (Another copy of the same translation is alluded to by Regiomontanus, and this is doubtless the Latin MS. 327 of 15th c. still extant at Venice.) From the fact that the translation of Jacobus Cremonensis has the same lacuna as that in F, B and C above referred to (Vol. 111, ed. Heiberg, p. 4), it seems clear that the translator had before him either the Valla MS. itself or (more likely) a copy of it, though the order of the books in the translation differs in one respect from that in our MSS., viz. that the arenarius comes after instead of before the guadratura parabolae.
It is probable that the Greek MS. used by Regiomontanus was V (= Codex Venetus Marcianus cccy. of the 15th c.), which is still extant and contains the same books of Archimedes and Eutocius with the same fragment of Heron as F has, and in the same order. If the above conclusion that F dates from 1491 or thereabouts is correct, then, as V belonged to Cardinal Bessarione who died in 1472, it cannot have been copied from F, and the simplest way of accounting for its similarity to F is to suppose that it too was derived from Valla’s MS.
* Tiraboschi, Storia della Letteratura Italiana, Vol. vi. Pt. 1 (p. 358 of the edition of 1807). Cantor (Vorlesungen iib. Gesch. d. Math., τα. p. 192) gives the full name and title as Jacopo da S. Cassiano Cremonese canonico regolare.
MANUSCRIPTS. XXV11
Regiomontanus mentions, in a note inserted later than the rest and in different ink, two other Greek MSS., one of which he calls “exemplar vetus apud magistrum Paulum.” Probably the monk Paulus (Albertini) of Venice is here meant, whose date was 1430 to 1475; and it is possible that the ‘exemplar vetus” is the MS. of Valla.
The two other inferior MSS., viz. A (=Codex Parisinus 2359, olim Mediceus) and D (=Cod. Parisinus 2362, Fonteblandensis), owe their origin to V.
It is next necessary to consider the probabilities as to the MSS. used by Nicolas Tartaglia for his Latin translation of certain of the works of Archimedes. The portion of this translation published at Venice in 1543 contained the books de centris gravium vel de aequerepentibus I-II, tetragonismus | parabolae|, dimensio circuli and de insidentibus aquae I; the rest, consisting of Book II de insidentibus aquae, was published with Book I of the same treatise, after Tartaglia’s death in 1557, by Troianus Curtius (Venice, 1565), Now the last-named treatise is not extant in any Greek MS. and, as Tartaglia adds it, without any hint of a separate origin, to the rest of the books which he says he took from a mutilated and almost illegible Greek MS., it might easily be inferred that the Greek MS. contained that treatise also. But it is established, by a letter written by Tartaglia himself eight years later (1551) that he then had no Greek text of the Books de insidentibus aquae, and it would be strange if it had disappeared in so short a time without leaving any trace. Further, Commandinus in the preface to his edition of the same treatise (Bologna, 1565) shows that he had never heard of a Greek text of it. Hence it is most natural to suppose that it reached Tartaglia from some other source and in the Latin translation only*.
The fact that Tartaglia speaks of the old MS. which he used as “fracti et qui vix legi poterant libri,” at practically the same time as the writer of the preface to C was giving a similar de- scription of Valla’s MS., makes it probable that the two were
* The Greek fragment of Book 1., περὶ τῶν ὕδατι ἐφισταμένων ἢ περὶ τῶν ὀχουμένων, edited by A. Mai from two Vatican MSS. (Classici auct. 1. p. 426-30 ; Vol. τι. of Heiberg’s edition, pp. 356-8), seems to be of doubtful authenticity. Except for the first proposition, it contains enunciations only and no proofs. Heiberg is inclined to think that it represents an attempt at retranslation into Greek made by some mediaeval scholar, and he compares the similar attempt made by Rivault.
XXV1i1 INTRODUCTION.
identical ; and this probability is confirmed by a considerable agree- ment between the mistakes in Tartaglia and in Valla’s versions.
But in the case of the quadratura parabolae and the dimensio circuli Tartaglia adopted bodily, without alluding in any way to the source of it, another Latin translation published by Lucas’ Gauricus “Tuphanensis ex regno Neapolitano” (Luca Gaurico of Gifuni) in 1503, and he copied it so faithfully as to reproduce most obvious errors and perverse punctuation, only filling up a few gaps and changing some figures and letters. This translation by Gauricus is seen, by means of a comparison with Valla’s readings and with the translation of Jacobus Cremonensis, to have been made from the same MS. as the latter, viz. that of Pope Nicolas V.
Even where Tartaglia used the Valla MS. he does not seem to have taken very great pains to decipher it when it was not easily legible—it may be that he was unused to deciphering MSS.—-and in such cases he did not hesitate to draw from other sources. In one place (de planor. equilib. 1. 9) he actually gives as the Archimedean proof a paraphrase of Eutocius some- what retouched and abridged, and in many other instances he has inserted corrections and interpolations from another Greek MS. which he once names. This MS. appears to have been a copy made from F, with interpolations due to some one not unskilled in the subject-matter; and this interpolated copy of F was ap- parently also the source of the Nirnberg MS. now to be mentioned.
N* (= Codex Norimbergensis) was written in the 16th century and brought from Rome to Niirnberg by Wilibald Pirckheymer. It contains the same works of Archimedes and Eutocius, and in the same order, as F, but was evidently not copied from F direct, while, on the other hand, it agrees so closely with Tartaglia’s version as to suggest a common origin. N* was used by Vena- torius in preparing the editio princeps, and Venatorius corrected many mistakes in it with his own hand by notes in the margin or on slips attached thereto; he also made many alterations in the body of it, erasing the original, and sometimes wrote on it directions to the printer, so that it was probably actually used to print from. The character of the MS. shows it to belong to the same class as the others; it agrees with them in the more important errors and in having a similar lacuna at the beginning. Some mistakes common to it and F alone show that its source was F, though at second hand, as above indicated.
EDITIONS AND TRANSLATIONS. XX1X
It remains to enumerate the principal editions of the Greek text and the published Latin versions which are based, wholly or partially, upon direct collation of the MSS. These are as follows, in addition to Gaurico’s and Tartaglia’s translations.
1. The editio princeps published at Basel in 1544 by Thomas Gechauff Venatorius under the title Archimedis opera quae quidem exstant omnia nunc primum graece et latine in lucem edita. Advecta quoque sunt Eutocit Ascalonitae commentaria item graece et latine nunquam antea excusa. The Greek text and the Latin version in this edition were taken from different sources, that of the Greek text being N*, while the translation was Joannes Regiomontanus’ revised copy (N?) of the Latin version made by Jacobus Cremo- nensis from the MS. of Pope Nicolas V. The revision by Regiomontanus was effected by the aid of (1) another copy of the same translation still extant, (2) other Greek MSS., one of which was probably V, while another may have been Valla’s MS. itself.
2. A translation by F. Commandinus (containing the following works, corcult dimensio, de lineis sprralibus, quadratura parabolae, de conoidibus et sphaerordibus, de arenae numero) appeared at Venice in 1558 under the title Archimedis opera nonnulla in latinum conversa et commentaris illustrata. For this translation several MSS. were used, among which was V, but none preferable to those which we now possess.
3. D. Rivault’s edition, Archimedis opera quae exstant graece et latine novis demonstr. et comment. rllustr. (Paris, 1615), gives only the propositions in Greek, while the proofs are in Latin and somewhat retouched. Rivault followed the Basel editio princeps with the assistance of B.
4, Torelli’s edition (Oxford, 1792) entitled ᾿Αρχιμήδους τὰ σω- ζόμενα peta τῶν Εὐτοκίου ᾿Ασκαλωνίτου ὑπομνημάτων, Archimedis quae supersunt omnia cum Eutoci Ascalonitae commentariis ex recensione J. Torelli Veronensis cum nova versione latina. <Acced- unt lectiones variantes ex codd. Mediceo et Parisiensibus. Torelli followed the Basel editio princeps in the main, but also collated V. The book was brought out after Torelli’s death by Abram Robertson, who added the collation of five more MSS., F, A, B, C, D, with the Basel edition. The collation however was not well done, and the edition was not properly corrected when in the press.
XXX INTRODUCTION.
5. Last of all comes the definitive edition of Heiberg (Archi- medis opera omnia cum commentariis Eutoci. E codice Florentino recensuit, Latine uertit notisque ilustrauit J. L. Heiberg. Leipzig, 1880—1).
The relation of all the MSS. and the above editions and trans- lations is well shown by Heiberg in the following scheme (with the omission, however, of his own edition) :
Codex Uallae saec. 1x—x
Cod. Nicolai V F Tartalea Vv B ee 6. 1453 e. 1491 a. 1543 saec. Xv c. 1500 a. 1544 ΕΞ 5555." 55. Cod. Tartaleae 11 | Ed. Riualti a. 1615 N® saec. xvi mee 2 ΕΘΘΕ = | Age) Commandinus Ed. Basil. 1544 saec. XVI 1558 Torellius 1792 a oS Gauricus Cremonensis 6. 1460 fp Re ee Cod. Uenet. 827 ΝΡ, c. 1461 5866. Xv
The remaining editions which give portions of Archimedes in Greek, and the rest of the translations of the complete works or parts of them which appeared before Heiberg’s edition, were not based upon any fresh collation of the original sources, though some excellent corrections of the text were made by some of the editors, notably Wallis and Nizze. The following books may be mentioned.
Joh, Chr. Sturm, Des unvergleichlichen Archimedis Kunstbiicher, tibersetzt und erldutert (Nirnberg, 1670). This translation em- braced all the works extant in Greek and followed three years after the same author’s separate translation of the Sand-reckoner. It appears from Sturm’s preface that he principally used the edition of Rivault.
Is. Barrow, Opera Archimedis, Apollonii Pergaei conicorum libri, Theodosii sphaerica methodo novo illustrata et demonstrata (London,
1675).
Wallis, Archimedis arenarius et dimensio circuli, Eutocii in hanc commentary cum versione et notis (Oxford, 1678), also given in Wallis’ Opera, Vol. ur. pp. 509—546,
Karl Friedr. Hauber, Archimeds zwei Biicher viber Kugel und
Cylinder. Ebendesselben Kreismessung. Uebersetzt mit Anmerkungen u. 8. w. begleitet (Tubingen, 1798).
TRANSLATIONS—ORDER OF WORKS. ἘΣΣῚ
F. Peyrard, Quvres αἱ Archiméde, traduites littéralement, avec un commentaire, suivies dun mémoire du traducteur, sur un nouveau miroir ardent, et d'un autre mémoire de M. Delambre, sur Varith- métique des Grecs. (Second edition, Paris, 1808.)
Ernst Nizze, Archimedes von Syrakus vorhandene Werke, aus dem Griechischen tibersetzt und mit erlauternden und kritischen Anmer- kungen begleitet (Stralsund, 1824).
The MSS. give the several treatises in the following order.
1. περὶ σφαίρας καὶ κυλίνδρου α΄ β΄, two Books On the Sphere and Cylinder. κύκλου μέτρησις ἢ, Measurement of a Circle. περὶ κωνοειδέων Kal σφαιροειδέων, On Conoids and Spheroids.
περὶ ἑλίκων, On Spirals.
or HR ὧν bo
ἐπιπέδων ἰσορροπιῶν a Bt, two Books On the Hquwilibrium of Planes.
6. ψαμμίτης, The Sand-reckoner.
7. τετραγωνισμὸς παραβολῆς (a name substituted later for that given to the treatise by Archimedes himself, which must undoubtedly have been τετραγωνισμὸς τῆς τοῦ ὀρθογωνίου
κώνου τομῆς 1), Quadrature of the Parabola. To these should be added
8. περὶ ὀχουμένων ὃ, the Greek title of the treatise On floating bodies, only preserved in a Latin translation.
* Pappus alludes (1. p. 312, ed. Hultsch) to the κύκλου μέτρησις in the words ἐν τῷ περὶ τῆς TOO κύκλου περιφερείας.
+ Archimedes himself twice alludes to properties proved in Book 1. as demonstrated ἐν τοῖς μηχανικοῖς (Quadrature of the Parabola, Props. 6, 10). Pappus (στα. p. 1034) quotes τὰ ᾿Αρχιμήδους περὶ ἰσορροπιῶν. The beginning of Book 1. is also cited by Proclus in his Commentary on Eucl. 1., Ὁ. 181, where the reading should be τοῦ ἃ ἰσορροπιών, and not τῶν ἀνισορροπιῶν (Hultsch).
+ The name ‘ parabola’ was first applied to the curve by Apollonius. Archi- medes always used the old term ‘section of a right-angled cone.’ Cf. Eutocius (Heiberg, vol. 111., p. 342) δέδεικται ἐν τῷ περὶ τῆς TOD ὀρθογωνίου κώνου τομῆς.
§ This title corresponds to the references to the book in Strabo 1. p. 54 (Ἀρχιμήδης ἐν τοῖς περὶ τῶν ὀχουμένων) and Pappus vir. p. 1024 (ὡς ᾿Αρχιμήδης ὀχουμένοι5). The fragment edited by Mai has a longer title, περὶ τῶν ὕδατι ἐφισταμένων ἢ περὶ τῶν ὀχουμένων, where the first part corresponds to Tartaglia’s version, de insidentibus aquae, and to that of Commandinus, de iis quae vehun- tur in aqua. But Archimedes intentionally used the more general word ὑγρόν (fluid) instead of ὕδωρ; and hence the shorter title περὶ ὀχουμένων, de iis quae in humido vehuntur (Torelli and Heiberg), seems the better.
ΧΧΧΙΙ INTRODUCTION.
The books were not, however, written in the above order; and Archimedes himself, partly through his prefatory letters and partly by the use in later works of properties proved in earlier treatises, gives indications sufficient to enable the chronological sequence to be stated approximately as follows :
On the equilibrium of planes, I. 2. Quadrature of the Parabola. 3. On the equilibrium of planes, 11. 4. On the Sphere and Cylinder, I, ΤΊ. 5. On Spirals. 6. On Conoids and Spheroids. 7. On floating bodies, I, 11. 8. Measurement of a circle. 9. The Sand-reckoner.
It should however be observed that, with regard to (7), no more is certain than that it was written after (6), and with regard to (8) no more than that it was later than (4) and before (9).
In addition to the above we have a collection of Lemmas (Liber Assumptorum) which has reached us through the Arabic. The collection was first edited by 8. Foster, Atscellanea (London, 1659), and next by Borelli in a book published at Florence, 1661, in which the title is given as Liber assumptorum Archimedis interprete Thebit ben Kora et exponente doctore Almochtasso Abilhasan. The Lemmas cannot, however, have been written by Archimedes in their present form, because his name is quoted in them more than once. The probability is that they were propositions collected by some Greek writer* of a later date for the purpose of elucidating some ancient work, though it is quite likely that some of the propositions were of Archimedean origin, e.g. those concerning the geometrical figures called respectively ἀἄρβηλος ἡ (literally
* Tt would seem that the compiler of the Liber Assumptorum must have drawn, to a considerable extent, from the same sources as Pappus. The number of propositions appearing substantially in the same form in both collections is, I think, even greater than has yet been noticed. Tannery (La Géométrie grecque, p. 162) mentions, as instances, Lemmas 1, 4, 5, 6; but it will be seen from the notes in this work that there are several other coin- cidences.
+ Pappus gives (p. 208) what he calls an ‘ancient proposition’ (ἀρχαία mporacis) about the same figure, which he describes as χωρίον, ὃ δὴ καλοῦσιν ἄρβηλον. Cf. the note to Prop. 6 (p. 308). The meaning of the word is gathered
WORKS ASCRIBED TO ARCHIMEDES. XXX111
‘shoemaker’s knife’) and σάλινον (probably a ‘salt-cellar’*), and Prop. 8 which bears on the problem of trisecting an angle.
from the Scholia to Nicander, Theriaca, 423: ἄρβηλοι λέγονται τὰ κυκλοτερῆ σιδήρια, οἷς οἱ σκυτοτόμοι τέμνουσι καὶ ξύουσι τὰ δέρματα. Cf. Hesychius, ἀνάρβηλα, τὰ μὴ ἐξεσμένα δέρματα" ἄρβηλοι γὰρ τὰ σμιλία.
* The best authorities appear to hold that in any case the name σάλινον was not applied to the figure in question by Archimedes himself but by some later writer. Subject to this remark, I believe σάλινον to be simply a Graecised form of the Latin word salinwm. We know that a salt-cellar was an essential part of the domestic apparatus in Italy from the early days of the Roman Republic. ‘All who were raised above poverty had one of silver which descended from father to son (Hor., Carm. τι. 16, 13, Liv. xxvr. 36), and was accompanied by a silver patella which was used together with the salt- cellar in the domestic sacrifices (Pers. 11. 24, 25). These two articles of silver were alone compatible with the simplicity of Roman manners in the early times of the Republic (Plin., H. N. xxxiu. § 153, Val. Max. tv. 4, ὃ 3). ...In shape the salinwm was probably in most cases a round shallow bowl” [Dict. of Greek and Roman Antiquities, article salinum]. Further we have in the early chapters of Mommsen’s History of Rome abundant evidence of similar transferences of Latin words to the Sicilian dialect of Greek. Thus (Book 1., ch. xiii.) it is shown that, in consequence of Latino-Sicilian com- merce, certain words denoting measures of weight, libra, triens, quadrans, seatans, uncia, found their way into the common speech of Sicily in the third century of the city under the forms λίτρα, τριᾶς, τετρᾶς, ἑξᾶς, οὐγκία. Similarly Latin law-terms (ch. xi.) were transferred; thus mutwum (a form of loan) became μοῖτον, carcer (a prison) kdpxapov. Lastly, the Latin word for lard, arvina, became in Sicilian Greek ἀρβίνη, and patina (a dish) πατάνη. The last word is as close a parallel for the supposed transfer of salinwm as could be wished. Moreover the explanation of σάλινον as salinum has two obvious advantages in that (1) it does not require any alteration in the word, and
(2) the resemblance of the lower curve to an ordinary type of salt-cellar is
evident. I should add, as confirmation of my hypothesis, that Dr A. S. Murray,
of the British Museum, expresses the opinion that we cannot be far wrong in
accepting as a salinuwm one of the small silver bowls in the Roman ministerium Hi. A. c
XXX1V INTRODUCTION.
Archimedes is further credited with the authorship of the Cattle-problem enunciated in the epigram edited by Lessing in 1773. According to the heading prefixed to the epigram it was communicated by Archimedes to the mathematicians at Alexandria in a letter to Eratosthenes*. There is also in the Scholia to Plato’s Charmides 165 πὶ ἃ reference to the problem “called by Archimedes the Cattle-problem” (τὸ κληθὲν ὑπ᾽ ᾿Αρχιμήδους βοεικὸν πρόβλημα). The question whether Archimedes really propounded the problem, or whether his name was only prefixed to it in order to mark the extraordinary difficulty of it, has been much debated. A complete account of the arguments for and against is given in an article by Krumbiegel in the Zeitschrift fiir Mathematik und Physik (Hist. litt, Abtheilung) xxv. (1880), p. 121 sq., to which Amthor added (ibid. p. 153 sq.) a discussion of the problem itself. The . general result of Krumbiegel’s investigation is to show (1) that
at the Museum which was found at Chaourse (Aisne) in France and is of a section sufficiently like the curve in the Salinon.
The other explanations of σάλινον which have been suggested are as follows.
(1) Cantor connects it with σάλος, ‘‘das Schwanken des hohen Meeres,” and would presumably translate it as wave-line. But the resemblance is not altogether satisfactory, and the termination -ἰνον would need explanation.
(2) Heiberg says the word is ‘‘sine dubio ab Arabibus deprauatum,” and suggests that it should be σέλινον, parsley (‘‘ex similitudine frondis apii’’). But, whatever may be thought of the resemblance, the theory that the word is corrupted is certainly not supported by the analogy of ἄρβηλος which is correctly reproduced by the Arabs, as we know from the passage of Pappus referred to in the last note.
(3) Dr Gow suggests that σάλινον may be a ‘sieve,’ comparing σάλαξ. But this guess is not supported by any evidence.
* The heading is, Πρόβλημα ὅπερ ᾿Αρχιμήδης ἐν ἐπιγράμμασιν εὑρὼν τοῖς ἐν ᾿Αλεξανδρείᾳ περὶ ταῦτα πραγματευομένοις ζητεῖν ἀπέστειλεν ἐν τῇ πρὸς ᾿Ἐρατοσθένην τὸν Κυρηναῖον ἐπιστολῇ. Heiberg translates this as ‘‘the problem which Archimedes discovered and sent in an epigram...in a letter to Eratosthenes.” He admits however that the order of words is against this, as is also the use of the plural ἐπιγράμμασιν. It is clear that to take the two expressions ἐν ἐπιγράμμασιν and ἐν ἐπιστολῇ as both following ἀπέστειλεν is very awkward. In fact there seems to be no alternative but to translate, as Krumbiegel does, in accordance with the order of the words, ‘‘a problem which Archimedes found among (some) epigrams and sent...in his letter to Eratosthenes”’ ; and this sense is certainly unsatisfactory. Hultsch remarks that, though the mistake πραγ- ματουμένοις for πραγματευομένοις and the composition of the heading as a whole betray the hand of a writer who lived some centuries after Archimedes, yet he must have had an earlier source of information, because he could hardly have invented the story of the letter to Eratosthenes.
WORKS ASCRIBED TO ARCHIMEDES. XXXV
the epigram can hardly have been written by Archimedes in its present form, but (2) that it is possible, nay probable, that the problem was in substance originated by Archimedes. Hultsch* has an ingenious suggestion as to the occasion of it. It is known that Apollonius in his ὠκυτόκιον had calculated a closer approximation to the value of z than that of Archimedes, and he must therefore have worked out more difficult multiplications than those contained in the Measurement of a circle. Also the other work of Apollonius on the multiplication of large numbers, which is partly preserved in Pappus, was inspired by the Sand-reckoner of Archimedes ; and, though we need not exactly regard the treatise of Apollonius as polemical, yet it did in fact constitute a criticism of the earlier book. Accordingly, that Archimedes should then reply with a problem which involved such a manipulation of immense numbers as would be difficult even for Apollonius is not altogether outside the bounds of possibility. And there is an unmistakable vein of satire in the opening words of the epigram “Compute the number of the oxen of the Sun, giving thy mind thereto, if thou hast a share of wisdom,” in the transition from the first part to the second where it is said that ability to solve the first part would entitle one to be regarded as “not unknowing nor unskilled in numbers, but still not yet to be numbered among the wise,” and again in the last lines. Hultsch concludes that in any case the problem is not much later than the time of Archimedes and dates from the beginning of the 2nd century B.c. at the latest.
Of the extant books it is certain that in the 6th century A.D. only three were generally known, viz. On the Sphere and Cylinder, the Measurement of a circle, and On the equilibrium of planes. Thus Eutocius of Ascalon who wrote commentaries on these works only knew the Quadrature of the Parabola by name and had never seen it nor the book On Spirals. Where passages might have been elucidated by references to the former book, Eutocius gives ex- planations derived from Apollonius and other sources, and he speaks vaguely of the discovery of a straight line equal to the circumference of a given circle “by means of certain spirals,” whereas, if he had known the treatise On Spirals, he would have quoted Prop. 18. There is reason to suppose that only the three treatises on which Eutocius commented were contained in the
* Pauly-Wissowa’s Real-Encyclopidie, τι. 1, pp. 534, 5.
XXXV1 INTRODUCTION.
ordinary editions of the time such as that of Isidorus of Miletus, the teacher of Eutocius, to which the latter several times alludes.
In these circumstances the wonder is that so many more books have survived to the present day. As it is, they have lost to a considerable extent their original form. Archimedes wrote in the Dorie dialect*, but in the best known books (On the Sphere and Cylinder and the Measurement of a circle) practically all traces of that dialect have disappeared, while a partial loss of Doric forms has taken place in other books, of which however the Sand- reckoner has suffered least. Moreover in all the books, except the Sand-reckoner, alterations and additions were first of all made by an interpolator who was acquainted with the Doric dialect, and then, at a date subsequent to that of Eutocius, the book On the Sphere and Cylinder and the Measurement of a circle were completely recast.
Of the lost works of Archimedes the following can be identified.
1. Investigations relating to polyhedra are referred to by Pappus who, after alluding (v. p. 352) to the five regular polyhedra, gives a description of thirteen others discovered by Archimedes which are semi-regular, being contained by polygons equilateral and equiangular but not similar.
2. A book of arithmetical content, entitled ἀρχαί Principles and dedicated to Zeuxippus. We learn from Archimedes himself that the book dealt with the naming of numbers (κατονόμαξις τῶν ἀριθμῶν) ἃπα expounded a system of expressing numbers higher
* Thus Eutocius in his commentary on Prop. 4 of Book 11. On the Sphere and Cylinder speaks of the fragment, which he found in an old book and which appeared to him to be the missing supplement to the proposition referred to, as ‘‘preserving in part Archimedes’ favourite Doric dialect’? (ἐν μέρει δὲ τὴν ᾿Αρχιμήδει φίλην Δωρίδα γλῶσσαν ἀπέσωζον). From the use of the expression ἐν μέρει Heiberg concludes that the Doric forms had by the time of Eutocius begun to disappear in the books which have come down to us no less than in the fragment referred to.
+ Observing that in all the references to this work in the Sand-reckoner Archimedes speaks of the naming of numbers or of numbers which are named or have their names (ἀριθμοὶ κατωνομασμένοι, τὰ ὀνόματα ἔχοντες, τὰν κατονομαξίαν ἔχοντες), Hultsch (Pauly-Wissowa’s Real-Encyclopidie, ττ. 1, p. 511) speaks οὗ κατονό- pasts τῶν ἀριθμῶν as the name of the work; and he explains the words τινὰς τῶν ἐν ἀρχαῖς <dpiOuGv> τῶν κατονομαξίαν ἐχόντων aS meaning ‘‘some of the numbers mentioned at the beginning which have a special name,” where ‘at the beginning” refers to the passage in which Archimedes first mentions τῶν
LOST WORKS. XXXVil
than those which could be expressed in the ordinary Greek no- tation. This system embraced all numbers up to the enormous figure which we should now represent by a 1 followed by 80,000 billion ciphers ; and, in setting out the same system in the Sand- reckoner, Archimedes explains that he does so for the benefit of those who had not had the opportunity of seeing the earlier work addressed to Zeuxippus.
3. περὶ ζυγῶν, On balances or levers, in which Pappus says (VIII. p. 1068) that Archimedes proved that “ greater circles overpower (κατακρατοῦσι) lesser circles when they revolve about the same centre.” It was doubtless in this book that Archimedes proved the theorem assumed by him in the Quadrature of the Parabola, Prop. 6, viz. that, if a body hangs at rest from a point, the centre of gravity of the body and the point of suspension are in the same vertical line.
4, xevtpoBapixa, On centres of gravity. This work is mentioned by Simplicius on Aristot. de caelo 11. (Scholia in Arist. 508 a 30). Archimedes may be referring to it when he says (On the equilibrium of planes 1. 4) that it has before been proved that the centre of gravity of two bodies taken together lies on the line joining the centres of gravity of the separate bodies. In the treatise On floating bodies Archimedes assumes that the centre of gravity of a segment of a paraboloid of revolution is on the axis of the segment at a distance from the vertex equal to Zrds of its length. This may perhaps have been proved in the κεντροβαρικά, if it was not made the subject of a separate work.
Doubtless both the περὶ ζυγῶν and the κεντροβαρικά preceded the extant treatise On the equilibrium of planes.
5. κατοπτρικά, an optical work, from which Theon (on Ptolemy, Synt. 1. p. 29, ed. Halma) quotes a remark about refraction. Cf. Olympiodorus in Aristot. Meteor., τι. p. 94, ed. Ideler.
ὑφ᾽ ἁμῶν κατωνομασμένων ἀριθμῶν καὶ ἐνδεδομένων ἐν τοῖς ποτὶ Ζεύξιππον γεγραμ- μένοις. But ἐν ἀρχαῖς seems a less natural expression for ‘‘at the beginning” than ἐν ἀρχῇ or κατ᾽ ἀρχάς would have been. Moreover, there being no participial expression except κατονομαξίαν ἐχόντων to be taken with ἐν ἀρχαῖς in this sense, the meaning would be unsatisfactory ; for the numbers are not named at the beginning, but only referred to, and therefore some word like εἰρημένων should have been used. For these reasons I think that Heiberg, Cantor and Susemihl are right in taking ἀρχαί to be the name of the treatise.
XXXVIl1 INTRODUCTION.
6. περὶ σφαιροποιΐας, On sphere-making, a mechanical work on the construction of a sphere representing the motions of the heavenly bodies as already mentioned (p. xxi).
7. ἐφόδιον, a Method, noticed by Suidas, who says that Theo- dosius wrote a commentary on it, but gives no further information about it.
8. According to Hipparchus Archimedes must have written on the Calendar or the length of the year (cf. p. xxi).
Some Arabian writers attribute to Archimedes works (1) On a heptagon in a circle, (2) On circles touching one another, (3) On parallel lines, (4) On triangles, (5) On the properties of right- angled triangles, (6) a book of Data; but there is no confirmatory evidence of his having written such works. A book translated into Latin from the Arabic by Gongava (Louvain, 1548) and en- titled antiqui scriptoris de speculo comburente concavitatis parabolae cannot be the work of Archimedes, since it quotes Apollonius.
CHAPTER III. THE RELATION OF ARCHIMEDES TO HIS PREDECESSORS.
An extraordinarily large proportion of the subject matter of the writings of Archimedes represents entirely new discoveries of his own. Though his range of subjects was almost encyclopaedic, embracing geometry (plane and solid), arithmetic, mechanics, hydro- statics and astronomy, he was no compiler, no writer of text- books ; and in this respect he differs even from his great successor Apollonius, whose work, like that of Euclid before him, largely consisted of systematising and generalising the methods used, and the results obtained, in the isolated efforts of earlier geometers. There is in Archimedes no mere working-up of existing materials ; g, some definite addition to the sum of knowledge, and his complete originality cannot fail
his objective is always some new thin
to strike any one who reads his works intelligently, without any corroborative evidence such as is found in the introductory letters prefixed to most of them. These introductions, however, are emi- nently characteristic of the man and of his work ; their directness and simplicity, the complete absence of egoism and of any effort to magnify his own achievements by comparison with those of others or by emphasising their failures where he himself succeeded : all these things intensify the same impression. . Thus his manner is to state simply what particular discoveries made by his pre- decessors had suggested to him the possibility of extending them in new directions; e.g. he says that, in connexion with the efforts of earlier geometers to square the circle and other figures, it occurred to him that no one had endeavoured to square a parabola, and he accordingly attempted the problem and finally solved it. In like manner, he speaks, in the preface of his treatise On the
xl] INTRODUCTION.
Sphere and Cylinder, of his discoveries with reference to those solids as supplementing the theorems about the pyramid, the cone and the cylinder proved by Eudoxus. He does not hesitate to say that certain problems baffled him for a long time, and that the solution of some took him many years to effect; and in one place (in the preface to the book On Spirals) he positively insists, for the sake of pointing a moral, on specifying two propositions which he had enunciated and which proved on further investigation to be wrong. The same preface contains a generous eulogy of Conon, declaring that, but for his untimely death, Conon would have solved certain problems before him and would have enriched geometry by many other discoveries in the meantime.
In some of his subjects Archimedes had no fore-runners, e.g. in hydrostatics, where he invented the whole science, and (so far as mathematical demonstration was concerned) in his me- chanical investigations. In these cases therefore he had, in laying the foundations of the subject, to adopt a form more closely re- sembling that of an elementary textbook, but in the later parts he at once applied himself to specialised investigations.
Thus the historian of mathematics, in dealing with Archimedes’ obligations to his predecessors, has a comparatively easy task before him. But it is necessary, first, to give some description of the use which Archimedes made of the general methods which had found acceptance with the earlier geometers, and, secondly, to refer to some particular results which he mentions as having been previously discovered and as lying at the root of his own investigations, or which he tacitly assumes as known.
81. Use of traditional geometrical methods.
In my edition of the Conics of Apollonius*, I endeavoured, following the lead given in Zeuthen’s work, Die Lehre von den Kegelschnitten im Altertum, to give some account of what has been fitly called the geometrical algebra which played such an important part in the works of the Greek geometers. The two main methods included under the term were (1) the use of the theory of pro- portions, and (2) the method of application of areas, and it was shown that, while both methods are fully expounded in the Elements of Euclid, the second was much the older of the two, being attributed by the pupils of Eudemus (quoted by Proclus) to the
* Apollonius of Perga, pp. ci sqq.
RELATION OF ARCHIMEDES TO HIS PREDECESSORS. xli
Pythagoreans. It was pointed out that the application of areas, as set forth in the second Book of Euclid and extended in the sixth, was made by Apollonius the means of expressing what he takes as the fundamental properties of the conic sections, namely the properties which we express by the Cartesian equations y? = px, y? = pa + Ea
referred to any diameter and the tangent at its extremity as axes ; and the latter equation was compared with the results obtained in the 27th, 28th and 29th Props. of Euclid’s Book v1, which are equivalent to the solution, by geometrical means, of the quadratic equations
b an+ —2’?=D. (5
It was also shown that Archimedes does not, as a rule, connect his description of the central conics with the method of application of areas, as Apollonius does, but that Archimedes generally expresses the fundamental property in the form of a proportion
2 12
LEME:
CDi.) eae
and, in the case of the ellipse, y? b? =u
2.0, (ἢ
where a, x, are the abscissae measured from the ends of the diameter of reference. .
It results from this that the application of areas is of much less frequent occurrence in Archimedes than in Apollonius. It is however used by the former in all but the most general form. The simplest form of ‘applying a rectangle” to a given straight line which shall be equal to a given area occurs e.g. in the proposition On the equilibrium of Planes τι. 1; and the same mode of expression is used (as in Apollonius) for the property y’= pz in the parabola, px being described in Archimedes’ phrase as the rectangle “applied to” (παραπίπτον παρά) a line equal to p and “having at its width” (πλάτος ἔχον) the abscissa (x). Then in Props. 2, 25, 26, 29 of the book On Conoids and Spheroids we have the complete expression which is the equivalent of solving the equation
ax+ οἷ = 6’,
“let a rectangle be applied (to a certain straight line) exceeding by
xii INTRODUCTION.
a square figure (παραπεπτωκέτω χωρίον ὑπερβάλλον εἴδει τετραγώνῳ) and equal to (a certain rectangle).” Thus a rectangle of this sort has to be made (in Prop. 25) equal to what we have above called x.«, in the case of the hyperbola, which is the same thing as a(a+a) or ax+a*, where ὦ is the length of the transverse axis. But, curiously enough, we do not find in Archimedes the application of a rectangle “falling short by a square figure,” which we should obtain in the case of the ellipse if we substituted «(a—~) for x. a. In the case of the ellipse the area x.a, is represented (On Conoids and Spheroids, Prop. 29) as a gnomon which is the difference between the rectangle h.h, (where h, h, are the abscissae of the ordinate bounding a segment of an ellipse) and a rectangle applied to h,—h and exceeding by a square figure whose side is ἢ -- α ; and the rectangle h.h, is simply constructed from the sides ἡ, h,. Thus Archimedes avoids* the application of a rectangle falling short by a square, using for x. a, the rather complicated form
h.h,—{(hy—h) (ἃ -- αν) + (h—-«)’}. It is easy to see that this last expression is equal to ἃ). a, for it reduces to h.h,— th, (h—«)—a(h—-2)} =a(h, +h)-2’, =ax—«x’, since h,+h=a, nee
It will readily be understood that the transformation of rectangles and squares in accordance with the methods of Euclid, Book 11, is just as important to Archimedes as to other geometers, and there is no need to enlarge on that form of geometrical algebra.
The theory of proportions, as expounded in the fifth and sixth Books of Euclid, including the transformation of ratios (denoted by the terms componendo, dividendo, etc.) and the composition or multiplication of ratios, made it possible for the ancient geometers to deal with magnitudes in general and to work out relations between them with an effectiveness not much inferior to that of modern algebra. Thus the addition and subtraction of ratios could be effected by procedure equivalent to what we should in algebra
* The object of Archimedes was no doubt to make the Lemma in Prop. 2 (dealing with the summation of a series of terms of the form a.rx + (rzx)?, where r successively takes the values 1, 2, 3,...) serve for the hyperboloid of revolution and the spheroid as well.
RELATION OF ARCHIMEDES TO HIS PREDECESSORS. ΧΙΠῚ
call bringing to a common denominator. Next, the composition or multiplication of ratios could be indefinitely extended, and hence the algebraical operations of multiplication and division found easy and convenient expression in the geometrical algebra. As a par- ticular case, suppose that there is a series of magnitudes in continued proportion (i.e. in geometrical progression) as dy, 4), @2, ... G,, 80 that
GH % ὄἄ,-
We have then, by multiplication,
a ONE a ἕ a
1 ; a *) 5 ὉΓ — = ch τεῦ 3 AX X X hy
It is easy to understand how powerful such a method as that of proportions would become in the hands of an Archimedes, and a few instances are here appended in order to illustrate the mastery with which he uses it.
1. A good example of a reduction in the order of a ratio after the manner just shown is furnished by On the equilibrium of Planes u1.10. Here Archimedes has a ratio which we will call a?/b*, where a?/6°=c/d; and he reduces the ratio between cubes to a ratio between straight lines by taking two lines a, y such that
Ce τὰ "ἢ. (ἢ 2 2 It follows from this that (<) ee τς x Gao or te GS ὃ x’ a? ON oe a ὁ and hence m= (5) Seta:
2. In the last example we have an instance of the use of auxiliary fixed lines for the purpose of simplifying ratios and thereby, as it were, economising power in order to grapple the more successfully with a complicated problem. With the aid of such auxiliary lines or (what is the same thing) auxiliary fixed points in a figure, combined with the use of proportions, Archimedes is able to effect some remarkable eliminations.
Thus in the proposition On the Sphere and Cylinder τι. 4 he obtains three relations connecting three as yet undetermined points, and
xliv INTRODUCTION.
proceeds at once to eliminate two of the points, so that the problem is then reduced to finding the remaining point by means of one equation. Expressed in an algebraical form, the three original relations amount to the three equations
θα--α y
and the result, after the elimination of y and 2, is stated by Archimedes in a form equivalent to
M+n A+2 4a? n @ (2a—2)°
Again the proposition On the equilibrium of Planes τι. 9 proves by the same method of proportions that, if a, ὦ, c, d, x, y, are straight lines satisfying the conditions
“ἘΠ: Φ (a>b>e>d)
Ben ἢ
@ ῸΝ x
a-d 3(a-c)’ 2a+4b+6c+3d = y δα - 106 Ὁ 106- δα a—c’
then x+y = 2a.
and
The proposition is merely brought in as a subsidiary lemma to the proposition following, and is not of any intrinsic importance ; but a glance at the proof (which again introduces an auxiliary line) will show that it is a really extraordinary instance of the manipulation of proportions.
3. Yet another instance is worth giving here. It amounts to the proof that, if
ey —+ 5=1 Gen dar oe 2a+a 2a—a2 then .y? (a—a)+ . 3 (a+2) — 4ab?. a+n ὅτι ) α--α γι )
A, A’ are the points of contact of two parallel tangent planes to a spheroid ; the plane of the paper is the plane through Ad’ and the
RELATION OF ARCHIMEDES TO HIS PREDECESSORS. xlv
axis of the spheroid, and PP’ is the intersection of this plane with another plane at right angles to it (and therefore parallel to the tangent planes), which latter plane divides the spheroid into two segments whose axes are AV, A’. Another plane is drawn through
the centre and parallel to the tangent plane, cutting the spheroid into two halves. Lastly cones are drawn whose bases are the sections of the spheroid by the parallel planes as shown in the figure.
Archimedes’ proposition takes the following form [On Conoids and Spheroids, Props. 31, 32].
APP’ being the smaller segment of the two whose common base is the section through PP’, and x, y being the coordinates of P, he has proved in preceding propositions that
(volume of) segment APP’ 2a+¢a
(volume of) cone APP’ = Aes a ae (a), half spheroid ABB’ = 26) and eae ABB’ Dy reece e rece eeccorece nae (3);
and he seeks to prove that segment A’PP! _ 2a -« cone APE aa.” The method is as follows. cone ABB’ a δ a a
We have : SS —— : : cone APP’ α-α y a-x @-# ΖΦ a If we suppose See ΓΕ IS αὐ 0 Oates eater PP ee (7), za
the ratio of the cones becomes
ea
xlvi INTRODUCTION.
Next, by hypothesis (a), cone APP’ _ a+a segemt. APP’ 2a+a°
Therefore, ex aequali, cone ABB’ 2a
segmt. APP’ (α-- α) (2a+2)’ It follows from (8) that
spheroid 4ξα
segmt. APP’ (a—2) (2a+2)’
Mee segmt. A'PP’ 4za—(a—2x) (2a+ 2) segmt. APP’ Ἕώ(α-- &) (2a +2)
2 (2a—a)+(2a+a)(z-a—«x)
(a—«) (2a + 2) ; Now we have to obtain the ratio of the segment A’PP’ to the cone A’PP', and the comparison between the segment APP’ and the cone A’PP’ is made by combining two ratios ex aequali. Thus
segmt. APP’ Ἢ 9ᾳ +a (a) Τα ΣΡ a) aki a
1 cone APP a—@ and ——— : cone A’PP’ a+2
Thus combining the last three proportions, ex aequali, we have
segmt. A’PP' 2z(2a—ax)+(2a+a) (6 --α -- αἡ cone A’PP’ — a? + 2ax + x
_ 2#(2a— w)+(2a+2)(2-a-a a)
z(a—x)+(2a+x) α :
since a =2z(a—2), by (y).
[The object of the transformation of the numerator and denominator of the last fraction, by which z(2a—«) and z(a—«) are made the
a is the fraction which
2a first terms, is now obvious, because
Archimedes wishes to arrive at, and, in order to prove that the required ratio is equal to this, it is only necessary to show that 2a-a@_ 6- z—(a—2) ἢ a—x a
RELATION OF ARCHIMEDES TO HIS PREDECESSORS. xlvii
Now SE ways zz a—x α--α z Ses: by (y), τᾷ τ ἢ ἘΞ Ὁ -
ΠῚ
ΠΕ segmt. A’PP’ 2ᾳ.--ὸὑψ ὴ ΘΟ ΠΟ a= ae
4, One use by Euclid of the method of proportions deserves mention because Archimedes does not use it in similar circumstances. Archimedes (Quadrature of the Parabola, Prop. 23) sums a particular geometric series
a+a(4+)+a(4)+...4+a(4)" in a manner somewhat similar to that of our text-books, whereas Euclid (1x. 35) sums any geometric series of any number of terms by means of proportions thus.
Suppose aj, dy, ...@n, Gn, to be (n+1) terms of a geometric series in which a@,,, is the greatest term. Then
nit my Uma On An_1 An_» Ay Therefore TEL Oe es an On, Gn _1 ay
Adding all the antecedents and all the consequents, we have
Anyi — GY Ay — Ay
Ay + Ay + Ag + ...°+ Ay ay
?
which gives the sum of 7 terms of the series.
