Thomas Little Heath

“ ARCHIMEDES (c. 287–212 B.C.) , Greek mathematician and inventor, was born at Syracuse, in Sicily. He was the son of Pheidias, an astronomer, and was on intimate terms with, if not related to, Hiero, king of Syracuse, and Gelo his son. He studied at Alexandria and doubtless met there Conon of Samos, whom he admired as a mathematician and cherished as a friend, and to whom he was in the habit of communicating his discoveries before publication. On his return to his native city he devoted himself to mathematical research. ”

“ Greek geometer, probably not of Antissa but of Antinoeia or Antinoupolis, a city in Egypt founded by Hadrian, lived, as may be safely inferred from the character and contents of his writings, long after the golden age of Greek geometry, most probably in the 4th century, between Pappus and Theon of Alexandria. ”

“ A remarkable feature is the bare statement of a number of very close approximations to the square roots of numbers which are not complete squares. Others occur in the Metrica where also a method of finding such approximate square, and even approximate cube, roots is shown. Hero’s expressions for the areas of regular polygons of from 5 to 12 sides in terms of the squares of the sides show interesting approximations to the values of trigonometrical ratios. ”

“ Among the great variety of problems solved are problems leading to determinate equations of the first degree in one, two, three or four variables, to determinate quadratic equations, and to indeterminate equations of the first degree in one or more variables, which are, however, transformed into determinate equations by arbitrarily assuming a value for one of the required numbers, Diophantus being always satisfied with a rational, even if fractional, result and not requiring a solution in integers. ”

“ For Books i.-iv. he claims only that the generation of the curves and their fundamental properties in Book i. are worked out more fully and generally than they were in earlier treatises, and that a number of theorems in Book iii. and the greater part of Book iv. are new. That he made the fullest use of his predecessors’ works, such as Euclid’s four Books on Conics, is clear from his allusions to Euclid, Conon and Nicoteles. The generality of treatment is indeed remarkable ”

“ ERATOSTHENES OF ALEXANDRIA (c. 276–c. 194 B.C.) , Greek scientific writer, was born at Cyrene. He studied grammar under Callimachus at Alexandria, and philosophy under the Stoic Ariston and the Academic Arcesilaus at Athens. He returned to Alexandria at the summons of Ptolemy III. Euergetes, by whom he was appointed chief librarian in place of Callimachus. He is said to have died of voluntary starvation, being threatened with total blindness. Eratosthenes was one of the most learned men of antiquity, and wrote on a great number of subjects. ”

“ With the mention of the Porisms of Euclid we have an account of the relation of porism to theorem and problem. In the same preface is included (a) the famous problem known by Pappus’s name, often enunciated thus: Having given a number of straight lines, to find the geometric locus of a point such that the lengths of the perpendiculars upon, or {more generally) the lines drawn from it obliquely at given inclinations to, the given lines satisfy the condition that the product of certain of them may bear a constant ratio to the product of the remaining ones ”

“ The lemmas which Pappus gives in connexion with the
porisms are interesting historically, because he gives (1) the fundamental theorem that the cross or an harmonic ratio of a pencil of four straight lines meeting in a point is constant for all transversals; (2) the proof of the harmonic properties of a complete quadrilateral; (3) the theorem that, if the six vertices of a hexagon lie three and three on two straight lines, the three points of concourse of opposite sides lie on a straight line. ”

“ With one symbol for an unknown, it will easily be understood what scope there is for adroit assumptions, for the required numbers, of expressions in the one unknown which are at once seen to satisfy some of the conditions, leaving only one or two to be satisfied by the particular value of x to be determined. ”

“ It is in this book that Hero proves the expression for the area of a triangle in terms of its sides. The Pneumatica in two books is also extant in Greek as is also the Automatopoietica. In the former will be found such things as siphons, “Hero’s fountain,” “penny-in-the-slot” machines, a fire-engine, a water-organ, and arrangements employing the force of steam. Pappus quotes from three books of Mechanics and from a work called Barulcus, both by Hero. The three books on Mechanics survive in an Arabic translation which, however, bears a title “On the lifting of heavy objects.” ”