§2. Harlier discoveries affecting quadrature and cuba- ture.
Archimedes quotes the theorem that circles are to one another as the squares on their diameters as having being proved by earlier geometers, and he also says that it was proved by means of a certain lemma which he states as follows: “Of unequal lines, unequal surfaces, or unequal solids, the greater exceeds the less by such a magnitude as is capable, if added [continually] to itself, of exceeding
xlvi INTRODUCTION.
any given magnitude of those which are comparable with one another (τῶν πρὸς ἄλληλα λεγομένων). We know that Hippocrates of Chios proved the theorem that circles are to one another as the squares on their diameters, but no clear conclusion can be established as to the method which he used. On the other hand, Eudoxus (who is mentioned in the preface to The Sphere and Cylinder as having proved two theorems in solid geometry to be mentioned presently) is generally credited with the invention of the method of exhaustion by which Euclid proves the proposition in question in x11. 2. The lemma stated by Archimedes to have been used in the original proof is not however found in that form in Euclid and is not used in the proof of xit. 2, where the lemma used is that proved by him in x. 1, viz. that “Given two unequal magnitudes, if from the greater [a part] be subtracted greater than the half, if from the remainder [ἃ part] greater than the half be subtracted, and so on continually, there will be left some magnitude which will be less than the lesser given magnitude.” This last lemma is frequently assumed by Archimedes, and the application of it to equilateral polygons in- scribed in a circle or sector in the manner of x11. 2 is referred to as having been handed down in the Hlements*, by which it is clear that only Euclid’s Hlements can be meant. The apparent difficulty caused by the mention of éwo lemmas in connexion with the theorem in question can, however, I think, be explained by reference to the proof of x. 1 in Euclid. He there takes the lesser magnitude and says that it is possible, by multiplying it, to make it some time exceed the greater, and this statement he clearly bases on the 4th definition of Book vy. to the effect that ‘‘ magnitudes are said to bear a ratio to one another, which can, if multiplied, exceed one another.” Since then the smaller magnitude in x. 1 may be regarded as the difference between some two unequal magnitudes, it is clear that the lemma first quoted by Archimedes is in substance used to prove the lemma in x. 1 which appears to play so much larger a part in the in- vestigations in quadrature and cubature which have come down to us.
The two theorems which Archimedes attributes to Eudoxus by namet are
(1) that any pyramid is one third part of the prism which has
the same base as the pyramid and equal height, and
* On the Sphere and Cylinder, τ. 6. + ibid. Preface.
RELATION OF ARCHIMEDES TO HIS PREDECESSORS. xlix
(2) that any cone is one third part of the cylinder which has the same base as the cone and equal height.
The other theorems in solid geometry which Archimedes quotes as having been proved by earlier geometers are*:
(3) Cones of equal height are in the ratio of their bases, and conversely.
(4) If a cylinder be divided by a plane parallel to the base, cylinder is to cylinder as axis to axis.
(5) Cones which have the same bases as cylinders and equal height with them are to one another as the cylinders.
(6) The bases of equal cones are reciprocally proportional to their heights, and conversely.
(7) Cones the diameters of whose bases have the same ratio as their axes are in the triplicate ratio of the diameters of their bases.
In the preface to the Quadrature of the Parabola he says that earlier geometers had also proved that
(8) Spheres have to one another the triplicate ratio of their diameters ; and he adds that this proposition and the first of those which he attributes to Eudoxus, numbered (1) above, were proved by means of the same lemma, viz. that the difference between any two unequal magnitudes can be so multiplied as to exceed any given magnitude, while (if the text of Heiberg is right) the second of the propositions of Eudoxus, numbered (2), was proved by means of “a lemma similar to that aforesaid.” As a matter of fact, all the propositions (1) to (8) are given in Euclid’s twelfth Book, except (5), which, however, is an easy deduction from (2) ; and (1), (2), (3), and (7) all depend upon the same lemma [x. 1] as that used in Eucl. xi. 2.
The proofs of the above seven propositions, excluding (5), as given by Euclid are too long to quote here, but the following sketch will show the line taken in the proofs and the order of the propo- sitions. Suppose ABCD to be a pyramid with a triangular base, and suppose it to be cut by two planes, one bisecting AB, AC, AD in F, G, FE respectively, and the other bisecting BC, BD, BA in H, K, F respectively. These planes are then each parallel to one face, and they cut off two pyramids each similar to the original
* Lemmas placed between Props. 16 and 17 of Book 1. On the Sphere and Cylinder.
H. A. d
] INTRODUCTION.
pyramid and equal to one another, while the remainder of the pyramid is proved to form two equal prisms which, taken together,
A
ef ee a πὰ
are greater than one half of the original pyramid [x1. 3]. It is next proved [x11. 4] that, if there are two pyramids with triangular bases and equal height, and if they are each divided in the manner shown into two equal pyramids each similar to the whole and two prisms, the sum of the prisms in one pyramid is to the sum of the prisms in the other in the ratio of the bases of the whole pyramids respectively. Thus, if we divide in the same manner the two pyramids which remain in each, then all the pyramids which remain, and so on continually, it follows on the one hand, by x. 1, that we shall ultimately have pyramids remaining which are together less than any assigned solid, while on the other hand the sums of all the prisms resulting from the successive subdivisions are in the ratio of the bases of the original pyramids. Accordingly Euclid is able to use the regular method of exhaustion exemplified in xu. 2, and to establish the proposition [x11. 5] that pyramids with the same height and with triangular bases are to one another as their bases. The proposition is then extended [x1 6] to pyramids with the same height and with polygonal bases. Next [xt 7] a prism with a triangular base is divided into three pyramids which are shown to be equal by means of x11. 5; and it follows, as a corollary, that any pyramid is one third part of the prism which has the same base and equal height. Again, two similar and similarly situated pyramids are taken and the solid parallelepipeds are completed, which are then seen to be six times as large as the pyramids respectively ; and, since (by x1. 33) similar parallelepipeds are in the triplicate ratio of corresponding sides, it follows that the same
D
RELATION OF ARCHIMEDES TO HIS PREDECESSORS. li
is true of the pyramids [x1. 8]. A corollary gives the obvious extension to the case of similar pyramids with polygonal bases, The proposition [xu. 9] that, in equal pyramids with triangular bases, the bases are reciprocally proportional to the heights is proved by the same method of completing the parallelepipeds and using x1. 34; and similarly for the converse. It is next proved [x11. 10] that, if in the circle which is the base of a cylinder a square be described, and then polygons be successively described by bisecting the arcs remaining in each case, and so doubling the number of sides, and if prisms of the same height as the cylinder be erected on the square and the polygons as bases respectively, the prism with the square base will be greater than half the cylinder, the next prism will add to it more than half of the remainder, and so on. And each prism is triple of the pyramid with the same base and altitude. Thus the same method of exhaustion as that in ΧΙ. 2 proves that any cone is one third part of the cylinder with the same base and equal height. Exactly the same method is used to prove [x1. 11] that cones and cylinders which have the same height are to one another as their bases, and [xi1. 12] that similar cones and cylinders are to one another in the triplicate ratio of the diameters of their bases (the latter proposition depending of course on the similar proposition xu. 8 for pyramids). The next three propositions are proved without fresh recourse to x. 1. Thus the criterion of equimultiples laid down in Def. 5 of Book v. is used to prove [x11. 13] that, if a cylinder be cut by a plane parallel to its bases, the resulting cylinders are to one another as their axes. It is an easy deduction [x11. 14] that cones and cylinders which have equal bases are proportional to their heights, and [xu. 15] that in equal cones and cylinders the bases are reciprocally proportional to the heights, and, conversely, that cones or cylinders having this property are equal. Lastly, to prove that spheres are to one another in the triplicate ratio of their diameters [xu. 18], a new procedure is adopted, involving two preliminary propositions. In the first of these [xu. 16] it is proved, by an application of the usual lemma x. 1, that, if two concentric circles are given (however nearly equal), an equilateral polygon can be inscribed in the outer circle whose sides do not touch the inner ; the second proposition [x11. 17] uses the result of the first to prove that, given two concentric spheres, it is possible to inscribe a certain polyhedron in the outer
ad 2
ΠῚ INTRODUCTION.
so that it does not anywhere touch the inner, and a corollary adds the proof that, if a similar polyhedron be inscribed in a second sphere, the volumes of the polyhedra are to one another in the triplicate ratio of the diameters of the respective spheres, This last property is then applied [x11. 18] to prove that spheres are in the triplicate ratio of their diameters.
§ 3. Conic Sections.
In my edition of the Conics of Apollonius there is a complete account of all the propositions in conics which are used by Archi- medes, classified under three headings, (1) those propositions which he expressly attributes to earlier writers, (2) those which are assumed without any such reference, (3) those which appear to represent new developments of the theory of conics due to Archi- medes himself. As all these properties will appear in this volume in their proper places, it will suffice here to state only such propositions as come under the first heading and a few under the second which may safely be supposed to have been previously known.
Archimedes says that the following propositions “are proved in the elements of conics,” i.e. in the earlier treatises of Euclid and Aristaeus.
1, In the parabola
(a) if PV be the diameter of a segment and @Vq the chord parallel to the tangent at P, then YV=Vq;
(Ὁ) if the tangent at ὦ meet VP produced in 7, then ey he ;
(c) if two chords QVq, Q'V’q’ each parallel to the tangent at P meet the diameter PV in V, V’ respectively,
PV: ΡΞ ΘΙ:
2. If straight lines drawn from the same point touch any conic section whatever, and if two chords parallel to the respective tangents intersect one another, then the rectangles under the segments of the chords are to one another as the squares on the parallel tangents respectively.
3. The following proposition is quoted as proved “in the conics.” If in a parabola p, be the parameter of the principal ordinates,
RELATION OF ARCHIMEDES TO HIS PREDECESSORS. hii
QQ’ any chord not perpendicular to the axis which is bisected in 1 by the diameter PV, p the parameter of the ordinates to PV, and if QD be drawn perpendicular to PV, then
OV? 3 OD =p: pa. [On Conoids and Spheroids, Prop. 3, which see. |
The properties of a parabola, PV?=p,.AWN, and QV*=p.PYV, were already well known before the time of Archimedes. In fact the former property was used by Menaechmus, the discoverer of conic sections, in his duplication of the cube.
It may be taken as certain that the following properties of the ellipse and hyperbola were proved in the Conices of Euclid.
1. For the ellipse PN*?: AN. A'N=P'N": AN’. A’N'=CB? : CA’ and OV PVP Va OV ΡΣ ΒΕ OD) ΟἿ", (Either proposition could in fact be derived from the proposition
about the rectangles under the segments of intersecting chords above referred to.)
2. For the hyperbola AN PAW. 2 Vie PANN 1) and OW? PY OP! Viz Vite EV SOV though in this case the absence of the conception of. the double hyperbola as one curve (first found in Apollonius) prevented Euclid,
and Archimedes also, from equating the respective ratios to those of the squares on the parallel semidiameters.
3. In a hyperbola, if P be any point on the curve and PA, PL be each drawn parallel to one asymptote and meeting the other,
PK. PL=(eonst.) This property, in the particular case of the rectangular hyperbola, was known to Menaechmus.
It is probable also that the property of the subnormal of the parabola (VG'=4p,) was known to Archimedes’ predecessors. It is tacitly assumed, On floating bodies, τι. 4, etc.
From the assumption that, in the hyperbola, 47’< AN (where N is the foot of the ordinate from P, and 7’ the point in which the
liv INTRODUCTION.
tangent at P meets the transverse axis) we may perhaps infer that the harmonic property
GLY ES bat LT cee wal aria all or at least the particular case of it, DA NA
was known before Archimedes’ time.
Lastly, with reference to the genesis of conic sections from cones and cylinders, Euclid had already stated in his Phaenomena that, “if a cone or cylinder be cut by a plane not parallel to the base, the resulting section is a section of an acute-angled cone {an ellipse] which is similar to a @vpeds.” Though it is not probable that Euclid had in mind any other than a right cone, the statement should be compared with On Conoids and Spheroids, Props. 7, 8, 9.
84. Surfaces of the second degree.
Prop. 11 of the treatise On Conoids and Spheroids states without proof the nature of certain plane sections of the conicoids of revo- lution. Besides the obvious facts (1) that sections perpendicular to the axis of revolution are circles, and (2) that sections through the axis are the same as the generating conic, Archimedes asserts the following.
1. In a paraboloid of revolution any plane section parallel to the axis is a parabola equal to the generating parabola.
2. In a hyperboloid of revolution any plane section parallel
to the axis is a hyperbola similar to the generating hyperbola.
3. Ina hyperboloid of revolution a plane section through the vertex of the enveloping cone is a hyperbola which is not similar to the generating hyperbola.
4. In any spheroid a plane section parallel to the axis is an ellipse similar to the generating ellipse.
Archimedes adds that “the proofs of all these propositions are manifest (φανεραί). The proofs may in fact be supplied as follows.
1. Section of a paraboloid of revolution by a plane parallel to the axis.
RELATION OF ARCHIMEDES TO HIS PREDECESSORS. lv
Suppose that the plane of the paper represents the plane section through the axis AV which intersects the given plane section at right angles, and let A’O be the line of intersection. Let POP’ be any double ordinate to AW in the section through the axis, meeting A’O and AV at right angles in O, WV respectively. Draw A’ perpendicular to AX.
Suppose a perpendicular drawn from O to A'O in the plane of the given section parallel to the axis, and let y be the length intercepted by the surface on this perpendicular.
Then, since the extremity of y is on the circular section whose diameter is PP’,
y= PO.OP",
If A’O =a, and if p is the principal parameter of the generating parabola, we have then
y? = PN? ON? = PN? A'M? =p(AN-Al) = ρα,
so that the section is a parabola equal to the generating parabola.
2. Section of a hyperboloid of revolution by a plane parallel to the axis, Take, as before, the plane section through the axis which intersects
the given plane section at right angles in A’O. Let the hyperbola
lvi INTRODUCTION.
PAP’ in the plane of the paper represent the plane section through the axis, and let C be the centre (or the vertex of the enveloping cone). Draw CC’ perpendicular to CA, and produce OA’ to meet it in C’. Let the rest of the construction be as before. Suppose that CA τ Ae" CO: and let y have the same meaning as before. Then P=PO OP ΞΟ =A: And, by the property of the original hyperbola, PN* : CN?-CA?=A'M? : CM? — CA? (which is constant). Thus «45.275: CM?—CA?= PWN? :CN?-CA? =PN?—-A'M’ : CN’-Cl? = Ὁ a οἷ - Ἴ
whence it appears that the section is a hyperbola similar to the original one.
3. Section of a hyperboloid of revolution by a plane passing through the centre (or the vertex of the enveloping cone).
I think there can be no doubt that Archimedes would have proved his proposition about this section by means of the same general property of conics which he uses to prove Props. 3 and 12—14 of the same treatise, and which he enunciates at the beginning of Prop. 3 as a known theorem proved in the ‘elements of conics,” viz. that the rectangles under the segments of intersecting chords are as the squares of the parallel tangents.
Let the plane of the paper represent the plane section through the axis which intersects the given plane passing through the centre at right angles. Let (/A’O be the line of intersection, C being the centre, and A’ being the point where ('A’O meets the surface. Suppose CAIN to be the axis of the hyperboloid, and POp, P'O'p' two double ordinates to it in the plane section through the axis, meeting C'A’O in O, O’ respectively ; similarly let 4΄ be the ordinate from A’. Draw the tangents at A and A’ to the section through the axis meeting in 7’, and let QOq, Q’O'q’ be the two double ordinates in the same section which are parallel to the tangent at A’ and pass through Ὁ, O’ respectively.
Suppose, as before, that y, y’ are the lengths cut off by the
RELATION OF ARCHIMEDES TO HIS PREDECESSORS. lvil
surface from the perpendiculars at O and 0’ to OC in the plane of the given section through C'A’O, and that
CO—a C0 =e. CA — nc A=
Then, by the property of the intersecting chords, we have, since QO = 09, PO! Op 3Q0° — Ae PA
Ξσ ca OGD; 2) OL OE Also PaO ρ ΞΡ Ὁ" and, by the property of the hyperbola, 00? =r — a A= OF 5 — a: It follows, ex aequali, that Cia ert) an) ROY Pc nts el a ee ο (a), and therefore that the section is a hyperbola.
To prove that this hyperbola is not similar to the generating hyperbola, we draw CC’ perpendicular to C/A, and C’A’ parallel to CA meeting CC’ in C’ and Pp in U.
If then the hyperbola (a) is similar to the original hyperbola, it must by the last proposition be similar to the hyperbolic section made by the plane through C'’'A’U at right angles to the plane of the paper.
Now C0O?-—CA"=(C'U*?—C'A”)+(CC'+0U)?— CC”
> C'U?—C'A”, and PO .Op<PU. Up.
lvili INTRODUCTION.
Therefore PO. Op:CO?—CA"%<PU. Up: C'U?—C'A4
and it follows that the hyperbolas are not similar*.
4, Section of a spheroid by a plane parallel to the axis.
That this is an ellipse similar to the generating ellipse can of course be proved in exactly the same way as theorem (2) above for the hyperboloid.
* T think Archimedes is more likely to have used this proof than one on the lines suggested by Zeuthen (p. 421). The latter uses the equation of the hyperbola simply and proceeds thus. If y haye the same meaning as above, and if the coordinates of P referred to CA, CC’ as axes be 2, x, while those of O referred to the same axes are z, x’, we have, for the point P,
x°=« (27—a?), where x is constant. Also, since the angle 4’C4 is given, «’=az, where a is constant.
Thus Yea i} g?= (« be a?) 22 r καϑὶ Now z is proportional to CO, being in fact equal to Vice and the equation becomes ᾿Ξ ΞΕ pat ᾿ YP Tat 6795 - καθ.......νυννννννννννννννννννον (1),
which is clearly a hyperbola, since a2<k.
Now, though the Greeks could have worked out the proof in a geometrical form equivalent to the above, I think that it is alien from the manner in which Archimedes regarded the equations to central conics, These he always expressed in the form of a proportion
2 "2 2 Ἢ a [ = = in the case of the ellipse | ᾽
wna? αἰ α and never in the form of an equation between areas like that used by Apollonius, viz. e—ps af τς
Moreover the occurrence of the two different constants and the necessity of expressing them geometrically as ratios between areas and lines respectively would have made the proof very long and complicated ; and, as a matter of fact, Archimedes never does express the ratio y?/(x? -- a?) in the case of the hyperbola in the form of a ratio between constant areas like b?/a?. Lastly, when the equation of the given section through C'4’O was found in the form (1), assuming that the Greeks had actually found the geometrical equivalent, it would still have been held necessary, I think, to verify that ee) nar
K-ay
before it was finally pronounced that the hyperbola represented by the equation and the section made by the plane were one and the same thing.
CAC
RELATION OF ARCHIMEDES TO HIS PREDECESSORS. ΠΣ
We are now in a position to consider the meaning of Archimedes’ remark that “the proofs of all these properties are manifest.” In the first place, it is not likely that “manifest” means “known” as having been proved by earlier geometers ; for Archimedes’ habit is to be precise in stating the fact whenever he uses important propositions due to his immediate predecessors, as witness his references to Eudoxus, to the Hlements [of Euclid], and to the “elements of conics.” When we consider the remark with reference to the cases of the sections parallel to the axes of the surfaces respectively, a natural interpretation of it is to suppose that Archimedes meant simply that the theorems are such as can easily be deduced from the fundamental properties of the three conics now expressed by their equations, coupled with the consideration that the sections by planes perpendicular to the axes are circles. But I think that this particular explanation of the ‘“‘ manifest” character of the proofs is not so applicable to the third of the theorems stating that any plane section of a hyperboloid of revolution through the vertex of the enveloping cone but not through the axis is a hyperbola. This fact is indeed no more “manifest” in the ordinary sense of the term than is the like theorem about the spheroid, viz. that any section through the centre but not through the axis is an ellipse. But this latter theorem is not given along with the other in Prop. 11 as being “manifest” ; the proof of it is included in the more general proposition (14) that any section of a spheroid not perpendicular to the axis is an ellipse, and that parallel sections are similar. Nor, seeing that the propositions are essen- tially similar in character, can I think it possible that Archimedes wished it to be understood, as Zeuthen suggests, that the proposition about the hyperboloid alone, and not the other, should be proved directly by means of the geometrical equivalent of the Cartesian equation of the conic, and not by means of the property of the rectangles under the segments of intersecting chords, used earlier [Prop. 3] with reference to the parabola and later for the case of the spheroid and the elliptic sections of the conoids and spheroids generally. This is the more unlikely, I think, because the proof by means of the equation of the conic alone would present much more difficulty to the Greek, and therefore could hardly be called “‘ manifest.”
It seems necessary therefore to seek for another explanation, and I think it is the following. The theorems, numbered 1, 2, and
lx INTRODUCTION.
4 above, about sections of conoids and spheroids parallel to the axis are used afterwards in Props. 15—i7 relating to tangent planes ; whereas the theorem (3) about the section of the hyperboloid by a plane through the centre but not through the axis is not used in connexion with tangent planes, but only for formally proving that a straight line drawn from any point on a hyperboloid parallel to any transverse diameter of the hyperboloid falls, on the convex side of the surface, without it, and on the concave side within it. Hence it does not seem so probable that the four theorems were collected in Prop. 11 on account of the use made of them later, as that they were inserted in the particular place with special reference to the three propositions (12—14) immediately following and treating of the elliptic sections of the three surfaces. The main object of the whole treatise was the determination of the volumes of segments of the three solids cut off by planes, and hence it was first necessary to determine all the sections which were ellipses or circles and therefore could form the bases of the segments. Thus in Props. 12-14 Archimedes addresses himself to finding the elliptic sections, but, before he does this, he gives the theorems grouped in Prop. 11 by way of clearing the ground, so as to enable the propositions about elliptic sections to be enunciated with the utmost precision. Prop. 11 contains, in fact, explanations directed to defining the scope of the three following propositions rather than theorems definitely enunciated for their own sake; Archimedes thinks it necessary to explain, before passing to elliptic sections, that sections perpen- dicular to the axis of each surface are not ellipses but circles, and that some sections of each of the two conoids are neither ellipses nor circles, but parabolas and hyperbolas respectively. It is as if he had said, ‘‘ My object being to find the volumes of segments of the three solids cut off by circular or elliptic sections, I proceed to consider the various elliptic sections ; but I should first explain that sections at right angles to the axis are not ellipses but circles, while sections of the conoids by planes drawn in a certain manner are neither ellipses nor circles, but parabolas and hyperbolas respectively. With these last sections I am not concerned in the next propositions, and I need not therefore cumber my book with the proofs ; but, as some of them can be easily supplied by the help of the ordinary properties of conics, and others by means of the methods illustrated in the propositions now about to be given, I leave them as an exercise for the reader.” This will, I think, completely explain the assumption
RELATION OF ARCHIMEDES TO HIS PREDECESSORS. Ixi
of all the theorems except that concerning the sections of a spheroid parallel to the axis; and I think this is mentioned along with the others for symmetry, and because it can be proved in the same way as the corresponding one for the hyperboloid, whereas, if mention of it had been postponed till Prop. 14 about the elliptic sections of a spheroid generally, it would still require a proposition for itself, since the axes of the sections dealt with in Prop. 14 make an angle with the axis of the spheroid and are not parallel to it.
At the same time the fact that Archimedes omits the proofs of the theorems about sections of conoids and spheroids parallel to the axis as “manifest” is in itself sufficient to raise the presumption that contemporary geometers were familiar with the idea of three dimensions and knew how to apply it in practice. This is no matter for surprise, seeing that we find Archytas, in his solution of the problem of the two mean proportionals, using the intersection of a certain cone with a curve of double curvature traced on a right circular cylinder*. But, when we look for other instances of early investigations in geometry of three dimensions, we find practically nothing except a few vague indications as to the contents of a lost treatise of Euclid’s consisting of two Books entitled Swrface-loci (τόποι πρὸς ἐπιφανείᾳ). This treatise is mentioned by Pappus among other works by Aristaeus, Euclid and Apollonius grouped as forming the so-called τόπος ἀναλυόμενος. As the other works in the list which were on plane subjects dealt only with straight lines, circles and conic sections, it is a prior? likely that the swrface-loci of
* Cf. Eutocius on Archimedes (Vol. 11. pp. 98—102), or Apollonius of Perga, pp. Xxii.—xxilii.
+ By this term we conclude that the Greeks meant ‘‘loci which are surfaces ” as distinct from loci which are lines. Cf. Proclus’ definition of a locus as ‘*a position of a line or a surface involving one and the same property” (γραμμῆς ἢ ἐπιφανείας θέσις ποιοῦσα ἕν καὶ ταὐτὸν σύμπτωμα), p. 394. Pappus (pp. 660—2) gives, quoting from the Plane Loci of Apollonius, a classification of loci according to their order in relation to that of which they are the loci. Thus, he says, loci are (1) ἐφεκτικοί, i.e. fixed, e.g. in this sense the locus of a point is a point, of a linea line, and so on; (2) διεξοδικοί or moving along, a line being in this sense the locus of a point, a surface of a line, and a solid of a surface; (3) ἀναστροφικοί, turning backwards, i.e., presumably, moving backwards and forwards, a surface being in this sense the locus of a point, and a solid of a line. Thus a surface-locus might apparently be either the locus of a point or the locus of a line moving in space.
+ Pappus, pp. 634, 636.
lxil INTRODUCTION.
Euclid included at least such loci as were cones, cylinders and spheres. Beyond this, all is conjecture based upon two lemmas given by Pappus in connexion with the treatise.
First lemma to the Surface-loci of Huclid*.
The text of this lemma and the attached figure are not satisfac- tory as they stand, but they have been explained by Tannery in a way which requires a change in the figure, but only the very slightest alteration in the text, as follows;.
“Tf AB be a straight line and CD be parallel to a straight line given in position, and if the ratio AD. DB: DC’ be [given], the point C lies on a conic section. If now AB be no longer given in position and A, B be no longer given but lie on straight lines AE, EB given in positiont, the point C raised above [the plane, B containing AH, HB] is on a surface given in position. And this was proved.”
According to this interpretation, it is asserted that, if A moves with one extremity on each of the lines AH, HB which are fixed, while DC is in a fixed direction and AD. DB: 2653 is constant, then Οὐ lies on a certain surface. So far as the first sentence is concerned, A remains of constant length, but it is not made precisely clear whether, when AB is no longer given in position, its length may also vary§. If however AB remains of constant length for all positions which it assumes, the surface which is the locus of C would be a complicated one which we cannot suppose that Euclid could have profitably investigated. It may, therefore, be that Pappus purposely left the enunciation somewhat vague in order to make it appear to cover several surface-loci which, though belonging to the same type, were separately discussed by Euclid as involving
E
* Pappus, p. 1004.
+ Bulletin des sciences math., 2° Série, v1. 149.
t+ The words of the Greek text are γένηται δὲ πρὸς θέσει εὐθεῖα ταῖς AE, EB, and the above translation only requires εὐθείαις instead of εὐθεῖα. The figure in the text is so drawn that ADB, AEB are represented as two parallel lines, and CD is represented as perpendicular to ADB and meeting AEB in E.
§ The words are simply “if AB be deprived of its position (στερηθῇ τῆς θέσεως) and the points 4, B be deprived of their [character of] being given” (στερηθῇ τοῦ δοθέντος εἶναι).
RELATION OF ARCHIMEDES TO HIS PREDECESSORS. [ΧΠῚ
in each case somewhat different sets of conditions limiting the generality of the theorem.
It is at least open to conjecture, as Zeuthen has pointed out*, that two cases of the type were considered by Euclid, namely, (1) that in which AB remains of constant length while the two fixed straight lines on which A, B respectively move are parallel instead of meeting in a point, and (2) that in which the two fixed straight lines meet in a point while A moves always parallel to itself and varies in length accordingly.
(1) In the first case, where the length of AB is constant and the two fixed lines parallel, we should have a surface described by a conic moving bodily+. This surface would be a cylindrical surface, though it would only have been called a “ cylinder” by the ancients in the case where the moving conic was an ellipse, since the essence of a “cylinder” was that it could be bounded between two parallel circular sections. If then the moving conic was an ellipse, it would not be difficult to find the circular sections of the cylinder ; this could be done by first taking a section at right angles to the axis, after which it could be proved, after the manner of Archimedes, On Conoids and Spheroids, Prop. 9, first that the section is an ellipse or a circle, and then, in the former case, that a section made by a plane drawn at a certain inclination to the ellipse and passing through, or parallel to, the major axis is a circle. There was nothing to prevent Euclid from investigating the surface similarly generated by a moving hyperbola or parabola; but there would be no circular sections, and hence the surfaces might perhaps not have been considered as of very great importance,
(2) In the second case, where AH, GH meet at a point and AB moves always parallel to itself, the surface generated is of course a cone. Some particular cases of this sort may easily have been discussed by Euclid, but he could hardly have dealt with the general case, where DC has any direction whatever, up to the point of showing that the surface was really a cone in the sense in which the Greeks understood the term, or (in other words) of finding the circular sections. To do this it would have been necessary to determine the principal planes, or to solve the dis-
* Zeuthen, Die Lehre von den Kegelschnitten, pp. 425 sqq. + This would give a surface generated by a moving line, διεξοδικὸς γραμμῆς as Pappus has it.
lxiv INTRODUCTION.
criminating cubic, which we cannot suppose Euclid to have done. Moreover, if Euclid had found the circular sections in the most general case, Archimedes would simply have referred to the fact instead of setting himself to do the same thing in the particular case where the plane of symmetry is given. These remarks apply to the case where the conic which is the locus of C is an ellipse ; there is still less ground for supposing that Euclid could have proved the existence of circular sections where the conic was a hyperbola, for there is no evidence that Euclid even knew that hyperbolas and parabolas could be obtained by cutting an oblique circular cone.
Second lemma to the Surface-loci.
In this Pappus states, and gives a complete proof of the propo- sition, that the locus of a point whose distance from a given point is in ὦ given ratio to its distance from a fixed line is a conic section, which is un ellipse, a parabola, or a hyperbola according as the given ratio is less than, equal to, or greater than unity*. Two conjectures are possible as to the application of this theorem by Euclid in the treatise referred to.
(1) Consider a plane and a straight line meeting it at any angle. Imagine any plane drawn at right angles to the straight line and meeting the first plane in another straight line which we will call X. If then the given straight line meets the plane at right angles to it in the point S, a conic can be described in that plane with S for focus and X for directrix ; and, as the perpendicular on α΄ from any point on the conic is in a constant ratio to the per- pendicular from the same point on the original plane, all points on the conic have the property that their distances from S are in a given ratio to their distances from the given plane respectively. Similarly, by taking planes cutting the given straight line at right angles in any number of other points besides S, we see that the locus of a point whose distance from a given straight line is in a given ratio to its distance from a given plane is a cone whose vertex is the point in which the given line meets the given plane, while the plane of symmetry passes through the given line and is at right angles to the given plane. If the given ratio was such that the guiding conic was an ellipse, the circular sections of the surface
* See Pappus, pp. 1006—1014, and Hultsch’s Appendix, pp. 1270—1273 ; or cf, Apollonius of Perga, pp. Xxxvi.—xxxviii,
RELATION OF ARCHIMEDES TO HIS PREDECESSORS. lxv
could, in that case at least, be found by the same method as that used by Archimedes (On Conoids and Spheroids, Prop. 8) in the rather more general case where the perpendicular from the vertex of the cone on the plane of the given elliptic section does not necessarily pass through the focus.
(2) Another natural conjecture would be to suppose that, by means of the proposition given by Pappus, Euclid found the locus of a point whose distance from a given point is in a given ratio to its distance from a fixed plane. This would have given surfaces identical with the conoids and spheroids discussed by Archimedes excluding the spheroid generated by the revolution of an ellipse about the minor axis. We are thus brought to the same point as Chasles who conjectured that the Surfuce-loci of Euclid dealt with surfaces of revolution of the second degree and sections of the same*, Recent writers have generally regarded this theory as improbable. Thus Heiberg says that the conoids and spheroids were without any doubt discovered by Archimedes himself; other- wise he would not have held it necessary to give exact definitions of them in his introductory letter to Dositheus ; hence they could not have been the subject of Euclid’s treatiset. I confess I think that the argument of Heiberg, so far from being conclusive against the probability of Chasles’ conjecture, is not of any great weight. To suppose that Euclid found, by means of the theorem enunciated and proved by Pappus, the locus of a point whose distance from a given point is in a given ratio to its distance from a fixed plane does not oblige us to assume either that he gave a name to the loci or that he investigated them further than to show that sections through the perpendicular from the given point on the given plane were conics, while sections at right angles to the same perpendicular were circles ; and of course these facts would readily suggest them- selves. Seeing however that the object of Archimedes was to find the volumes of segments of each surface, it is not surprising that he should have preferred to give a definition of them which would indicate their form more directly than a description of them as loci would have done; and we have a parallel case in the dis- tinction drawn between conics as such and conics regarded as loci, which is illustrated by the different titles of Euclid’s Conics and the Solid Loci of Aristaeus, and also by the fact that Apollonius,
* Apercu historique, pp. 273, 4. + Litterargeschichtliche Studien tiber Euklid, Ὁ. 79.
Ixvi_ INTRODUCTION.
though he speaks in his preface of some of the theorems in his Conics as useful for the synthesis of ‘solid loci’ and goes on to mention the ‘locus with respect to three or four lines,’ yet enun- ciates no proposition stating that the locus of such and such a point is a conic. There was a further special reason for defining the conoids and spheroids as surfaces described by the revolution of a conic about its axis, namely that this definition enabled Archi- medes to include the spheroid which he calls ‘flat’ (ἐπιπλατὺ σφαιροειδές), 1.6. the spheroid described by the revolution of an ellipse about its minor axis, which is not one of the loci which the hypothesis assumes Euclid to have discovered. Archimedes’ new definition had the incidental effect of making the nature of the sections through and perpendicular to the axis of revolution even more obvious than it would be from Euclid’s supposed way of treating the surfaces; and this would account for Archimedes’ omission to state that the two classes of sections had been known before, for there would have been no point in attributing to Euclid the proof of propositions which, with the new definition of the surfaces, became self-evident. The further definitions given by Archimedes may be explained on the same principle. Thus the axis, as defined by him, has special reference to his definition of the surfaces, since it means the axis of revolution, whereas the axis of a conic is for Archimedes a diameter. The enveloping cone of the hyperboloid, which is generated by the revolution of the asymptotes about the axis, and the centre regarded as the point of intersection of the asymptotes were useful to Archimedes’ dis- cussion of the surfaces, but need not have been brought into Euclid’s description of the surfaces as loci. Similarly with the axis and vertex of a segment of each surface. And, generally, it seems to me that all the definitions given by Archimedes can be explained in like manner without prejudice to the supposed dis- covery of three of the surfaces by Euclid.
I think, then, that we may still regard it as possible that Euclid’s Swrface-loct was concerned, not only with cones, cylinders and (probably) spheres, but also (to a limited extent) with three other surfaces of revolution of the second degree, viz. the paraboloid, the hyperboloid and the prolate spheroid. Unfortunately however we are confined to the statement of pussibilities; and certainty can hardly be attained unless as the result of the discovery of fresh documents.
RELATION OF ARCHIMEDES TO HIS PREDECESSORS. lxvil
§ 5. Two mean proportionals in continued proportion.
Archimedes assumes the construction of two mean proportionals in two propositions (On the Sphere and Cylinder u. 1, 5). Perhaps he was content to use the constructions given by Archytas, Menaechmus*, and Eudoxus. It is worth noting, however, that Archimedes does not introduce the two geometric means where they are merely convenient but not necessary ; thus, when (On the.
1
Sphere and Cylinder τ. 34) he has to substitute for a ratio (f) : δ where β: γγ, a ratio between lines, and it is sufficient for his purpose that the required ratio cannot be greater than (Fy but 7
may be less, he takes two arithmetic means between β, y, as ὃ, ε, and then assumesy as a known result that ΒΕ: ey" * The constructions of Archytas and Menaechmus are given by Eutocius [Archimedes, Vol. 111. pp. 92—102]; or see Apollonius of Perga, pp. xix—xxiil.
+ The proposition is proved by Eutocius; see the note to On the Sphere and Cylinder τ. 34 (p, 42).
62
CHAPTER IV.
ARITHMETIC IN ARCHIMEDES.
Two of the treatises, the Measurement of a circle and the Sand-reckoner, are mostly arithmetical in content. Of the Sand- reckoner nothing need be said here, because the system for expressing numbers of any magnitude which it unfolds and applies cannot be better described than in the book itself; in the Weaswrement of a circle, however, which involves a great deal of manipulation of numbers of considerable size though expressible by means of the ordinary Greek notation for numerals, Archimedes merely gives the results of the various arithmetical operations, multiplication, extrac- tion of the square root, etc., without setting out any of the operations themselves. Various interesting questions are accordingly involved, and, for the convenience of the reader, I shall first give a short account of the Greek system of numerals and of the methods by which other Greek mathematicians usually performed the various operations included under the general term λογιστική (the art of calculating), in order to lead up to an explanation (1) of the way in which Archimedes worked out approximations to the square roots of large numbers, (2) of his method of arriving at the two approximate
values of V3 which he simply sets down without any hint as to how they were obtained*.
* In writing this chapter I have been under particular obligations to Hultsch’s articles Arithmetica and Archimedes in Pauly-Wissowa’s Real-Encyclopiddie, τι. 1, as well as to the same scholar’s articles (1) Die Néherungswerthe irrationaler Quadratwurzeln bei Archimedes in the Nachrichten von der kgl. Gesellschaft der Wissenschajten zu Gittingen (1893), pp. 367 sqq., and (2) Zur Kreismessung des Archimedes in the Zeitschrift fiir Math. u. Physik (Hist. litt. Abtheilung) xxxix. (1894), pp. 121 sqq. and 161 sqq. I have also made use, in the earlier part of the chapter, of Nesselmann’s work Die Algebra der Griechen and the histories of Cantor and Gow.
ARITHMETIC IN ARCHIMEDES, lxix
§ 1. Greek numeral system.
It is well known that the Greeks expressed all numbers from 1 to 999 by means of the letters of the alphabet reinforced by the addition of three other signs, according to the following scheme, in which however the accent on each letter might be replaced by a short horizontal stroke above it, as @.
a, 8, y', ὃ, €,'s", ὁ ἢ, 0 are 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. MEU μον, δ, Oty G 55 LO, BOOK τῷ. 90 ες ΠΡΟ τὸ 6 OX, Ψ ὦ, 4 ,, 100, 200, 300,...... 900 95
Intermediate numbers were expressed by simple juxtaposition (representing in this case addition), the largest number being placed on the left, the next largest following it, and so on in order. Thus the number 153 would be expressed by ρνγ' or pvy. There was no sign for zero, and therefore 780 was ψπ', and 306 rs" simply. .
Thousands (χιλιάδες) were taken as units of a higher order, and 1,000, 2,000, ... up to 9,000 (spoken of as χίλιοι, δισχίλιοι, κιτ.λ.} Were represented by the same letters as the first nine natural numbers but with a small dash in front and below the line; thus e.g. 6’ was 4,000, and, on the same principle of juxtaposition as before, 1,823 was expressed by awxy’ or awxy, 1,007 by af’, and so on.
Above 9,999 came a myriad (μυριάς), and 10,000 and higher numbers were expressed by using the ordinary numerals with the substantive μυριάδες taken as a new denomination (though the words μύριοι, δισμύριοι, τρισμύριοι, κιτ.λ. are also found, following the analogy of χίλιοι, δισχίλιοι and so on). Various abbreviations were used for the word μυριάς, the most common being M or Mv; and, where this was used, the number of myriads, or the multiple of 10,000, was generally written over the abbreviation, though some-
times before it and even after it. Thus 349,450 was Μθυν' ὩΣ Fractions (λεπτά) were written in a variety of ways. The most usual was to express the denominator by the ordinary numeral with two accents affixed. When the numerator was unity, and it was therefore simply a question of a symbol for a single word such as * Diophantus denoted myriads followed by thousands by the ordinary signs for numbers of units, only separating them by a dot from the thousands. Thus
for 3,069,000 he writes rs. 0, and λγ. ἀψος for 331,776. Sometimes myriads were represented by the ordinary letters with two dots above, as p =100 myriads (1,000,000), and myriads of myriads with two pairs of dots, as ¢ for 10 myriad- myriads (1,000,000,000).
Ixx INTRODUCTION.
τρίτον, 1, there was no need to express the numerator, and the
symbol was γ΄; similarly ς΄ -- 1, ιε΄ -- 4,, and so on. When the
numerator was not unity and a certain number of fourths, fifths, c., had to be expressed, the ordinary numeral was used for the
numerator; thus θ΄ ια΄ = %, ι΄ οα΄ --19. In Heron’s Geometry the denominator was written twice in the latter class of fractions ; thus
ents, 23
δύο πέμπτα) was β' εἴ ε $3 (λεπτὰ τ ιακοστότριτα κγ΄ ΟΥ εἰκοσιτρία μ ᾽ ρ ρ γ ρ
σαι
τριακοστότριτα) Was Ky’ dy” λγ΄. The sign for 3, ἥμισυ, is in Archimedes, Diophantus and Eutocius _”, in Heron C or a sign similar to a capital S*.
A favourite way of expressing fractions with numerators greater than unity was to separate them into component fractions with numerator unity, when juxtaposition as usual meant addition. Thus # was written ἰ.. δ΄ - Ε1, 18 was Cd’yis”=4+44+44+3; Eutocius writes £6” or $+, for 23, and so on. Sometimes the same fraction was separated into several different sums; thus in Heron (p. 119, ed. ey 363 is variously expressed as
224
(0) $+7+30+a:2+2Rn (6) 3+3+ fan ay and (0) ἐ Ἐπ τ οὖν Ἐτῖσ Ἐ αὖτ.
Seaagesimal fractions. This system has to be mentioned because the only instances of the working out of some arithmetical operations which have been handed down to us are calculations expressed in terms of such fractions; and moreover they are of special interest as having much in common with the modern system of decimal fractions, with the difference of course that the submultiple is 60 instead of 10. The scheme of sexagesimal fractions was used by the Greeks in astronomical calculations and appears fully developed in the σύνταξις of Ptolemy. The circumference of a circle, and along with it the four right angles subtended by it at the centre, are divided into 360 parts (τμήματα or μοῖραι) or as we should say degrees, each μοῖρα into 60 parts called (first) sixtieths, (πρῶτα) ἑξηκοστά, or minutes (λεπτά), each of these again into δεύτερα ἑξηκοστά (seconds), and so on. <A similar division of the radius of the circle into 60
* Diophantus has a general method of expressing fractions which is the
exact reverse of modern practice; the denominator is written above the Ύ κε a. wis
numerator, thus ¢=5/3, κα = 21/25, and pxf. φξη =1,270,568/10,816. Some- times he writes down the numerator and then introduces the denominator
with ἐν μορίῳ or μορίου, e.g. Ts .,8 wop. Ny. αψος = 3,069,000/331,776.
ARITHMETIC IN ARCHIMEDES. xxi
parts (τμήματα) was also made, and these were each subdivided into sixtieths, and so on. Thus a convenient fractional system was available for general arithmetical calculations, expressed in units of any magnitude or character, so many of the fractions which we should represent by εἶσ» so many of those which we should write (25), (gs), and so on to any extent. It is therefore not surprising that Ptolemy should say in one place “In general we shall use the method of numbers according to the sexagesimal manner because of the inconvenience of the [ordinary] fractions.” For it is clear that the successive submultiples by 60 formed a sort of frame with fixed compartments into which any fractions whatever could be located, and it is easy to see that e.g. in additions and subtractions the sexagesimal fractions were almost as easy to work with as decimals are now, 60 units of one denomination being equal to one unit of the next higher denomination, and “carrying” and “borrowing” being no less simple than it is when the number of units of one denomination necessary to make one of the next higher is 10 instead of 60. In expressing the units of the circumference, degrees, μοῖραι or the symbol 4 was generally used along with the ordinary numeral which had a stroke above it ; minutes, seconds, etc. were expressed by one, two, etc. accents affixed to the numerals. Thus fi B=2°, μοιρῶν pl μβ΄ μ' =47° 42’ 40". Also where there was no unit in any particular denomination O was used, signifying οὐδεμία μοῖρα, οὐδὲν ἑξηκοστόν and the like; thus Oa’ β΄ O’’=0° 1΄ 2" 0". Similarly, for the units representing the divisions of the radius the word τμήματα or some equivalent was used, and the fractions were represented as
before ; thus τμημάτων ἕζ δ' ve” = 67 (units) 4’ 55”. § 2. Addition and Subtraction.
There is no doubt that, in writing down numbers for these purposes, the several powers of 10 were kept separate in a manner corresponding practically to our system of numerals, and the hundreds, thousands, etc., were written in separate vertical rows, The following would therefore be a typical form of a sum in addition ;
αυκδ' = 1424 pon 103
Mora’ 12281 M 30030 ὃ
M yor η΄ 43838
Ixxil INTRODUCTION.
and the mental part of the work would be the same for the Greek as for us.
Similarly a subtraction would be represented as follows: θ M yxAs’ = 93636 B My θ΄ 23409
M σκζ Τ0227
§ 8. Multiplication.
A number of instances are given in Eutocius’ commentary on the Measurement of a circle, and the similarity to our procedure is just as marked as in the above cases of addition and subtraction. The multiplicand is written first, and below it the multiplier preceded by ἐπί (=“‘into”). Then the highest power of 10 in the multiplier is taken and multiplied into the terms containing the separate multiples of the successive powers of 10, beginning with the highest and descending to the lowest ; after which the next highest power of 10 in the multiplier is multiplied into the various denominations in the multiplicand in the same order. The same procedure is followed where either or both of the numbers to be multiplied contain fractions. Two instances from Eutocius are appended from which the whole procedure will be understood.
(1) ie 780 ἐπὶ wr’ x 780 Oe MM ς΄ 490000 56000 Mosul 56000 6400 é ὁμοῦ M nv’ sum 608400 (2) By 8? 30133 } [=30133] ἐπὶ γιγ' "8" x 30133 4 ει Χ MM 6 adyv’ 9,000,000 30,000 9,000 1500 750 Μρλέβ' 1." 30,000 100 ὃ. Ὁ - Dre ea igh 9,000 30 9 ὙΠ αφ' εα' ἔδυ 1,500 5 11 1 = yr’ β΄ hee Eo sy 750 24 4 ἊΣ 1 ἑ τῷ
Ἂ [ὁμοῦ] Μβχαθιις" [9,041,250 + 30,1374 + 9,041} + 1506+4+}42 +7534+4434+ = 9,082,689,),.
ARITHMETIC IN ARCHIMEDES. Ixxili
One instance of a similar multiplication of numbers involving fractions may be given from Heron (pp. 80, 81). It is only one of many, and, for brevity, the Greek notation will be omitted. Heron has to find the product of 423 and 782, and proceeds as follows:
64) 1.7 58 Δ 5 4. ee ΞΞ πες
9) 510 62 H lng ene 62 6 1 28 +e t+ ee ec = 28 ἢ Ὁ 51 ot tz 5+ a
The multiplication of 37° 4’ 55” (in the sexagesimal system) by itself is performed by Theon of Alexandria in his commentary on Ptolemy’s σύνταξις in an exactly similar manner.
§ 4. Division.
The operation of dividing by a number of one digit only was easy for the Greeks as for us, and what we call “long division” was with them performed, mutatis mutandis, in the same way as now with the help of multiplication and subtraction. Suppose, for instance, that the operation in the first case of multiplication given
above had to be reversed and that Μην (608,400) had to be divided by Wz’ (780). The terms involving the different powers of 10 would be mentally kept separate as in addition and subtraction, and the first question would be, how many times will 7 hundreds go into 60 myriads, due allowance being made for the fact that the 7 hundreds have 80 behind them and that 780 is not far short of 8 hundreds ? The answer is 7 hundreds or y’, and this multiplied by the divisor
vd ξ Wx’ (780) would give Ms’ (546,000) which, subtracted from M ηυ'
(608,400), leaves the remainder M Bu’ (62,400). This remainder has then to be divided by 780 or a number approaching 8 hundreds, and 8 tens or π' would have to be tried. In the particular case the result would then be complete, the quotient being ψπ' (780), and there being no remainder, since π΄ (80) multiplied by wz’ (780) gives
the exact figure M βυ' (62,400).
lxxiv INTRODUCTION.
An actual case of long division where the dividend and divisor contain sexagesimal fractions is described by Theon. The problem is to divide 1515 20’ 15” by 25 12’ 10”, and Theon’s account of the process comes to this.
Divisor Dividend Quotient 25-12) 10? 515 20% 15” First term 60 a 25. 60 = 1500
Remainder 15 = 900’ Sum πο 127160 — 720’ Remainder 200’ 10%. Cor 10’ Remainder 190’ Second term 7’ ae (= 175. τ 15’ = 900 Sum Oth” ibe ay fe 84” Remainder Sole LO 1.7. Oe Remainder 829” 50” |Third term 33” 2,7) 1 ΘΠ, 825" Remainder 4" 50" = 290” 19 1955 996"
(too great by) 106”
Thus the quotient is something less than 60 7’ 33”. It will be observed that the difference between this operation of Theon’s and
that followed in dividing Mee (608,400) by ψπ' (780) as above is that Theon makes three subtractions for one term of the quotient, whereas the remainder was arrived at in the other case after one subtraction. The result is that, though Theon’s method is quite clear, it is longer, and moreover makes it less easy to foresee what will be the proper figure to try in the quotient, so that more time would be apt to be lost in making unsuccessful trials.
§ 5. Extraction of the square root.
We are now in a position to see how the operation of extracting the square root would be likely to be attacked, First, as in the case of division, the given whole number whose square root is required would be separated, so to speak, into compartments each containing
ARITHMETIC IN ARCHIMEDES. Ixxv
such and such a number of units and of the separate powers of 10. Thus there would be so many units, so many tens, so many hundreds, etc., and it would have to be borne in mind that the squares of numbers from 1 to 9 would lie between 1 and 99, the squares of numbers from 10 to 90 between 100 and 9900, and so on. Then the first term of the square root would be some number of tens or hundreds or thousands, and so on, and would have to be found in much the same way as the first term of a quotient in a “long division,” by trial if necessary. If A is the number whose square root is required, while a represents the first term or denomination of the square root and « the next term or denomination still to be found, it would be necessary to use the identity (a + #)?=a? + 2aa +a and to find « so that 2ax+a? might be somewhat less than the remainder 4—a?, Thus by trial the highest possible value of a satisfying the condition would be easily found. If that value were ὁ, the further quantity 2ab +6? would have to be subtracted from the first remainder A — a’, and from the second remainder thus left a third term or denomination of the square root would have to be derived, and so on. That this was the actual procedure adopted is clear from a simple case given by Theon in his commentary on the σύνταξις. Here the square root of 144 is in question, and it is obtained by means of Eucl. 1. 4. The highest possible denomina- tion (i.e. power of 10) in the square root is 10 ; 10? subtracted from 144 leaves 44, and this must contain not only twice the product of 10 and the next term of the square root but also the square of that next term itself. Now, since 2.10 itself produces 20, the division of 44 by 20 suggests 2 as the next term of the square root; and this turns out to be the exact figure required, since
2. 20+ 2° = 44.
The same procedure is illustrated by Theon’s explanation of Ptolemy’s method of extracting square roots according to the sexagesimal system of fractions. The problem is to find approxi- mately the square root of 4500 μοῖραι or degrees, and a geometrical figure is used which makes clear the essentially Euclidean basis of the whole method. Nesselmann gives a complete reproduction of the passage of Theon, but the following purely arithmetical represen- tation of its purport will probably be found clearer, when looked at side by side with the figure.
Ptolemy has first found the integral part of /4500 to be 67.
lxxvl INTRODUCTION.
Now 67? = 4489, so that the remainder is 11. Suppose now that the rest of the square root is expressed by means of the usual sexagesimal fractions, and that we may therefore put
45002 67? s LL Sen ee ποῦ
60 * 60?’ 2.67x where a, y are yet to be found. Thus ὦ must be such that 60 : 11.60 is somewhat less than 11, or x must be somewhat less than 267
or which is at the same time greater than 4. On trial, it
oT turns out that 4 will satisfy the conditions of the problem, namely
2
that (67 + a) must be less than 4500, so that a remainder will
be left by means of which y may be found.
a n K ὃ | 67° 4! 55” 4489 268’ | & = a | 3 | of εἶ - § 4’ 268' 16” g λ 55” 3688” 40" | β Y 2.67.4 AN? ; ΔΝ Now 11 — ————_ — & is the remainder, and this is equal to 60 60 11; 60'—2.67.4560—16 Ὁ 7155 60° ἡ oo 4\ y : 1424 ᾿ : G7 acess baie - Thus we must suppose that 2 (67 + a) 602 approximates to 60F ?
or that 8048y is approximately equal to 7424. 60.
ARITHMETIC IN ARCHIMEDES. ΙΧΧΥΙ
Therefore y is approximately equal to 55. We have then to subtract
: Ὁ ΕΣ BB\? Ἀ442640 805 2( 60) 60? τε. oF 60° * 60F”
: 7424 from the remainder 0" above found.
442640 1424. 2800 46 40
The subtraction of $0? from 602 δ “Gos > Of Boe δ’ 3025
OS
but Theon does not go further and subtract the remaining
ἼΩΝ 60? As a matter of fact, if we deduct the
instead of which he merely remarks that the square of
40 60?” 60%" 3025 £ 2800 : ; ΐ bee Gor [πὰ gor» 80 as to obtain the correct remainder, it is 164975
Coe” To show the power of this method of extracting square roots by means of sexagesimal fractions, it is only necessary to mention that
δ 28 60 60) 60° approximation is equivalent to 1:7320509 in the ordinary decimal notation and is therefore correct to 6 places.
But it is now time to pass to the question how Archimedes
approximates to
found to be
Ptolemy gives as an approximation to \/3, which
obtained the two approximations to the value of /3 which he assumes in the Measwrement of a circle. In dealing with this subject I shall follow the historical method of explanation adopted by Hultsch, in preference to any of the mostly ὦ priori theories which the ingenuity of a multitude of writers has devised at different times.
§ 6. Early investigations of surds or incommensurables.
From a passage in Proclus’ commentary on Eucl. 1.* we learn that it was Pythagoras who discovered the theory of irrationals (ἡ τῶν ἀλόγων πραγματεία). Further Plato says (Theaetetus 147 Ὁ), “On square roots this Theodorus [of Cyrene] wrote a work in
* p. 65 (ed. Friedlein).
lxxvili INTRODUCTION.
which he proved to us, with reference to those of 3 or 5 [square] feet that they are incommensurable in length with the side of one square foot, and proceeded similarly to select, one by one, each [of the other incommensurable roots] as far as the root of 17 square feet, beyond
which for some reason he did not go.” The reason why ν 3 is not mentioned as an incommensurable square root must be, as Cantor says, that it was before known to be such. We may therefore conclude that it was the square root of 2 which was geometrically constructed by Pythagoras and proved to be incommensurable with the side of a square in which it represented the diagonal. A clue to the method by which Pythagoras investigated the value of J2 is found by Cantor and Hultsch in the famous passage of Plato (Rep. vit. 546 B, c) about the ‘geometrical’ or ‘nuptial’ number. Thus, when Plato contrasts the ῥητὴ and ἄρρητος διάμετρος τῆς πεμπάδος, he is referring to the diagonal of a square whose side contains five units of length ; the appyros διάμετρος, or the irrational diagonal, is then /50 itself, and the nearest rational number is J/50—1, which is the ῥητὴ διάμετρος. We have herein the explanation of the way in which Pythagoras must have made the first and most readily comprehensible approximation to /2; he must have taken, instead of 2, an improper fraction equal to it but such that the denominator was a square in any case, while the numerator was as near as possible to a complete square. Thus
Pythagoras chose and the first approximation to /2 was
50
fas : - ; accordingly 5? it being moreover obvious that / I>. Again,
Pythagoras cannot have been unaware of the truth of the proposition, proved in Eucl. τι. 4, that (a+ δ)" Ξ- αὐ + 2ab +b’, where a, b are any two straight lines, for this proposition depends solely upon propositions in Book 1. which precede the Pythagorean proposition 1. 47 and which, as the basis of 1. 47, must necessarily have been in substance known to its author. <A slightly different geometrical proof would give the formula (α -- ὁ)" -- αὖ -- 3αὖ - δ᾽, which must have been equally well known to Pythagoras. It could not therefore have escaped the discoverer of the first approximation
J/50—1 for /50 that the use of the formula with the positive sign
: ΕΠ : 1 ean would give a much nearer approximation, viz. 7 + ——, which is only
14’
ARITHMETIC IN ARCHIMEDES. lxxix
14 assign to Pythagoras the discovery of the fact represented by
greater than /50 to the extent of (=): Thus we may properly
ron ἢ EN {πω 50> 7.
The consequential result that /2> ; /50—1 is used by
Aristarchus of Samos in the 7th proposition of his work On the size and distances of the sun and moon*.
With reference to the investigations of the values of /3, V5,
Gon. J17 by Theodorus, it is pretty certain that /3 was geometrically represented by him, in the same way as it appears
* Part of the proof of this proposition was a sort of foretaste of the first part of Prop. 3 of Archimedes’ Measurement of a circle, and the substance of it is accordingly A appended as reproduced by Hultsch.
ABEK is a square, KB a diagonal, 2 HBH =i4KBE, 2 FBE=3°,and AC is perpendicu- lar to BF so that the triangles ACB, BEF are similar.
Aristarchus seeks to prove that
Ald8Y & TeXOF se 1S} 2 1
If R denote a right angle, the angles KBE, HBE, FBE are respectively 3910, 15}, 2,R. B =
Then HE: FE > ZHBE: Z£FBE.
[This is assumed as a known lemma by Aristarchus as well as Archimedes. ]
K
Therefore γε BD ag) 2) ee ee See ee (a). Now, by construction, BK?=2BE?, Also [Eucl. vr. 3] ΒΚ 2 ois HOPI SED) ®
whence KH=N2HE.
And, since N2 =x γ33Ξ: Ρ 25
HiGal G83) S75 15%, so that ἜΗΙ Ds Re) CEN | a Race (g). From (a) and (8), ex aequali, KE: FE > 18:1. Therefore, since BF > BE (or KE),
BF: FE > 18: 1, so that, by similar triangles, ABT BC = Nis le
Ixxx INTRODUCTION.
afterwards in Archimedes, as the perpendicular from an angular point of an equilateral triangle on the opposite side. It would thus be readily comparable with the side of the “1 square foot” mentioned by Plato. The fact also that it is the side of three square feet (τρίπους δύναμις) which was proved to be incommensurable suggests that there was some special reason in Theodorus’ proof for specifying feet, instead of units of length simply; and the ex- planation is probably that Theodorus subdivided the sides of his triangles in the same way as the Greek foot was divided into halves, fourths, eighths and sixteenths. Presumably therefore,
exactly as Pythagoras had approximated to /2 by putting =
for 2, Theodorus started from the identity 3= Ἐξ It would then
16 = ἴεν, ἢ Mie ae Ae
To investigate ./48 further, Theodorus would put it in the form /49—1, as Pythagoras put ./50 into the form J/49+1, and the result would be
be clear that
/48 (= eee
We know of no further investigations into incommensurable square roots until we come to Archimedes.
5.7. Archimedes’ approximations to V3.
Seeing that Aristarchus of Samos was still content to use the first and very rough approximation to ν 2 discovered by Pythagoras, it is all the more astounding that Aristarchus’ younger contemporary Archimedes should all at once, without a word of explanation, give out that
1351 - 265 780 > V5 T53° as he does in the Measurement of a circle.
In order to lead up to the explanation of the probable steps by which Archimedes obtained these approximations, Hultsch adopts the same method of analysis as was used by the Greek geometers in solving problems, the method, that is, of supposing the problem solved and following out the necessary consequences. ΤῸ compare
ARITHMETIC IN ARCHIMEDES. ΙΧΧΧῚ
265 1351 sey : ς the two fractions -- 153 ὃ and 7B” we first divide both denominators
* into their smallest factors, and we obtain ΠΟ ΞΟ 229.52 139,
fod =o. 1 We observe also that 2.2.13 =52, while 3.17 =51, and we may therefore show the Ἔν between the numbers thus, (80'= 3.5. 2, Ξε οἷς
For convenience of comparison we multiply the numerator and
denominator of = by 5; the two original fractions are then 1351 ἃ 1325 15. Sou dae ol
so that we can put Archimedes’ assumption in the form
1351 1325 ΠΡΌΣ ΙΝ:
> 15/3 > and this is seen to be equivalent to
ieee Die Oras
1 : Now 26-5 = / 2 6°—1+ (ss) , and the latter expression
is an approximation to /26?—1.
We have then 26 = > /26? -- 1. = jl.
As La was compared with 15,/3, and we want an ap-
proximation to ./3 itself, we divide by 15 and so obtain
ἡ (28- 5) > τὸ 28-1.
15 52 LO ere 676-1 Bi ΤΤς Ὁ ͵ Bae 15 V26?-1= \/ 995. ~ A/ 995. ~ /3, and it follows
that - (26 ~55)> "5"
The lower limit for /3 was given by
1 1
. xxxii INTRODUCTION.
and a glance at this suggests that it may have been arrived at by simply substituting (52 -- 1) for 52.
Now as a matter of fact the following proposition is true. Jf a’ +b is a whole number which is not a square, while a? is the nearest square number (above or below the first number, as the case may be), then b
ὖ Cis -- Jato +b>at 2a 2a+1°
Hultsch proves this pair of inequalities in a series of propositions formulated after the Greek manner, and there can be little doubt that Archimedes had discovered and proved the same results in substance, if not in the same form. ‘The following circumstances confirm the probability of this assumption.
(1) Certain approximations given by Heron show that he knew and frequently used the formula
ρος b V@tbwats, «Ὁ
(where the sign οὦ denotes ‘‘is approximately equal to”).
Thus he gives /50 07 + = : Vopese= 16’ 11 V75 οὐ 8 Ὁ 16° (2) The formula Ja*+boa+ ἘΠ i is used by the Arabian Alkarkhi (11th century) who drew from Greek sources (Cantor, Ῥ 719'sq.).
It can therefore hardly be accidental that the formula b = b + — a+b>at+-— Be ko eas = 9a41 gives us what we want in order to obtain the two Archimedean approximations to ν 9, and that in direct connexion with one another*.
* Most of the a priori theories as to the origin of the approximations are open to the serious objection that, as a rule, they give series of approximate values in which the two now in question do not follow consecutively, but are separated by others which do not appear in Archimedes. Hultsch’s explanation
is much preferable as being free from this objection. But it is fair to say that the actual formula used by Hultsch appears in Hunrath’s solution of the puzzle
ARITHMETIC IN ARCHIMEDES. lxxxili
We are now in a position to work out the synthesis as follows.
From the geometrical representation of /3 as the perpendicular from an angle of an equilateral triangle on the opposite side we
obtain /2?—1= J3 and, as a first approximation,
2 on : > 3. Using our formula we can transform this at once into = il 1 2 —-_—. 2---. ν8-:- resins 3
1 5 Archimedes would then square (2 - 3) or 3 and would obtain
; Le. he would put
which he would compare with 3, or = :
3 (5+ 5)> V3, ie. 2 > 0/8.
3 To obtain a still nearer approximation, he would proceed in the nee 26\? 676 h3, 675 h : same manner and compare (=) 1 OF 558» wit or 555,» Whence it 5 Adel would appear that /3 = 995? Mites rore that 1. ὦ and therefore tha iB (2 -- 5) Say : 1351
that is, 780 > J3.
The application of the formula would then give the result
5] 1 =. (eer V3 > τὸ (26 a1)
1326 —1 265 Tosnie 7 + 153" The complete result would therefore be
1351 aS 265
780 153° (Die Berechnung irrationaler Quadratwurzeln vor der Herrschaft der Decimal- briiche, Kiel, 1884, p, 21; ef. Ueber das Ausziehen der Quadratwurzel bei Griechen und Indern, Hadersleben, 1883), and the same formula is implicitly used in one of the solutions suggested by Tannery (Sur la mesure du cercle ad Archiméde in Mémoires de la société des sciences physiques et naturelles de Bordeaux, 2° série, 1v. (1882), p. 313-337).
that is, V3 >
f2
lxxxiv INTRODUCTION.
Thus Archimedes probably passed from the first approximation a to from 1 to BBS and from =
directly to the closest
Lape + 3) 3, a kp 5 780 ’ approximation of all, from which again he derived the less close
imation 28} approxima 153° nearer approximation than | is probably that the squaring of this fraction would have brought in numbers much too large to be
conveniently used in the rest of his calculations. A similar reason
The reason why he did not proceed to a still
‘will account for his having started from 5 instead of Ἷ ; if he had
used the latter, he would first have obtained, by the same method,
ea | - τ πς U lee ἢ a Ne 5° ane /3, or 567 /3; the squaring Oy pee of ea would have given ν ὃ-- oe , and the corresponding 18817
approximation would have given 56.194? where again the numbers are inconveniently large for his purpose.
§ 8. Approximations to the square roots of large numbers.
Archimedes gives in the Measurement of a circle the following approximate values :
(1) 30132 > /9082321, (2) 18382, > 3380929, (3) 10093 > 1018405, (4) 20174 > /4069284,1,, (5) 5911 </349450,
(6) 11721 < se (7) 23394 < /54721322.,
There is no doubt that in obtaining the integral portion of the square root of these numbers Archimedes used the method based on the Euclidean theorem (a+b)? = a?+ 2ab +6? which has
ARITHMETIC IN ARCHIMEDES. Ixxxv
already been exemplified in the instance given above from Theon, where an approximation to /4500 is found in sexagesimal fractions. The method does not substantially differ from that now followed; but whereas, to take the first case, 9082321, we can at once see what will be the number of digits in the square root by marking off pairs of digits in the given number, beginning from the end, the absence of a sign for Ὁ in Greek made the number of digits in the square root less easy to ascertain because, as written in Greek, the number
Μ᾿ Brea! only contains six signs representing digits instead of seven. Even in the Greek notation however it would not be difficult to see that, of the denominations, units, tens, hundreds, etc. in the square root, the units would correspond to xa’ in the original number, the
Ul Ἂ tens to Br, the hundreds to M, and the thousands to M. Thus it would be clear that the square root of 9082321 must be of the form
1000a + 100y + 10z + w,
where «, y, z, w can only have one or other of the values 0, 1, 2,...9. Supposing then that x is found, the remainder V —(1000z)’, where WV is the given number, must next contain 2.1000x”.100y and (100y)’, then 2(1000xz+100y).10z and (10z)’, after which the remainder must contain two more numbers similarly formed.
In the particular case (1) clearly e=3, The subtraction of (3000)? leaves 82321, which must contain 2.3000.100y. But, even if y is as small as 1, this product would be 600,000, which is greater than 82321. Hence there is no digit representing hundreds in the square root. To find z, we know that 82321 must contain
2.3000. 102 + (10z)’,
and z has to be obtained by dividing 82321 by 60,000. Therefore z=1. Again, to find w, we know that the remainder
(82321 — 2.3000. 10 -- 105),
or 22221, must contain 2.3010w+w*, and dividing 22221 by 2.3010 we see that w=3. Thus 3013 is the integral portion of the square root, and the remainder is 22221 —(2.3010.3+ 3°), or 4152.
The conditions of the proposition now require that the approxi- mate value to be taken for the square root must not be less than
lxxxvi INTRODUCTION.
the real value, and therefore the fractional part to be added to 3013 must be if anything too great. Now it is easy to see that the
2 fraction to be added is greater than : because 2.3013. _ (5) is
less than the remainder 4152. Suppose then that the number required (which is nearer to 3014 than to 3013) is 3014,
and - has to be if anything too small.
Now (3014)? = (3013)? + 2. 3013 + 1 = (3013)? + 6027 = 9082321 -- 4152 + 6027,
whence 9082321 = (3014)? -- 1875. By applying Archimedes’ formula Ja? +b<a+ 5 , we obtain 3014 -- oe 9082321. The required value / Ξ has therefore to be not greater than δ ae : It remains to be explained why Archimedes put for δ the value 4 which is equal to oe In the first place, he evidently preferred
fractions with unity for numerator and some power of 2 for denominator because they contributed to ease in working, e.g. when two such fractions, being equal to each other, had to be added.
The exceptions, the fractions me and τ are to be explained Ὁ " Tyas ᾿ ἦ
exceptional circumstances presently to be mentioned.) Further, in the particular case, it must be remembered that in the subsequent
work 2911 had to be added to 3014 —F and the sum divided by 780,
or 2.2.3.5.13. It would obviously lead to simplification if a
factor could be divided out, e.g. the best for the purpose, 13. Now,
dividing 2911 + 3014, or 5925, by 13, we obtain the quotient 455, P
and a remainder 10, so that ae remains to be divided by 13. Therefore ᾧ has to be so chosen that 10g -- Ὁ is divisible by 13, while
i approximates to, but is not greater than, ee The solution
p=1,q=4 would therefore be natural and easy.
ARITHMETIC IN ARCHIMEDES. lxxxvll
(2) »/3380929.
The usual process for extraction of the square root gave as the integral part of it 1838, and as the remainder 2685. As before, it was easy to see that the exact root was nearer to 1839 than to 1838, and that
/3380929 = 1838? + 2685 = 1839" -- 2. 1838 —1 + 2685
= 1839? — 992. The Archimedean formula then gave 992 ππαπασς sane 2 1839 — στο. 9" V3380929.
It could not have escaped Archimedes that :
992 , 1984 : 1 1839 1 3678 Of 73567 1508 G = 735° and Z would have satisfied
the necessary condition that the fraction to be taken ἘΠΕ be less
was a near approxima-
tion to
than the real value. Thus it is clear that, in taking es as the
approximate value of the fraction, Archimedes had in view the simplification of the we. work by the elimination of a factor,
If the fraction be denoted by Ὁ , the sum of 1839-7 and 1823, or
3662 -- a had to be divided by en ie. by 6.40. Division of 3662 by 40 gave 22 as remainder, and then », ῳ had to be so chosen that
22-7 was conveniently divisible by 40, while 7 was less than but
9 approximately equal to ee The solution p= 2, g=11 was easily
3678 ° seen to satisfy the conditions.
(3) »/1018405.
The usual procedure gave 1018405=1009°+324 and the ap- proximation
sts
24 It was here necessary that the fraction to replace — should be
: : bY (Oy Ve greater but approximately equal to it, and δ satisfied the conditions,
while the subsequent work did not require any change in it.
Ixxxvlil INTRODUCTION.
(4) »/40692842,. The usual process gave 4069284. = 2017? + 995, ; it followed that 36.995 +1] ποσπσςτπτ a ao py > /4069284,2,, and 20174 was an obvious value to take as an approximation somewhat greater than the left side of the inequality.
(5) /349450.
In the case of this and the two following roots an approximation had to be obtained which was /ess, instead of greater, than the true value. Thus Archimedes had to use the second part of the formula
b
b at 5,70 tb>aty iy:
In the particular case of ./349450 the integral part of the root is 591, and the remainder is 169. This gave the result
16 169 B91 +a 501 2.59141’
and since 160-- 18", while 2.591+1=7.13°, it resulted without further calculation that
> /349450 > 5914
/349450 > 6911.
Why then did Archimedes take, instead of this approximation, another which was not so close, viz. 59142 The answer which the subsequent working and the other approximations in the first part of the proof suggest is that he preferred, for convenience of calculation,
1
to use for his approximations fractions of the form = δὴ only. But he could not have failed to see that to take the nearest fraction of this aes iL ae : : form, 3? instead of z might conceivably affect his final result and make it less near the truth than it need be. As a matter of fact, as Hultsch shows, it does not affect the result to take 5914 and to work onwards from that figure. Hence we must suppose that Archimedes had satisfied himself, by taking 5914 and proceeding on that basis for some distance, that he would not be introducing any appreciable error in taking the more convenient though less accurate approximation 5911.
ARITHMETIC IN ARCHIMEDES. Ixxxix
(6) ./137394333,
In this case the integral portion of the root is 1172, and the remainder 35923. Thus, if & denote the root,
37 Se 79 64 AMORA. 359 τς τ τος τ lore. >117 Ἐς Τ|75 11’ a fortiorr Now 2.1172+1=2345; the fraction accordingly becomes 9345? and : (= ima) satisfies the necessary conditions, viz. that it must
be approximately equal to, but not greater than, the given fraction. Here again Archimedes would have taken 11721 as the approximate value but that, for the same reason as in the last case, 11721 was more convenient.
(7) ./547213255
The integral portion of the root is here 2339, and the remainder 1211,4,, so that, if & is the exact root,
12113, 2. 233941 > 23391, a fortiori.
A few words may be added concerning Archimedes’ ultimate reduction of the inequalities
6671 284 »π:: 9-
* 46734 2017
to the simpler result 3 : >1r>3 τ ‘ 1 6674
As a matter of fact ἘΝ 46725"
only necessary to make the small change of diminishing the de-
R> 2339 +
so that in the first fraction it was
nominator by | in order to obtain the simple ae
2844 1137, 2017} ~ 8069° Hultsch ingeniously suggests the method of trying the effect of increasing the denominator of the latter fraction by 1. This
As regards the lower limit for 7, we see that and
ΧΟ INTRODUCTION.
τς a ; and, if we divide 2690 by 379, the quotient is between 7 and 8, so that Le ἢ 7 ~ 2690 ~ 8° Now it is a known proposition (proved in Pappus vil. p. 689)
a@ at+e that, if 5 then ho bad
Similarly it may be proved that
produces
ate 6 b+d a’ It follows in the above case that 919... (STO il 2690 2690. 8 8” which exactly gives = > !
10; 379 1. and -- 7] is very much nearer to 5,5. 3690 than Ὁ
Note on alternative hypotheses with regard to the
approximations to /3.
For a description and examination of all the various theories put forward, up to the year 1882, for the purpose of explaining Archimedes’
approximations to 1/3 the reader is referred to the exhaustive paper by Dr Siegmund Giinther, entitled Die guadratischen Irrationalitdten der Alten und deren Entwickelungsmethoden (Leipzig, 1882). The same author gives further references in his Abriss der Geschichte der Mathematik und der Natur- wissenschaften im Altertum forming an Appendix to Vol. v. Pt. 1 of Iwan von Miiller’s Handbuch der klassischen Altertums-wissenschaft (Miinchen, 1894). Giinther groups the different hypotheses under three general heads :
(1) those which amount to a more or less disguised use of the method of continued fractions and under which are included the solutions of De Lagny, Mollweide, Hauber, Buzengeiger, Zeuthen, P. Tannery (first solution), Heilermann ;
(2) those which give the approximations in the form of a series
of fractions such as ὦ + : -- : a 1 +...; under this class come the
Gt 919s Sass solutions of Radicke, v. Pessl, Rodet (with reference to the Qulvasiitras),
Tannery (second solution) ;
ARITHMETIC IN ARCHIMEDES. XCl
(3) those which locate the incommensurable surd between a greater and lesser limit and then proceed to draw the limits closer and closer. This class includes the solutions of Oppermann, Alexejeff, Schénborn, Hunrath, though the first two are also connected by Giinther with the method of continued fractions.
Of the methods so distinguished by Giinther only those need be here referred to which can, more or less, claim to rest on a historical basis in the sense of representing applications or extensions of principles laid down in the works of Greek mathematicians other than Archimedes which have come down to us. Most of these quasi-historical solutions connect themselves with the system of side- and diagonal-numbers (πλευρικοὶ and διαμετρικοὶ ἀριθμοί) explained by Theon of Smyrna (c. 130 A.D.) in a work which was intended to give so much of the principles of mathematics as was necessary for the study of the works of Plato.
The stde- and diagonal-numbers are formed as follows. We start with two units, and (a) from the sum of them, (Ὁ) from the sum of twice the first unit and once the second, we form two new numbers ; thus
1.141=2, 2.141=3.
Of these numbers the first is a side- and the second a diagonal-number respectively, or (as we may say) ας ΞΞ2, d,.=3. In the same way as these numbers were formed from a,=1, d,=1, suc- cessive pairs of numbers are formed from a,, d,, and so on, in accordance with the formula An+1=AIntn, Ay 41 = 20, +n, whence we have d,=1.243=5, d,5=2.24+3=7, =1.5+7=12, dy=2.54+7=17, and so on. Theon states, with reference to these numbers, the general proposition which we should express by the equation dy? =D? +1. The proof (no doubt omitted because it was well-known) is simple. For we have Oy? — Qty? = (2d, 1 + Ay 1)? — 2 (Gin - +A)?
Ξε2α,.. i a An"
Sle (d,-?= 2a,_1”)
= + (d,_.? — 2a,-,”), and so on,
while d,?—2a,?= --Ἰ ; whence the proposition is established.
Cantor has pointed out that any one familiar with the truth of this proposition could not have failed to observe that, as the numbers were successively formed, the value of d,?/a,? would approach more and more nearly to 2, and consequently the successive fractions d,/a, would give
xcll INTRODUCTION.
nearer and nearer approximations to the value of 4/2, or in other words that ego ert i Medi ΤᾺ 2? 5? 12° 29’ eeeece are successive approximations to »/2. It is to be observed that the third of these approximations, . is the Pythagorean approximation which
appears to be hinted at by Plato, while the above scheme of Theon, amounting to a method of finding all the solutions in positive integers of the indeterminate equation 272 — y= +1,
and given in a work designedly introductory to the study of Plato, distinctly suggests, as Tannery has pointed out, the probability that even in Plato’s lifetime the systematic investigation of the said equation had already begun in the Academy. In this connexion Proclus’ commentary on Eucl. 1. 47 is interesting. It is there explained that in isosceles right-angled triangles “it is not possible to find numbers corresponding to the sides; for there is no square number which is double of a square except in the sense of approaimately double, e.g. 7? is double of 5? less 1.” When it is remembered that Theon’s process has for its object the finding of any number of squares differing only by unity from double the squares of another series of numbers respectively, and that the sides of the two sets of squares are called diagonal- and side-numbers respectively, the conclusion becomes almost irresistible that Plato had such a system in mind when he spoke of ῥητὴ διάμετρος (rational diagonal) as compared with ἄρρητος διάμετρος (irrational diagonal) τῆς πεμπάδος (cf. p. xxviii above).
One supposition then is that, following a similar line to that by which successive approximations to »/2 could be obtained from the successive solutions, in rational numbers, of the indeterminate equations 247 —y?= +1, Archimedes set himself the task of finding all the solutions, in rational numbers, of the two indeterminate equations bearing a similar relation to 3, viz.
v*—37?=1, et —3y?= -- 2.
Zeuthen appears to have been the first to connect, eo nomine, the ancient approximations to /3 with the solution of these equations, which are also made by Tannery the basis of his first method. But, in substance, the same method had been used as early as 1723 by De Lagny, whose hypothesis will be, for purposes of comparison, described after Tannery’s which it so exactly anticipated.
Zeuthen’s solution.
After recalling the fact that, even before Euclid’s time, the solution of the indeterminate equation #?+72=2 by means of the substitutions m? —n* m+n
L=MN, eae 5
ARITHMETIC IN ARCHIMEDES, xclli
was well known, Zeuthen concludes that there could have been no difficulty in deducing from Eucl. τι. 5 the identity
m?—3n?\?__ (m?+3n*\* 3 (nn)?+( 3 ) - 5) ) ) from which, by multiplying up, it was easy to obtain the formula 3 (Qmn)? + (m? — 8n?)? = (m? + 3x7).
If therefore one solution m?—3n?=1 was known, a second could at once be found by putting
L=m?+3n7, y=2mn. Now obviously the equation m —3n?=1
is satisfied by the values m=2, n=1; hence the next solution of the equation xv? — 3y?=1
is =274+3.1=7, yy=2.2.1=4; and, proceeding in like manner, we have any number of solutions as Bye=P4+3.4=97, youd.7.4=56,
24=9724+3.562=18817, yg=2.97.56=10864, and so on.
Next, addressing himself to the other equation
ue? — 3y?= — 2, Zeuthen uses the identity
(m+3n)?—3 (m+n)?= — 2 (m?—3n?). Thus, if we know one solution of the equation m?—3n?=1, we can proceed
to substitute L=M+3n, Y=M+N.
Suppose m=2, n=1, as before ; we then have
a,=5, Yy=3. If we put #,=2,+3y,=14, y,=27,+4,=8, we obtain a es
ee ὃ 4 (and m=7, n=4 is seen to be a solution of m?—3n?=1).
Starting again from 7, y., we have ,=38, Y3= 22,
and zs
whence “ἜΞΕ -
XClv INTRODUCTION.
(and m=26, n=15 satisfies m?-3n?=1), ,=284, y,=164,
or ὅς εἶθ Ye ΧΑ Similarly “= ee Ete ΞΘΟ and isos 00:
Ys 56’ y 153’ This method gives all the successive approximations to /3, taking
account as it does of both the equations
v—3y=1,
φῇ — 3y?= — 2.
Tannery's first solution.
Tannery asks himself the question how Diophantus would have set about solving the two indeterminate equations. He takes the first equation in the generalised form
v?—ay*=1, and then, assuming one solution (p, 4) of the equation to be known, he supposes »)ιτεηιῶ-- , H=r+q-
Then py — ag? =m" — Impx +p? — ax? —2age—ag?=1, whence, since p?—aq?=1, by hypothesis, o=2,—P ΞΙ 51 ᾿ m — a , _ (m7? +a) p+2amq _ 2mp+(m?+a)q so that i=" gee Vie
and p,?—aq,?=1. The values of p,, g, so found are rational but not necessarily integral ; if integral solutions are wanted, we have only to put
P= (+ av?) p+ 2aurg, 9, =2pu + (uv? + av") g, where (εν, v) is another integral solution of #? -- ay?=1. Generally, if (p, g) be a known solution of the equation v—ay’=r, suppose p,=ap+q, g,=yp + 6g, and “il suffit pour déterminer a, β, y, ὃ de connaitre les trois groupes de solutions les plus simples et de résoudre
deux couples d’équations du premier degré ἃ deux inconnues.” Thus
(1) for the equation v?—3y?=1, the first three solutions are
(p=, q=9), (p=2, g=1), (p=7, q=4),
2=a 7=2a+fB whence 152. and ray Ae
so that a=2, B=3, y=1, 8=2,
ARITHMETIC IN ARCHIMEDES. XCV
and it follows that the fourth solution is given by p=2.7+3.4=26, g=1.7+2.4=15; (2) for the equation x? —37?= --2, the first three solutions being (1, 1), (5, 3), (19, 11), we have = 19=5a+3 oat and iy ene , whence a=2, B=3, y=1, 5=2, and the next solution is given by p=2.19+3.11=71,
g=1.19+2.11=41, and so on. Therefore, by using the two indeterminate equations and proceeding as shown, all the successive approximations to V3 can be found. Of the two methods of dealing with the equations it will be seen that Tannery’s has the advantage, as compared with Zeuthen’s, that it can be applied to the solution of any equation of the form x?—ay*=r.
De Lagny’s method.
The argument is this. If /3 could be exactly expressed by an im- proper fraction, that fraction would fall between 1 and 2, and the square of its numerator would be three times the square of its denominator. Since this is impossible, two numbers have to be sought such that the square of the greater differs as little as possible from 3 times the square of the smaller, though it may be either greater or less. De Lagny then evolved the following successive relations,
2=3.124+1, 57=3.3?-2 72=3.4?+1, 19%=3.11?-2, 267=3.157+1, 712=3. 412-2, ete.
From these relations were derived a series of fractions greater than V3,
viz. = a: = , etc., and another series of fractions less than V3, viz. 5 19 7] : : 3 3°11’ 41’ etc. The law of formation was found in each case to be that, if
2 was one fraction in the series and x the next, then
ΤΡ Ἐ50 q 5) This led to the results ST F2G τον 1862 1351 fe 1747 15. 56> 9090. 780 «τὸ ονο, δ. 19 71 905 989 369] = and 5. τὶ <a < 153 “671 9151" <3
xevl INTRODUCTION.
while the law of formation of the successive approximations in each series is precisely that obtained by Tannery as the result of treating the two indeterminate equations by the Diophantine method.
Heilermann’s method.
This method needs to be mentioned because it also depends upon a generalisation of the system of séde- and diagonal-numbers given by Theon of Smyrna.
Theon’s rule of formation was
Sn=Sn-1+Da—1; Dy, =28p-1+ Dn-13 and Heilermann simply substitutes for 2 in the second relation any arbitrary number a, developing the following scheme,
S,=Sot+ Do; D, = aSy+ Dy; S,=+D,, D,=aS,+D,, S3=S,+D,, D;=aS,.+D,,
ΘΙ ΞΡ ϑΞδ ΓΕ It follows that AS? = aS yy? + 2a8,_; Dn_ + aD y_4*, De =O Byatt Oe By subtraction, D,2 — aS,2= (1 — a) (Dy_ 12 — αϑ...2) (1 —a)? (Dy? — an”), similarly,
=(1—a)" (D,?- a8,?). This corresponds to the most general form of the “ Pellian” equation 43 — ay? = (const.). If now we put D)=S)=1, we have D,?2 (. -- αὐ. 11 ἜΣ ὍΣ from which it appears that, where the fraction on the right-hand side
. ἌΣ : aie approaches zero as ” increases, y 15 an approximate value for Va. n
Clearly in the case where a=3, Dy)=2, Sp=1 we have
Dy2 Τὶ δος, DO aes
Bi? & 3? Sy) Gee? 6, gale Ib? D, Te VD: let or ae seen
ΠῚ
S, ἀπ δ. dle 60” 82 v3"
and so on,
ARITHMETIC IN ARCHIMEDES. X¢eV1l But the method is, as shown by Heilermann, more rapid if it is used to find, not γα, but b/a, where ὁ is so chosen as to make 0?a (which takes
the place of a) somewhat near to unity. Thus suppose a=2, so that
Va=2 V3, and we then have (putting D)=S)=1)
52 5 26 26 S,=2, “Ὁ and V30 5. Ree ge
102 54452 106 106, 265 mon ar ean π᾿ and 4/8 os 3 102? * 153° 208 102.27 106 5404 Ss=o5 Ys 95.95 +95 85. 58’
- 5404 ὅ 1351 SIG regener or 780 ° This is one of the very few instances of success in bringing out the two Archimedean approximations in immediate sequence without any foreign values intervening. No other methods appear to connect the two values in this direct way except those of Hunrath and Hultsch depending on the formula
and
b τ ὖ ats >V/a+b HOS ab Ἐπ
We now pass to the second class of solutions which develops the approximations in the form of the sum of a series of fractions, and under this head comes
Tannery's second method.