“ Halley added in his edition (1710) a restoration of Book viii., in which he was guided by the fact that Pappus gives lemmas “to the seventh and eighth books” under that one heading, as well as by the statement of Apollonius himself that the use of the seventh book was illustrated by the problems solved in the eighth.
The degree of originality of the Conics can best be judged from Apollonius’ own prefaces. Books i.-iv. form an “elementary introduction,” i.e. contain the essential principles; the rest are specialized investigations in particular directions. ”

“ Pappus gives several solutions of this problem, including a method of making successive approximations to the solution, the significance of which he apparently failed to appreciate; he adds his own solution of the more general problem of finding geometrically the side of a cube whose content is in any given ratio to that of a given one. (2) On the arithmetic, geometric and harmonic means between two straight lines, and the problem of representing all three in one and the same geometrical figure. ”

“ If about two fixed points P, Q we make turn two straight lines meeting on a given straight line L, and if one of them cut off a segment AM from a fixed straight line AX, given in position, we can determine another fixed straight line BY, and a point B fixed on it, such that the segment BM' made by the second moving line on this second fixed line measured from B has a given ratio λ to the first segment AM. The rest of the enunciation's given by Pappus are incomplete, and he merely says that he gives thirty-eight lemmas for the three books of porisms ”

“ Alexander became at Rome one of the most celebrated medical men of his time; Olympius was deeply versed in Roman jurisprudence; and Metrodorus was one of the distinguished grammarians of the great Eastern capital. It is related of Anthemius that, having a quarrel with his next-door neighbour Zeno, he annoyed him in two ways. First, he made a number of leathern tubes the ends of which he contrived to fix among the joists and flooring of a fine upper-room in which Zeno entertained his friends, and then subjected it to a miniature earthquake by sending steam through the tubes. ”

“ He is doubtless the same as Theodosius the mathematician, who is mentioned by Strabo amongst the natives of Bithynia distinguished for their learning, and whose sons were also mathematicians, the same, too, as the inventor of a universal sun-dial (horologium πρὸς πᾶν κλίμα) of that name who is praised by Vitruvius (De Architectura, ix. 9) . His date, therefore, could not have been later than the 1st century B.C. ”

“ He was also called Beta as being second in all branches of learning, though not actually first in any. In mathematics he wrote two books On means (Περὶ μεσοτήτων) which are lost, but appear, from a remark of Pappus, to have dealt with “loci with reference to means.” ”

“ It is clearly the form of the fundamental property (expressed in the terminology of the “application of areas”) which led him to call the curves for the first time by the names parabola, ellipse, hyperbola. ”

“ Heiberg) , shows in Prop. 20 that “it is possible to exhibit a cone and a cylinder cutting one another in one and the same ellipse.” He then solves problems such as—“given a cone (cylinder) and an ellipse on it, to find the cylinder (cone) which is cut in the same ellipse as the cone (cylinder) ” (Props. 21, 22) ; “given a cone (cylinder) to find a cylinder (cone) , and to cut both by one and the same plane so that the sections thus formed shall be similar ellipses” (Props. 23, 24) . In Props. 27, 28 he deals with sub contrary and other similar sections of a scalene cylinder or cone. ”

“ A Latin version of the complete work, with ancient scholia and figures, was given by Joseph Auria (Rome, 1591, 4to) . Pappus has given a pretty full commentary on the first book of this work of Theodosius. His work περὶ οἰκήσεων (On Habitations) also was published by Auria (Rome, 1588) . It gives an account of how, for every inhabitant of the earth from the equator to the pole, the starry firmament presents itself in the course of a year. ”

“ We proceed to summarize briefly the contents of that portion of the Collection which has survived, mentioning separately certain propositions which seem to be among the most important.
We can only conjecture that the lost book i., as well as book ii., was concerned with arithmetic, book iii. being clearly introduced as beginning a new subject. ”