This may be exhibited by means of its application (1) to the case of the square root of a large number, e.g. /349450 or 5712+ 23409, the first of the kind appearing in Archimedes, (2) to the case of 4/3.
(1) Using the formula
ΠΕΣ ΤΟΝ 2a
we try the effect of putting for /571?+ 23409 the expression
23409 1142 "
It turns out that this gives correctly the integral part of the root, and we now suppose the root to be
571 + ———
1 571+20+—. m Squaring and regarding = as negligible, we have
7124400 + 22840+ ae ar Ὁ 5717+ 23409,
xXe@Vlil INTRODUCTION.
whence Lipet 169, m and ΘΗΝ Ss he 1859. 7 so that /349450 >591 τ (2) Bearing in mind that b 2 V@+beate ἘΠ ἈΞ 2 pee 2 aie we have V/3="/1 +2e01+5 τὶ 2 5 ~ 1.5» or 3°
: = 1 ς Assuming then that /3 = G + = , Squaring and neglecting = , we obtain
engl St sa whence m=15, and we get as the second approximation Bey re ae 3. 15’ ES We have now 26?-3.15?=1,
and can proceed to find other approximations by means of Tannery’s first method.
ΘΠ = eal NS Or we can also put (+3+ists) =)
and, neglecting =) we get 267 ὅδ᾽ ΤΙΣ. Abia: whence n= —15.52= — 780, and
Ξ Ἢ 1 1351 v3.09 (145 +35 - 7607 780)”
It is however to be observed that this method only connects a with
and not with the intermediate approximation τ to obtain which
Tannery implicitly uses a particular case of the formula of Hunrath and Hultsch.
Rodet’s method was apparently invented to explain the approximation in the Culvasitras* 1 1
VB 1+5 3134 39,4584"
* See Cantor, Vorleswngen tiber Gesch. d. Math. Ὁ. 600 sq.
ARITHMETIC IN ARCHIMEDES. ΧΟΙΧ
but, given the approximation 5 2 , the other two successive approximations indicated by the formula can be obtained by the method of squaring just described* without such elaborate work as that of Rodet, which, when applied to 4/3, only gives the same results as the simpler method. Lastly, with reference to the third class of solutions, it may be mentioned (1) that Oppermann used the formula at 2ab —>V/ab >— τη}
which gave successively
ΙΝ but only led to one of the Archimedean approximations, and that by combining the last two ratios, thus
97+168 265
56497 153? (2) that Schénborn came somewhat near to the formula successfully used by Hunrath and Hultsch when he proved t te
ate PNG EIS a+
ai
* Cantor had already pointed this out in his first edition of 1880. + Zeitschrift fiir Math. u. Physik (Hist. litt, Abtheilung) xxviu. (1883), p. 169 sq.
CHAPTER V. ON THE PROBLEMS KNOWN AS ΝΕΥΣΕΙΣ.
THE word νεῦσις, commonly znclinatio in Latin, is difficult to translate satisfactorily, but its meaning will be gathered from some general remarks by Pappus having reference to the two Books of Apollonius entitled νεύσεις (now lost). Pappus says*, “A line is said to verge (vevew) towards a point if, being produced, it reach the point,” and he gives, among particular cases of the general form of the problem, the following.
“Two lines being given in position, to place between them a straight line given in length and verging towards a given point.”
“Tf there be given in position (1) a semicircle and a straight line at right angles to the base, or (2) two semicircles with their bases in a straight line, to place between the two lines a straight line given in length and verging towards a corner (ywviav) of a semicircle.”
Thus a straight line has to be laid across two lines or curves so that it passes through a given point and the intercept on it between the lines or curves is equal to a given lengthf.
81. The following allusions to particular vevoes are found in Archimedes. The proofs of Props. 5, 6, 7 of the book On Spirals use respectively three particular cases of the general theorem that,
* Pappus (ed. Hultsch) vit. p. 670.
+ In the. German translation of Zeuthen’s work, Die Lehre von den Kegélschnitten im Altertum, νεῦσις is translated by “ Kinschiebung,”’ or as we might say “insertion,” but this fails to express the condition that the required line must pass through a given point, just as inclinatio (and for that matter the Greek term itself) fails to express the other requirement that the intercept on the line must be of given length.
ON THE PROBLEMS KNOWN AS ΝΕΥΣΕΙΣ, cl
if A be any point on a circle and BC any diameter, ἐξ is possible to draw through A a straight line, meeting the circle again in P and BC produced in R, such that the intercept PR is equal to any given
length. In each particular case the fact is merely stated as true without any explanation or proof, and
(1) Prop. 5 assumes the case where the tangent at A is parallel to BC,
(2) Prop. 6 the case where the points A, P in the figure are interchanged,
(3) Prop. 7 the case where A, P are in the relative positions shown in the figure.
Again, (4) Props. 8 and 9 each assume (as before, without proof, and without giving any solution of the implied problem) that, if AZ, BC be two chords of a circle intersecting at right angles in a point D such that BD> DC, then it is possible to draw through A another line ARP, meeting BC in R and the circle again in P, such that PR = DE.
Lastly, with the assumptions in Props. ὅ, 6, 7 should be compared Prop. 8 of the Liber Assumptorum, which may well be due to Archimedes, whatever may be said of the composition of the whole book. This proposition proves that, if in the first figure APR is so drawn that PR is equal to the radius OP, then the are AB is three times the are PC. In other words, if an arc AB of a circle be taken subtending any angle at the centre QO, an arc equal to one-third of the given arc can be found, ἐ.6. the given angle can be trisected, uf only APR can be drawn through A in such a manner
cll INTRODUCTION.
that the intercept PR between the circle and BO produced is equal to the radius of the circle. Thus the trisection of an angle is reduced to a νεῦσις exactly similar to those assumed as possible in Props. 6, 7 of the book On Spirals.
The νεύσεις so referred to by Archimedes are not, in general, capable of solution by means of the straight line and circle alone, as may be easily shown. Suppose in the first figure that x represents the unknown length OF, where O is the middle point of BOC, and that & is the given length to which PF is to be equal ; also let OD=a, AD=b, BC =2c. Then, whether BC be a diameter or (more generally) any chord of the circle, we have
Ah te — Bi, and therefore kJ? + (α-- α}"- οὗ -- ὁ,
The resulting equation, after rationalisation, is an equation of the fourth degree in x; or, if we denote the length of AA by y, we have, for the determination of x and y, the two equations
y? = (α -- a)? + δ᾽
ky = α -- οὗ } In other words, if we have a rectangular system of coordinate axes, the values of x and y satisfying the conditions of the problem can be determined as the coordinates of the points of intersection of a certain rectangular hyperbola and a certain parabola.
In one particular case, that namely in which D coincides with O the middle point of BC, or in which A is one extremity of the diameter bisecting BC at right angles, a=0, and the equations reduce to the single equation
y’ —ky=b' -- οὗ, which is a quadratic and can be geometrically solved by the
traditional method of application of areas; for, if οὐ be substituted for y—k, so that «= AP, the equation becomes
αν (ἢ - τι) τε δ᾽ +e’,
and we have simply “to apply to a straight line of length & a rectangle exceeding by a square figure and equal to a given area (b+ c?).”
The other νεῦσις referred to in Props. 8 and 9 can be solved in the more general form where ὦ, the given length to which PA is to be equal, has any value within a certain maximum and is not
ON THE PROBLEMS KNOWN AS ΝΕΎΣΕΙΣ. ΟἿ
necessarily equal to DZ, in exactly the same manner ; and the two equations corresponding to (a) will be for the second figure
x
y? -- (α -- αν)" +b? ly =e Ὡς j AERA ERE λας (B).
Here, again, the problem can be solved by the ordinary method of application of areas in the particular case where AJL is the diameter bisecting BC at right angles; and it is interesting to note that this particular case appears to be assumed in a fragment of Hippocrates’ Quadrature of lunes preserved in a quotation by Simplicius* from Eudemus’ History of Geometry, while Hippo- crates flourished probably as early as 450 B.c.
Accordingly we find that Pappus distinguishes different classes of νεύσεις corresponding to his classification of geometrical problems in general. According to him, the Greeks distinguished three kinds of problems, some being plane, others solid, and others linear. He proceeds thus7: ‘‘ Those which can be solved by means of a straight line and a circumference of a circle may properly be called plane (ἐπίπεδα) ; for the lines by means of which such problems are solved have their origin in a plane. Those however which are solved by using for their discovery (εὕρεσιν) one or more of the sections of the cone have been called solid (στερεά); for the construction requires the use of surfaces of solid figures, namely, those of cones. There remains a third kind of problem, that — which is called linear (γραμμικόν) ; for other lines [curves] besides ἡ those mentioned are assumed for the construction whose origin is more complicated and less natural, as they are generated from more irregular surfaces and intricate movements.” Among other instances of the linear class of curves Pappus mentions spirals, the curves known as quadratrices, conchoids and cissoids. He adds that “it seems to be a grave error which geometers fall into whenever any one discovers the solution of a plane problem by means of conics or linear curves, or generally solves it by means of a foreign kind, as is the case, for example, (1) with the problem in the fifth Book of the Conics of Apollonius relating to the parabola},
* Simplicius, Comment. in Aristot. Phys. pp. 61—68 (ed. Diels). The whole quotation is reproduced by Bretschneider, Die Geometrie und die Geometer vor Euklides, pp. 109—121. As regards the assumed construction see particularly p- 64 and p. xxiv of Diels’ edition; cf. Bretschneider, pp. 114,115, and Zeuthen, Die Lehre von den Kegelschnitten im Altertum, pp. 269, 270.
+ Pappus rv. pp. 270—272. 1 Cf. Apollonius of Perga, pp. exxvili. exxix.
οἷν INTRODUCTION.
and (2) when Archimedes assumes in his work on the spiral a νεῦσις Of a solid character with reference to a circle; for it is possible without calling in the aid of anything solid to find the [proof of the] theorem given by the latter [Archimedes], that is, to prove that the circumference of the circle arrived at in the first revolution is equal to the straight line drawn at right angles to the initial line to meet the tangent to the spiral.”
The “solid vetous” referred to in this passage is that assumed to be possible in Props. 8 and 9 of the book On Spirals, and is mentioned again by Pappus in another place where he shows how to solve the problem by means of conics*. This solution will be given later, but, when Pappus objects to the procedure of Archimedes as unorthodox, the objection appears strained if we consider what precisely it is that Archimedes assumes. It is not the actual solution which is assumed, but only its possibility; and its possibility can be perceived without any use of conics. For in the particular case it is only necessary, as a condition of possibility, that DZ# in the second figure above should not be the maximum length which the intercept PR could have as APR revolves about A from the position ADF in the direction of the centre of the circle; and that DE is not the maximum length which PA can have is almost self-evident. In fact, if P, instead of moving along the circle, moved along the straight line through £ parallel to LC, and if ARP moved from the position ADF in the direction of the centre, the length of P# would continually increase, and ὦ fortiori, so long as P is on the are of the circle cut off by the parallel through # to BC, PR must be greater in length than D#; and on the other hand, as A4#P moves further in the direction of 5, it must sometime intercept a length PR equal to DE before P reaches 4, when P# vanishes. Since, then, Archimedes’ method merely depends upon the theoretical possibility of a solution of the νεῦσις, and this possibility could be inferred from quite elementary considerations, he had no occasion to use conic sections for the purpose immediately in view, and he cannot fairly be said to have solved a plane problem by the use of conics.
At the same time we may safely assume that Archimedes was in possession of a solution of the νεῦσις referred to. But there is no evidence to show how he solved it, whether by means of conics, or otherwise. That he would have been able to effect the solution,
* Pappus Iv. p. 298 sq.
ON THE PROBLEMS KNOWN AS ΝΕΥΣΕΙΣ, CV
as Pappus does, by the use of conics cannot be doubted. A precedent for the introduction of conics where a “solid problem” had to be solved was at hand in the determination of two mean proportionals between two unequal straight lines by Menaechmus, the inventor οὗ the conic sections, who used for the purpose the intersections of a parabola and a rectangular hyperbola. The solution of the cubic equation on which the proposition On the Sphere and Cylinder τι. 4 depends is also effected by means of the intersections of a parabola with a rectangular hyperbola in the fragment given by Eutocius and by him assumed to be the work of Archimedes himself*.
Whenever a problem did not admit of solution by means of the straight line and circle, its solution, where possible, by means of conics was of the greatest theoretical importance. First, the possibility of such a solution enabled the problem to be classified as a “solid problem”; hence the importance attached by Pappus to solution by means of conics. But, secondly, the method had other great advantages, particularly in view of the requirement that the solution of a problem should be accompanied by a διορισμός giving the criterion for the possibility of a real solution. Often too the διορισμός involved (as frequently in Apollonius) the determination of the number of solutions as well as the limits for their possibility. Thus, in any case where the solution of a problem depended on the intersections of two conics, the theory of conics afforded an effective means of investigating διορισμοί.
§ 2. But though the solution of “solid problems” by means of conics had such advantages, it was not the only method open to Archimedes. An alternative would be the use of some mechanical construction such as was often used by the Greek geometers and is recognised by Pappus himself as a legitimate substitute for conics, which are not easy to draw in a planet. Thus in Apollonius’ solution of the problem of the two mean proportionals as given by Eutocius a ruler is supposed to be moved about a point until the points at which the ruler crosses two given straight lines at right angles are equidistant from a certain other fixed point; and the same construction is also given under Heron’s name. Another version of Apollonius’ solution is that given by Ioannes Philoponus, which assumes that, given a circle with diameter OC and two
* See note to On the Sphere and Cylinder, τι. 4. + Pappus ut. p. 54,
ronal INTRODUCTION.
straight lines OD, OF through O and at right angles to one another, a line can be drawn through C, meeting the circle again in #/ and the two lines in D, £ respectively, such that the in- tercepts CD, FE are equal. This solution was no doubt discovered by means of the intersection of the circle with a rectangular hyper- bola drawn with OD, OF as asymptotes and passing through C ; and this supposition accords with Pappus’ statement that Apollonius solved the problem by means of the sections of the cone*. The equivalent mechanical construction is given by Eutocius as that of Philo Byzantinus, who turns a ruler about C until CD, FEF are equal 7.
Now clearly a similar method could be used for the purpose of effecting a νεῦσις. We have only to suppose a ruler (or any object with a straight edge) with two marks made on it at a distance equal to the given length which the problem requires to be intercepted between two curves by a line passing through the fixed point; then, if the ruler be so moved that it always passes through the fixed point, while one of the marked points on it follows the course of one of the curves, it is only necessary to move the ruler until the second marked point falls on the other curve. Some such operation as this may have led Nicomedes to the discovery of his curve, the conchoid, which he introduced (according to Pappus) into his doubling of the cube, and by which he also trisected an angle (according to the same authority). From the fact that Nicomedes is said to have spoken disrespectfully of Eratosthenes’ mechanical solution of the duplication problem, and therefore must have lived later than Eratosthenes, it is concluded that his date must have been subsequent to 200 B.c., while on the other hand he must have written earlier than 70 B.c., since Geminus knew the name of the curve about that date; Tannery places him between Archimedes and Apolloniust. While therefore there appears to be no evidence of the use, before the time of Nicomedes, of such a mechanical method of solving a νεῦσις, the interval between Archimedes and the discovery of the conchoid can hardly have been very long. As a matter of fact, the conchoid of Nicomedes can be used to solve not only all the νεύσεις mentioned in Archimedes but any case of such a problem where one of the curves is a straight
* Pappus 111. p. ὅθ. + For fuller details see Apollonius of Perga, pp. ¢XXV—CXxvil. 1 Bulletin des Sciences Mathématiques, 2° série vit. p. 284,
ON THE PROBLEMS KNOWN AS ΝΕΎΣΕΙΣ. evil
line. Both Pappus and Eutocius attribute to Nicomedes the inven- tion of a machine for drawing his conchoid. AP is supposed to be
a ruler with a slot in it parallel to its length, /# a second ruler at right angles to the first with a fixed peg in it, C. This peg moves in a slot made in a third ruler parallel to its length, while this ruler has a fixed peg on it, D, in a straight line with the slot in which (Οὐ moves ; and the peg D can move along the slot in Ab. Τί then the ruler PD moves so that the peg D describes the length of the slot in AB on each side of /, the extremity of the ruler, P, describes the curve which is called a conchoid. Nicomedes called the straight line AB the ruler (κανών), the fixed point C the pole (πόλος), and the length PD the distance (διάστημα) ; and the fundamental property of the curve, which in polar coordinates would now be denoted by the equation r=a+bsec θ, is that, if any radius vector be drawn from C to the curve, as CP, the length intercepted on the radius vector between the curve and the straight line AB is constant. Thus any νεῦσις in which one of the two given lines is a straight line can be solved by means of the intersection of the other line with a certain conchoid whose pole is the fixed point to which the required straight line must verge (vevew). In practice Pappus tells us that the conchoid was not always actually drawn, but that “some,” for greater convenience, moved the ruler about the fixed point until by trial the intercept was made equal to the given length*.
§ 3. The following is the way in which Pappus applies conic sections to the solution of the «νεῦσις referred to in Props. ὃ, 9 of the book On Spirals. He begins with two lemmas.
* Pappus Iv. p. 246.
ΟΥ̓ΠῚ INTRODUCTION.
(1) If from a given point A any straight line be drawn meeting a straight line BC given in position in #, and if RY be drawn perpendicular to BC and bearing a given ratio to AR, the locus of Q is a hyperbola.
For draw AD perpendicular to BC, and on AD produced take A’ such that Qk: RA=A'D: DA =(the given ratio). Measure DA” along DA equal to DA’. Then, if QV be perpendicular to AW, (AR? — AD?) ; (QR? — A'D*) = (const.), or QN* : A'N . A” N = (const.)
(2) If BC be given in length, and if RQ, a straight line drawn at right angles to BC from any point 1 on it, be such that BR. RC=k. RQ, where & is a straight line of given length, then the locus of Q isa parabola.
Let O be the middle point of BC, and let OX be drawn at right angles to it and of such length that
OC* =k. KO. Draw QN’ perpendicular to OX. Then ON? = OR? ΞΟ BE ne
=k.(KO— RQ), by hypothesis, = Kea piel
ON THE PROBLEMS KNOWN AS ΝΕΥΣΕΙ͂Σ. ΟΙΧ
In the particular case referred to by Archimedes (with the slight generalisation that the given length & to which P£ is to be equal is not necessarily equal to D#) we have
(1) the given ratio RQ: AF is unity, or RY=AR, whence A” coincides with A, and, by the first lemma, ON? = AN ΑΝ, so that Q lies on a rectangular hyperbola.
(2) BR. RC=AR. RP=k. AR=k. RQ, and, by the second lemma, Q lies on a certain parabola.
If now we take O as origin, OC as axis of # and OK as axis of y, and if we put OD=a, AD=b, BC = 2c, the hyperbola and parabola determining the position of ᾧ are respectively denoted by the equations
(α -- α)" -Ξ- γ΄ — δ',
οὗ --αὐ -- ky, which correspond exactly to the equations (8) above obtained by purely algebraical methods.
Pappus says nothing of the διορισμός which is necessary to the complete solution of the generalised problem, the διορισμός namely which determines the maximwm value of & for which the solution is possible, This maximum value would of course correspond to the case in which the rectangular hyperbola and the parabola touch one another. Zeuthen has shown* that the corresponding value of & can be determined by means of the intersection of two other hyperbolas or of a hyperbola and a parabola, and there is no doubt that Apollonius, with his knowledge of conics, and in accordance with his avowed object in giving the properties useful and necessary for διορισμοί, would have been able to work out this particular διορισμός by means of conies; but there is no evidence to show that Archimedes investi- gated it by the aid of conics, or indeed at all, it being clear, as shown above, that it was not necessary for his immediate purpose.
This chapter may fitly conclude with a description of (1) some important applications of νεύσεις given by Pappus, and (2) certain particular cases of the same class of problems which are plane, that is, can be solved by the aid of the straight line and circle only, and which were (according to Pappus) shown by the Greek geometers to be of that character.
* Zeuthen, Die Lehre von den Kegelschnitten im Altertum, pp. 273—5.
cx INTRODUCTION.
8.4, One of the two important applications of ‘solid’ νεύσεις was discovered by Nicomedes, the inventor of the conchoid, who intro- duced that curve for solving a νεῦσις to which he reduced the problem of doubling the cube* or (what amounts to the same thing) the finding of two mean proportionals between two given unequal straight lines.
Let the given unequal straight lines be placed at right angles as CL, LA. Complete the parallelogram 4 BCL, and bisect AB at D, and BC at H. Join LD and produce it to meet CB produced in Z. From £ draw EF at right angles to BC, and take a point / on £F such that CF is equal to AD. Join HF, and through C draw CG& parallel to HF. If we produce BC to λ΄, the straight lines CG, CK
form an angle, and we now draw from the given point / a straight line FGK, meeting CG, CK in G, K respectively, such that the intercept G'K is equal to AD or FC. (This is the νεῦσις to which the problem is reduced, and it can be solved by means of a conchoid with F as pole.) Join KL and produce it to meet LA produced in 77. Then shall C.K, AM be the required mean proportionals between CL, LA, or Cl: CK =CK : AM=AM AE: We have, by Eucl. 11. 6, BK ΚΟ + CE*= ER’. If we add EF? to each side, BK . KC +CF?=FR’.
Now, by parallels, MA:AB=ML: LK
= BC CK; * Pappus rv. p. 242 sq. and m1. p. 58 sq. ; Eutocius on Archimedes, On the Sphere and Cylinder, τι. 1 (Vol. 111. p. 114 sq.)
ON THE PROBLEMS KNOWN AS ΝΕΥΣΕΙΣ. Cx1
and, since AB=2AD, and BC =+HC, MA AD=HO: CK
= FG : GK, by parallels, whence, componendo, MD:AD=FEK : GK. But GK = AD; therefore VD = ΚΑ, and MD? = ΚΞ".
Again, MD*?=BM .MA+ AD’, and FK°=BK . KC + CF’, from above, while MD?* — FK*, and AD*=CF’,; therefore BM .MA=BK . KC. Hence CK: MA=BM: BK =MA:AL
= OMEN OG that is, DOVCKA~CK MA MA: AL.
§ 5. The second important problem which can be reduced to a ‘solid’ νεῦσις is the trisection of any angle. One method of reducing it to ἃ νεῦσις has been mentioned above as following from Prop. 8 of the Liber Asswmptorum, This method is not mentioned by Pappus, who describes (tv. p. 272 sq.) another way of effecting the reduction, introducing it with the words, ‘The earlier geometers, when they sought to solve the aforesaid problem about the [trisection of the] angle, a problem by nature ‘solid,’ by ‘plane’ methods, were unable to discover the solution; for they were not yet accustomed to the use of the sections of the cone, and were for that reason at a loss. Later, however, they trisected an angle by means of conics, having used for the discovery of it the following νεῦσις."
The νεῦσις is thus enunciated: Given a rectangle ABCD, let it be required to draw through A a straight line AVR, meeting CD in Q and BC produced in R, such that the intercept QA is equal to a given length, & suppose.
Suppose the problem solved, QR being equal to k. Draw DP parallel to Q# and FP parallel to CD, meeting in P. Then, in the parallelogram DR, DP = QR =k.
Hence P lies on a circle with centre D and radius k.
Again, by Eucl. 1. 43 relating to the complements of the parallelograms about the diagonal of the complete parallelogram,
BC ..CD=BEK. QD ΞΡ ΡΒ:
} , by parallels,
CXil INTRODUCTION.
and, since BC. CD is given, it follows that P lies on a rectangular hyperbola with BR, BA as asymptotes and passing through D.
B Cc R
Therefore, to effect the construction, we have only to draw this rectangular hyperbola and the circle with centre D and radius equal to k. The intersection of the two curves gives the point P, and & is determined by drawing PR parallel to DC. Thus AQF is found.
[Though Pappus makes ABCD a rectangle, the construction applies equally if ABCD is any parallelogram. |
Now suppose 4 &C to be any acute angle which it is required to trisect. Let AC be perpendicular to BC. Complete the parallelo- gram ADBC, and produce DA.
Suppose the problem solved, and let the angle CBZ be one-third of the angle ASC. Let BH meet AC in αὶ and DA produced in F. Bisect HF in H, and join AH.
Then, since the angle ABZ is equal to twice the angle HBC and, by parallels, the angles HBC, HFA are equal,
LABE=22£AFH= 1 AHB,
Therefore Ais A= A and ELF=2HF = 24}.
B Cc
Hence, in order to trisect the angle ABC, we have only to solve the following νεῦσις : Given the rectangle ADBC whose diagonal
ON THE PROBLEMS KNOWN AS ΝΕΥΣΕΙΣ. cxill
is AB, to draw through B a straight line BEF, meeting AC in E and DA produced in F, such that EF may be equal to twice AB; and this νεῦσις is solved in the manner just shown.
These methods of doubling the cube and trisecting any acute angle are seen to depend upon the application of one and the same νεῦσις, Which may be stated in its most general form thus. Given any two straight lines forming an angle and any fixed point which is not on either line, it is required to draw through the fixed point a straight line such that the portion of it intercepted between the fixed lines is equal to a given length. If AEH, AC be
the fixed lines and & the fixed point, let the parallelogram ACLD be completed, and suppose that LYRA, meeting CA in Q and AZ in f, satisfies the conditions of the problem, so that QR is equal to the given length. If then the parallelogram CQRP is completed, we may regard P as an auxiliary point to be determined in order that the problem may be solved ; and we have seen that P can be found as one of the points of intersection of (1) a circle with centre C and radius equal to &, the given length, and (2) the hyperbola which passes through C’ and has DH, DB for its asymptotes.
It remains only to consider some particular cases of the problem which do not require conics for their solution, but are ‘plane’ problems requiring only the use of the straight line and circle.
§ 6. We know from Pappus that Apollonius occupied him- self, in his two Books of vetoes, with problems of that type which were capable of solution by ‘plane’ methods. As a matter of fact, the above νεῦσις reduces to a ‘plane’ problem in the particular case where £& lies on one of the bisectors of the angle between the two given straight lines, or (in other words) where the parallelogram ACBD is a rhombus or a square. Accordingly we find Pappus enunciating, as one of the ‘plane’ cases which had
ἘΠῚ A. h
ΟΧΙΥ INTRODUCTION.
been singled out for proof on account of their greater utility for many purposes, the following*: Given a rhombus with one side produced, to fit into the exterior angle a straight line given in length and verging to the opposite angle ; and he gives later on, in his lemmas to Apollonius’ work, a theorem bearing on the problem with regard to the rhombus, and (after a preliminary lemma) a solution of the νεῦσις with reference to a square.
The question therefore arises, how did the Greek geometers discover these and other particular cases, where a problem which is in general ‘solid,’ and therefore requires the use of conics (or a mechanical equivalent), becomes ‘plane’? Zeuthen is of opinion that they were probably discovered as the result of a study of the general solution by means of conicst. I do not feel convinced of this, for the following reasons.
(1) The authenticated instances appear to be very rare in which we should be justified in assuming that the Greeks used the properties of conics, in the same way as we should combine and transform two Cartesian equations of the second degree, for the purpose of proving that the intersections of two conics also lie on certain circles or straight lines. It is true that we may reasonably infer that Apollonius discovered by a method of this sort his solution of the problem of doubling the cube where, in place of the parabola and rectangular hyperbola used by Menaechmus, he employs the same hyperbola along with the cicle which passes through the points common to the hyperbola and parabola{; but in the only propositions contained in his conics which offer an opportunity for making a similar reduction§, Apollonius does not make it, and is blamed by Pappus for not doing so. In the pro- positions referred to the feet of the normals to a parabola drawn from a given point are determined as the intersections of the parabola with a certain rectangular hyperbola, and Pappus objects
* Pappus vil. p. 670.
+ ‘Mit dieser selben Aufgabe ist naimlich ein wichtiges Beispiel dafiir verknitipft, dass man bemiiht war solche Falle zu entdecken, in denen Aufgaben, zu deren Lésung im allgemeinen Kegelschnitte erforderlich sind, sich mittels Zirkel und Lineal lésen lassen. Da nun das Studium der allgemeinen Lésung durch Kegelschnitte das beste Mittel gewihrt solche Fille zu entdecken, so ist es ziemlich wahrscheinlich, dass man wirklich diesen Weg eingeschlagen hat.” Zeuthen, op. cit. p. 280.
+ Apollonius of Perga, p. xxv, CXxvi.
§ Ibid. p. cxxvili and pp. 182, 186 (Conics, v. 58, 62
ON THE PROBLEMS KNOWN AS ΝΕΎΥΣΕΙΣ. CXV
to this method as an instance of discovering the solution of a ‘plane’ problem by means of conics*, the objection having reference to the use of a hyperbola where the same points could be obtained as the intersections of the parabola with a certain circle. Now the proof of this latter fact would present no difficulty to Apollonius, and Pappus must have been aware that it would not; if therefore he objects in the circumstances to the use of the hyperbola, it is at least arguable that he would equally have objected had Apollonius brought in the hyperbola and used its properties for the purpose of proving the problem to be ‘plane’ in the particular case.
(2) The solution of the general problem by means of conics brings in the auxiliary point P and the straight line CP. We should therefore naturally expect to find some trace of these in the particular solutions of the νεῦσις for a rhombus and square; but they do not appear in the corresponding demonstrations and figures given by Pappus.
Zeuthen considers that the νεῦσις with reference to a square was probably shown to be ‘plane’ by means of the same investigation which showed that the more general case of the rhombus was also capable of solution with the help of the straight line and circle only, i.e. by a systematic study of the general solution by means of conics. This supposition seems to him more probable than the view that the discovery of the plane construction for the square may have been accidental ; for (he says) if the same problem is treated solely by the aid of elementary geometrical expedients, the discovery that it is ‘plane’ is by no means a simple mattery. Here, again, I am not convinced by Zeuthen’s argument, as it seems to me that a simpler explanation is possible of the way in which the Greeks were led to the discovery that the particular νεύσεις were plane. They knew in the first place that the trisection of a right angle was a ‘plane’ problem, and therefore that half a right angle could be trisected by means of the straight line and circle. It followed
* Pappus tv. p. 270. Cf. p. cili above.
+ ‘Die Ausfiihrbarkeit kann dann auf die zuerst angedeutete Weise gefunden sein, die den allgemeinen Fall, wo der Winkel zwischen den gegebenen Geraden beliebig ist, in sich begreift. Dies scheint mir viel wahrscheinlicher als die Annahme, dass die Entdeckung dieser ebenen Konstruction zufallig sein sollte ; denn wenn man dieselbe Aufgabe nur mittels rein elementar-geometrischer Hiilfsmittel behandelt, so liegt die Entdeckung, dass sie eben ist, ziemlich fern.” Zeuthen, op. cit. p. 282.
h2
CXV1 INTRODUCTION.
therefore that the corresponding νεῦσις, 1.6. that for a square, was a ‘plane’ problem in the particular case where the given length to which the required intercept was to be equal was double of the diagonal of the square. This fact would naturally suggest the question whether the problem was still plane if ᾧ had any other value; and, when once this question was thoroughly investigated, the proof that the problem was ‘plane,’ and the solution of it, could hardly have evaded for long the pursuit of geometers so ingenious as the Greeks. This will, I think, be clear when the solution given by Pappus and reproduced below is examined, Again, after it had been proved that the νεῦσις with reference to a square was ‘plane,’ what more natural than the further inquiry as to whether the intermediate case between that of the square and parallelogram, that of the rhombus, might perhaps be a ‘plane’ problem ?
As regards the actual solution of the plane νεύσεις with respect to the rhombus and square, i.e. the cases in general where the fixed point & lies on one of the bisectors of the angles between the two given straight lines, Zeuthen says that only in one of the cases have we a positive statement that the Greeks solved the νεῦσις by means of the circle and ruler, the case, namely, where AC LD is a square*. This appears to be a misapprehension, for not only does Pappus mention the case of the rhombus as one of the plane νεύσεις which the Greeks had solved, but it is clear, from a proposition given by him later, how it was actually solved. The proposition is stated by Pappus to be “involved” (παραθεωρούμενον, meaning presumably “the subject of concurrent investigation”) in the 8th problem of Apollonius’ first Book of vetoes, and is enunciated in the following formy+. Given a rhombus AD with diameter BC produced to LE, if EF be a mean proportional between BE, EC, and if a circle be described with centre E and radius EF cutting CD in K and AC produced in H, BKH shall be a straight line. The proof is as follows.
Let the circle cut AC in Z, and join H#, KE, LE. Let LK meet BC in 777.
* “Tndessen besitzen wir doch nur in einem einzelnen hierher gehérigen Falle eine positive Angabe dariiber, dass die Griechen die Einschiebung mittels Zirkel und Lineal ausgefiihrt haben, wenn nimlich die gegebenen Geraden zugleich rechte Winkel bilden, AIBC also ein Quadrat wird.” Zeuthen, op. cit. p. 281.
+ Pappus vit. p. 778.
ON THE PROBLEMS KNOWN AS NEYSEIS. eCXVll
Since, from the property of the rhombus, the angles LOM, KCM are equal, and therefore CL, Οἱ Καὶ make equal angles with the diameter FG of the circle, it follows that CL = CK.
B D
Also Hk = EL, and CE is common to the triangles HCK, LCL. Therefore the said triangles are equal in all respects, and £CKE=LCLE=cCHE. Now, by hypothesis, EB: EF=EF: EC, or EB: EK=EK: EC (since HF = EK),
and the angle CZK is common to the triangles BEX, KEC; there- fore the triangles BEX, KEC are similar, and
TER OTE me OUD = CHE, from above.
Again, RHCE=LACE— 2 BCK.
Thus in the triangles CBX, CHE two angles are equal re- spectively ; therefore i CEH=2CKB.
But, since . CKE=2 CHE, from above, the points K, C, #, 1 are concyclic. Hence 4 CEH + ἐ CKH = (two right angles). Accordingly, since LCEH = CKB, . CKH+2CKB = (two right angles), and LXH is a straight line.
CXVill INTRODUCTION.
Now the form of the proposition at once suggests that, in the 8th problem referred to, Apollonius had simply given a construction involving the drawing of a circle cutting CD and AC produced in the points Α΄, H respectively, and Pappus’ proof that BAH is a straight line is intended to prove that HK verges towards B, or (in other words) to verify that the construction given by Apollonius solves a certain νεῦσις requiring BKH to be drawn so that KH ἐξ equal to a given length.
The analysis leading to the construction must have been worked out somewhat as follows.
Suppose ΒΑ drawn so that AH is equal to the given length ὦ. Bisect KH at NV, and draw NE at right angles to KH meeting BC produced in ἢ.
Draw KW perpendicular to BC and produce it to meet CA in ὦ. Then, from the property of the rhombus, the triangles KCM, LCM are equal in all respects.
Therefore KM= ML; and accordingly, if IN be joined, JN, LH are parallel.
Now, since the angles at 1, WV are right, a circle can be described about ΚΑ.
Therefore .CEK=2zMNK&, in the same segment, = CHK, by parallels. Hence a circle can be described about CHHK. It follows that LBCD=.CEK+LCKE =.CHK+LCHE = 2 oe Therefore the triangles HAH, DBC are similar. Lastly, LCKN=LCBK+2BCK;
and, subtracting from these equals the equal angles HAN, BCK
respectively, we have LEKC =c EBK.
Hence the triangles HBX, FKC are similar, and BE: HR=EK : EC, or BE. EC = Ek’. But, by similar triangles, 7K: KH = DC: CB, and the ratio DC : CB is given, while KH is also given (= 4).
ON THE PROBLEMS KNOWN AS NETZEI®, CX1X
Therefore HX is given, and, in order to find #, we have only, in the Greek phrase, to ‘‘apply to BC a rectangle exceeding by a square figure and equal to the given area HK.”
Thus the construction given by Apollonius was clearly the
following*. If k be the given length, take a straight line p such that
pik AB BC.
Apply to BC a rectangle exceeding by a square figure and equal to the area p’. Let BE. EC be this rectangle, and with E as centre and radius equal to p describe a circle cutting AC produced in H and
CD in K. HK is then equal to ἄ, and verges towards Bb, as proved by Pappus; the problem is therefore solved.
The construction used by Apollonius for the ‘plane’ νεῦσις with reference to the rhombus having been thus restored by means of the theorem given by Pappus, we are enabled to understand the purpose
* This construction was suggested to me by a careful examination of Pappus’ proposition without other aid; but it is no new discovery. Samuel Horsley gives the same construction in his restoration of Apollonii Pergaei Inclinationum libri duo (Oxford, 1770); he explains, however, that he went astray in consequence of a mistake in the figure given in the mss., and was unable to deduce the construction from Pappus’s proposition until he was recalled to the right track by a solution of the same problem by Hugo @Omerique. This solution appears in a work entitled, Analysis geometrica, sive nova et vera methodus resolvendi tam problemata geometrica quam arithmeticas quaestiones, published at Cadiz in 1698. D’Omerique’s construction, which is practically identical with that of Apollonius, appears to have been evolved by means of an independent analysis of his own, since he makes no reference to Pappus, as he does in other cases where Pappus is drawn upon (e.g. when giving the construction for the case of the square attributed by Pappus to one Heraclitus). The construction differs from that given above only in the fact that the circle is merely used to determine the point K, after which BK is joined and produced to meet AC in H. Of other solutions of the same problem two may here be mentioned. (1) The solution contained in Marino Ghetaldi’s posthumous work De Resolutione et Compositione Mathematica Libri quinque (Rome, 1630), and included among the solutions of other problems all purporting to be solved ‘‘ methodo qua antiqui utebantur,” is, though geometrical, entirely different from that above given, being effected by means of a reduction of the problem to a simpler plane νεῦσις of the same character as that assumed by Hippocrates in his Quadrature of lunes. (2) Christian Huygens (De circuli magnitudine inventa; accedunt problematum quorundam illustrium constructiones, Lugduni Batavorum, 1654) gave a rather complicated solution, which may be described as a generalisation of Heraclitus’ solution in the case of a square.
CXX INTRODUCTION.
for which Pappus, while still on the subject of the “8th problem ” of Apollonius, adds a solution for the particular case of the square (which he calls a ‘‘ problem after Heraclitus”) with an introductory lemma. It seems clear that Apollonius did not treat the case of the square separately from the rhombus because the solution for the rhombus was equally applicable to the square, and this supposition is confirmed by the fact that, in setting out the main problems discussed in the νεύσεις, Pappus only mentions the rhombus and not the square. Being however acquainted with a solution by one Heraclitus of the νεῦσις relating to a square which was not on the same lines as that of Apollonius, while it was not applicable to the case of the rhombus, Pappus adds it as an alternative method for the square which is worth noting*. This is no doubt the explanation of the heading to the lemma prefixed to Heraclitus’ problem which Hultsch found so much difficulty in explaining and put in brackets as an interpolation by a writer who misunderstood the figure and the object of the theorem. The words mean ‘“‘ Lemma useful for the [problem] with reference to squares taking the place of the rhombus” (literally “having the same property as the rhombus”), 1.6. a lemma useful for Heraclitus’ solution of the
* This view of the matter receives strong support from the following facts. In Pappus’ summary (p. 670) of the contents of the νεύσεις of Apollonius ‘two cases” of the νεῦσις with reference to the rhombus are mentioned last among the particular problems given in the first of the two Books. As we have seen, one case (that given above) was the subject of the ‘‘8th problem” of Apollonius, and it is equally clear that the other case was dealt with in the “9th problem.” The other case is clearly that in which the line to be drawn through B, instead of crossing the ἢ A [Ὁ exterior angle of the rhombus at C, lies across the angle C itself, i.e. meets CA, CD both produced. In the former case the solution of the problem is always possible what- ever be the length of k; but in the second case clearly the problem is not capable of solution if k, the given length, is less than a certain minimum. Hence the problem requires a διορισμός to determine the minimum K length of k. Accordingly we find Pappus giving, after the interposition of the case of the square, a ‘‘lemma useful for the διορισμός of the 9th problem,” which proves that, if CH=CK and B be the middle point of HK, then HK is the least straight line which can be drawn through B to meet CH, CK. Pappus adds that the διορισμός for the rhombus is then evident; if HK be the line drawn through B perpendicular to CB and meeting CA, CD produced in H, K, then, in order that the problem may admit of solution, the given length k must be not less than HK.
ON THE PROBLEMS KNOWN AS ΝΕΥΣΕΙΣ. CXX1
νεῦσις in the particular case of a square*. The lemma is as follows.
ABCD being a square, suppose BHE drawn so as to meet CD in H and AD produced in E, and let EF be drawn perpendicular to BE meeting BC produced in F. To prove that
CF’ = BC? + ΜΕ".
Suppose HG drawn parallel to DC meeting CF in G. Then
since BEF is a right angle, the angles HBC, FEE are equal.
hee ae ae B
ἜΣ ΠΩΣ
c G F
Therefore the triangles BCH, HGF are equal in all respects, and
EF=BH. Now BF? = BE? + ΕΓ", or BC. BF+BF.FC=BH. BE +BE.EH + EF".
But, the angles HCF, HEF being right, the points C, H, Z, F are concyclic, and therefore BC, BY = BH, BH. Subtracting these equals, we have BF. FC=BE.EH+ EF?’ = BE. EH + BH’ = BH. HE + EH’ + BH’ =HB.BH + HH? = FB. BC+ EF’.
* Hultsch translates the words λῆμμα χρήσιμον eis τὸ ἐπὶ τετραγώνων ποιούντων τὰ αὐτὰ τῷ ῥόμβῳ (p. 780) thus, “‘ Lemma utile ad problema de quadratis quorum summa rhombo aequalis est,” and has a note in his Appendix (p. 1260) explaining what he supposes to be meant. The ‘squares’ he takes to be the given square and the square on the given length of the intercept, and the rhombus to be one for which he indicates a construction but which is not shown in Pappus’ figure. Thus he is obliged to translate τῷ ῥόμβῳ as ‘a rhombus,” which is one objec- tion to his interpretation, while “whose squares are equal” scarcely seems a possible rendering of ποιούντων τὰ αὐτά.
ΟΧΧΙΙ INTRODUCTION.
Take away the common part BC. CF, and Cl? = BC + LR", Heraclitus’ analysis and construction are now as follows. Suppose that we have drawn BHE so that HE has a given length &.
Since CF* = BC’? + EH’, or BC? + k’, and BC and k are both given, CF is given, and therefore BF is given.
Thus the semicircle on BF as diameter is given, and therefore also #, its intersection with the given line ADE; hence BE is given.
To effect the construction, we first find a square equal to the sum of the given square and the square on &. We then produce BC to F so that CF is equal to the side of the square so found. If a semicircle be now described on BF as diameter, it will pass above D (since CF'> CD, and therefore BC. CF > CD”), and will therefore meet AD produced in some point £.
Join BE meeting CD in H.
Then HE =k, and the problem is solved.
CHAPTER VI.
CUBIC EQUATIONS.
Ir has often been explained how the Greek geometers were able to solve geometrically all forms of the quadratic equation which give positive roots; while they could take no account of others because the conception of a negative quantity was unknown to them. The quadratic equation was regarded as a simple equation connecting areas, and its geometrical expression was facilitated by the methods which they possessed of transforming any rectilineal areas whatever into parallelograms, rectangles, and ultimately squares, of equal area ; its solution then depended on the principle of application of areas, the discovery of which is attributed to the Pythagoreans. Thus any plane problem which could be reduced to the geometrical equivalent of a quadratic equation with a positive root was at once solved. <A particular form of the equation was the pure quadratic, which meant for the Greeks the problem of finding a square equal to a given rectilineal area. This area could be transformed into a rectangle, and the general form of the equation thus became a? = ab, so that it was only necessary to find a mean proportional between ὦ and ὁ. In the particular case where the area was given as the sum of two or more squares, or as the difference of two squares, an alternative method depended on the Pythagorean theorem of Eucl. 1. 47 (applied, if necessary, any number of times successively). The connexion between the two methods is seen by comparing Eucl. vi. 13, where the mean proportional between a and ὦ is found, and Eucl. τι. 14, where the same problem is solved without the use of proportions by means of 1. 47, and where in fact the
formula used is ae GONG a—b\2 ἘΠῚ ΞΞ - (4) -( 5) ) A
CXX1V INTRODUCTION.
The choice between the two methods was equally patent when the equation to be solved was «*=pa*, where p is any integer; hence the ‘multiplication’ of squares was seen to be dependent on the finding of a mean proportional. The equation x= 2a’ was the simplest equation of the kind, and the discovery of a geometrical construction for the side of a square equal to twice a given square was specially important, as it was the beginning of the theory of incommensurables or ‘irrationals’ (ἀλόγων πραγματεία) which was invented by Pythagoras. There is every reason to believe that this successful doubling of the square was what suggested the question whether a construction could not be found for the doubling of the cube, and the stories of the tomb erected by Minos for his son and of the oracle bidding the Delians to double a cubical altar were no doubt intended to invest the purely mathematical problem with an element of romance. It may then have been the connexion between the doubling of the square and the finding of one mean proportional which suggested the reduction of the doubling of the cube to the problem of finding two mean proportionals between two unequal straight lines. This reduction, attributed to Hippocrates of Chios, showed at the same time the possibility of multiplying the cube by any ratio. Thus, if «, y are two mean proportionals between a, ὦ, we have τ ΞΘ τ ya sD.
and we derive at once
G20 es
whence a cube (2*) is obtained which bears to οὐ the ratio ὦ : a, P
while any fraction ~ can be transformed into a ratio between lines qY
of which one (the consequent) is equal to the side a of the given cube. Thus the finding of two mean proportionals gives the solution of any pure cubic equation, or the equivalent of extracting the cube root, just as the single mean proportional is equivalent to extracting the square root. For suppose the given equation to be 2’ = bed. We have then only to tind a mean proportional ὦ between ὁ and d,
: ᾿ ὐ ΝΕ and the equation becomes #*=a?.b=a*.— which is exactly the a
multiplication of a cube by a ratio between lines which the two mean proportionals enable us to effect.
As a matter of fact, we do not find that the great geometers were in the habit of reducing problems to the multiplication of the
CUBIC EQUATIONS. CXXV
cube eo nomine, but to the equivalent problem of the two mean proportionals ; and the cubic equation x* -- αὐὖ is not usually stated in that form but as a proportion. Thus in the two propositions On the Sphere and Cylinder τι. 1, 5, where Archimedes uses the two mean proportionals, it is required to find « where (op Hp ane
he does not speak of finding the side of a cube equal to a certain parallelepiped, as the analogy of finding a square equal to a given rectangle might have suggested. So far therefore we do not find any evidence of a general system of adding and subtracting solids by transforming parallelepipeds into cubes and cubes into parallel- epipeds which we should have expected to see in operation if the Greeks had systematically investigated the solution of the general form of the cubic equation by a method analogous to that of the application of areas employed in dealing with quadratic equations.
The question then arises, did the Greek geometers deal thus generally with the cubic equation
αὐ τ ααὐ-π Be+T=0,
which, on the supposition that it was regarded as an independent problem in solid geometry, would be for them a simple equation between solid figures, ὦ and a both representing linear magnitudes, B an area (a rectangle), and I a volume (a parallelepiped)? And was the reduction of a problem of an order higher than that which could be solved by means of a quadratic equation to the solution of a cubic equation in the form shown above a regular and recognised method of dealing with such a problem? The only direct evidence pointing to such a supposition is found in Archimedes, who reduces the problem of dividing a sphere by a plane into two segments whose volumes are in a given ratio (On the Sphere and Cylinder τι. 4) to the solution of a cubic equation which he states in a form equivalent to
where ὦ is the radius of the sphere, m:n the given ratio (being a ratio between straight lines of which m>n), and x the height of the greater of the required segments. Archimedes explains that this is a particular case of a more general problem, to divide a straight line (a) into two parts (ὦ, a—«) such that one part (a—«) is to an- other given straight line (c) as a given area (which for convenience’
ΟΧΧΝῚ INTRODUCTION,
sake we suppose transformed into a square, ὦ) is to the square on the other part («’), i.e. so that
Gudea πε σεις" Βρω |)
He further explains that the equation (2) stated thus generally requires a διορισμός, i.e. that the limits for the possibility of a real solution, etc., require to be investigated, but that the particular case (with the conditions obtaining in the particular proposition) requires no διορισμός, i.e. the equation (1) will always give a real solution. He adds that ‘“‘the analysis and synthesis of both these problems will be given at the end.” That is, he promises to give separately a complete investigation of the equation (2), which is equivalent to the cubic equation
and to apply it to the particular case (1).
Wherever the solution was given, it was temporarily lost, having apparently disappeared even before the time of Dionysodorus and Diocles (the latter of whom lived, according to Cantor, not later than about 100 B.c.); but Eutocius describes how he found an old fragment which appeared to contain the original solution of Archimedes, and gives it in full. It will be seen on reference to Eutocius’ note (which I have reproduced immediately after the proposition to which it relates, On the Sphere and Cylinder τι. 4) that the solution (the genuineness of which there seems to be no reason to doubt) was effected by means of the intersection of a parabola and a rectangular hyperbola whose equations may re- spectively be written thus,
ΤΉ τ w= τ (ὦ -- α) y=ac,
The διορισμός takes the form of investigating the maximum possible value of x? (a—2), and it is proved that this maximum
2 value for a real solution is that corresponding to the value x= 5 a.
3 This is established by showing that, if b’c == a’, the curves touch 2 at the point for which «= 3 ὦ. If on the other hand δ᾽ < = a’, it
is proved that there are two real solutions. In the particular case (1) it is clear that the condition for a real solution is satisfied, for
CUBIC EQUATIONS. CXxVil
ne
the expression in (1) corresponding to 6’c in (2) is i 4a’, and it is only necessary that
m
4a? > = (3a)*, or 4%,
m+n
which is obviously true.
Hence it is clear that not only did Archimedes solve the cubic equation (3) by means of the intersections of two conics, but he also discussed completely the conditions under which there are 0, 1 or 2 roots lying between 0 and a. It is to be noted further that the διορισμός is similar in character to that by which Apollonius investigates the number of possible normals that can be drawn to a conic from a given point*. Lastly, Archimedes’ method is seen to be an extension of that used by Menaechmus for the solution of the pure cubic equation. This can be put in the form
ὩΣ ΞΘ Ds which can again be put in Archimedes’ form thus, Ce τ ἢ and the conics used by Menaechmus are respectively x = ay, cy = ab, which were of course suggested by the two mean proportionals satisfying the equations Ge ΞΡ τυ ΞΡ: 0
The case above described is not the only one where we may assume Archimedes to have solved a problem by first reducing it to a cubic equation and then solving that. At the end of the preface to the book On Conoids and Spheroids he says that the results therein obtained may be used for discovering many theorems and problems, and, as instances of the latter, he mentions the following, “from a given spheroidal figure or conoid to cut off, by a plane drawn parallel to a given plane, a segment which shall be equal to a given cone or cylinder, or to a given sphere.” Though Archimedes does not give the solutions, the following considerations may satisfy us as to his method.
(1) The case of the ‘right-angled conoid’ (the paraboloid of revolution) is a ‘plane’ problem and therefore does not concern us here.
* Cf. Apollonius of Perga, p. 168 sqq.
CXXVill INTRODUCTION.
(2) In the case of the spheroid, the volume of the whole spheroid could be easily ascertained, and, by means of that, the ratio between the required segment and the remaining segment ; after which the problem could be solved in exactly the same way as the similar one in the case of the sphere above described, since the results in On Conoids and Spheroids, Props. 29—32, correspond to those of On the Sphere and Cylinder τι. 2. Or Archimedes may have proceeded in this case by a more direct method, which we may represent thus. Let a plane be drawa through the axis of the spheroid perpendicular to the given plane (and therefore to the base of the required segment). This plane will cut the elliptical base of the segment in one of its axes, which we will call 2y. Let α be the length of the axis of the segment (or the length intercepted within the segment of the diameter of the spheroid passing through the centre of the base of the segment). Then the area of the base of the segment will vary as y” (since all sections of the spheroid parallel to the given plane must be similar), and therefore the volume of the cone which has the same vertex and base as the required segment will vary as yx. And the ratio of the volume of the segment to that of the cone is (On Conoids and Spheroids, Props. 29—32) the ratio (3a — a): (2a—a), where 2a is the length of the diameter of the spheroid which passes through the vertex of the segment. There- fore
96 — ὦ
Yn . = Maia Pee :
where C is a known volume. Further, since x, y are the coordinates of a point on the elliptical section of the spheroid made by the plane through the axis perpendicular to the cutting plane, referred to a diameter of that ellipse and the tangent at the extremity of the diameter, the ratio γῆ: ὦ (3α -- α) is given. Hence the equation can be put in the form a? (3a — 2) = b*e,
and this again is the same equation as that solved in the fragment given by Eutocius. A διορισμός is formally necessary in this case, though it only requires the constants to be such that the volume to which the segment is to be equal must be less than that of the whole spheroid.
(3) For the ‘obtuse-angled conoid’ (hyperboloid of revolution) it would be necessary to use the direct method just described for
CUBIC EQUATIONS. ΟΧΧΙΧ
the spheroid, and, if the notation be the same, the corresponding equations will be found, with the help of On Conoids and Spheroids, Props. 25, 26, to be
eas
7 γα. = “ ϑατα
?
and, since the ratio 7” : a (2a + 2) is constant, αὐ (3a + x) = b’e.
Tf this equation is written in the form of a proportion like the similar one above, it becomes ὅν Ὁ = (Sa'+ia).2 δ.
There can be no doubt that Archimedes solved this equation as well as the similar one with a negative sign, i.e. he solved the two equations
αὐ + au’ ¥ b’c=0, obtaining all their positive real roots. In other words, he solved completely, so far as the real roots are concerned, a cubic equation in which the term in α is absent, although the determination of the positive and negative roots of one and the same equation meant for him two separate problems. And it is clear that all cubic equations can be easily reduced to the type which Archimedes solved.
We possess one other solution of the cubic equation to which the division of a sphere into segments bearing a given ratio to one another is reduced by Archimedes. This solution is by Dionysodorus, and is given in the same note of Eutocius*. Dionysodorus does not generalise the equation, however, as is done in the fragment quoted above ; he merely addresses himself to the particular case,
4a’: x? = (Ξα-- αν : Ξ ΤῊ 26
m+n’
thereby avoiding the necessity for a διορισμός. The curves which he uses are the parabola
m ire m+n el and the rectangular hyperbola m —— 2a? = xy. m+n r Ἵ
When we turn to Apollonius, we find him emphasising in his
* On the Sphere and Cylinder τι. 4 (note at end).
CXXX INTRODUCTION,
preface to Book tv. of the Conics* the usefulness of investigations of the possible number of points in which conics may intersect one another or circles, because “they at all events afford a more ready means of observing some things, e.g. that several solutions are possible, or that they are so many in number, and again that no solution is possible”; and he shows his mastery of this method of investigation in Book v., where he determines the number of normals that can be drawn to a conic through any given point, the condition that two normals through it coincide, or (in other words) that the point lies on the evolute of the conic, and so on. For these purposes he uses the points of intersection of a certain rectangular hyperbola with the conic in question, and among the cases we find (v. 51, 58, 62) some which can be reduced to cubic equations, those namely in which the conic is a parabola and the axis of the parabola is parallel to one of the asymptotes of the hyperbola. Apollonius however does not bring in the cubic equation ; he addresses himself to the direct geometrical solution of the problem in hand without reducing it to another. This is after all only natural, because the solution necessitated the drawing of the rectangular hyperbola in the actual figure containing the conic in question ; thus, e.g. in the case of the problem leading to a cubic equation, Apollonius can, so to speak, compress two steps into one, and the introduction of the cubic as such would be mere surplusage. The case was different with Archimedes, when he had no conic in his original figure ; and the fact that he set himself to solve a cubic somewhat more general than that actually involved in the problem made separate treatment with a number of new figures necessary. Moreover Apollonius was at the same time dealing, in other propositions, with cases which did not reduce to cubics, but would, if put in an algebraical form, lead to biquadratic equations, and these, expressed as such, would have had no meaning for the Greeks ; there was therefore the less reason in the simpler case to introduce a subsidiary problem.
As already indicated, the cubic equation, as a subject of syste- matic and independent study, appears to have been lost sight of within a century or so after the death of Archimedes. Thus Diocles, the discoverer of the cissoid, speaks of the problem of the division of the sphere into segments in a given ratio as having been reduced by Archimedes ‘to another problem, which he does not solve in his work on the sphere and cylinder”; and he then proceeds to
* Apollonius of Perga, p. 1xxiii.
CUBIC EQUATIONS. CXXX1
solve the original problem directly, without in any way bringing in the cubic. This circumstance does not argue any want of geometrical ability in Diocles; on the contrary, his solution of the original problem is a remarkable instance of dexterity in the use of conics for the solution of a somewhat complicated problem, and it proceeds on independent lines in that it depends on the intersection of an ellipse and a rectangular hyperbola, whereas the solutions of the cubic equation have accustomed us to the use of the parabola and the rectangular hyperbola. I have reproduced Diocles’ solution in its proper place as part of the note of Eutocius on Archimedes’ proposition ; but it will, I think, be convenient to give here its equivalent in the ordinary notation of analytical geometry, in accordance with the plan of this chapter. Archimedes had proved [On the Sphere and Cylinder τι. 2] that, if & be the height of a segment cut off by a plane from a sphere of radius a, and if h be the height of the cone standing on the same base as that of the segment and equal in volume to the segment, then
(ϑα -- ἀ) : (Qa—-—k)=h : 1.
Also, if h’ be the height of the cone similarly related to the remaining segment of the sphere,
(a+k):k=h': (2a—h). From these equations we derive
(h—k):k=a: (2a—h), and (h'—2a+k): (2a --ἀγ τεῳ: ἢ.
Slightly generalising these equations by substituting for ὦ in the third term of each proportion another length ὦ, and adding the condition that the segments (and therefore the cones) are to bear to each other the ratio m:n, Diocles sets himself to solve the three equations
(h—k):k=6: (2a—hk) (h’—Qa+k):(2Q4—h)=b:b& | ceeveevsesee νον (A).
and A:h’=m:n
Suppose m>n, so that k>a. The problem then is to divide a straight line of length 2a into two parts k and (2a—h) of which & is the greater, and which are such that the three given equations are all simultaneously satisfied.
Imagine two coordinate axes such that the origin is the middle point of the given straight line, the axis of y is at right angles to it,
42
“ἰ
exxxil INTRODUCTION.
and a is positive when measured along that half of the given straight line which is to contain the required point of division. Then the conics drawn by Diocles are
(1) the ellipse represented by the equation (y+a-2)*=— {(a+ ba,
and (2) the rectangular hyperbola
(a + a) (y + 6) = 2ab. One intersection between these conics gives a value of x between 0 and a, and leads to the solution required. Treating the equations algebraically, and eliminating y by means of the second equation which gives
we obtain from the first equation
(a—x)? (: ἡγοῦ ᾿Ξ 5. sq +b) ὧι,
a+ x m
that is, (a+ a)? (α -- ὃ -- α) ΞΞ ~ (ὦ -- 2)" (ὦ Ὁ 542) .τ. {τς (B).
In other words Diocles’ method is the equivalent of solving a complete cubic equation containing all the three powers of α and a constant, though no mention is made of such an equation.
To verify the correctness of the result we have only to remember that, « being the distance of the point of division from the middle point of the given straight line,
k=a+a, 2a-k=a-x. Thus, from the first two of the given equations (A) we obtain respectively
᾿ a+x
h=a+x+——.6, α--α
h'=a-2+—.}, a+a
whence, by means of the third equation, we derive (a+2)'(a+b—-a)=™ (a-2) (a +b+«),
which is the same equation as that found by elimination above (B).
CUBIC EQUATIONS. CXXXill
I have purposely postponed, until the evidence respecting the Greek treatment of the cubic equation was complete, any allusion to an interesting hypothesis of Zeuthen’s* which, if it could be accepted as proved, would explain some difficulties involved in Pappus’ account of the orthodox classification of problems and loci. I have already quoted the passage in which Pappus distinguishes the problems which are plane (ἐπίπεδα), those which are solid (στερεά) and those which are linear (ypappuxa)}. Parallel to this division of problems into three orders or classes is the distinction between three classes of /oci{. The first class consists of plane loci (τόποι ἐπίπεδοι) which are exclusively straight lines and circles, the second of solid loct (τόποι στερεοί) which are conic sections§, and the third of linear loci (τόποι γραμμικοί). It is at the same time clearly implied by Pappus that problems were originally called plane, solid or linear respectively for the specific reason that they required for their solution the geometrical loci which bore the corresponding names. But there are some logical defects in the classification both as regards the problems and the loci.
(1) Pappus speaks of its being a serious error on the part of geometers to solve a plane problem by means of conics (i.e. ‘solid loci’) or ‘linear’ curves, and generally to solve a problem “ by means of a foreign kind” (ἐξ ἀνοικείου γένους). If this principle were applied strictly, the objection would surely apply equally to the solution of a ‘solid’ problem by means of a ‘linear’ curve. Yet, though e.g. Pappus mentions the conchoid and the cissoid as being ‘linear’ curves, he does not object to their employment in the solution of the problem of the two mean proportionals, which is a ‘solid’ problem.
(2) The application of the term ‘solid loci’ to the three conic sections must have reference simply to the definition of the curves as sections of a solid figure, viz. the cone, and it was no doubt in contrast to the ‘solid locus’ that the ‘plane locus’ was so called. This agrees with the statement of Pappus that ‘ plane’ problems may
* Die Lehre von den Kegelschnitten, p. 226 sqq.
+ Be Ciil.
+ Pappus vit. pp. 652, 662.
8 It is true that Proclus (p. 394, ed. Friedlein) gives a wider definition of ‘‘ solid lines” as those which arise ‘‘ from some section of a solid figure, as the cylindrical helix and the conic curves’’; but the reference to the cylindrical helix would seem to be due to some confusion.
CXXX1V INTRODUCTION.
properly be so called because the lines by means of which they are solved “have their origin in a plane.” But, though this may be regarded as a satisfactory distinction when ‘plane’ and ‘solid’ loci are merely considered in relation to one another, it becomes at once logically defective when the third or ‘linear’ class is also brought in. For, on the one hand, Pappus shows how the ‘ quadratrix’ (a ‘linear’ curve) can be produced by a construction in three dimensions (“by means of surface-loci,” διὰ τῶν πρὸς ἐπιφανείαις τόπων) ; and, on the other hand, other ‘linear’ loci, the conchoid and cissoid, have their origin in a plane. If then Pappus’ account of the origin of the terms ‘plane’ and ‘solid’ as applied to problems and loci is literally correct, it would seem necessary to assume that the third name of ‘linear’ problems and loci was not invented until a period when the terms ‘plane’ and ‘solid loci’ had been so long recognised and used that their origin was forgotten.
To get rid of these difficulties, Zeuthen suggests that the terms ‘plane’ and ‘solid’ were first applied to problems, and that they came afterwards to be applied to the geometrical loci which were used for the purpose of solving them. On this interpretation, when problems which could be solved by means of the straight line and circle were called ‘plane,’ the term is supposed to have had reference, not to any particular property of the straight line or circle, but to the fact that the problems were such as depend on an equation of a degree not higher than the second. The solution of a quadratic equation took the geometrical form of application of areas, and the term ‘plane’ became a natural one to apply to the class of problems so soon as the Greeks found themselves confronted with a new class of problems to which, in contrast, the term ‘solid’ could be applied. This would happen when the operations by which problems were reduced to applications of areas were tried upon problems which depend on the solution of a cubic equation. Zeuthen, then, supposes that the Greeks sought to give this equation a similar shape to that which the reduced ‘plane’ problem took, that is, to form a simple equation between solids corresponding to the cubic equation
αὐ +ae+ Bu+T=0; the term ‘solid’ or ‘plane’ being then applied according as it had been reduced, in the manner indicated, to the geometrical equivalent of a cubic or a quadratic equation.
Zeuthen further explains the term ‘linear problem’ as having
CUBIC EQUATIONS. CXXXV
been invented afterwards to describe the cases which, being equivalent to algebraical equations of an order higher than the third, would not admit of reduction to a simple relation between lengths, areas and volumes, and either could not be reduced to an equation at all or could only be represented as such by the use of compound ratios. The term ‘linear’ may perhaps have been applied because, in such cases, recourse was had to new classes of curves, directly and without any intermediate step in the shape of an equation. Or, possibly, the term may not have been used at all until a time when the original source of the names ‘plane’ and ‘solid’ problems had been forgotten.
On these assumptions, it would still be necessary to explain how Pappus came to give a more extended meaning to the term ‘solid problem,’ which according to him equally includes those problems which, though solved by the same method of conics as was used to solve the equivalent of cubics, do not reduce to cubic equations but to biquadratics. This is explained by the supposition that, the cubic equation having by the time of Apollonius been obscured from view owing to the attention given to the method of solution by means of conics and the discovery that the latter method was one admitting of wider application, the possibility of solution by means of conics came itself to be regarded as the criterion deter- mining the class of problem, and the name ‘solid problem’ came to be used in the sense given to it by Pappus through a natural misapprehension. A similar supposition would account, in Zeuthen’s view, for a circumstance which would otherwise seem strange, viz. that Apollonius does not use the expression ‘solid problem,’ though it might have been looked for in the preface to the fourth Book of the Conics. The term may have been avoided by Apollonius because it then had the more restricted meaning attributed to it by Zeuthen and therefore would not have been applicable to all the problems which Apollonius had in view.
It must be admitted that Zeuthen’s hypothesis is in several respects attractive. I cannot however feel satisfied that the positive evidence in favour of it is sufficiently strong to outweigh the authority of Pappus where his statements tell the other way. To make the position clear, we have to remember that Menaechmus, the discoverer of the conic sections, was a pupil of Eudoxus who flourished about 365 B.c.; probably therefore we may place the discovery of conics at about 350 B.c. Now Aristaeus ‘the elder’
CXXXV1 INTRODUCTION.
wrote a book on solid loci (στερεοὶ τόποι) the date of which Cantor concludes to have been about 320 B.c. Thus, on Zeuthen’s hypo- thesis, the ‘solid problems’ the solution of which by means of conics caused the latter to be called ‘solid loci’ must have been such as had been already investigated and recognised as solid problems before 320 B.c., while the definite appropriation, so to speak, of the newly discovered curves to the service of the class of problems must have come about in the short period between their discovery and the date of Aristaeus’ work. It is therefore important to consider what particular problems leading to cubic equations appear to have been the subject of speculation before 320 B.c. We have certainly no ground for assuming that the cubic equation used by Archimedes (On the Sphere and Cylinder τι. 4) was one of these problems ; for the problem of cutting a sphere into segments bearing a given ratio to one another could not have been investigated by geometers who had not succeeded in finding the volume of a sphere and a segment of a sphere, and we know that Archimedes was the first to discover this. On the other hand there was the duplication of the cube, or the solution of a pure cubic equation, which was a problem dating from very early times. Also it is certain that the trisection of an angle had Jong exercised the minds of the Greek geometers. Pappus says that “the ancient geometers” considered this problem and first tried to solve it, though it was by nature a solid problem (πρόβλημα τῇ φύσει στερεὸν ὑπάρχον), by means of plane considerations (διὰ τών ἐπιπέδων) but failed; and we know that Hippias of Elis invented, about 420 B.c,, a transcendental curve which was capable of being used for two purposes, the trisection of an angle, and the quadrature of a circle*. This curve came to be called the Quadratrixt, but, as Deinostratus, a brother of Menaechmus, was apparently the first to apply the curve to the quadrature of the circle{, we may no doubt. conclude that it was originally intended for the purpose of trisecting
* Proclus (ed. Friedlein), p. 272.
+ The character of the curve may be described as follows. Suppose there are two rectangular axes Oy, Ox and that a straight line OP of a certain length (a) revolves uniformly from a position along Oy to a position along Ox, while a straight line remaining always parallel to Ox and passing through P in its original position also moves uniformly and reaches Ox in the same time as the moving radius OP. The point of intersection of this line and OP describes the Quadratrix, which may therefore be represented by the equation
y/a=20/7. + Pappus tv. pp. 250—2.
CUBIC EQUATIONS. CXXXV1l
an angle. Seeing therefore that the Greek geometers had used their best efforts to solve this problem before the invention of conics, it may easily be that they had succeeded in reducing it to the geo- metrical equivalent of a cubic equation. They would not have been unequal to effecting this reduction by means of the figure of the νεῦσις given above on p. ΟΧΙ]. with a few lines added. The proof would of course be the equivalent of eliminating « between the two equations
ay = ab
αὐ ον eae, ἜΤΙ een (a) where «= DF, y=FP= EC, a=DA, b= DB. The second equation gives
(a + a) (ὦ -- 3a) = (y +b) (3b -- ψ). From the first equation it is easily seen that
(c+a):(y+b)=a:y,
and that (x — 3a) y=a(b — 3y) ; we have therefore GO 3) a OD — Of) eens ere -- (8) [or γ᾽ — 3by? — 3a*y + ab = ΟἹ.
If then the trisection of an angle had been reduced to the geo- metrical equivalent of this cubic equation, it would be natural for the Greeks to speak of it as a solid problem. In this respect it would be seen to be similar in character to the simpler problem of the duplication of the cube or the equivalent of a pure cubic equation; and it would be natural to see whether the transformation of volumes would enable the mixed cubic to be reduced to the form of the pure cubic, in the same way as the transformation of areas enabled the mixed quadratic to be reduced to the pure quadratic. The reduction to the pure cubic would soon be seen to be impossible, and the stereometric line of investigation would prove unfruitful and be abandoned accordingly.
The two problems of the duplication of the cube and the trisection of an angle, leading in one case to a pure cubic equation and in the other to a mixed cubic, are then the only problems leading to cubic equations which we can be certain that the Greeks had occupied themselves with up to the time of the discovery of the conic sections. Menaechmus, who discovered these, showed that they could be successfully used for finding the two mean propor- tionals and therefore for solving the pure cubic equation, and the
CXXXVI1Li INTRODUCTION.
next question is whether it had been proved before the date of Aristaeus’ Solid Loci that the trisection of an angle could be effected by means of the same conics, either in the form of the νεῦσις above described directly and without the reduction to a cubic equation, or in the form of the subsidiary cubic (8). Now (1) the solution of the cubic would be somewhat difficult in the days when conics were still a new thing. The solution of the equation () as such would involve the drawing of the conics which we should represent by the equations xy =a’, ba = 3a’ + 3by — y’,
and the construction would be decidedly more difficult than that used by Archimedes in connexion with his cubic, which only requires the construction of the conics
2
6? C= — a
Y)
(a- x) y=ac;
hence we can hardly assume that the trisection of an angle in the form of the subsidiary cwbic equation was solved by means of conics before 320 B.c. (2) The angle may have been trisected by means of conics in the sense that the νεῦσις referred to was effected by drawing the curves (a), ie. a rectangular hyperbola and a circle. This could easily have been done before the date of Aristaeus ; but if the assignment of the name ‘solid loci’ to conics had in view their applicability to the direct solution of the problem in this manner without any reference to the cubic equation, or simply because the problem had been before proved to be ‘solid’ by means of the reduction to that cubic, then there does not appear to be any reason why the Quadratrix, which had been used for the same- purpose, should not at the time have been also regarded as a ‘solid locus,’ in which case Aristaews could hardly have appropriated the latter term, in his work, to conics alone. (3) The only remaining alternative consistent with Zeuthen’s view of the origin of the name ‘solid locus’ appears to be to suppose that conics were so called simply because they gave a means of solving one ‘solid problem,’ viz. the doubling of the cube, and not a problem of the more general character corresponding to a mixed cubic equation, in which case the justification for the general name ‘solid locus’ could only be admitted on the assumption that it was adopted at a time
CUBIC EQUATIONS. CXXX1X
when the Grecks were still hoping to be able to reduce the general cubic equation to the pure form. I think however that the traditional explanation of the term is more natural than this would be. Conics were the first curves of general interest for the description of which recourse to solid figures was necessary as distinct from the ordinary construction of plane figures in a plane*; hence the use of the term ‘solid locus’ for conics on the mere ground of their solid origin would be a natural way of describing the new class of curves in the first instance, and the term would be likely to remain in use, even when the solid origin was no longer thought of, just as the individual conics continued to be called “ sections of a right-angled, obtuse-angled, and acute-angled” cone respectively. While therefore, as I have said, the two problems mentioned might naturally have been called ‘solid problems’ before the dis- covery of ‘solid loci,’ I do not think there is sufficient evidence to show that ‘solid problem’ was then or later a technical term for a problem capable of reduction to a cubic equation in the sense of implying that the geometrical equivalent of the general cubic equation was investigated for its own sake, independently of its applications, and that it ever occupied such a recognised position in Greek geometry that a problem would be considered solved so soon as it was reduced to a cubic equation. If this had been so, and if the technical term for such a cubic was ‘solid problem,’ I find it hard to see how Archimedes could have failed to imply some- thing of the kind when arriving at his cubic equation. Instead of this, his words rather suggest that he had attacked it as res integra. Again, if the general cubic had been regarded over any length of time as a problem of independent interest which was solved by means of the intersections of conics, the fact could hardly have been unknown to Nicoteles who is mentioned in the preface to Book Iv. of the Conics of Apollonius as having had a controversy with Conon respecting the investigations in which the latter discussed the maxi- mum number of points of intersection between two conics. Now Nicoteles is stated by Apollonius to have maintained that no use
* Tt is true that Archytas’ solution of the problem of the two mean propor- tionals used a curve of double curvature drawn on a cylinder; but this was not such a curve as was likely to be investigated for itself or even to be regarded as a locus, strictly speaking; hence the solid origin of this isolated curve would not be likely to suggest objections to the appropriation of the term ‘solid locus’ to conics.
ex! INTRODUCTION.
could be made of the discoveries of Conon for διορισμοί; but it seems incredible that Nicoteles could have made such a statement, even for controversial purposes, if cubic equations then formed a recognised class of problems for the discussion of which the intersections of conics were necessarily all-important.
I think therefore that the positive evidence available will not justify us in accepting the conclusions of Zeuthen except to the following extent.
1. Pappus’ explanation of the meaning of the term ‘plane problem’ (ἐπίπεδον πρόβλημα) as used by the ancients can hardly be right. Pappus says, namely, that “problems which can be solved by means of the straight line and circle may properly be called plane (λέγοιτ᾽ ἂν εἰκότως ἐπίπεδα) ; for the lines by means of which such problems are solved have their origin in a plane.” The words ‘may properly be called” suggest that, so far as plane problems were concerned, Pappus was not giving the ancient definition of them, but his own inference as to why they were called ‘plane.’ The true significance of the term is no doubt, as Zeuthen says, not that straight lines and circles have their origin in a plane (which would be equally true of some other curves), but that the problems in question admitted of solution by the ordinary plane methods of transformation of areas, manipulation of simple equations between areas, and in particular the application of areas. In other words, plane problems were those which, if expressed algebraically, depend on equations of a degree not higher than the second.
2. When further problems were attacked which proved to be beyond the scope of the plane methods referred to, it would be found that some of such problems, in particular the duplication of the cube and the trisection of an angle, were reducible to simple . equations between volwmes instead of equations between areas ; and it is quite possible that, following the analogy of the distinction existing in nature between plane figures and solid figures (an analogy which was also followed in the distinction between numbers as ‘plane’ and ‘solid’ expressly drawn by Euclid), the Greeks applied the term ‘solid problem’ to such a problem as they could reduce to an equation between volumes, as distinct from a ‘plane problem’ reducible to a simple equation between areas.
3. The first ‘solid problem’ in this sense which they succeeded
CUBIC EQUATIONS. exli
in solving was the multiplication of the cube, corresponding to the solution of a pure cubic equation in algebra, and it was found that this could be effected by means of curves obtained by making plane sections of a solid figure, namely the cone. Thus curves having a solid origin were found to solve one particular solid problem, which could not but seem an appropriate result ; and hence the conic, as being the simplest curve so connected with a solid problem, was considered to be properly termed a ‘solid locus,’ whether because of its application or (more probably) because of its origin.
4. Further investigation showed that the general cubic equation could not be reduced, by means of stereometric methods, to the simpler form, the pure cubic; and it was found necessary to try the method of conics directly either (1) upon the derivative cubic equation or (2) upon the original problem which led to it. In practice, as e.g. in the case of the trisection of an angle, it was found that the cubic was often more difficult to solve in that manner than the original problem was. Hence the reduction of it to a cubic was dropped as an unnecessary complication, and the geometrical equivalent of a cubic equation stated as an in- dependent problem never obtained a permanent footing as the ‘solid problem’ par excellence.
5. It followed that solution by conics came to be regarded as the criterion for distinguishing a certain class of problem, and, as conics had retained their old name of ‘solid loci,’ the corresponding term ‘solid problem’ came to be used in the wider sense in which Pappus interprets it, according to which it includes a problem depending on a biquadratic as well as a problem reducible to a cubic equation.
6. The terms ‘linear problem’ and ‘linear locus’ were then invented on the analogy of the other terms to describe respectively a problem which could not be solved by means of straight lines, circles, or conics, and a curve which could be used for solving such a problem, as explained by Pappus.
CHAPTER VII. ANTICIPATIONS BY ARCHIMEDES OF THE INTEGRAL CALCULUS.
Ir has been often remarked that, though the method of exhaustion exemplified in Euclid xu. 2 really brought the Greek geometers face to face with the infinitely great and the infinitely small, they never allowed themselves to use such conceptions. It is true that Antiphon, a sophist who is said to have often had disputes with Socrates, had stated* that, if one inscribed any regular polygon, say a square, in a circle, then inscribed an octagon by constructing isosceles triangles in the four segments, then inscribed isosceles triangles in the remaining eight segments, and so on, ‘until the whole area of the circle was by this means exhausted, a polygon would thus be inscribed whose sides, in consequence of their small- ness, would coincide with the circumference of the circle.” But as against this Simplicius remarks, and quotes Eudemus to the same effect, that the inscribed polygon will never coincide with the circumference of the circle, even though it be possible to carry the division of the area to infinity, and to suppose that it would is to set aside a geometrical principle which lays down that magni- tudes are divisible ad infinitum}. The time had, in fact, not come for the acceptance of Antiphon’s idea, and, perhaps as the result of - the dialectic disputes to which the notion of the infinite gave rise, the Greek geometers shrank from the use of such expressions as infinitely great and infinitely small and substituted the idea of things greater or less than any assigned magnitude. Thus, as Hankel says t, they never said that a circle is a polygon with an infinite number of
* Bretschneider, p. 101.
+ Bretschneider, p. 102.
+ Hankel, Zur Geschichte der Mathematik im Alterthum und Mittelalter, Ρ. 123.
ARCHIMEDES ANTICIPATIONS OF THE INTEGRAL CALCULUS, exlili
infinitely small sides ; they always stood still before the abyss of the infinite and never ventured to overstep the bounds of clear con- ceptions. They never spoke of an infinitely close approximation or a limiting value of the sum of a series extending to an infinite number of terms. Yet they must have arrived practically at such a conception, e.g., in the case of the proposition that circles are to one another as the squares on their diameters, they must have been in the first instance led to infer the truth of the proposition by the idea that the circle could be regarded as the limit of an inscribed regular polygon with an indefinitely increased number of corre- spondingly small sides, They did not, however, rest satisfied with such an inference ; they strove after an irrefragable proof, and this, from the nature of the case, could only be an indirect one. <Ac- cordingly we always find, in proofs by the method of exhaustion, a demonstration that an impossibility is involved by any other assumption than that which the proposition maintains. Moreover this stringent verification, by means of a double reductio ad ab- surdum, is repeated in every individual instance of the use of the method of exhaustion ; there is no attempt to establish, in lieu of this part of the proof, any general propositions which could be simply quoted in any particular case.
The above general characteristics of the Greek method of exhaustion are equally present in the extensions of the method found in Archimedes. To illustrate this, it will be convenient, before passing to the cases where he performs genuine integrations, to mention his geometrical proof of the property that the area of a parabolic segment is four-thirds of the triangle with the same base and vertex. Here Archimedes exhausts the parabola by continually drawing, in each segment left over, a triangle with the same base and vertex as the segment. If A be the area of the triangle so inscribed in the original segment, the process gives a series of areas
Ag An {ΠΕ eas. and the area of the segment is really the sum of the infinite series A {1444 (D8+ (+. But Archimedes does not express it in this way. He first proves
that, if A,, A,,...4,, be any number of terms of such a series, so that A, =4A,, A,=4A,,..., then
A,+A,+A,+...+4,+44, = $4), or Aa GY oe HG) SSG) 4 - τὶ
exliv INTRODUCTION.
Having obtained this result, we should nowadays suppose n to increase indefinitely and should infer at once that (4)"-? becomes indefinitely small, and that the limit of the sum on the left-hand side is the area of the parabolic segment, which must therefore be equal to 44. Archimedes does not avow that he inferred the result in this way; he merely states that the area of the segment is equal to 4A, and then verifies it in the orthodox manner by proving that it cannot be either greater or less than $A.
I pass now to the extensions by Archimedes of the method of exhaustion which are the immediate subject of this chapter. It will be noticed, as an essential feature of all of them, that Archimedes takes both an inscribed figure and a circumscribed figure in relation to the curve or surface of which he is investigating the area or the solid content, and then, as it were, compresses the two figures into one so that they coincide with one another and with the curvilinear figure to be measured; but again it must be understood that he does not describe his method in this way or say at any time that the given curve or surface is the limiting form of the circumscribed or inscribed figure. I will take the cases in the order in which they come in the text of this book.
1. Surface of a sphere or spherical segment.
The first step is to prove (On the Sphere and Cylinder 1. 21, 22) that, if in a circle or a segment of a circle there be inscribed polygons, whose sides AB, BC, CD, ... are all equal, as shown in the respective figures, then
(a) for the circle (BB τοῦδέ +..:) AA HAR 1.83. (Ὁ) for the segment (BB'+CC'+...+ KK'+ IM): AM=A'B: BA.
Next it is proved that, if the polygons revolve about the diameter AA’, the surface described by the equal sides of the polygon in a complete revolution is [1. 24, 35]
(a) equal to a circle with radius Ὄ AB(BB’+CC gray YY)
or (ὦ) equal to a circle with radius JAB (BB +CC' +, ASL
Therefore, by means of the above proportions, the surfaces described by the equal sides are seen to be equal to
ARCHIMEDES’ ANTICIPATIONS OF THE INTEGRAL CALCULUS. exlv
(a) a circle with radius AAR AB,
and (Ὁ) a circle with radius ,/ AM. 4΄8;
they are therefore respectively [1. 25, 37] less than (a) acircle with radius 4 Μ΄, (6) a circle with radius AL.
Archimedes now proceeds to take polygons circumscribed to the circle or segment of a circle (supposed in this case to be less than a semicircle) so that their sides are parallel to those of the inscribed polygons before mentioned (cf. the figures on pp. 38, 51); and he proves by like steps [1. 30, 40] that, if the polygons revolve about the diameter as before, the surfaces described by the equal sides during a complete revolution are greater than the same circles respectively.
Lastly, having proved these results for the inscribed and circumscribed figures respectively, Archimedes concludes and proves [1. 33, 42, 43] that the surface of the sphere or the segment of the sphere is equal to the first or the second of the circles respectively.
In order to see the effect of the successive steps, let us express the several results by means of trigonometry. If, in the figures on pp. 33, 47 respectively, we suppose 4n to be the number of sides in the polygon inscribed in the circle and 2n the number of the equal sides in the polygon inscribed in the segment, while in the latter case the angle AOL is denoted by a, the proportions given above are respectively equivalent to the formulae *
9
= T - ΧΙ - T T sin =— + sin — +...+sin(2n—1) — =cot-, Qn Qn Qn 4n α΄ ἢ τ ὭΧΩ : a : 2 4sin—+sin —+...+sin(n—1)—}+sina vi 1 γὺ α and ΞΟ l—cosa Qn
Thus the two proportions give in fact a summation of the series sin 6+ sin 26+ ...+sin (n—1)6
both generally where 70 is equal to any angle a less than z, and in the particular case where 7 is even and 6=7/n.
Again, the areas of the circles which are equal to the surfaces described by the revolution of the equal sides of the inscribed
* These formulae are taken, with a slight modification, from Loria, Il periodo aureo della geometria greca, p. 108.
Ἐΐ A. k
exlvi INTRODUCTION.
polygons are respectively (if ὦ be the radius of the great circle of the sphere)
9
5 T Ξ π a ate - T π
47a? sin - Ἐπ =—+sin5— +... 51π0ὸ ἰδ or Ἵππον 4n 2n 2n 2n 4n
“:
and
9) . α τος ἰοῦ . «α . α . wa’. 2 sin =| 2 jsin—+sin — +... + sin(n—1)—} +sina |, 2n n n n
a or πα". 2 cos ae (1 — cos a). In
The areas of the circles which are equal to the surfaces described by the equal sides of the circwmscribed polygons are obtained from the areas of the circles just given by dividing them by cos’ z/4n and cos’ a/2n respectively.
Thus the results obtained by Archimedes are the same as would be obtained by taking the limiting value of the above trigonometri- cal expressions when 7 is indefinitely increased, and when therefore cos 7/4n and cos αἰ 27. are both unity.