“ His treatise on Conics gained him the title of The Great Geometer, and is that by which his fame has been transmitted to modern times. All his numerous other treatises have perished, save one, and we have only their titles handed down, with general indications of their contents, by later writers, especially Pappus. After the Conics in eight Books had been written in a first edition, Apollonius brought out a second edition, considerably revised as regards Books i.-ii., at the instance of one Eudemus of Pergamum ”

“ Akin to the geometrical works is that On the Dioptra, a remarkable book on land-surveying, so called from the instrument described in it, which was used for the same purposes as the modern theodolite. ”

“ The great work of Pappus, in eight books and entitled συναγωγή or Collection, we possess only in an incomplete form, the first book being lost, and the rest having suffered considerably. Suidas enumerates other works of Pappus as follows: Χωρογραφία οἰκουμενική, εἰς τὰ τέσσαρα βιβλία τῆς Πτολεμαίου μεγάλης ουντάξεως ὑπόμνημα, ποταμοὺς τοὺς ἐν Λιβύῃ, ὀνειροκριτικά. The question of Pappus’s commentary on Ptolemy’s work is discussed by Hultsch Pappi collectio (Berlin, 1878) , vol. iii. p. xiii. seq. ”

“ No blame attaches to the Roman general, Marcellus, since he had given orders to his men to spare the house and person of the sage; and in the midst of his triumph he lamented the death of so illustrious a person, directed an honourable burial to be given him, and befriended his surviving relatives. ”

“ A locus (says Simson) is a species of porism. Then follows a Latin translation of Pappus's note on the porisms, and the propositions which form the bulk of the treatise. These are Pappus's thirty-eight lemmas relating to the porisms, ten cases of the proposition concerning four straight lines, twenty-nine porisms, two problems in illustration and some preliminary lemmas. ”

“ Arenarius, or sand reckoner) , a small treatise, addressed to Gelo, the eldest son of Hiero, expounding, as applied to reckoning the number of grains of sand that could be contained in a sphere of the size of our “universe,” a system of naming large numbers according to “orders” and “periods” which would enable any number to be expressed up to that which we should write with 1 followed by 80,000 ciphers!
(9) A Collection of Lemmas, consisting of fifteen propositions in plane geometry. This has come down to us through a Latin version of an Arabic manuscript ”

“ The second book is in nine propositions, eight of which deal with segments of spheres and include the problems of cutting a given sphere by a plane so that (a) the surfaces, (b) the volumes, of the segments are in a given ratio (Props. 3, 4) , and of constructing a segment of a sphere similar to one given segment and having (a) its volume, (b) its surface, equal to that of another (5, 6) .
(2) The Measurement of the Circle (Κύκλου μέτρησις) is a short book of three propositions, the main result being obtained in Prop. ”

“ The Mechanics was first published by Carra de Vaux in the Journal asiatique (ix. série, ii., 1893) . In 1899 began the publication in Teubner’s series of Heronis Alexandrini opera quae supersunt omnia. Vol. i. and Supplement (by W. Schmidt) contains the Pneumatica and Automata, the fragment on Water Clocks, the De ingeniis spiritualibus of Philon of Byzantium and extracts on Pneumatics by Vitruvius. ”

“ Praemittitur, &c., Opera et Studio Edmundi Halley (Oxoniae, 1706) , 4to; (5) a German translation of the Conics by H. Balsam (Berlin, 1861) ; (6) the definitive Greek text of Heiberg (Apollonii Pergaei quae Graece exstant Opera, Leipzig, 1891–1893) ; (7) T. L. Heath, Apollonius, Treatise on Conic Sections (Cambridge, 1896) ”

“ Theodosii Tripolitae, Sphaericorum, libri iii.; Euclidis, Data, Optica, Catoptrica, ac Phaenomena; Theodosii Tripolitae, De Hobitationibus et Noctibus ac Diebus, libri ii.; Autolyci Pitanaei, De Sphaera Mota, et libri ii. ”