But the first expressions for the areas of the circles are (when 7 is indefinitely increased) exactly what we represent by the integrals
Tr 47a”. 4 i sin 6d6, or 47a’, 0 9 vd 2 9 and πα΄. | 2 sin @d6, or 27a’ (1 -- 008 a). 0
Thus Archimedes’ procedure is the equivalent of a genuine integration in each case.
2. Volume of a sphere or a sector of a sphere.
The method does not need to be separately set out in detail here, because it depends directly on the preceding case: The investiga- tion proceeds concurrently with that of the surface of a sphere or a segment of a sphere. The same inscribed and circumscribed figures are used, the sector of a sphere being of course compared with the solid figure made up of the figure inscribed or circumscribed to the segment and of the cone which has the same base as that figure and has its vertex at the centre of the sphere. It is then proved, (1) for the figure inscribed or circumscribed to the sphere, that its volume is equal to that of a cone with base equal to the surface of the figure and height equal to the perpendicular from the centre of the sphere on any one of the equal sides of the revolving polygon, (2) for the figure inscribed or circumscribed to the sector, that the
ARCHIMEDES ANTICIPATIONS OF THE INTEGRAL CALCULUS. exlvii
volume is equal to that of a cone with base equal to the surface of the portion of the figure which is inscribed or circumscribed to the segment of the sphere included in the sector and whose height is the perpendicular from the centre on one of the equal sides of the polygon.
Thus, when the inscribed and circumscribed figures are, so to speak, compressed into one, the taking of the limit is practically the same thing in this case as in the case of the surfaces, the resulting volumes being simply the before-mentioned surfaces multiplied in each case by 4a.
3. Area of an ellipse.
This case again is not strictly in point here, because it does not exhibit any of the peculiarities of Archimedes’ extensions of the method of exhaustion. That method is, in fact, applied in the same manner, mutatis mutandis, as in Eucl. xu. 2. There is no simultaneous use of inscribed and circumscribed figures, but only the simple exhaustion of the ellipse and auxiliary circle by increasing to any desired extent the number of sides in polygons inscribed to each (On Conoids and Spheroids, Prop. 4).
4. Volume of a segment of a paraboloid of revolution.
Archimedes first states, as a Lemma, a result proved incidentally in a proposition of another treatise (On Spirals, Prop. 11), viz. that, if there be » terms of an arithmetical progression h, 2h, 3h, ..., then
h+2h+ 3h+...+nh>tnh and Pe SE aa
Next he inscribes and circumscribes to the segment of the paraboloid figures made up of small cylinders (as shown in the figure of On Conoids and Spheroids, Props. 21, 22) whose axes lie along the axis of the segment and divide it into any number of equal parts. If ὁ is the length of the axis 4D of the segment, and if there are 7 cylinders in the circumscribed figure and their axes are each of length h, so that e=nh, Archimedes proves that
1 cylinder CE _ wh (1) inscribed fig. h+2h+3h+...+(n-1p)h > 2, by the Lemma, cylinder CH nh
2 ee nen eae (2) circumscribed fig, h+2h+3h+...+nh
«Ὁ.
k2
exlvill INTRODUCTION.
Meantime it has been proved [Props. 19, 20] that, by increasing m sufficiently, the inscribed and circumscribed figure can be made to differ by less than any assignable volume. It is accordingly concluded and proved by the usual rigorous method that
(cylinder CZ) = 2 (segment), so that (segment ABC) = 3 (cone ABC).
The proof is therefore equivalent to the assertion, that if h is indefinitely diminished and m indefinitely increased, while nh remains equal to ὁ,
limit of h{h + 2h+3h+...+(n—-1)h} =he';
Cc i ada = 1c. ᾿ 2
Thus the method is essentially the same as ours when we express the volume of the segment of the paraboloid in the form
that is, in our notation,
eo K | ydx, Jo
where « is a constant, which does not appear in Archimedes’ result for the reason that he does not give the actual content of the segment of the paraboloid but only the ratio which it bears to the circumscribed cylinder.
5. Volume of a segment of a hyperboloid of revolution.
The first step in this case is to prove [On Conoids and Spheroids, Prop. 2] that, if there be a series of m terms,
ah+h?, a.2h+(2h), a.3h+ (3h), ... a.nh+ (nhy’,
and if (ah+h?)+{'a.2h+(2h)+...4+fa.nh+(nh)} =S8,,
then fa. wh + (nl), <(a+nh)| (+S) ee (8). nh ane nia. nh + (nh)?}/Sy— -17 (a te mt) | (5+ 2)
Next [Props. 25, 26] Archimedes draws inscribed and cireum- scribed figures made up of cylinders as before (figure on p. 137), and
ARCHIMEDES’ ANTICIPATIONS OF THE INTEGRAL CALCULUS. cxlix proves that, if AD is divided into m equal parts of length ἡ, so that nh = AD, and if AA’ =a, then
cylinder HB’ πίω. nh + (nh)? inscribed figure | Spon
> (a+ nh) | (5 + =) :
cylinder Β΄ —s n{a. nh + (nh)’} circumscribed fig. | δ
a nh <(a+nh)/(=-+—)- (a+nhy/(5 +5) The conclusion, arrived at in the same manner as before, is that
cylinder LB’ Ψ a nh segment ABB’ — (Ὁ un) (5 3 ) ;
and
This is the same as saying that, if nh =b, and if h be indefinitely diminished while x is indefinitely increased,
limit of n (ab +.6°)/S,=(a+8) / ( + 3) eee b tie a or limit of 5 ἊΣ ΞΡ (ξ -Ὲ 5)" Now 8S,=a(h+2h+...+nh) + {h?+ (2h)? +...+(nhy}, sothat AS,=ah(h+2h+...+nh)+h th? + (2h)? +... + (nh). The limit of the last expression is what we should write as b i (ax + x) da, 0 which is equal to ὑ" (5 + >) :
and Archimedes has given the equivalent of this integration.
6. Volume of a segment of a spheroid. Archimedes does not here give the equivalent of the integration (ax — x”), 0
presumably because, with his method, it would have required yet another lemma corresponding to that in which the results (@) above are established.
el INTRODUCTION.
Suppose that, in the case of a segment less than half the spheroid (figure on p. 142), AA’=a, CD=4c, AD=b; and let AD be divided into n equal parts of length h.
The gnomons mentioned in Props. 29, 30 are then the differences between the rectangle cb + 6” and the successive rectangles
ch+h?, ¢.2h+(2h)’, ... e.(n—1)h+{(n—- 1) Ay, and in this case we have the conclusions that (if S,, be the sum of n terms of the series representing the latter rectangles) cylinder HB’ π(οδ- δὴ) inscribed figure ἡ (cb +b) — S,
76 2b τ χὰ Goa)
_cylinder EB' n (cb + 6°) circumscribed fig. ἡ (cb + 6°) — δ᾽...
Bt
: sane cylinder LB’ _ and in the limit eon Gia (e+ »| (5+
and
Accordingly we have the limit taken of the expression n (οὗ + δ) —S, την, δ πὶ (οὗ -- 6) ᾿ γι (οὗ + 65)
and the integration performed is the same as that in the case of the hyperboloid above, with ὁ substituted for a,
Archimedes discusses, as a separate case, the volume of half. a spheroid [Props. 27, 28]. It differs from that just given in that ὁ vanishes and b= 4a, so that it is necessary to find the limit of h? + (2h)? + (3h)? +... + (nh)? | n (nh)? et
and this is done by means of a corollary to the lemma given on pp. 107—9 [On Spirals, Prop. 10] which proves that
h? + (2h)? +... + (nh)? > En (nh)?, and h? + (2h)? + "τὸ 1) h}? «1 (nh)’.
The limit of course vor. to the integral
b i a’dx = 46°. 0
ARCHIMEDES’ ANTICIPATIONS OF THE INTEGRAL CALCULUS. cli
7. Area of a spiral.
(1) Archimedes finds the area bounded by the first complete turn of a spiral and the initial line by means of the proposition just quoted, viz.
Δ + (2h)? +... + (nh) > kn (nh), h? + (2h)? +... + {(n—1) hi? < dn (nh)?.
He proves [Props. 21, 22, 23] that a figure consisting of similar sectors of circles can be circumscribed about any arc of a spiral such that the area of the circumscribed figure exceeds that of the spiral by less than any assigned area, and also that a figure of the same kind can be inscribed such that the area of the spiral exceeds that of the inscribed figure by less than any assigned area. Then, lastly, he circumscribes and inscribes figures of this kind [ Prop. 24]; thus e.g. in the circumscribed figure, if there are ~ similar sectors, the radii will be m lines forming an arithmetical progression, as h, 2h, 3h, ... nh, and nh will be equal to a, where a is the length inter- cepted on the initial line by the spiral at the end of the first turn. Since, then, similar sectors are to one another as the square of their radii, and 7 times the sector of radius nh or a is equal to the circle with the same radius, the first of the above formulae proves that
(circumscribed fig.) > 47a’.
A similar procedure for the inscribed figure leads, by the use of the second formula, to the result that
(inscribed fig.) < 47a’. The conclusion, arrived at in the usual manner, is that (area of spiral) = 47a’ ;
and the proof is equivalent to taking the limit of T 9 9 A [5 + (2h)? +... + τοὺ — 1) A}? ] f re 21. f 12 or 0 - Lh + (2h +... + {(n— 1) A}? ], which last limit we should express as
a T —| 2°dx=t7a" (10
eli INTRODUCTION.
{It is clear that this method of proof equally gives the area bounded by the spiral and any radius vector of length 6 not being greater than a; for we have only to substitute wb/a for z, and to remember that in this case nh=b. We thus obtain for the area
T
b -- | ada, or 47b*/a. | ao
(2) To find the area bounded by an are on any turn of the spiral (not being greater than a complete turn) and the radii vectores to its extremities, of lengths ὦ and ὁ say, where 6: ὦ, Archimedes uses the proposition that, if there be an arithmetic progression consisting of the terms
b, b+h, b+ 2h, ... δ- (ἡ -- 1),
and if δ΄, Ξε 0+ (b+h)? + (b+ 2h)? +... 4+ {b+ (ἡ -- 1)λ},
(n—1){b+(m—1)h}? ἐδ + (m— 1) hj? nee S,-8 ει ὦ —1){b+(n-1)h'? {b+ (n—1)h}? d (ays es al νἀ ππ" τὰ ἘΣ * + (m—1)yhib+ki(n—1) hp
[On Spirals, Prop. 11 and note. ]
Then in Prop. 26 he circumscribes and inscribes figures consisting of similar sectors of circles, as before. There are n—1 sectors in each figure and therefore 7 radii altogether, including both ὦ and ὁ, so that we can take them to be the terms of the arithmetic progres- sion given above, where {b+(u—1)h}=c. It is thus proved, by means of the above inequalities, that
__ sector OB'C ¥ {b+ (n—1)h}? _ sector OBC circumscribed fig. — {b+(n—1)h}b+44(n—1)h\? inser. fig.”
and it is concluded after the usual manner that
sector OB'C {b+ (n—1)h?
spiral OBU ~ {b+(n—1)htb +4 f(n—1) hi?
9
— σ' δ -Ἐ1 (6-- δ)"
Remembering that ἡ -- 1 -- (ὁ -- ὁὴ),}, we see that the result is the
ARCHIMEDES’ ANTICIPATIONS OF THE INTEGRAL CALCULUS. ΟΠ
same thing as proving that, in the limit, when x becomes indefinitely great and ἢ indefinitely small, while ὦ + (n—1)h=c,
limit of A[b°+(b+A)? +... + {b+ (m— 2)}}}} Ξε (ὁ -- ὁ) {ch +4 (e—b)"}
Ξε ἡ (οὗ -- δ); that is, with our notation,
(3) Archimedes works out separately [Prop. 25], by exactly the same method, the particular case where the area is that described in any one complete turn of the spiral beginning from the initial line. This is equivalent to substituting (x—1)a for ὦ and na for ¢, where ὦ is the radius vector to the end of the first complete turn of the spiral.
It will be observed that Archimedes does not use the result
corresponding to 6 6 b i wda— | eda = | x da. 0 b 0
8. Area of a parabolic segment.
Of the two solutions which Archimedes gives of the problem of squaring a parabolic segment, it is the mechanical solution which gives the equivalent of a genuine integration. In Props. 14, 15 of the Quadrature of the Parabola it is proved that, of two figures inscribed and circumscribed to the segment and consisting in each case of trapezia whose parallel sides are diameters of the parabola, the inscribed figure is less, and the circumscribed figure greater, than one-third of a certain triangle (ZqQ in the figure on p. 242). Then in Prop. 16 we have the usual process which is equivalent to taking the limit when the trapezia become infinite in number and their breadth infinitely small, and it is proved that
(area of segment) =} A £qQ.
The result is the equivalent of using the equation of the parabola referred to Qg as axis of « and the diameter through Q as axis of ae VIZ:
py =x (2a — 2), which can, as shown on p. 236, be obtained from Prop. 4, and finding
2a
yda, 0
cliv INTRODUCTION.
where y has the value in terms of # given by the equation ; and of
course 1 [2α =f (2ax —x*) dx = P Jo The equivalence of the method to an integration can also be seen thus. It is proved in Prop. 16 (see figure on p. 244) that, if gE be divided into 2 equal parts and the construction of the proposition be made, QYq is divided at O,, O,,... into the same number of equal parts. The area of the circumscribed figure is then easily seen to be the sum of the areas of the triangles
QqF, QR,F,, QR,F,, tee
403 3p ;
that is, of the areas of the triangles QqF, QO,R,, 0. ;, ...
Suppose now that the area of the triangle qf is denoted by A, and it follows that
- 7): ΞΕ ΟἿΣ (circumscribed fig.) = Δ {1 + @ it) + ine ) eee =
» 9
nm ne 7.0
Similarly we obtain
. . 1 9AQ (inscribed fig.) = PORE A fA? + 27A? +... + (n— 1) At.
Taking the limit we have, if A denote the area of the triangle ZqQ, so that A=nA,
1 4 (area of segment) = Ἵ: | A’dA Jo
=
If the conclusion be regarded in this manner, the integration is the same as that which corresponds to Archimedes’ squaring of the spiral.
CHAPTER VIII. THE TERMINOLOGY OF ARCHIMEDES.
So far as the language of Archimedes is that of Greek geometry in general, it must necessarily have much in common with that of Euclid and Apollonius, and it is therefore inevitable that the present chapter should repeat many of the explanations of terms of general application which I have already given in the corresponding chapter of my edition of Apollonius’ Conics*. But I think it will be best to make this chapter so far as possible complete and self- contained, even at the cost of some slight repetition, which will however be relieved (1) by the fact that all the particular phrases quoted by way of illustration will be taken from the text of Archimedes instead of Apollonius, and (2) by the addition of a large amount of entirely different matter corresponding to the great variety of subjects dealt with by Archimedes as compared with the limitation of the work of Apollonius to the one subject of conics.
One element of difficulty in the present case arises out of the circumstance that, whereas Archimedes wrote in the Doric dialect, the original language has been in some books completely, and in others partially, transformed into the ordinary dialect of Greek. Uni- formity of dialect cannot therefore be preserved in the quotations about to be made; but I have thought it best, when explaining single words, to use the ordinary form, and, when illustrating their use by quoting phrases or sentences, to give the latter as they appear in Heiberg’s text, whether in Doric or Attic in the particular case. Lest the casual reader should imagine the paroxytone words εὐθείαι, διαμέτροι, πεσείται, πεσούνται, ἐσσείται, δυνάνται, ἁπτέται, καλείσθαι, κείσθαι and the like to be misprints, I add that the quotations in Doric from Heiberg’s text have the unfamiliar Doric accents.
Τ shall again follow the plan of grouping the various technical
* Apollonius of Perga, pp. clvii—clxx.
elvi INTRODUCTION.
terms under certain general headings, which will enable the Greek term corresponding to each expression in the ordinary mathematical phraseology of the present day to be readily traced wherever such a Greek equivalent exists.
Points and lines.
A point is σημεῖον, the point B τὸ Β σημεῖον or τὸ B simply; a point on (a line or curve) σημεῖον ἐπί (with gen.) or ἐν; ὦ point raised above (a plane) σημεῖον μετέωρον ; any two points whatever being taken δύο σημείων λαμβανομένων ὁποιωνοῦν.
At a point (e.g. of an angle) πρός (with dat.), having its vertex at the centre of the sphere κορυφὴν ἔχων πρὸς TO κέντρῳ τῆς σφαίρας ; of lines meeting in a point, touching or dividing αὐ a point, etc., κατά (with acc.), thus AE is bisected at Zis a AE δίχα τεμνέται κατὰ τὸ Z; of a point falling on or being placed on another ἐπί or κατά (with acc.), thus Z will fall on T, τὸ μὲν Z ἐπὶ τὸ Τ' πεσείται, so that ἘΣ lies on Δ, ὥστε τὸ μὲν E κατὰ τὸ A κείσθαι.
Particular points are eatremity πέρας, vertex κορυφή, centre κέντρον, point of division διαίρεσις, point of meeting σύμπτωσις, pornt of section τομή, point of bisection διχοτομία, the middle point τὸ μέσον ; the points of division H, 1, K, ta τῶν διαιρεσίων σαμεῖα τὰ H, I, K; let B be its middle point μέσον δὲ αὐτᾶς ἔστω τὸ B; the point of section in which (a circle) cuts a toa, καθ᾽ ἃν τέμνει.
A line is γραμμή, a curved line καμπύλη γραμμή, a straight line εὐθεῖα with or without γραμμή. The straight line @IKA, a OIKA εὐθεῖα ; but sometimes the older expression is used, the straight line on which (ἐπί with gen. or dat. of the pronoun) are placed certain letters, thus let it be the straight line M, ἔστω ἐφ᾽ ἃ τὸ M, other straight lines K, A, ἄλλαι γραμμαί, ἐφ᾽ av ta K, A. The straight lines between the points at μεταξὺ τῶν σημείων εὐθεῖαι, of the lines which have the same extremities the straight line is the least τῶν τὰ αὐτὰ πέρατα ἐχουσῶν γραμμῶν ἐλαχίστην εἶναι τὴν εὐθεῖαν, straight lines cutting one another εὐθείαι τεμνούσαι ἀλλάλας.
For points in relation to lines we have such expressions as the following: the points T, ©, M are on a straight line ἐπ᾽ εὐθείας ἐστὶ τὰ Τ', @, M σαμεῖα, the point of bisection of the straight line containing the centres of the middle magnitudes ἃ διχοτομία τᾶς εὐθείας τᾶς ἐχούσας τὰ κέντρα τῶν μέσων μεγεθέων. A very characteristic phrase for αὐ a point which divides the straight line in such a proportion that... is ἐπὶ τᾶς εὐθείας διαιρεθείσας wore...; similarly ἐπὶ τᾶς XE
THE TERMINOLOGY OF ARCHIMEDES. elvil
τμαθείσας οὕτως, ὦστε. A certain point will be on the straight line... dividing it so that... ἐσσείται ἐπὶ ras εὐθείας...διαιρέον οὕτως τὰν εἰρημέναν εὐθεῖαν, Wore....
The middle point of a line is often elegantly denoted by an adjective in agreement; thus at the middle point of the segment ἐπὶ μέσου τοῦ τμάματος, (ὦ line) drawn from T to the middle pot of EB, ἀπὸ τοῦ T ἐπὶ μέσαν τὰν EB ἀχθεῖσα, drawn to the middle point of the base ἐπὶ μέσαν τὰν βάσιν ἀγομένα.
A straight line produced is the (straight line) in the same straight line with it ἡ ἐπ᾽ εὐθείας αὐτῇ. In the same straight line with the avis ἐπὶ tas αὐτᾶς εὐθείας τῷ ἄξονι. Of a straight line falling on another line κατά (with gen.) is used, e.g. πίπτουσι κατ᾽ αὐτῆς ; ἐπί (with acc.) is also used of a straight line placed on another, thus ¢f EH be placed on BA, τεθείσας tas EH ἐπὶ τὰν BA.
For lines passing through points we find the following ex- pressions: will pass through N, ἥξει διὰ τοῦ N ; will pass through the centre διὰ τοῦ κέντρου πορεύσεται, will fall through @ πεσείται διὰ Tod Θ, verging towards B νεύουσα ἐπὶ τὸ B, pass through the same point ἐπὶ τὸ αὐτὸ σαμεῖον ἐρχόνται ; the diagonals of the parallelogram fall (i.e. meet) at Θ, κατὰ δὲ τὸ Θ at διαμέτροι τοῦ παραλληλογράμμου πίπτοντι ; EZ (passes) through the points bisecting AB, TA, ἐπὶ δὲ τὰν διχοτομίαν τἂν AB, TA ἁ EZ. The verb εἰμί is also used of passing through, thus ἐσσείται δὴ αὐτὰ διὰ τοῦ Θ.
For lines in relation to other lines we have perpendicular to κάθετος ἐπί (with acc.), parallel to παράλληλος with dat. or παρά (with acc.) ; leé KA be (drawn) from K parallel to TA, ἀπὸ τοῦ Καὶ παρὰ τὰν TA ἔστω a KA,
Lines meeting one another συμπίπτουσαι ἀλλήλαις ; the point in which ZH, MN produced meet one another and AV, τὸ σημεῖον, καθ᾽ ὃ συμβάλλουσιν ἐκβαλλόμεναι ai ZH, MN ἀλλήλαις τε καὶ τῇ AT; so as to meet the tangent wore ἐμπεσεῖν τᾷ ἐπιψαυούσᾳ, let straight lines be drawn parallel to AT to meet the section of the cone ἄχθων εὐθείαι παρὰ τὰν AT ἔστε ποτὶ τὰν τοῦ κώνου τομάν, to draw a straight line to meet its circwmference ποτὶ τὰν περιφέρειαν αὐτοῦ ποτιβαλεῖν εὐθεῖαν, the line drawn to meet ἃ ποτιπεσοῦσα, let AE, AA be drawn from the point A to meet the spiral and produced to meet the circumference of the circle ποτιπιπτόντων ἀπὸ τοῦ A σαμείου ποτὶ τὰν ἕλικα ai AE, AA καὶ ἐκπιπτόντων ποτὶ τὰν τοῦ κύκλου περιφέρειαν ; wntil it meets OA in O, ἔστε κα συμπέσῃ τᾷ ΘΑ κατὰ τὸ O (οὗ a circle).
elvill INTRODUCTION.
(The straight line) wili fall outside (i.e. will extend beyond) P, ἐκτὸς τοῦ P πεσεῖται ; will fall within the section of the figure ἐντὸς πεσούνται TAS τοῦ σχήματος TOMAS.
The (perpendicular) distance between (two parallel lines) AZ, BH, τὸ διάστημα τᾶν AZ, BH. Other ways of expressing distances are the following: the magnitudes equidistant from the middle one τὰ ἴσον ἀπέχοντα ἀπὸ τοῦ μέσου μεγέθεα, are at equal distances from one another ἴσα am ἀλλάλων διέστακεν ; the segments (lengths) on AH equal to N, τὰ ἐν τᾷ AH τμάματα ἰσομεγέθεα τᾷ N; greater by one segment ἑνὶ τμάματι μείζων.
The word εὐθεῖα itself is also often used in the sense of distance ; cf. the terms πρώτη εὐθεῖα etc. in the book On Spirals, also ἃ εὐθεῖα ἅ μεταξὺ τοῦ κέντρου τοῦ ἁλίου Kal τοῦ κέντρου τᾶς yas the distance between the centre of the sun and the centre of the earth.
The word for join is ἐπιζευγνύω or ἐπιζεύγνυμι ; the straight line joining the points of contact ἃ τὰς ἁφὰς ἐπιζευγνύουσα εὐθεῖα, BA when joined & BA ἐπιζευχθεῖσα ; let EZ join the points of bisection of AA, BI, a δὲ EZ ἐπιζευγνυέτω τὰς διχοτομίας τάν AA, BI. In one case the word seems to be used in the sense of drawing simply, εἴ κα
εὐθεῖα ἐπιζευχθῇ ypappa ἐν ἐπιπέδῳ.
Angles.
An angle is γωνία, the three kinds of angles are right ὀρθή, acute ὀξεῖα, obtuse ἀμβλεῖα ; right-angled etc. ὀρθογώνιος, ὀξυγώνιος, ἀμβλυ- γώνιος ; equiangular ἰσογώνιος ; with an even number of angles ἀρτιόγωνος Or ἀρτιογώνιος.
At right angles to ὀρθὸς πρός (with acc.) or πρὸς ὀρθάς (with dat. following); thus if a line be erected at right angles to the plane ypappas ἀνεστακούσας ὀρθᾶς ποτὶ τὸ ἐπίπεδον, the planes are at right angles to one another ὀρθὰ wot ἀλλαλά ἐντι τὰ ἐπίπεδα, being at right angles to ABI, zpos ὀρθὰς ὧν τῷ ABI; KT, ZA are at right angles to one another ποτ᾽ ὀρθάς ἐντι ἀλλάλαις αἱ KT, BA, to cut at right angles τέμνειν πρὸς ὀρθάς. The expression making right angles with is also used, e.g. ὀρθὰς ποιοῦσα γωνίας ποτὶ τὰν ΑΒ.
The complete expression for the angle contained by the lines AH, AT is ἅ γωνία & περιεχομένα ὑπὸ τἂν AH, AT’; but there are a great variety of shorter expressions, ywvia itself being often understood ; thus the angles A, E, A, B, αἱ A, E, A, Β γωνίαι; the angle at Θ, ἃ ποτὶ τῷ Θ; the angle contained by AA, AZ, ἃ γωνία ἁ ὑπὸ τᾶν AA, AZ; the angle AHT, ἡ ὑπὸ τῶν AHL γωνία, ἡ ὑπὸ AHT (with or without γωνία).
THE TERMINOLOGY OF ARCHIMEDES. clix
Making the angle K equal to the angle ©, γωνίαν ποιοῦσα τὰν Καὶ ἴσαν τᾷ @; the angle into which the sun fits and which has its vertex at the eye γωνία, εἰς av ὃ ἅλιος ἐναρμόζει τὰν κορυφὰν ἔχουσαν ποτὶ τᾷ ὄψει; of the sides subtending the right angle (hypotenuses) τᾶν ὑπὸ τὰν ὀρθὰν γωνίαν ὑποτεινουσᾶν, they subtend the same angle ἐντὶ ὑπὸ τὰν αὐτὰν γωνίαν.
If a line through an angular point of a polygon divides it exactly symmetrically, the opposite angles of the polygon, at ἀπεναντίον γωνίαι τοῦ πολυγώνου, are those answering to each other on each side of the bisecting line.
Planes and plane figures.
A plane ἐπίπεδον; the plane through BA, τὸ ἐπίπεδον τὸ κατὰ τὴν BA, or τὸ διὰ τῆς BA, plane of the base ἐπίπεδον τῆς βάσεως, plane (ie. base) of the cylinder ἐπίπεδον τοῦ κυλίνδρου : cutting plane ἐπί. medov τέμνον, tangent plune ἐπίπεδον ἐπιψαῦον ; the intersection of planes is their common section κοινὴ Tomy.
In the same plane as the circle ἐν τῷ αὐτῷ ἐπιπέδῳ τῷ κύκλῳ.
Let a plane be erected on UZ at right angles to the plane in which AB, TA are ἀπὸ tas ΠΖ ἐπίπεδον ἀνεστακέτω ὀρθὸν ποτὶ τὸ ἐπίπεδον τό, ἐν ᾧ ἐντι αἱ ΑΒ, TA.
The plane surface ἡ ἐπίπεδος (ἐπιφάνεια), a plane segment ἐπίπεδον τμῆμα, a plane figure σχῆμα ἐπίπεδον.
A rectilineal figure εὐθύγραμμον (σχῆμα), a side πλευρά, perimeter n περίμετρος, similar ὅμοιος, similarly sitwated ὁμοίως κείμενος.
To coincide with (when one figure is applied to another), ἐφαρμόζειν followed by the dative or ἐπί (with acc.); one part coincides with the other ἐφαρμόζει τὸ ἕτερον μέρος ἐπὶ τὸ ἕτερον ; the plane through NZ coincides with the plane through AT’, τὸ ἐπίπεδον τὸ κατὰ τὰν NZ ἐφαρμόζει τῷ ἐπιπέδῳ τῷ κατὰ τὰν Α΄. The passive is also used ; if equal and similar plane figures coincide with one another
na FF WO ΄, / > fd 3 / ΓΝ ΔῈ τῶν ἴσων καὶ ὁμοίων σχημάτων ἐπιπέδων ἐφαρμοζομένων ἐπ᾽ ἀἄλλαλα.
Triangles.
A triangle is τρίγωνον, the triangles bounded by (their three sides) τὰ περιεχόμενα τρίγωνα ὑπὸ τῶν.... A right-angled triangle τρίγωνον ὀρθογώνιον, one of the sides about the right angle μία τῶν περὶ τὴν ὀρθήν. The triangle through the aais (of a cone) τὸ διὰ τοῦ ἄξονος τρίγωνον.
elx INTRODUCTION.
Quadrilaterals.
A quadrilateral is a fowr-sided figure (τετράπλευρον) as dis- tinguished from a fowr-angled figure, τετράγωνον, which means a square. A trapezium, τραπέζιον, is in one place more precisely described as a trapeziwm having its two sides parallel τραπέζιον τὰς δύο πλευρὰς ἔχον παραλλάλους ἀλλάλαις.
A parallelogram παραλληλόγραμμον ; for a parallelogram on a straight line as base ἐπί (with gen.) is used, thus the parallelograms on them are of equal height ἐστὶν ἰσούψη τὰ παραλληλόγραμμα τὰ ἐπ᾽ αὐτῶν. A diagonal of a parallelogram is διάμετρος, the opposite sides of the parallelogram ai κατ᾽ ἐναντίον ToD παραλληλογράμμου πλευραί.
Rectangles.
The word generally used for a rectangle is χωρίον (space or area) without any further description. As in the case of angles, the rectangles contained by straight lines are generally expressed more shortly than by the phrase τὰ περιεχόμενα χωρία ὑπό ; either χωρίον may be omitted or both χωρίον and περιεχόμενον, thus the rectangle AT, TE may be any of the following, τὸ ὑπὸ τῶν AT, TE, τὸ ὑπὸ AT, TE, τὸ ὑπὸ ATE, and the rectangle under ΘΚ, AH is τὸ ὑπὸ τῆς ΘΚ καὶ τῆς AH. Rectangles Θ, 1, K, A, χωρία ἐν οἷς τὰ (or ἐφ᾽ ὧν ἕκαστον τῶν) Θ,1, K, A.
To apply a rectangle to a straight line (in the technical sense) is παραβάλλειν, and παραπίπτω is generally used in place of the passive; the participle παρακείμενος is also used in the sense of applied to. In each case applying fo a straight line is expressed by παρά (with acc.). Examples are, areas which we can apply to a given straight line (i.e. which we can transform into a rectangle of the same area) χωρία, ἃ δυνάμεθα παρὰ τὰν δοθεῖσαν εὐθεῖαν παραβαλεῖν, let ὦ rectangle be applied to each of them παραπεπτωκέτω παρ᾽ ἑκάσταν αὐτᾶν χωρίον : if there be applied to each of them a rectangle exceeding by a square figure, and the sides of the excesses exceed each other by an equal amount (1.6. form an arithmetical progression) εἴ κα παρ᾽ ἑκάσταν αὐτᾶν παραπέσῃ τι χωρίον ὑπερβάλλον εἴδει τετραγώνῳ, ἔωντι δὲ αἱ πλευραὶ τῶν ὑπερβλημάτων τῷ ἴσῳ ἀλλάλαν ὑπερεχούσαι.
The rectangle applied is παράβλημα.
Squares.
A square is τετράγωνον, a square on a straight line is a square
—
(erected) from it (ἀπό). The square on TH, τὸ ἀπὸ τᾶς VE τετράγωνον,
THE TERMINOLOGY OF ARCHIMEDES. elxi
is shortened into τὸ ἀπὸ τᾶς ΓΈ, or τὸ ἀπὸ TE simply. The square next in order to it (when there are a number of squares in a row) is τὸ Tap αὐτῷ τετράγωνον OF τὸ ἐχόμενον τετράγωνον.
With reference to squares, a most important part is played by the word δύναμις πα the various parts of the verb δύναμαι. δύναμις expresses a square (literally a power); thus in Diophantus it is used throughout as the technical term for the square of the unknown quantity in an algebraical equation, i.e. for αὖ. In geometrical language it is the dative singular δυνάμει which is mostly used ; thus a straight line is said to be potentially equal, δυνάμει ἴσα, to a certain rectangle where the meaning is that the sgware on the straight line is equal to the rectangle ; similarly for the square.on BA is less than double the square on AK we have ἡ BA ἐλάσσων ἐστὶν ἢ διπλασίων δυνάμει τῆς AK. The verb δύνασθαι (with or without ἴσον) has the sense of being δυνάμει ἴσα, and, when δύνασθαι is used alone, it is followed by the accusative ; thus the square (on a straight line) is equal to the rectangle contained by... is (εὐθεῖα) ἴσον δύναται τῷ περιεχομένῳ ὑπό... ; let the square on the radius be equal to the rectangle BA, AZ, ἡ ἐκ τοῦ κέντρου δυνάσθω τὸ ὑπὸ τών BAZ, (the difference) by which the square on ZV is greater than the square on half the other diameter ᾧ μεῖζον δυνάται ἃ ZT τᾶς ἡμισείας τᾶς ἑτέρας διαμέτρου.
A gnomon is γνώμων, and its breadth (πλάτος) is the breadth of each end; a gnomon of breadth equal to BI, γνώμων πλάτος ἔχων ἴσον τᾷ BI, (a gnomon) whose breadth is greater by one segment than the breadth of the gnomon last taken away οὗ πλάτος ἑνὶ τμάματι μεῖζον
lal 4 “ Ν 3 a 3 / he τοῦ πλάτεος τοῦ TPO αὐτοῦ ἀφαιρουμένου γνωμονος.
Polygons.
A polygon is πολύγωνον, an equilateral polygon is ἰσόπλευρον, a polygon of an even number of sides or angles ἀρτιόπλευρον or ἀρτιόγωνον ; a polygon with all its sides equal except BA, AA, ἴσας ἔχον Tas πλευρὰς χωρὶς τῶν BAA; a polygon with its sides, excluding the base, equal and even in number τὰς πλευρὰς ἔχον χωρὶς τῆς βάσεως ἴσας καὶ ἀρτίους : an equilateral polygon the number of whose sides is measured by four πολύγωνον ἰσόπλευρον, οὗ ai πλευραὶ ὑπὸ τετράδος μετροῦνται, let the number of its sides be measured by four τὸ πλῆθος τῶν πλευρῶν μετρείσθω ὑπὸ τετράδος. A chiliagon χιλιάγωνον.
The straight lines subtending two sides of the polygon (i.e. joining angles next but one to each other) αἱ ὑπὸ δύο πλευρὰς τοῦ πολυγώνου
H. A. 1
elxil INTRODUCTION.
ὑποτείνουσαι, the straight line subtending one less than half the . ine. 4 ἃς ADs / ~ ε , number of the sides n ὑποτείνουσα τὰς μιᾷ ἐλάσσονας τῶν ἡμίσεων.
Circles.
A circle is κύκλος, the circle Ψ is 6 Ψ κύκλος or ὃ κύκλος ἐν ᾧ τὸ Ψ, let the given circle be that drawn below ἔστω ὃ δοθεὶς κύκλος ὃ ὑποκείμενος.
The centre is κέντρον, the circumference περιφέρεια, the former word having doubtless been suggested by something stwck im and the latter by something, e.g. a cord stretched tight, carried round the centre as a fixed point and describing a circle with its other extremity. Accordingly περιφέρεια is used for a circular are as well as for the whole circumference ; thus the are BA is ἡ BA περιφέρεια, the (part of the) circumference of the circle cut off by the same (straight line) ἡ τοῦ κύκλου περιφέρεια ἡ ὑπὸ τῆς αὐτῆς ἀποτεμνομένη. Though the circumference of a circle is also sometimes called its perimeter (ἢ περίμετρος) in the treatises On the Sphere and Cylinder and on the Measwrement of a Circle, the word does not seem to have been used by Archimedes himself in this sense ; he speaks, however, in the Sand-reckoner of the perimeter of the earth (περίμετρος Tas yas).
The radius is y ἐκ τοῦ κέντρου simply, and this expression without the article is used as a predicate as if it were one word ; thus the circle whose radius is ΘῈΣ is ὃ κύκλος ot ἐκ τοῦ κέντρου a OE; BE is a radius of the circle ἡ δὲ BE ἐκ τοῦ κέντρου ἐστὶ τοῦ κύκλου.
A diameter is διάμετρος, the circle on AE as diameter ὃ περὶ διάμετρον τὴν AE κύκλος.
For drawing a chord of a circle there is no special technical term, but we find such phrases as the following: ἐὰν eis τὸν κύκλον εὐθεῖα γραμμὴ ἐμπέσῃ if in a circle ὦ straight line be placed, and the chord is then the straight line so placed ἡ ἐμπεσοῦσα, or quite commonly 7 ἐν τῷ κύκλῳ (εὐθεῖα) simply. For the chord subtending one 656th part of the circumference of a circle we have the following interesting phrase, ἃ ὑποτείνουσα ἕν τμᾶμα διαιρεθείσας Tas τοῦ ABI κύκλου περιφερείας ἐς χνς΄.
A segment of a circle is τμῆμα κύκλου ; sometimes, to distinguish it from a segment of a sphere, it is called a plane segment τμῆμα ἐπίπεδον. A semicircle is ἡμικύκλιον ; a segment less than a semicircle cut off by AB, τμῆμα ἔλασσον ἡμικυκλίου ὃ ἀποτέμνει ἡ AB. The segments on AE, EB (as bases) are ta ἐπὶ τῶν AE, EB τμήματα; but the semicircle on ZH as diameter is τὸ
THE TERMINOLOGY OF ARCHIMEDES. elxili
ἡμικύκλιον TO περὶ διάμετρον τὰν ZH or τὸ ἡμικύκλιον τὸ περὶ τὰν ΖΗ simply. The expression the angle of the semicircle, ἃ τοῦ ἡμικυκλίου (ywvia), is used of the (right) angle contained by the diameter and the arc (or tangent) at one extremity of it.
A sector of a circle is τομεύς or, when it is necessary to distinguish it from what Archimedes calls a ‘solid sector,’ ἐπίπεδος τομεὺς κύκλου a plane sector of a circle. The sector including the right angle (at the centre) is ὃ τομεὺς ὃ τὰν ὀρθὰν γωνίαν περιέχων. Either of the radii bounding a sector is called a side of it, πλευρά ; each of the sectors (is) equal to the sector which has a side common (with it) ἕκαστος τῶν τομέων ἴσος τῷ κοινὰν ἔχοντι πλευρὰν τομεῖ ; a sector is sometimes regarded as described on one of the bounding radii as a side, thus similar sectors have been described on all (the straight lines) ἀναγεγραφάται ἀπὸ πασᾶν ὁμοίοι τομέες.
Of polygons inscribed in or circumscribed about a circle ἐγγράφειν eis or ἐν and περιγράφειν περί (with acc.) are used; we also find περιγεγραμμένος used with the simple dative, thus τὸ περιγε- γραμμένον σχῆμα τῷ τομεῖ is the figure circumscribed to the sector. A polygon is said to be inscribed in a segment of w circle when the base of the segment is one side and the other sides subtend ares making up the circumference ; thus let a polygon be inscribed on AY in the segment ABI, ἐπὶ τῆς AT πολύγωνον ἐγγεγράφθω eis τὸ ABT τμῆμα. A regular polygon is said to be wnseribed in a sector when the two radii are two of the sides and the other sides are all equal to one another, and a similar polygon is said to be circumscribed about a sector when the equal sides are formed by the tangents to the arc which are respectively parallel to the equal sides of the inscribed polygon and the remaining two sides are the bounding radii produced to meet the adjacent tangents. Of a circle circwmscribed to a polygon περιλαμβάνειν is also used; thus πολύγωνον κύκλος περιγεγραμμένος περιλαμβανέτω περὶ TO αὐτὸ κέντρον γινόμενος, as we might say let ὦ circumscribed circle be drawn with the same centre going round the polygon. Similarly the circle ABTA containing the polygon ὃ ABTA κύκλος ἔχων τὸ πολύγωνον.
When a polygon is inscribed in a circle, the segments left over between the sides of the polygon and the subtended arcs are περιλειπόμενα τμήματα; when a polygon is circumscribed to the circle, the spaces between the two are variously called τὰ περι- λειπόμενα τῆς περιγραφῆς τμήματα, τὰ περιλειπόμενα σχήματα, τὰ περιλείμματα Or τὰ ἀπολείμματα.
clxiv INTRODUCTION.
Spheres, etc.
In connexion with a sphere (σφαῖρα) a number of terms are used on the analogy of the older and similar terms connected with the circle. Thus the centre is κέντρον, the radius ἡ ἐκ τοῦ κέντρου, the diameter y διάμετρος. Two segments, τμήματα σφαίρας or τμήματα σφαιρικά, are formed when a sphere is cut by a plane; a hemisphere is ἡμισφαίριον ; the segment of the sphere at 1, τὸ κατὰ τὸ Τ' τμῆμα τῆς σφαίρας ; the segment on the side of ABT, τὸ ἀπὸ ABT τμῆμα; the segment including the circumference BAA, τὸ κατὰ τὴν BAA περιφέρειαν τμῆμα. The curved surface of a sphere or segment is ἐπιφάνεια ; thus of spherical segments bounded by equal surfaces the hemisphere is greatest is τῶν τῇ ion ἐπιφανείᾳ περιεχομένων σφαιρικῶν τμημάτων μεῖζόν ἐστι τὸ ἡμισφαίριον. The terms base (βάσις), vertex (κορυφή) and height (ὕψος) are also used with reference to a segment of a sphere.
Another term borrowed from the geometry of the circle is the word sector (τομεύς) qualified with the adjective στερεός (solid). A solid sector (τομεὺς στερεός) is defined by Archimedes as the figure bounded by a cone which has its vertex at the centre of a sphere and the part of the surface of the sphere within the cone. The segment of the sphere included wn the sector is τὸ τμῆμα τῆς σφαίρας τὸ ἐν τῷ τομεῖ OF TO κατὰ TOV τομέα.
A great circle of a sphere is ὃ μέγιστος κύκλος τῶν ἐν τῇ σφαίρᾳ and often ὃ μέγιστος κύκλος alone.
Let a sphere be cut by a plane not through the centre τετμήσθω σφαῖρα μὴ διὰ τοῦ κέντρου ἐπιπέδῳ ; a sphere cut by a plane through the centre in the circle EZH®, σφαῖρα ἐπιπέδῳ τετμημένη διὰ τοῦ
κέντρου κατὰ τὸν ΕΖΗΘ κύκλον.
Prisms and pyramids,
A prism is πρῖσμα, a pyramid πυραμίς. As usual, ἀναγράφειν ἀπό is used of describing a prism or pyramid on a rectilineal figure as base; thus let a prism be described on the rectilineal figure (as base) ἀναγεγράφθω ἀπὸ τοῦ εὐθυγράμμου πρῖσμα, on the polygon circumscribed about the circle A let a pyramid be set wp ἀπὸ τοῦ περὶ τὸν A κύκλον περιγεγραμμένου πολυγώνου πυραμὶς ἀνεστάτω ἀναγεγραμ- μένη. A pyramid with an equilateral base ABT is πυραμὶς ἰσόπλευρον ἔχουσα βάσιν τὸ ABY.
The surface is, as usual, ἐπιφάνεια and, when any particular face or a base is excluded, some qualifying phrase has to be used.
THE TERMINOLOGY OF ARCHIMEDES. elxv
Thus the surface of the prism consisting of the parallelograms (ie. excluding the bases) ἡ ἐπιφάνεια τοῦ πρίσματος ἡ ἐκ τῶν παραλληλογράμμων συγκειμένη ; the sunface (of a pyramid) excluding the base or the triangle AET, ἡ ἐπιφάνεια χωρὶς τῆς βάσεως or τοῦ AED τριγώνου.
The triangles bounding the pyramid τὰ περιέχοντα τρίγωνα τὴν πυραμίδα (as distinct from the base, which may be polygonal).
Cones and solid rhombi.
The Elements of Euclid only introduce right cones, which are simply called cones without the qualifying adjective. A cone is there defined as the surface described by the revolution of a right- angled triangle about one of the sides containing the right angle. Archimedes does not define a cone, but generally describes a right cone as an isosceles cone (κῶνος ἰσοσκελής), though once he calls it right (ὀρθός). J. H. T. Miiller rightly observes that the term isosceles applied to a cone was suggested by the analogy of the isosceles triangle, but I doubt whether such a cone was thought of (as he supposes) as one which could be described by making an isosceles triangle revolve about the perpendicular from the vertex on the base; it seems more natural to connect it with the use of the word side (πλευρά) by which Archimedes designates a generator of the cone, a right cone being thus directly regarded as a cone having all its legs equal. The latter supposition would also accord better with the term scalene cone (κῶνος σκαληνός) by which Apollonius denotes an oblique circular cone; such a cone could not of course be described by the revolution of a scalene triangle. An oblique circular cone is simply a cone for Archimedes, and he does not define it; but, while he speaks of finding a cone with a given vertex and passing through every point on a given ‘section of an acute-angled cone’ [ellipse], he regards the finding of the cone as being equivalent to finding the circular sections, and we may therefore conclude that he would have defined the cone in practically the same way as Apollonius does, namely as the surface described by a straight line always passing through a fixed point and moving round the circumference of any circle not in the same plane with the point.
The vertex of a cone is, as usual, κορυφή, the base βάσις, the axis ἄξων and the height ὕψος ; the cones are of the same height εἰσὶν οἱ κῶνοι ὑπὸ TO αὐτὸ ὕψος. A generator is called a side (πλευρά); of α
clxvi INTRODUCTION.
cone be cut by a plane meeting all the generators of the cone εἴ κα κῶνος ἐπιπέδῳ τμαθῇ συμπίπτοντι πάσαις ταῖς TOD κώνου πλευραῖς.
The surface of the cone excluding the base ἡ ἐπιφάνεια τοῦ κώνου χωρὶς τῆς βάσεως ; the conical surface between (two generators) AA, AB, κωνικὴ ἐπιφάνεια ἡ μεταξὺ τῶν ΑΔΒ.
There is no special name for what we call ἃ frustwm of a cone or the portion intercepted between two planes parallel to the base ; the surface of such a frustum is simply the surface of the cone between the parallel planes ἢ ἐπιφάνεια τοῦ κώνου μεταξὺ τῶν παραλλήλων ἐπιπέδων.
A curious term is segment of a cone (ἀπότμαμα κώνου), which is used of the portion of any circular cone, right or oblique, cut off towards the vertex by any plane which makes an elliptic and not a circular section. With reference to a segment of a cone the axis (ἄξων) is defined as the straight line drawn from the vertex of the cone to the centre of the elliptic base.
As usual, ἀναγράφειν ἀπὸ is used of describing a cone on a circle as base. Similarly, a very common phrase is ἀπὸ τοῦ κύκλου κῶνος ἔστω let there be a cone on the circle (as base).
A solid rhombus (ῥόμβος στερεός) is the figure made up of two cones having their base common, their vertices on opposite sides of it, and their axes in one straight line. A rhombus made up of isosceles cones ῥόμβος ἐξ ἰσοσκελῶν κώνων συγκείμενος, and the two cones are spoken of as the cones bounding the rhombus ot κῶνοι ot περιέχοντες TOV ῥόμβον.
Cylinders.
A right cylinder is κύλινδρος ὀρθός, and the following terms apply to the cylinder as to the cone: base βάσις, one base or the other ἡ ἑτέρα βάσις, of which the circle AB is a base and TA opposite to it οὗ βάσις μὲν ὃ AB κύκλος, ἀπεναντίον δὲ ὁ TA; axis ἄξων, height ὕψος, generator πλευρά. The cylindrical surface cut off by (two generators) AT, BA, ἡ ἀποτεμνομένη κυλινδρικὴ ἐπιφάνεια ὑπὸ τῶν AT, ΒΔ; the surface of the cylinder adjacent to the circumference ABY, ἡ ἐπιφάνεια τοῦ κυλίνδρου ἡ κατὰ τὴν ABI περιφέρειαν denotes the surface of the cylinder between the two generators drawn through the extremities of the arc.
A frustum of a cylinder topos κυλίνδρου is a portion of a cylinder intercepted between two parallel sections which are elliptic and not circular, and the axis (ἄξων) of it is the straight line
THE TERMINOLOGY OF ARCHIMEDES. elxvil
joining the centres of the two sections, which is in the same straight line with the axis of the cylinder.
Conic Sections,
General terms are κωνικὰ στοιχεῖα, elements of conics, τὰ κωνικά (the theory of) conics. Any conic section κώνου τομὴ ὅποιαοῦν. Chords are simply εὐθείαι ἐν τᾷ τοῦ κώνου τομᾷ ἀγμέναι. Archimedes never uses the word awis (aéwv) with reference to a conic ; the axes are with him diameters (διάμετροι), and διάμετρος, when it has reference to a complete conic, is used in this sense exclusively. A tangent is ἐπιψαύουσα or ἐφαπτομένη (with gen.).
The separate conic sections are still denoted by the old names ; a parabola is a section of a right-angled cone ὀρθογωνίου κώνου τομή, a hyperbola a section of an obtuse-angled cone ἀμβλυγωνίου κώνου τομή, and an ellipse a section of an acute-angled cone ὀξυγωνίου κώνου
τομή.
The parabola.
Only the axis of a complete parabola is called a diameter, and the other diameters are simply lines parallel to the diameter. Thus parallel to the diameter or itself the diameter is παρὰ τὰν διάμετρον ἢ αὐτὰ διάμετρος; AZ is parallel to the diameter ἃ AZ παρὰ τὰν διάμετρόν ἐστι. Once the term principal or original (diameter) is used, ἀρχικά (sc. διάμετρος).
A. segment of a parabola is τμῆμα, which is more fully described as the segment bounded by a straight line and a section of a right- angled cone τμᾶμα τὸ περιεχόμενον ὑπό τε εὐθείας Kal ὀρθογωνίου κώνου τομᾶς. The word διάμετρος is again used with reference to ἃ segment of a parabola in the sense of our word awis; Archimedes defines the diameter of any segment as the line bisecting all the straight lines (chords) drawn parallel to its base τὰν δίχα τέμνουσαν τὰς εὐθείας πάσας Tas παρὰ τὰν βάσιν αὐτοῦ ἀγομένας.
The part of a parabola included between two parallel chords is called a frustwm τόμος (ἀπὸ ὀρθογωνίου κώνου Tomas ἀφαιρούμενος), the two chords are its lesser and greater base (ἐλάσσων and μείζων βάσις) respectively, and the line joining the middle points of the two chords is the diameter (διάμετρος) of the frustum.
What we call the Jatus rectum of a parabola is in Archimedes the line which is double of the line drawn as far as the axis ἃ διπλασία τᾶς μέχρι τοῦ ἄξονος. In this expression the awis (aéwv) is the axis
elxvill INTRODUCTION.
of the right-angled cone from which the curve was originally derived by means of a section perpendicular to a generator*. Or, again, the equivalent of our word parameter (παρ᾽ ἃν δυνάνται at ἀπὸ τᾶς Topas) is used by Archimedes as by Apollonius, meaning the straight line to which the rectangle which has its breadth equal to the abscissa of a point and is equal to the square of the ordinate must be applied as base. The full phrase states that the ordinates have their squares equal to the rectangles applied to the line equal to N (or the parameter) which have as their breadth the lines which they (the ordinates) cut off from AZ (the diameter) towards the extremity A, δυνάνται τὰ παρὰ τὰν ἴσαν τᾷ N παραπίπτοντα πλάτος ἔχοντα, as αὐταὶ ἀπολαμβάνοντι ἀπὸ τᾶς ΔΖ ποτὶ τὸ Δ πέρας.
Ordinates are the lines drawn from the section to the diameter (of the segment) parallel to the base (of the segment) ai ἀπὸ τᾶς Topas ἐπὶ τὰν AZ ἀγομέναι παρὰ τὰν AE, or simply ai ἀπὸ τᾶς touds. Once also the regular phrase drawn ordinate-wise τεταγμένως κατηγμένη 15 used to describe an ordinate, as in Apollonius.
The hyperbola.
What we call the asymptotes (ai ἀσύμπτωτοι in Apollonius) are in Archimedes the lines (approaching) nearest to the section of the obtuse-angled cone ai ἔγγιστα τᾶς τοῦ ἀμβλυγωνίου κώνου τομάς.
The centre is not described as such, but it is the point at which the lines nearest (to the curve) meet τὸ σαμεῖον, καθ᾽ ὃ ai ἔγγιστα συμπιίιπτοντι.
This is ὦ property of the sections of obtuse-angled cones τοῦτο γάρ
“ na ’ a , ἐστιν ἐν ταῖς Tov ἀμβλυγωνίου κώνου τομαῖς σύμπτωμα.
The ellipse.
The major and minor axes are the greater and lesser diameters μείζων and ἐλάσσων διάμετρος. Let the greater diameter be AT, διάμετρος δὲ (αὐτᾶς) ἃ μὲν μείζων ἔστω ἐφ᾽ ἃς τὰ A,T. The rectangle contained by the diameters (axes) τὸ περιεχόμενον ὑπὸ τᾶν διαμέτρων. One axis is called conjugate (συζυγής) to the other: thus let the straight line N be equal to half of the other diameter which is conjugate to AB, ἁ δὲ Ν εὐθεῖα ἴσα ἔστω τᾷ ἡμισείᾳ τᾶς ἑτέρας διαμέτρου, ἅ ἐστι συζυγὴς τᾷ ΑΒ.
The centre is here κέντρον.
* Cf. Apollonius of Perga, pp. xxiv, xxv.
THE TERMINOLOGY OF ARCHIMEDES. clxix
Conoids and Spheroids.,
There is a remarkable similarity between the language in which Archimedes describes the genesis of his solids of revolution and that used by Euclid in defining the sphere. Thus Euclid says: when, the diameter of a semicircle remaining fixed, the semicircle revolves and returns to the same position from which it began to move, the included Jigure is a sphere σφαῖρά ἐστιν, ὅταν ἡμικυκλίου μενούσης τῆς διαμέτρου περιενεχθὲν τὸ ἡμικύκλιον εἰς τὸ αὐτὸ πάλιν ἀποκατασταθῇ, ὅθεν ἤρξατο φέρεσθαι, τὸ περιληφθὲν σχῆμα; and he proceeds to state that the axis of the sphere is the fixed straight line about which the semicircle turns ἄξων δὲ τῆς σφαίρας ἐστὶν ἡ μένουσα εὐθεῖα, περὶ ἣν τὸ ἡμικύκλιον στρέφεται. Compare with this e.g. Archimedes’ definition of the right-angled conoid (paraboloid of revolution): if ὦ section of a right-angled cone, with its diameter (axis) remaining fixed, revolves and returns to the position from which it started, the figure included by the section of the right-angled cone is called a right-angled conoid, and its axis is defined as the diameter which has remained fixed, εἴ Ka ὀρθογωνίου κώνου τομὰ μενούσας τᾶς διαμέτρου περιενεχθεῖσα ἀποκατασταθῇ πάλιν, ὅθεν ὥρμασεν, τὸ περιλαφθὲν σχῆμα ὑπὸ τᾶς τοῦ ὀρθογωνίου κώνου τομᾶς ὀρθογώνιον κωνοειδὲς καλείσθαι, καὶ ἄξονα μὲν αὐτοῦ τὰν μεμενακοῦσαν διάμετρον καλείσθαι, and it will be seen that the several phrases used are practically identical with those of Euclid, except that ὥρμασεν takes the place of ἤρξατο φέρεσθαι; and even the latter phrase occurs in Archimedes’ description of the genesis of the spiral later on.
The words conoid κωνοειδὲς (σχῆμα) and spheroid σφαιροειδὲς (σχῆμα) are simply adapted from κῶνος and σφαῖρα, meaning that the respective figures have the appearance (εἶδος) of, or resemble, cones and spheres; and in this respect the names are perhaps more satisfactory than paraboloid, hyperboloid and ellipsoid, which can only be said to resemble the respective conics in a different sense. But when xwvoedés is qualified by the adjective right-angled ὀρθογώνιον to denote the paraboloid of revolution, and by ἀμβλυ- γώνιον obtuse-angled to denote the hyperboloid of revolution, the expressions are less logical, as the solids do not resemble right- angled and obtuse-angled cones respectively; in fact, since the angle between the asymptotes of the generating hyperbola may be acute, a hyperboloid of revolution would in that case more resemble an acute-angled cone. The terms right-angled and obtuse-angled
elxx INTRODUCTION.
were merely transferred to the conoids from the names for the respective conics without any more thought of their meaning.
It is unnecessary to give separately the definition of each conoid and spheroid; the phraseology is in all cases the same as that given above for the paraboloid. But it may be remarked that Archimedes does not mention the conjugate axis of a hyperbola or the figure obtained by causing a hyperbola to revolve about that axis ; the conjugate axis of a hyperbola first appears in Apollonius, who was apparently the first to conceive of the two branches of a hyperbola as one curve. Thus there is only one obtuse-angled conoid in Archimedes, whereas there are two kinds of spheroids according as the revolution takes place about the greater diameter (axis) or lesser diameter of the generating section of an acute- angled cone (ellipse); the spheroid is in the former case oblong (παραμᾶκες σφαιροειδές) and in the latter case flat (ἐπιπλατὺ σφαιροειδές).
A special feature is, however, to be observed in the description of the obtuse-angled conoid (hyperboloid of revolution), namely that the asymptotes of the hyperbola are supposed to revolve about the axis at the same time as the curve, and Archimedes explains that they will include an isosceles cone (κῶνον ἰσοσκελέα περιλαψούνται), which he thereupon defines as the cone enveloping the conoid (περιέχων τὸ κωνοειδές). Also in a spheroid the term diameter (διάμετρος) is appropriated to the straight line drawn through the centre at right angles to the awis (ἃ διὰ τοῦ κέντρου ποτ᾽ ὀρθὰς ἀγομένα τῷ agovr). The centre of a spheroid is the middle point of the axis τὸ μέσον τοῦ a€ovos.
The following terms are used of all the conoids and spheroids. The vertex (κορυφή) is the point at which the axis meets the surface τὸ σαμεῖον, καθ᾽ ὃ ἁπτέται 6 ἀξων τᾶς ἐπιφανείας, the spheroid having of course two vertices. A segment (τμᾶμαλ) is a part cut off by a plane, and the base (βάσις) of the segment is defined as the plane ( figure) included by the section of the conoid (or spheroid) in the cutting plane τὸ ἐπίπεδον τὸ περιλαφθὲν ὑπὸ τᾶς τοῦ κωνοειδέος (Or σφαιροειδέος) τομάς ἐν τῷ ἀποτέμνοντι ἐπιπέδῳ. The vertex of a segment is the point at which the tangent plane parallel to the base of the segment meets the surface, τὸ capetov, καθ᾽ ὃ ἁπτέται τὸ ἐπίπεδον τὸ ἐπιψαῦον (τοῦ κωνοειδέος). The axis (agwv) of a segment is differently defined for the three surfaces ; (a) in the paraboloid it is the straight line cut off within the segment from the line drawn through the vertex of the
THE TERMINOLOGY OF ARCHIMEDES. elxxi
segment parallel to the axis of the conoid ἃ ἐναπολαφθεῖσα εὐθεῖα ἐν τῷ τμάματι ἀπὸ τᾶς ἀχθείσας διὰ τᾶς κορυφᾶς τοῦ τμάματος παρὰ TOV ἄξονα τοῦ κωνοειδέος, (6) in the hyperboloid it is the straight line cut off within the segment from the line drawn through the vertex of the segment and the vertex of the cone enveloping the conoid ἀπὸ tas ἀχθείσας διὰ Tas κορυφᾶς τοῦ τμάματος καὶ τᾶς κορυφᾶς τοῦ κώνου TOD περιέχοντος τὸ κωνοειδές, (6) in the spheroid it is the part similarly cut off from the straight line joining the vertices of the two segments into which the base divides the spheroid, ἀπὸ τᾶς εὐθείας τᾶς τὰς κορυφὰς αὐτῶν (τῶν τμαμάτων) ἐπιζευγνυούσας.
Archimedes does not use the word centre with respect to the hyperboloid of revolution, but calls the centre the vertex of the enveloping cone. Also the axis of a hyperboloid or a segment is only that part of it which is within the surface. The distance between the vertex of the hyperboloid or segment and the vertex of the enveloping cone is the line adjacent to the axis ἃ ποτεοῦσα τῷ ἀξονι. ᾿
The following are miscellaneous expressions. Zhe part inter- cepted within the conoid of the intersection of the planes ἃ ἐναπο- λαφθεῖσα ἐν τῷ κωνοειδεῖ Tas γενομένας τομᾶς τῶν ἐπιπέδων, (the plane) will have cut the spheroid through its axis τετμακὸς ἐσσείται τὸ σφαιροειδὲς διὰ τοῦ ἀξονος, so that the section it makes will be a conic section ὥστε τὰν τομὰν ποιήσει κώνου Tomar, let two segments be cut off in any manner ἀποτετμάσθω δύο τμάματα ws ἔτυχεν or by planes drawn in any manner ἐπιπέδοις ὁπωσοῦν aypévoss.
Half the spheroid τὸ ἁμίσεον tod σφαιροειδέος, half the line joining the vertices of the segments (of a spheroid), i.e. what we should call a semi-diameter, a ἡμισέα αὐτᾶς tas ἐπιζευγνυούσας τὰς κορυφὰς
“ , τῶν τμαμαάτων.
The spiral.
We have already had, in the conoids and spheroids, instances of the evolution of figures by the motion of curves about an axis. The same sort of motion is used for the construction of solid figures inscribed in and circumscribed about a sphere, a circle and an inscribed or circumscribed polygon being made to revolve about a diameter passing through an angular point of the polygon and dividing it and the circle symmetrically, In this case, in Archimedes’ phrase, the angular points of the polygon will move along the circwm- Jerences of circles, ai γωνίαι κατὰ κύκλων περιφερειῶν ἐνεχθήσονται (or
elxxl INTRODUCTION.
οἰσθήσονται) and the sides will move on certain cones, or on the surface of a cone κατά τινων κώνων ἐνεχθήσονται or κατ᾽ ἐπιφανείας κώνου ; and sometimes the angular points or the points of contact of the sides of a circumscribed polygon are said to describe circles γράφουσι κύκλους. The solid figure so formed is τὸ γενηθὲν στερεὸν σχῆμα, and let the sphere by its revolution make a figure περιενεχθεῖσα ἡ σφαῖρα ποιείτω σχῆμά τι.
For the construction of the spiral, however, we have a new element introduced, that of time, and we have two different uniform motions combined ; if a straight line in a plane turn uniformly about one extremity which remains fixed, and return to the position Jrom which ut started and if, at the same time as the line is revolving, a point move at a uniform rate along the line starting from the fixed extremity, the point will desoribe a spiral in the plane, εἴ κα εὐθεῖα... ἐν ἐπιπέδῳ... μένοντος τοῦ ἑτέρου πέρατος αὐτᾶς ἰσοταχέως περιενεχθεῖσα ἀποκατασταθῇ πάλιν, ὅθεν ὥρμασεν, ἅμα δὲ τᾷ γραμμᾷ περιαγομένᾳ φερήται τι σαμεῖον ἰσοταχέως αὐτὸ ἑαυτῷ κατὰ τᾶς εὐθείας ἀρξάμενον ἀπὸ τοῦ μένοντος πέρατος, τὸ σαμεῖον ἕλικα γράψει ἐν τῷ ἐπιπέδῳ.
The spiral (described) in the first, second, or any turn is & ἕλιξ ἃ ἐν τᾷ πρώτᾳ, δευτέρᾳ, OY ὁποιᾳοῦν περιφορᾷ γεγραμμένα, and the turns other than any particular ones are the other spirals at ἀλλαι ἕλικες.
The distance traversed by the point along the line in any time is a εὐθεῖα a διανυσθεῖσα, and the times in which the point moved over the distances οἱ χρόνοι, ἐν οἷς τὸ σαμεῖον Tas γραμμὰς ἐπορεύθη ; in the tume in which the revolving line reaches AT from AB, ἐν ᾧ χρόνῳ ἃ περιαγομένα γραμμὰ ἀπὸ τᾶς AB ἐπὶ τὰν AT ἀφικνείται.
The origin of the spiral is ἀρχὰ τᾶς ἕλικος, the intial line ἀρχὰ τᾶς περιφορᾶς. The distance described by the point along the line in the first complete revolution is εὐθεῖα πρώτα (first distance), that described during the second revolution the second distance εὐθεῖα δευτέρα, and so on, the distances being called by the number of the revolutions ὁμωνύμως ταῖς περιφοραῖς. The first area, χωρίον πρῶτον, is the area bounded by the spiral described in the first revolution and by the ‘first distance’ τὸ χωρίον τὸ περιλαφθὲν ὑπό τε Tas ἕλικος τᾶς ἐν τᾷ πρώτᾳ περιφορᾷ γραφείσας καὶ τᾶς εὐθείας, a ἐστιν πρώτα : the second area is that bounded by the spiral in the second turn and the ‘second distance,’ and so on. The area added by the spiral in any turn is τὸ χωρίον τὸ ποτιλαφθὲν ὑπὸ τᾶς ἕλικος ἔν τινι περιφορᾷ.
The first circle, κύκλος πρῶτος, is the circle described with the ‘first distance’ as radius and the origin as centre, the second circle
THE TERMINOLOGY OF ARCHIMEDES. elxxili
that with the origin as centre and twice the ‘first distance’ as radius, and so on.
Together with as many times the whole of the circumference of the circle as (is represented by) the number less by one than (that of) the revolutions μεθ᾽ ὅλας τᾶς τοῦ κύκλου περιφερείας τοσαυτάκις Aap- βανομένας, ὅσος ἐστὶν ὃ ἑνὶ ἐλάσσων ἀριθμὸς τᾶν περιφορᾶν, the cirele called by the number corresponding to that of the revolutions ὃ κύκλος 6 κατὰ τὸν αὐτὸν ἀριθμὸν λεγόμενος ταῖς περιφοραῖς.
With reference to any radius vector, the side which is in the direction of the revolution is forward τὰ προαγούμενα, the other backward τὰ ἑπόμενα.
Tangents, etc.
Though the word ἅπτομαι is sometimes used in Archimedes of a line touching a curve, its general meaning is not to touch but simply to meet; e.g. the axis of a conoid or spheroid meets (ἅπτεται) the surface in the vertex. (The word is also often used elsewhere than in Archimedes of points /ying on a locus ; e.g. in Pappus, p. 664, the point will lie on a straight line given in position ἅψεται τὸ σημεῖον θέσει δεδομένης εὐθείας.)
To touch a curve or surface is generally ἐφάπτεσθαι or ἐπιψαύειν (with gen.). A tangent is ἐφαπτομένη or ἐπιψαύουσα (sc. εὐθεῖα) and a tangent plane érupatov ἐπίπεδον. Let tangents be drawn to the circle ABD, τοῦ ΑΒΤ' κύκλου ἐφαπτόμεναι ἤχθωσαν; if straight lines be drawn touching the circles ἐὰν. ἀχθῶσίν τινες ἐπιψαύουσαι τῶν κύκλων. The full phrase of touching without cutting is sometimes found in Archimedes; if a plane touch (any of) the conoidal figures without cutting the conoid εἴ κα τῶν κωνοειδέων σχημάτων ἐπίπεδον ἐφαπτήται μὴ τέμνον τὸ κωνοειδές. The simple word ψαύειν is occasionally used (participially), the tangent planes τὰ ἐπίπεδα τὰ ψαύοντα. .
To touch αὐ a point is expressed by κατά (with acc.) ; the points at which the sides...touch (or meet) the circle σημεῖα, καθ᾽ ἃ ἅπτονται τοῦ κύκλου ai πλευραί.... Let them touch the circle at the middle points of the circumferences cut off by the sides of the inscribed polygon ἐπιψαυέτωσαν τοῦ κύκλου κατὰ μέσα τῶν περιφερειῶν τῶν ἀποτεμνομένων ὑπὸ τοῦ ἐγγεγραμμένου πολυγώνου πλευρών.
The distinction between ἐπιψαύειν and ἅπτομαι is well brought out in the following sentence; but that the planes touching the spheroid meet its surface at one point only we shall prove ὅτι δὲ
clxxiv INTRODUCTION.
τὰ ἐπιψαύοντα ἐπίπεδα τοῦ σφαιροειδέος καθ᾽ ἕν μόνον ἁπτόνται σαμεῖον τᾶς ἐπιφανείας αὐτοῦ δειξοῦμες.
The point of contact ἡ ἁφή.
Tangents drawn from (a point) ἀγμέναι ἀπό; we find also the elliptical expression ἀπὸ τοῦ Ξὶ ἐφαπτέσθω ἡ OT, let ΟΞΠ be the tangent from &, where, in the particular case, = is on the circle.
Constructions.
The richness of the Greek language in expressions for con-- structions is forcibly illustrated by the variety of words which may be used (with different shades of meaning) for drawing a line. Thus we have in the first place ayw and the compounds διάγω (of drawing a line through a figure, with εἰς or ἐν following, of producing a plane beyond a figure, or of drawing a line in ἃ plane), κατάγω (used of drawing an ordinate down from a point on a conic), προσάγω (of drawing a line ¢o meet another). As an alternative to προσάγω, προσβάλλω is also used; and προσπίπτω may take the place of the passive of either verb. To produce is ἐκβάλλω, and the same word is also used of a plane drawn through a point or through a straight line; an alternative for the passive 1s supplied by ἐκπίπτω. Moreover πρόσκειμαι is an alternative word for being produced (literally being added).
In the vast majority of cases constructions are expressed by the elegant use of the perfect imperative passive (with which may be classed such forms as γεγονέτω from γίγνομαι, ἔστω from εἰμί, and κείσθω from κεῖμαι), or occasionally the aorist imperative passive. The great variety of the forms used will be understood from the following specimens. Let BI be made (or supposed) equal to A, κείσθω τῷ Δ ἴσον τὸ BY; let it be drawn ἤχθω, let ὦ straight line be drawn in it (a chord of a circle) διήχθω τις εἰς αὐτὸν εὐθεῖα, let KM be drawn equal to... ton κατήχθω ἡ KM, let it be joined ἐπεζεύχθω, let KA be drawn to meet προσβεβλήσθω ἡ KA, let them be produced ἐκβεβλήσθωσαν, suppose them found εὑρήσθωσαν, let a circle be set out ἐκκείσθω κύκλος, let it be taken εἰλήφθω, let K, H be taken ἔστωσαν εἰλημμέναι at K, H, let a circle Ψ be taken χελάφθω κύκλος ἐν ᾧ τὸ Ψ, let it be cut τετμήσθω, let it be divided διαιρήσθω (διῃρήσθω) ; let one cone be cut by a plane parallel to the base and produce the section EZ, τμηθήτω 6 ἕτερος κῶνος ἐπιπέδῳ παραλλήλῳ τῇ βάσει καὶ ποιείτω τομὴν τὴν BZ, let TZ be cut off ἀπολελάφθω ἁ TZ; let (such an angle) be left and let rt be NHI, λελείφθω καὶ ἔστω ἡ ὑπὸ NHI, let a figure be made γεγενήσθω
THE TERMINOLOGY OF ARCHIMEDES. elxxv
σχῆμα, let the sector be made ἔστω γεγενημένος ὁ τομεύς, let cones be described on the circles (as bases) ἀναγεγράφθωσαν ἀπὸ τῶν κύκλων κῶνοι, ἀπὸ TOD κύκλου κῶνος ἔστω, let it be inscribed or circumscribed ἐγγεγράφθω (or ἐγγεγραμμένον ἔστω), περιγεγράφθω ; let an area (equal to that) of AB be applied to AH, παραβεβλήσθω rapa τὰν AH τὸ χωρίον τοῦ AB; let a segment of a circle be described on OK, ἐπὶ τῆς OK κύκλου τμῆμα ἐφεστάσθω, let the circle be completed ἀναπεπληρώσθω ὃ κύκλος, let NE (a parallelogram) be completed συμπεπληρώσθω τὸ NE, let it be made πεποιήσθω, let the rest of the construction be the same as before τὰ ἄλλα κατεσκευάσθω τὸν αὐτὸν τρόπον τοῖς πρότερον. Swppose it done γεγονέτω.
Another method is to use the passive imperative of νοέω (/et it be conceived). Let straight lines be conceived to be drawn νοείσθωσαν εὐθεῖαι nypevat, let the sphere be conceived to be cut νοείσθω ἡ σφαῖρα τετμημένη, let a figure (generated) from the inscribed polygon be conceived as inscribed in the sphere ἀπὸ τοῦ πολυγώνου τοῦ ἐγγραφο- μένου νοείσθω τι εἰς τὴν σφαῖραν ἐγγραφὲν σχῆμα. Sometimes the participle for drawn is left out; thus ἀπ᾽ αὐτοῦ νοείσθω ἐπιφάνεια Let a surface be concewved (generated) from tt.
The active is much more rarely used; but we find (1) ἐὰν with subjunctive, 1f we cut ἐὰν τέμωμεν, if we draw ἐὰν ἀγάγωμεν, if you produce ἐὰν ἐκβαλῇς ; (2) the participle, ἐέ is possible to inscribe...and (ultimately) to leave δυνατόν ἐστιν ἐγγράφοντα...λείπειν, if we con- tinually cirewmscribe polygons, bisecting the remaining circumferences and drawing tangents, we shall (ultimately) leave ἀεὶ δὴ περιγράφοντες πολύγωνα δίχα τεμνομένων TOV περιλειπομένων περιφερειῶν καὶ ἀγομένων ἐφαπτομένων λείψομεν, it is possible, if we take the area..., to inscribe λαβόντα (or λαμβάνοντα) τὸ Xwpiov...dvvarov ἐστιν... ἐγγράψαι ; (3) the first person singular, J take two straight lines λαμβάνω δύο εὐθείας, 1] took a straight line ἔλαβόν twa εὐθεῖαν ; 7 draw ΘΜ from ® parallel to AZ, ἄγω ἀπὸ τοῦ Θ τὰν ΘΜ παράλληλον τᾷ AZ, having drawn TK perpendicular, I cut off AK equal to TK ἀγαγὼν κάθετον τὰν TK τᾷ TK ἴσαν ἀπέλαβον τὰν AK, 7 inscribed a solid figure...and cirewm- scribed another ἐνέγραψα σχῆμα oTepeov...kat ἄλλο περιέγραψα.
The genitive of the passive participle is used absolutely, εὑρεθέντος δή it being supposed found, ἐγγραφέντος δή (the figure) being inscribed.
To make a figure similar to one (and equal to another) δμοιώσαι,
to find experimentally ὀργανικῶς λαβεῖν, to cut into unequal parts εἰς ἄνισα τέμνειν.
clxxvi INTRODUCTION.
Operations (addition, subtraction, etc.).
1. Addition, and sums, of magnitudes.
To add is προστίθημι, for the passive of which πρόσκειμαι is often used ; thus one segment being added ἑνὸς τμάματος ποτιτεθέντος, the added (straight line) ἃ ποτικειμένα, let the common HA, Ζ1 be added Kowal προσκείσθωσαν ai HA, ZI; the words are generally followed by πρός (with acc. of the thing added ἔθ), but sometimes by the dative, that to which the addition was made ᾧ ποτετέθη.
For being added together we have συντίθεσθαι ; thus being added to itself συντιθέμενον αὐτὸ ἑαυτῷ, added together és τὸ αὐτὸ συντεθέντα, added to itself (continually) ἐπισυντιθέμενον ἑαυτῷ.
Sums are commonly expressed for two magnitudes by συναμφό- repos used in the following different ways; the swum of BA, AA συναμφότερος ἡ BAA, the sum of AT, TB συναμφότερος ἡ AT, TB, the sum of the area and the circle τὸ συναμφότερον 6 τε κύκλος καὶ TO χωρίον. Again for swms in general we have such expressions as the line which is equal to both the radii ἡ ἴση ἀμφοτέραις ταῖς ἐκ τοῦ κέντρου, the line equal to (the swum of) all the lines joining ἡ ton πάσαις ταῖς ἐπιζευγνυούσαις. Also all the circles οἱ πάντες κύκλοι means the sum of all the circles; and σύγκειται ἐκ is used for is equal to the sum of (two other magnitudes).
To denote plus μετά (with gen.) and σύν are. used ; together with the bases μετὰ τῶν βάσεων, together with half the base of the segment σὺν τῇ ἡμισείᾳ τῆς TOD τμήματος βάσεως ; Te and καί also express the same thing, and the participle of προσλαμβάνω gives another way of describing having something added to it; thus the squares on (all) the lines equal to the greatest together with the square on the greatest... is τὰ τετράγωνα τὰ ἀπὸ τᾶν ἰσᾶἂν τᾷ μεγίστᾳ ποτιλαμβάνοντα τό τε ἀπὸ
τᾶς μεγίστας τετρ ἄγωνον.....
2. Subtraction and differences.
To subtract from is ἀφαιρεῖν ἀπό; if (the rhombus) be conceived as taken away ἐὰν νοηθῇ ἀφῃρημένος, let the segments be subtracted ἀφαιρεθέντων τὰ τμήματα. Terms common to each side in an equation are κοινᾶ ; the squares are common to both (sides) κοινά eVTL ἑκατέρων τὰ tetpaywva. Then let the common area be subtracted is κοινὸν ἀφῃρήσθω τὸ χωρίον, and so on; the remainder is denoted by the adjective λοιπός, e.g. the conical surface remaining λοιπὴ ἡ κωνικὴ ἐπιφάνεια.
The difference or excess is ὑπεροχή, or more fully the eacess by
THE TERMINOLOGY OF ARCHIMEDES. elxxvil
which (one magnitude) exceeds (another) ὑπεροχή, ἧ ὑπερέχει... OF ὑπεροχά, ἃ μείζων ἐστί.... The excess is also expressed by means of the verb ὑπερέχειν alone ; let the difference by which the said triangles exceed the triangle AAT be ©, ᾧ δὴ ὑπερέχει τὰ εἰρημένα τρίγωνα τοῦ AAT τριγώνου ἔστω τὸ Θ, to exceed by less than the excess of the cone W over the half of the spheroid ὑπερέχειν ἐλάσσονι ἢ ᾧ (or ἁλίκῳ) ὑπερέχει ὃ Ψ κῶνος τοῦ ἡμίσεος TOD σφαιροειδέος (Where ᾧ ὑπερέχει May also be omitted). Again the excess may be ᾧ μείζων ἐστί. The opposite to ὑπερέχει is λείπεται (with gen.).
Equal to twice a certain excess ἴσα δυσὶν ὑπεροχαῖς, with which equal to one excess, ἴσα μιᾷ ὑπεροχᾷ, is contrasted.
The following sentence practically states the equivalent of an algebraical equation ; the rectangle under ZH, BA exceeds the rect- angle under ZE, EA by the (sum of) the rectangle contained by BA, EH and the rectangle under ZE, BE, ὑπερέχει τὸ ὑπὸ τἂν ZH, BA τοῦ ὑπὸ τᾶν ΖΕ), EA τῷ τε ὑπὸ τἂν ΞΔ, EH περιεχομένῳ καὶ τῷ ὑπὸ τᾶν ΖΕ, HE. Similarly twice PH together with ILS is (equal to) the swum of ΣΡ, PH, δύο μὲν ai PH μετὰ τᾶς ΠΣ συναμφότερός ἐστιν ἃ PII.
3. Multiplication.
To multiply is πολλαπλασιάζω; multiply one another (of numbers) πολλαπλασιάζειν ἀλλάλους ; to multiply by a number is expressed by the dative ; let A be multiplied by Θ πεπολλαπλασιάσθω 6 A τῷ ©.
Multiplied into is sometimes ἐπί (with acc.); thus the rectangle HO, ΘΑ into ΘΑ (1.6. a solid figure) is τὸ ὑπὸ τῶν HO, ΘΑ ἐπὶ τὴν ΘΑ.
"4, Diwision.
To divide διαιρεῖν ; let it be divided into three equal parts at the points K, Θ, διῃρήσθω εἰς τρία ἴσα κατὰ τὰ K, © σαμεῖα ; to be divisible by μετρεῖσθαι ὑπό.
Proportions.
A ratio is λόγος, proportional is expressed by the phrase im proportion ἀνάλογον, and a proportion is ἀναλογία. We find in Archimedes some uses of the verb λέγω which seem to throw light on the definition found in Euclid of the relation or ratio between two magnitudes. One passage (On Conoids and Spheroids, Prop. 1) says if the terms similarly placed have, two and two, the same ratio and the first magnitudes are taken in relation to some other mag- nitudes in any ratios whatever εἴ κα κατὰ δύο τὸν αὐτὸν λόγον ἔχωντι
ἘΠ. Ἃ. m
elxxvili INTRODUCTION.
τὰ ὁμοίως τεταγμένα, λεγήται δὲ TA πρῶτα μεγέθεα ποτί τινα ἄλλα μεγέθεα...ἐν λόγοις ὁποιοισοῦν, if A, B... be in relation to N, Z... but Z be not in relation to anything (i.e. has no term corresponding to it) εἴ κα... τὰ μὲν A, B,... Aeywvrar ποτὶ τὰ N, By... τὸ δὲ Z μηδὲ ποθ᾽ ἕν λεγήται.
A mean proportional between is μέση ἀνάλογον τῶν..., is ὦ mean proportional between μέσον λόγον ἔχει THS...Kal THS..., two mean pro- portionals δύο μέσαι ἀνάλογον with or without κατὰ τὸ συνεχές in continued proportion.
If three straight lines be proportional ἐὰν τρεῖς εὐθεῖαι ἀνάλογον ὦσι, a fourth proportional τετάρτα ἀνάλογον, if four straight lines be proportional in continued proportion εἴ κα τέσσαρες γραμμαὶ ἀνάλογον ἔωντι ἐν τᾷ συνεχεῖ ἀναλογίᾳ, at the ‘point dividing (the line) in the said proportion κατὰ τὰν ἀνάλογον τομὰν τᾷ εἰρημένᾳ.
The ratio of one straight line to another is e.g. ὁ τῆς PA πρὸς AX λόγος or ὁ (λόγος), dv ἔχει ἡ PA πρὸς τὴν AX; the ratio of the bases ὃ τῶν βασίων λόγος ; has the ratio of 5 to 2 λόγον ἔχει, ὃν πέντε πρὸς δύο.
For having the same ratio as we find the following constructions. Have the same ratio to one another as the bases τὸν αὐτὸν ἔχοντι λόγον mot ἀλλάλους ταῖς βάσεσιν, as the squares on the radi ὃν ai ἐκ τῶν κέντρων δυνάμει ; TA has to PZ the (linear) ratio which the square on TA has to the square on H, ov ἔχει λόγον ἡ TA πρὸς τὴν H δυνάμει, τοῦτον ἔχει Tov λόγον ἡ TA πρὸς PZ μήκει. 18 dwided wn the same ratio εἰς τὸν αὐτὸν λόγον τέτμηται, Or Simply ὁμοίως ; will divide the diameter in the proportion of the successive odd numbers, unity corresponding to the (part) adjacent to the vertex of the segment τὰν διάμετρον τεμοῦντι εἰς τοὺς τῶν ἑξῆς περισσών ἀριθμῶν λόγους, ἑνὸς λεγομένου ποτὶ τᾷ κορυφᾷ τοῦ τμάματος.
To have a less (or greater) ratio than is ἔχειν λόγον ἐλάσσονα (or μείζονα) with the genitive of the second ratio or a phrase introduced by 7; to have a less ratio than the greater magnitude has to the less, ἔχειν λόγον ἐλάσσονα ἢ τὸ μεῖζον μέγεθος πρὸς TO ἔλασσον.
For duplicate, triplicate etc. ratios we have the following expressions: has the triplicate ratio of the same ratio τριπλασίονα λόγον ἔχει TOD αὐτοῦ λόγου, has the duplicate ratio of EA to AK διπλασίονα λόγον ἔχει ἥπερ ἡ EA πρὸς AK, are in the triplicate ratio of the diameters in the bases ἐν τριπλασίονι λόγῳ εἰσὶ τῶν ἐν ταῖς βάσεσι διαμέτρων, sesquialterate ratio ἡμιόλιος λόγος. With these expressions must be contrasted the use of double, quadruple οἷο.
THE TERMINOLOGY OF ARCHIMEDES. clxxix
ratio in the sense of a simple multiple by 2, 4 etc, eg. of any number of areas be placed in order, each being four times the next εἴ κα χωρία τεθέωντι ἑξῆς ὁποσαοῦν ἐν τῷ τετραπλασίονι λόγῳ.
The ordinary expression for a proportion is as A is to B so ws T to A, ws ἡ A πρὸς τὴν B, οὕτως ἡ Τὶ πρὸς τὴν A. Let AE be made so that AE is to TE as the sum of ΘΑ, AE ts to AE, πεποιήσθω, ὡς συναμφότερος ἡ ΘΑ, AE zpos τὴν AE, οὕτως ἡ AE zpos TE. The antecedents are τὰ ἡγούμενα, the consequents τὰ ἑπόμενα.
For reciprocally proportional the parts of ἀντιπέπονθα are used ; the bases are reciprocally proportional to the heights ἀντιπεπόνθασιν ai βάσεις ταῖς ὕψεσιν, to be reciprocally in the same proportion ἀντιπεπονθέμεν κατὰ τὸν αὐτὸν λόγον.
A ratio compounded of is λόγος συνημμένος (or συγκείμενος) ἔκ TE TOv...Kai Tov...; the ratio of PA to AX ἐδ equal to that compounded of 6 τῆς PA πρὸς AX λόγος συνῆπται éx.... Two other expressions for compounded ratios are 6 τοῦ ἀπὸ ΑΘ -πρὸς τὸ ἀπὸ BO καὶ ὃ (or προσλαβὼν τὸν) τῆς ΑΘ πρὸς ΘΒ, the ratio of the square on A® to the square on BO multiplied by the ratio of ΑΘ to ΘΒ.
The technical terms for transforming such a proportion as a:b=c:d are as follows:
1, ἐναλλάξ alternately (usually called permutando or alternando) means transforming the proportion into a :c=6: d.
2. ἀνάπαλιν reversely (usually invertendo), ὦ : ὦ -- (ὦ : 6.
9. σύνθεσις λόγου is composition of a ratio by which the ratio a:b becomes a+6:6. The corresponding Greek term to com- ponendo is συνθέντι, which means no doubt literally “to one who has compounded,” i.e. “if we compound,” the ratios. Thus συνθέντι denotes the inference thata+6b:b=c+d:d. κατὰ σύνθεσιν is also used in the same sense by Archimedes.
4. διαίρεσις λόγου signifies the division of a ratio in the sense of separation or subtraction by which a : ὦ becomes ὦ ---ὦ : ὦ. Similarly διελόντι (or κατὰ διαίρεσιν) denotes the inference that a—b:b= c—d:d. The translation diwidendo is therefore somewhat mis- leading.
5. ἀναστροφὴ λόγου conversion of a ratio and ἀναστρέψαντι correspond respectively to the ratio a : ὦ -- ὦ and to the inference thata:a—b=c:c-—d.
clxxx INTRODUCTION.
6. διέ ἴσου ex aequali (sc. distantia) is applied e.g. to the inference from the proportions
a:6:c:detec..=A:B:C:D etc. that a:a—A - ἢ.
When this dividing-out of ratios takes place between proportions with corresponding terms placed crosswise, it is described as δ ἴσου ἐν τῇ τεταραγμένῃ ἀναλογίᾳ, ex aequali im disturbed proportion or ἀνομοίως τῶν λόγων τεταγμένων the ratios being dissimilarly placed ; this is the case e.g. when we have two proportions
Cro=BsE eC Dc — ANG, and we infer that a:c=A:C.
Arithmetical terms.
Whole multiples of any magnitude are generally described as the double of, the triple of etc., ὃ διπλάσιος, ὃ τριπλάσιος κ-τ.λ., following the gender of the particular magnitude ; thus the (surface which is) four times the greatest circle in the sphere 4 τετραπλασία τοῦ μεγίστου κύκλου τῶν ἐν τῇ σφαίρᾳ ; five times the sum of AB, BE together with ten times the swum of ΓΒ, BA, ἃ πενταπλασία συναμφοτέρου tas AB, BE μετὰ τᾶς δεκαπλασίας συναμφοτέρου τᾶς TB, BA. The same multiple as τοσαυταπλασίων... ὁσαπλασίων ἐστί, or ἰσάκις πολλαπλασίων... καί. The general word for a multiple of is πολλαπλάσιος or πολλαπλασίων, which may be qualified by any expression denoting the number of times multiplied ; thus multiplied by the same number πολλαπλάσιος τῷ αὐτῷ ἀριθμῷ, multiples .according to the successive mumbers πολλαπλάσια κατὰ τοὺς ἑξῆς ἀριθμούς.
Another method is to use the adverbial forms twice dis, thrice τρίς, etc., which are either followed by the nominative, e.g. twice EA δὶς ἡ EA, or constructed with a participle, e.g. twice taken dis Nap- Bavopevos or δὶς εἰρημένος ; together with twice the whole circumference of the circle μεθ᾽ ὅλας τᾶς τοῦ κύκλου περιφερείας Sis λαμβανομένας. Similarly the same number of times (the said circumference) as is expressed by the number one less than (that of) the revolutions τοσαυτάκις λαμβανομένας, ὅσος ἐστὶν ὃ ἑνὶ ἐλάσσων ἀριθμὸς τᾶν περιφορᾶν. An interesting phrase is the following, as many times as the line Τ' Δ is contained (literally added together) in AA, so many temes let the time ZH be contained in the time AH, ὁσάκις συγκείται ἃ TA
THE TERMINOLOGY OF ARCHIMEDES. elxxx1
γραμμὰ ἐν τᾷ AA, τοσαυτάκις συγκείσθω ὃ χρόνος ὃ ZH ἐν τῷ χρόνῳ τῷ ΛΗ.
Submultiples are denoted by the ordinal number followed by μέρος ; one-seventh is ἕβδομον μέρος and so on, one-half being however ἥμισυς. When the denominator is a large number, a circumlocutory phrase is used; thus less than ;};th part of a right angle ἐλάττων ἢ διαιρεθείσας Tas ὀρθᾶς εἰς ρξδ΄ τούτων ἕν μέρος.
When the numerator of a fraction is not unity, it is expressed by the ordinal number, and the denominator by a compound substantive denoting such and such a submultiple; e.g. two-thirds δύο τριταμόρια, three-fifths τρία πεμπταμόρια.
There are two improper fractions which have special names, thus one-and-a-half of is ἡμιόλιος, one-and-a-third of ἐπίτριτος. Where a number is partly integral and partly fractional, the integer is first stated and the fraction follows introduced by καὶ ἔτι or καί and besides. The phrases used to express the fact that the cir- cumference of a circle is less than 31 but greater than 31° times its diameter deserve special notice ; (1) παντὸς κύκλου ἡ περίμετρος τῆς διαμέτρου τριπλασίων ἐστί, καὶ ἔτι ὑπερέχει ἐλάσσονι μὲν ἢ ἑβδόμῳ μέρει τῆς διαμέτρου, μείζονι δὲ ἢ δέκα ἑβδομηκοστομόνοις, and (2) τριπλασίων ἐστὶ καὶ ἐλάσσονι μὲν ἢ ἑβδόμῳ μέρει, μείζονι δὲ ἢ ( οα΄ μείζων. We also have the phrase for the first part ἐλάσσων ἢ τριπλασίων καὶ ἑβδόμῳ μέρει μείζων.
To measure μετρεῖν, common measure κοινὸν μέτρον, commensurable, incommensurable σύμμετρος, ἀσύμμετρος.
Mechanical terms.
Mechanics τὰ μηχανικά, weight βάρος ; centre of gravity κέντρον tov Bapeos with another genitive of the body or magnitude; in the plural we have either τὰ κέντρα αὐτῶν τοῦ Bapeos or τὰ κέντρα τῶν βαρέων. κέντρον is also used alone.
A lever ζυγός or ζύγιον, the horizon ὃ δρίζων ; in a vertical line is represented by perpendicularly κατὰ κάθετον, thus the point of suspension and the centre. of gravity of the body suspended are in a vertical line κατὰ καθετόν ἐστι TO τε σαμεῖον TOD κρεμαστοῦ καὶ TO κέντρον τοῦ βάρεος τοῦ κρεμαμένου. Of suspension from or αὐ ἐκ or ° κατά (with acc.) is used. Let the triangle be suspended from the points B, T, κρεμάσθω τὸ τρίγωνον ἐκ τῶν B, Τ' σαμείων ; if the suspension of the triangle BAT αὐ B, Τ' be set free, and it be suspended at B, the triangle remains in its position εἴ κα τοῦ BAT τριγώνου ἃ
elxxxul INTRODUCTION.
μὲν κατὰ τὰ B, Τ' κρέμασις λυθῇ, κατὰ δὲ τὸ E κρεμασθῇ, μένει τὸ τρίγωνον, ὡς νῦν ἔχει.
To incline towards ῥέπειν ἐπί (800.)}; to be im equilibrium ἰσορροπεῖν, they will be in equilibrium with A held fast κατεχομένου τοῦ A ἰσορροπήσει, they will be in equilibrium at A (1.6. will balance about A) κατὰ τὸ A ἰσορροπησοῦντι; AB is too great to balance Τ' μεῖζόν ἐστι τὸ AB ἢ ὥστε ἰσορροπεῖν τῷ 1. The adjective for wm equilibrium is ἰσορρεπής ; let it be in equilibrium with the triangle TAH, ἰσορρεπὲς ἔστω τῷ TAH τριγώνῳ: To balance at certain distances (from the point of support or the centre of gravity of a
system) is ἀπό τινων μακέων ἰσορροπεῖν.
Theorems, problems, etc.
A theorem θεώρημα (from θεωρεῖν to investigate); a problem πρόβλημα, with which the following expressions may be compared, the (questions) propounded concerning the figures τὰ προβεβλημένα περὶ τῶν σχημάτων, these things are propounded for investigation προβαλλέται τάδε θεωρήσαι ; also πρόκειμαι takes the place of the passive, which it was proposed (or required) to find ὅπερ προέκειτο εὑρεῖν.
Another similar word is ἐπίταγμα, direction or requirement ; thus the theorems and directions necessary for the proofs of them τὰ θεωρήματα καὶ τὰ ἐπιτάγματα τὰ χρείαν ἔχοντα εἰς Tas ἀποδειξίας αὐτῶν, in order that the requirement may be fulfilled ὅπως γένηται τὸ ἐπι- ταχθέν (or ἐπίταγμα). To satisfy the requirement is ποιεῖν τὸ ἐπίταγμα (either e.g. of lines in a figure, or of the person solving the problem).
After the setting out (ἔκθεσις) in any proposition there follows the short statement of what it is required to prove or to do. In the former case (that of a theorem) Archimedes uses one of three expressions δεικτέον it is required to prove, λέγω or φαμὶ δή 1 assert or say; and in the second case (that of a problem) δεῖ dy i ts required (to do so and so).
In a problem the analysis ἀνάλυσις and synthesis σύνθεσις are distinguished, the latter being generally introduced with the words the synthesis of the problem will be as follows συντεθήσεται τὸ πρόβλημα οὕτως. The parts of the verb ἀναλύειν are similarly used ; thus the analysis and synthesis of each of these (problems) will be given at the end ἑκάτερα δὲ ταῦτα ἐπὶ τέλει ἀναλυθήσεταί τε καὶ
, συντεθήσεται.
THE TERMINOLOGY OF ARCHIMEDES. elxxxiil
A notable term in connexion with problems is the διορισμός (determination), which means the determination of the limits within which a solution is possible*. If a solution is always possible, the problem does not involve a διορισμός, οὐκ ἔχει διορισμόν ; otherwise it does involve it, ἔχει διορισμόν.
Data and hypotheses.
For given some part of the verb δίδωμι is used, generally the participle δοθείς, but sometimes δεδομένος and once or twice διδόμενος. Let a circle be given δεδόσθω κύκλος, given two unequal magnitudes δύο μεγεθῶν ἀνίσων δοθέντων, each of the two lines TA, EZ is given ἐστὶν δοθεῖσα ἑκατέρα τῶν TA, EZ, the same ratio as the given one λόγος ὃ αὐτὸς τῷ δοθέντι. Similar expressions are the assigned ratio 6 ταχθεὶς λόγος, the given area τὸ προτεθὲν (Or προκείμενον) χωρίον.
Given in position θέσει simply (sc. δεδομένη).
Of hypotheses the parts of the verb ὑποτίθεμαι and (for the passive) ὑπόκειμαι are used; with the same suppositions τῶν αὐτῶν ὑποκειμένων, let the said suppositions be made ὑποκείσθω τὰ εἰρημένα, we make these suppositions ὑποτιθέμεθα τάδε.
Where in a reductio ad absurdum the original hypothesis is referred to, and generally where an earlier step is quoted, the past tense of the verb is used ; but it was not (so) οὐκ ἣν δέ, for it was less qv γὰρ ἐλάσσων, they were proved equal ἀπεδείχθησαν ἴσοι, for this has been proved to be possible δεδείκται yap τοῦτο δυνατὸν ἐόν. Where a hypothesis is thus quoted, the past tense of ὑπόκειμαι has various constructions after it, (1) an adjective or participle, AZ, BH were supposed equal ἴσαι ὑπέκειντο ai AZ, BH, it is by hypothesis a tangent ὑπέκειτο ἐπιψαύουσα, (2) an infinitive, for by hypothesis it does not cut ὑπέκειτο yap μὴ τέμνειν, the axis is by hypothesis not at right angies to the parallel planes ὑπέκειτο ὃ ἀξων μὴ εἶμεν ὀρθὸς ποτὶ τὰ παράλλαλα ἐπίπεδα, (3) the plane is supposed to have been drawn through the centre τὸ ἐπίπεδον ὑπόκειται διὰ τοῦ κέντρου ἄχθαι.
Supposing it found εὑρεθέντος absolutely. Swppose it done γεγονέτω.
The usual idiomatic use of εἰ δὲ μή after a negative statement may be mentioned ; i¢ will not meet the surface in another point, otherwise... ob yap ἁψέται κατ᾽ ἄλλο σαμεῖον τᾶς ἐπιφανείας: εἰ δὲ
* Cf. Apollonius of Perga, p. Ιχχ, note,
clxxxiv INTRODUCTION.
Inferences, and adaptation to different cases.
The usual equivalent for therefore is apa; οὖν and τοίνυν are generally used in a somewhat weaker sense to mark the starting- point of an argument, thus ἐπεὶ οὖν may be translated as since, then. Since is ἐπεί, because διότι.
πολλῷ μᾶλλον much more then is apparently not used in Archi- medes, who has πολλῷ alone; thus much less then is the ratio of the circumscribed figure to the inscribed than that of Καὶ to Ἡ πολλῷ apa τὸ περιγραφὲν πρὸς τὸ ἐγγραφὲν ἐλάσσονα λόγον ἔχει τοῦ, ὃν ἔχει ἡ Κ πρὸς Η.
διά with the accusative is a common way of expressing the reason why; because the cone is isosceles διὰ τὸ ἰσοσκελῆ εἶναι τὸν κῶνον, for the same reason διὰ ταὐτά.
διά with the genitive expresses the means by which a proposition is proved ; by means of the construction διὰ τῆς κατασκευῆς, by the same means διὰ τῶν αὐτῶν, by the same method διὰ τοῦ αὐτοῦ τρόπου.
Whenever this is the case, the surface is greater ὅταν τοῦτο ἢ, μείζων γίνεται ἡ ἐπιφάνεια..., if this is the case, the angle BA® is equal..., εἰ δὲ τοῦτο, ἴσα ἐστὶν ἃ ὑπὸ ΒΑΘ ywvia..., which is the same thing as showing that... 6 ταὐτόν ἐστι τῷ δεῖξαι, ὅτι...
Similarly for the sector ὁμοίως δὲ καὶ ἐπὶ τοῦ τομέως, the proof is the same as (that wsed to show) that ἃ αὐτὰ ἀπόδειξις ἅπερ καὶ ὅτι, the proof that...is the same ἃ αὐτὰ ἀπόδειξίς ἐντι καὶ διότι..., the same argument holds for all rectilineal figures inscribed in the segments in the recognised manner (see p. 204) ἐπὶ πάντων εὐθυγράμμων τῶν ἐγγραφομένων ἐς τὰ τμάματα γνωρίμως ὃ αὐτὸς λόγος ; it will be possible, having proved it for a circle, to transfer the same argument im the case of the sector ἔσται ἐπὶ κύκλου δείξαντα μεταγαγεῖν τὸν ὅμοιον λόγον καὶ ἐπὶ τοῦ τομέως ; the rest will be the same, but τέ will be the lesser of the diameters which will be intercepted within the spheroid (instead of the greater) τὰ μὲν ἄλλα τὰ αὐτὰ ἐσσείται, τᾶν δὲ διαμέτρων ἃ ἐλάσσων ἐσσείται ἃ ἐναπολαφθεῖσα ἐν τῷ σφαιροειδεῖ ; it will make
no difference whether...or...dvolcer δὲ οὐδέν, εἴτε... εἴτε....
Conclusions.
The proposition is therefore obvious, or is proved δῆλον οὖν ἐστι (or δέδεικται) τὸ προτεθέν ; similarly φανερὸν οὖν ἐστιν, ὃ ἔδει δείξαι, and ἔδει δὲ τοῦτο δείξα. Which is absurd, or impossible ὅπερ ἄτοπον, ΟΥ ἀδύνατον.
A curious use of two negatives is contained in the following:
THE TERMINOLOGY OF ARCHIMEDES. elxxxv
οὐκ apa οὔκ ἐστι κέντρον τοῦ Bapeos τοῦ ΔΕΖ τριγώνου τὸ N σαμεῖον. ἔστιν apa, therefore it is not possible that the point N should not be the centre of gravity of the triangle AEZ. It must therefore be so.
Thus a rhombus will have been formed ἔσται δὴ γεγονὼς ῥόμβος ; two unequal straight lines have been found satisfying the requirement
ὑ Ξ ἰσὶν ἄρα δύο εὐθεῖαι ἃ οιοῦσαι τὸ ἐπίταγμα εὑρημέναι εἰσὶν apa δύο εὐθεῖαι ἀνισοι 7 α γμα.
Direction, concavity, convexity.
In the same direction ἐπὶ τὰ αὐτά, in the other direction ἐπὶ τὰ ἕτερα, concave in the same direction ἐπὶ τὰ αὐτὰ κοίλη ; in the same direction as ἐπὶ τὰ αὐτά with the dative or ἐφ᾽ a, thus in the same direction as the vertex of the cone ἐπὶ τὰ αὐτὰ τᾷ TOD κώνου κορυφᾷ, drawn in the same direction as (that of) the convex side of it ἐπὶ τὰ αὐτὰ ἀγομέναι, ἐφ᾽ a ἐντι τὰ κυρτὰ αὐτοῦ. For on the same side of ἐπὶ τὰ αὐτά is followed by the genitive, they fall on the same side of the line ἐπὶ τὰ αὐτὰ πίπτουσι τῆς γραμμῆς.
On each side of ἐφ᾽ ἑκάτερα (with gen.); on each side of the plane of the base ἐφ᾽ ἑκάτερα τοῦ ἐπιπέδου τῆς βάσεως.
Miscellaneous.
Property σύμπτωμα. Proceeding thus continually, ἀεὶ τοῦτο ποιοῦντες, ἀεὶ τούτου γενομένου, OY τούτου ἑξῆς γινομένου. In the elements ἐν τῇ στοιχειώσει.
One special difference between our terminology and the Greek is that whereas we speak of any circle, any straight line and the like, the Greeks say every circle, every straight line, etc. Thus any pyramid is one third part of the prism with the same base as the pyramid and equal height πᾶσα πυραμὶς τρίτον μέρος ἐστὶ τοῦ πρίσματος τοῦ τὰν αὐτὰν βάσιν ἔχοντος τᾷ πυραμίδι καὶ ὕψος ἴσον. 7 define the diameter of any segment as διάμετρον καλέω παντὸς τμάματος. 70 exceed any assigned (magnitude) of those which are comparable with one another ὑπερέχειν παντὸς τοῦ προτεθέντος τῶν πρὸς ἄλληλα λεγομένων.
Another noteworthy difference is illustrated in the last sentence. The Greeks did not speak as we do of @ given area, ὦ given ratio etc., but of the given area, the given ratio, and the like. Thus J/¢ is possible...to leave certain segments less than a given area δυνατόν ἐστιν... λείπειν τινα τμήματα, ἅπερ ἔσται ἐλάσσονα τοῦ προκειμένου χωρίου ; to divide a given sphere by a plane so that the segments have to one another an assigned ratio τὰν δοθεῖσαν σφαῖραν ἐπιπέδῳ τεμεῖν,
7 lol ὥστε TA τμάματα αὐτᾶς ToT ἀλλαλα τὸν ταχθέντα λόγον ἔχειν.
H. A. γ7ὺ
elxxxvl INTRODUCTION.
Magnitudes in arithmetical progression are said to exceed each other by an equal (amount) ; of there be any number of magnitudes in arithmetical progression εἴ κα ἔωντι μεγέθεα ὁποσαοῦν τῷ ἴσῳ ἀλλάλων ὑπερέχοντα. The common difference is the eacess ὑπεροχά, and the terms collectively are spoken of as the magnitudes exceeding by the equal (difference) τὰ τῷ ἴσῳ ὑπερέχοντα. The least term is τὸ ἐλάχιστον, the greatest term τὸ μέγιστον. The swm of the terms is expressed by πάντα τὰ τῷ low ὑπερέχοντα. ;
Terms of a geometrical progression are simply im (continued) proportion ἀνάλογον, the series is then 9 ἀναλογία, the proportion, and a term of the series is τὶς τῶν ἐν τᾷ αὐτᾷ avadoyia. Numbers in geometrical progression beginning from unity are ἀριθμοὶ ἀνάλογον ἀπὸ μονάδος. Let the term A of the progression be taken which is distant the same number of terms from © as A is distant from unity λελάφθω ἐκ τᾶς ἀναλογίας ὁ A ἀπέχων ἀπὸ τοῦ Θ τοσούτους, ὅσους
ε > Ν Ψ' > , 6 A ἀπὸ μονάδος ἀπέχει.
THE WORKS OF
ARCHIMEDES.
ON THE SPHERE AND CYLINDER. BOOK I.
‘“ ARCHIMEDES to Dositheus greeting.
On a former occasion I sent you the investigations which I had up to that time completed, including the proofs, showing that any segment bounded by a straight line and a section of a right-angled cone [a parabola] is four-thirds of the triangle which has the same base with the segment and equal height. Since then certain theorems not hitherto demonstrated (ave- λέγκτων) have occurred to me, and I have worked out the proofs of them. They are these: first, that the surface of any sphere is four times its greatest circle (τοῦ μεγίστου κύκλου); next, that the surface of any segment of a sphere is equal to a circle whose radius (ἡ ἐκ τοῦ κέντρου) is equal to the straight line drawn from the vertex (κορυφή) of the segment to the circum- ference of the circle which is the base of the segment; and, further, that any cylinder having its base equal to the greatest circle of those in the sphere, and height equal to the diameter of the sphere, is itself [1.6. in content] half as large again as the sphere, and its surface also [including its bases] is half as large again as the surface of the sphere. Now these properties were all along naturally inherent in the figures referred to (αὐτῇ τῇ φύσει προυπῆρχεν περὶ τὰ εἰρημένα σχήματα), but remained unknown to those who were before my time engaged in the study of geometry. Having, however, now discovered that the properties are true of these figures, I cannot feel any hesitation
H. A, 1
2 ARCHIMEDES
in setting them side by side both with my former investiga- tions and with those of the theorems of Eudoxus on solids which are held to be most irrefragably established, namely, that any pyramid is one third part of the prism which has the same base with the pyramid and equal height, and that any cone is one third part of the cylinder which has the same base with the cone and equal height. For, though these properties also were naturally inherent in the figures all along, yet they were in fact unknown to all the many able geometers who lived before Eudoxus, and had not been observed by any one. Now, however, it will be open to those who possess the requisite ability to examine these discoveries of mine. They ought to have been published while Conon was still alive, for I should conceive that he would best have been able to grasp them and to pronounce upon them the appropriate verdict ; but, as I judge it well to communicate them to those who are conversant with mathematics, I send them to you with the proofs written out, which it will be open to mathematicians to examine. Farewell.
I first set out the axioms* and the assumptions which I have used for the proofs of my propositions.
DEFINITIONS.
1. There are in a plane certain terminated bent lines (καμπύλαι γραμμαὶ πεπερασμέναι), which either lie wholly on the same side of the straight lines joining their extremities, or have no part of them on the other side.
2. I apply the term concave in the same direction to a line such that, if any two points on it are taken, either all the straight lines connecting the points fall on the same side of the line, or some fall on one and the same side while others fall on the line itself, but none on the other side.
* Though the word used is ἀξιώματα, the “axioms” are more of the nature of definitions ; and in fact Eutocius in his notes speaks of them as such (ὅροι).
+ Under the term bent line Archimedes includes not only curved lines of continuous curvature, but lines made up of any number of lines which may be either straight or curved.
ON THE SPHERE AND CYLINDER I. 3
3. Similarly also there are certain terminated surfaces, not themselves being in a plane but having their extremities in a plane, and such that they will either be wholly on the same side of the plane containing their extremities, or have no part of them on the other side.
4. I apply the term concave in the same direction to surfaces such that, if any two points on them are taken, the straight lines connecting the points either all fall on the same side of the surface, or some fall on one and the same side of it while some fall upon it, but none on the other side.
5. I use the term solid sector, when a cone cuts a sphere, and has its apex at the centre of the sphere, to denote the figure comprehended by the surface of the cone and the surface of the sphere included within the cone.
6. I apply the term solid rhombus, when two cones with the same base have their apices on opposite sides of the plane of the base in such a position that their axes lie in a straight line, to denote the solid figure made up of both the cones.
ASSUMPTIONS.
1. Of all lines which have the same extremities the straight line is the least*.
* This well-known Archimedean assumption is scarcely, as it stands, a definition of a straight line, though Proclus says [p. 110 ed. Friedlein] ‘‘ Archi- medes defined (ὡρίσατο) the straight line as the least of those [lines] which have the same extremities. For because, as Euclid’s definition says, ἐξ ἴσου κεῖται τοῖς ἐφ᾽ ἑαυτῆς σημείοις, it is in consequence the least of those which have the same extremities.” Proclus had just before [p. 109] explained Euclid’s definition, which, as will be seen, is different from the ordinary version given in our text- books; a straight line is not ‘‘that which lies evenly between its extreme points,” but ‘‘that which ἐξ ἴσου τοῖς ἐφ᾽ ἑαυτῆς σημείοις κεῖται." The words of Proclus are, ‘‘ He [Euclid] shows by means of this that the straight line alone [of all lines] occupies a distance (κατέχειν διάστημα) equal to that between the points onit. For, as far as one of its points is removed from another, so great is the length (μέγεθος) of the straight line of which the points are the extremities ; and this is the meaning of τὸ ἐξ ἴσου κεῖσθαι τοῖς ἐφ᾽ ἑαυτῆς σημείοις. But, if you take two points on a circumference or any other line, the distance cut off between them along the line is greater than the interval separating them; and this is the case with every line except the straight line.’ It appears then from this that Euclid’s definition should be understood in a sense very like that of
1—2
4 i ARCHIMEDES
2. Of other lines in a plane and having the same extremi- ties, [any two] such are unequal whenever both are concave in the same direction and one of them is either wholly included between the other and the straight line which has the same extremities with it, or is partly included by, and is partly common with, the other; and that [line] which is included is the lesser [of the two].
3. Similarly, of surfaces which have the same extremities, if those extremities are in a plane, the plane is the least [in area].
4. Of other surfaces with the same extremities, the ex- tremities being in a plane, [any two] such are unequal when- ever both are concave in the same direction and one surface is either wholly included between the other and the plane which - has the same extremities with it, or is partly included by, and partly common with, the other; and that [surface] which is included is the lesser [of the two in area].
5. Further, of unequal lines, unequal surfaces, and unequal solids, the greater exceeds the less by such a magnitude as, when added to itself, can be made to exceed any assigned magnitude among those which are comparable with [it and with] one another*.
These things being premised, ἐγ a polygon be inscribed in a circle, it is plain that the perimeter of the inscribed polygon is less than the circumference of the circle; for each of the sides of the polygon is less than that part of the circumference of the circle which is cut off by it.”
Archimedes’ assumption, and we might perhaps translate as follows, ‘A straight line is that which extends equally (ἐξ ἔσου κεῖται) with the points on it,” or, to follow Proclus’ interpretation more closely, ‘‘A straight line is that which represents equal extension with [the distances separating] the points on it.”
* With regard to this assumption compare the Introduction, chapter 11. ὃ 2.
Or
ON THE SPHERE AND CYLINDER I.
Proposition 1.
If a polygon be circumscribed about a circle, the perimeter of the circumscribed polygon is greater A than the perimeter of the circle. 8
Let any two adjacent sides, meet- ing in A, touch the circle at P, Q respectively.
Then [Assumptions, 2]
PA+AQ>(are PQ).
A similar imequality holds for each angle of the polygon; and, by ad- dition, the required result follows.
Proposition 2.
Given two unequal magnitudes, it is possible to find two un- equal straight lines such that the greater straight line has to the less a ratio less than the greater magnitude has to the less.
Let AB, D represent the two unequal magnitudes, AB being the greater.
Suppose BC measured along BA equal to D, and let GH be any straight line. :
Then, if CA be added to itself a sufficient A number of times, the sum will exceed D. Let ἢ
AF be this sum, and take # on @H produced
such that GH is the same multiple of HZ that 7 AF is of AC. Thus BA HG — AC » AF. Ὁ But, since AF’ > D (or CB), AC: AF<AC: CB. Β Therefore, componendo, G
EG:GH<AB: D. Hence EG, GH are two lines satisfying the given condition.
6 ARCHIMEDES
Proposition 3.
Given two unequal magnitudes and a circle, it is possible to inscribe a polygon in the circle and to describe another about it so that the side of the circumscribed polygon may have to the side of the inscribed polygon a ratio less than that of the greater magnitude to the less.
Let A, B represent the given magnitudes, A being the
greater.
Find [Prop. 2] two straight lines F, KL, of which F is the greater, such that
BSNS .
L M
Draw LM perpendicular to LK and of such length that KM=F.
In the given circle let CH, DG be two diameters at right angles. Then, bisecting the angle DOC, bisecting the half again, and so on, we shall arrive ultimately at an angle (as
NOC) less than twice the angle LKM.
Join VC, which (by the construction) will be the side of a regular polygon inscribed in the circle. Let OP be the radius of the circle bisecting the angle NOC (and therefore bisecting NC at right angles, in H, say), and let the tangent at P meet OC, ON produced in S, 7’ respectively.
Now, since ZCON ΞΖ ΕΙΣ ZHOC <2 LKM,
ON THE SPHERE AND CYLINDER I. rf
and the angles at H, L are right ; therefore MK: LK >OC: OH SOP = OF. Hence Se CN a Mis LK ἜΝ: ΚΕ: therefore, a fortiori, by (1), po CN <A Bs, Thus two polygons are found satisfying the given condition.
Proposition 4.
Again, given two unequal magnitudes and ὦ sector, it 18 possible to describe a polygon about the sector and to inscribe another in it so that the side of the circumscribed polygon may have to the side of the inscribed polygon a ratio less than the greater magnitude has to the less.
[The “inscribed polygon” found in this proposition is one which has for two sides the two radii bounding the sector, while the remaining sides (the number of which is, by construction, some power of 2) subtend equal parts of the are of the sector ; the “circumscribed polygon” is formed by the tangents parallel to the sides of the inscribed polygon and by the two bounding radu produced. ]
In this case we make the same construction as in the last proposition except that we bisect the angle COD of the sector, instead of the right angle between two diameters, then bisect the half again, and so on. The proof is exactly similar to the preceding one.
8 ARCHIMEDES
Proposition 5.
Given a circle and two unequal magnitudes, to describe a polygon about the circle and inscribe another in it, so that the circumscribed polygon may have to the inscribed a ratio less than the greater magmtude has to the less.
Let A be the given circle and B, C the given magnitudes, B being the greater.
F
Take two unequal straight lines D, H, of which D is the greater, such that D: #<B:C [Prop. 2], and let # be a mean proportional between D, Μ΄, so that D is also greater than F.
Describe (in the manner of Prop. 3) one polygon about the circle, and inscribe another in it, so that the side of the former has to the side of the latter a ratio less than the ratio D : F.
Thus the duplicate ratio of the side of the former polygon to the side of the latter is less than the ratio D? : δ᾽"
But the said duplicate ratio of the sides is equal to the ratio of the areas of the polygons, since they are similar ;
therefore the area of the circumscribed polygon has to the area of the inscribed polygon a ratio less than the ratio D® : F”, or D: FE, and a fortiori less than the ratio B: C.
ON THE SPHERE AND CYLINDER I. 9
Proposition 6.
“Similarly we can show that, given two unequal magnitudes and a sector, it ts possible to circumscribe a polygon about the sector and inscribe in it another similar one so that the circum- scribed may have to the inscribed a ratio less than the greater magnitude has to the less.
And it is likewise clear that, if a circle or a sector, as well as ὦ certain area, be given, it is possible, by inscribing regular polygons in the circle or sector, and by continually imscribing such in the remaining segments, to leave segments of the circle or sector which are [together] less than the given area. For this is proved in the Hlements [Eucl. x11. 2].
But it is yet to be proved that, given a circle or sector and an area, it is possible to describe a polygon about the circle or sector, such that the area remaining between the circumference and the circumscribed figure is less than the given area.”
CSE
The proof for the circle (which, as Archimedes says, can be equally applied to a sector) is as follows.
Let A be the given circle and B the given area.
Now, there being two unequal magnitudes A + B and A, let a polygon (C) be circumscribed about the circle and a polygon (Z) inscribed in it [as in Prop. 5], so that
eed a LS a ti hae ee τος, (1). The circumscribed polygon (C) shall be that required.
10 ARCHIMEDES
For the circle (A) is greater than the inscribed polygon (JZ). Therefore, from (1), a fortiori, C:A<A+B:A, whence C<A+B8, or C-A<B.
Proposition 7.
If in an isosceles cone [1.6. a right circular cone] a pyramid be inscribed having an equilateral base, the surface of the pyramid excluding the base is equal to a triangle having tts base equal to the perimeter of the base of the pyramid and tts height equal to the perpendicular drawn from the apex on one side of the base.
Since the sides of the base of the pyramid are equal, it follows that the perpendiculars from the apex to all the sides of the base are equal; and the proof of the proposition is obvious.
Proposition 8.
If a pyramid be circumscribed about an isosceles cone, the surface of the pyramid excluding its base is equal to a triangle having its base equal to the perimeter of the base of the pyramid and its height equal to the side [1.e. a generator] of the cone.
The base of the pyramid is a polygon circumscribed about the circular base of the cone, and the line joining the apex of the cone or pyramid to the point of contact of any side of the polygon is perpendicular to that side. Also all these perpen- diculars, being generators of the cone, are equal; whence the proposition follows immediately.
ON THE SPHERE AND CYLINDER 1. 11
Proposition 9.
Lf in the circular base of an isosceles cone a chord be placed, and from its extrenuties straight lines be drawn to the apex of the cone, the triangle so formed will be less than the portion of the surface of the cone intercepted between the lines drawn to the apex.
Let ABC be the circular base of the cone, and O its apex.
Draw a chord AB in the circle, and join OA, OB. Bisect the are ACB in C, and join AC, BC, OC.
Then A OAC +A OBC> A OAB.
Let the excess of the sum of the first two triangles over the third be equal to the area D.
Then JD is either less than the sum of the segments 4 EC, CFB, or not less.
I. Let D be not less than the sum of the segments referred to.
We have now two surfaces
(1) that consisting of the portion OAEC of the surface of the cone together with the segment AZ#C, and
(2) the triangle OAC; and, since the two surfaces have the same extremities (the
perimeter of the triangle OAC), the former surface is greater than the latter, which is included by it [Asswmptions, 3 or 4].
12 ARCHIMEDES
Hence (surface OA EC)+ (segment AEC) >A OAC. Similarly (surface OCFB) + (segment CFB) >A OBC.
Therefore, since D is not less than the sum of the segments, we have, by addition,
(surface OAHCFB)+D>AOAC+ AOBC >A OAB+ D, by hypothesis. Taking away the common part D, we have the required result. II. Let D be less than the sum of the segments AEC, CFB.
If now we bisect the arcs AC, CB, then bisect the halves, and so on, we shall ultimately leave segments which are together less than D. [Prop. 6]
Let AGH, EHC, CKF, FLB be those segments, and join OF, OF.
Then, as before,
(surface OAGE) + (segment AGE) > A OAH and (surface OF HC) + (segment LHC) > A OKC. Therefore (surface OAGHC) + (segments AGH, EHC)
>AOAE+ AOEC > AOAC, a fortiori.
Similarly for the part of the surface of the cone bounded by OC, OB and the are CFB.
Hence, by addition, (surface OAGHHCKFLB)+ (segments AGE, EHC, CKF, FLB) >AOAC+ AOBC > AOAB + D, by hypothesis.
But the sum of the segments is less than D, and the re- quired result follows.
ON THE SPHERE AND CYLINDER I. 13
Proposition 10.
If in the plane of the circular base of an isosceles cone two tangents be drawn to the circle meeting in a point, and the points of contact and the point of concourse of the tangents be respectively joined to the apex of the cone, the sum of the two triangles formed by the joining lines and the two tangents are together greater than the included portion of the surface of the cone.
Let ABC be the circular base of the cone, O its apex, AD, BD the two tangents to the circle meeting in D. Join OA, OB, OD.
Let HCF be drawn touching the circle at C, the middle point of the arc ACB, and therefore parallel to AB. Join OE, OF.
Then ED+DF> EF, and, adding AH + FB to each side, AD+ DB>AE+ EF + FB.
Now OA, OC, OB, being generators of the cone, are equal, and they are respectively perpendicular to the tangents at A, Ὁ B.
14 ARCHIMEDES
It follows that AOAD+ AODB>AOAE+A OEF+A OFB.
Let the area G be equal to the excess of the first sum over the second.
G is then either less, or not less, than the sum of the spaces EAHC, FCKB remaining between the circle and the tangents, which sum we will call L.
I. Let G be not less than Z. We have now two surfaces
(1) that of the pyramid with apex O and base 4.0.8, excluding the face OAB,
(2) that consisting of the part OACB of the surface of the cone together with the segment ACB.
These two surfaces have the same extremities, viz. the perimeter of the triangle OAB, and, since the former includes the latter, the former is the greater [Assumptions, 4].
That is, the surface of the pyramid exclusive of the face OAB is greater than the sum of the surface OACB and the segment ACB.
Taking away the segment from each sum, we have
A OAE+A OEFF+A OFB +I >the surface OAHCKB.
And G is not less than L.
It follows that
AOAE+A OEF +A OFB+G,
which is by hypothesis equal to AOAD+A ODB, is greater than the same surface.
II. Let G be less than L.
If we bisect the ares AC, CB and draw tangents at their middle points, then bisect the halves and draw tangents, and so on, we shall lastly arrive at a polygon such that the sum of the parts remaining between the sides of the polygon and the circumference of the segment is less than G.
ON THE SPHERE AND CYLINDER I. 15
Let the remainders be those between the segment and the polygon APQRSB, and let their sum be M. Join OP WOQ, ete.
Then, as before,
AOAE+A OFF+A OFB>AOAP+A0PQ+...+A OSB.
Also, as before,
(surface of pyramid OAPQRSB excluding the face OAB)
>the part OACB of the surface of the cone together with the segment ACB.
Taking away the segment from each sum,
A OAP+AO0PQ+...+M> the part OACB of the surface of the cone.
Hence, a fortrore,
AOAE+ A OEF+ A OFB+G, which is by hypothesis equal to AOAD +A ODB, is greater than the part OACB of the surface of the cone.
Proposition 11.
If a plane parallel to the axis of a right cylinder cut the cylinder, the part of the surface of the cylinder cut off by the plane ws greater than the area of the parallelogram in which the plane cuts τί.
Proposition 12.
If at the extremities of two generators of any right cylinder tangents be drawn to the circular bases in the planes of those bases respectively, and if the pairs of tangents meet, the parallelograms formed by each generator and the two corre- sponding tangents respectively are together greater than the
imeluded portion of the surface of the cylinder between the two generators.
[The proofs of these two propositions follow exactly the methods of Props. 9, 10 respectively, and it is therefore un- necessary to reproduce them.]
16 ARCHIMEDES
“From the properties thus proved it is clear (1) that, 7f a pyramid be inscribed in an isosceles cone, the surface of the pyramid excluding the base is less than the surface of the cone [excluding the base], and (2) that, if a pyramid be circumscribed about an isosceles cone, the surface of the pyramid excluding the base is greater than the surface of the cone eacluding the base.
“Tt is also clear from what has been proved both (1) that, if a prism be inscribed in a right cylinder, the surface of the prism made up of its parallelograms [i.e. excluding its bases] 2s less than the surface of the cylinder excluding its bases, and (2) that, if a prism be circumscribed about a right cylinder, the surface of the prism made up of its parallelograms is greater than the surface of the cylinder excluding its bases.”
Proposition 13.
The surface of any right cylinder excluding the bases 15 equal to a circle whose radius is a mean proportional between the side [1.e. a generator] of the cylinder and the diameter of its base.
Let the base of the cylinder be the circle A, and make CD equal to the diameter of this circle, and #F equal to the height of the cylinder.
ON THE SPHERE AND CYLINDER I. ΠῚ
Let H be a mean proportional between CD, EF, and B a circle with radius equal to H.
Then the circle B shall be equal to the surface of the cylinder (excluding the bases), which we will call S.
For, if not, B must be either greater or less than S.
I. Suppose B< S.
Then it is possible to circumscribe a regular polygon about B, and to inscribe another in it, such that the ratio of the former to the latter is less than the ratio S : B.
Suppose this done, and circumscribe about A a polygon similar to that described about B; then erect on the polygon about A a prism of the same height as the cylinder. The prism will therefore be circumscribed to the cylinder,
Let KD, perpendicular to CD, and FL, perpendicular to EF, be each equal to the perimeter of the polygon about A. Bisect CD in M, and join MK.
Then A KDM =the polygon about A. Also 67 Ei.= surface of prism (excluding bases). Produce FE to N so that FE = EN, and join NL.
Now the polygons about A, B, being similar, are in the duplicate ratio of the radii of A, B.
Thus
A KDM : (polygon about B) = MD? : H* Ξε MIP CD. bE =D ONE
=AKDM: ALFN
(since DK = FT). Therefore (polygon about B)=A LFN
Ξ- EL = (surface of prism about A),
from above.
But (polygon about B) : (polygon in B)< S: B.
ἘΠ. A;
bo
18 ARCHIMEDES
Therefore (surface of prism about A): (polygon in B)<S : B, and, alternately, (surface of prism about A): S<(polygon in B): B; which is impossible, since the surface of the prism is greater than S, while the polygon inscribed in B is less than B.
Therefore Be€s.
II. Suppose B>S.
Let a regular polygon be circumscribed about B and another inscribed in it so that
(polygon about B): (polygon in B)< δ: 5.
Inscribe in A a polygon similar to that inscribed in B, and erect a prism on the polygon inscribed in A of the same height as the cylinder.
Again, let DK, FL, drawn as before, be each equal to the perimeter of the polygon inscribed in A.
Then, in this case,
A KDM > (polygon inscribed in A) (since the perpendicular from the centre on a side of the polygon is less than the radius of A).
Also ALFN = 67 EL =surface of prism (excluding bases).
Now (polygon in A) : (polygon in B)= MD* : H’,
= AKDM : ALFN, as before. And AKDM > (polygon in A). Therefore A LFN, or (surface of prism) > (polygon in B). But this is impossible, because (polygon about B) : (polygon in B)< Β : 5, < (polygon about B): S, a fortiors, so that (polygon in B) >S8, > (surface of prism), a fortiort. Hence B is neither greater nor less than S, and therefore Bs:
ON THE SPHERE AND CYLINDER I. 19
Proposition 14.
The surface of any isosceles cone excluding the base is equal to a circle whose radius is a mean proportional between the side of the cone [a generator] and the radius of the circle which is the base of the cone.
Let the circle A be the base of the cone; draw C equal to the radius of the circle, and D equal to the side of the cone, and let H be a mean proportional between C, D.
D
Draw a circle B with radius equal to £.
Then shall B be equal to the surface of the cone (excluding the base), which we will call S.
If not, B must be either greater or less than ΑΚ.
I. Suppose B< S.
Let a