the Greek Anthology, from which it appears that he died at the age of eighty-four years; but it is not certain if the Diophantus therein named is the author of the Arithmetics. Suidas, under the name "Hypatia," makes mention of Diophantus. His work is a singular production. There are not known to be extant any writings of an earlier date on the Greek arithmetic of a similar nature. It consists of a large collection of questions, some of them of considerable difficulty, with ingenious solutions. The work contains the solution both of determinate and indeterminate problems. The substance and details of the work itself appear to afford a strong presumption that, though this collection of problems was made and augmented by Diophantus himself, it is quite impossible to discriminate what he gathered from preceding writers, whose works are lost, and what additions and improvements were made by himself. The work originally consisted of thirteen books, and was dedicated to his friend Dionysius. Only the first six books are known to be extant, and an imperfect treatise on triangular and multangular numbers. These writings were first translated into Latin by Xylander, a professor at Heidelberg, and published in 1575. Both the original Greek, with a Latin version and a commentary, were published by Bachet at Paris in 1621. Another edition was edited by his son and Fermat, and published at Toulouse in 1670.1 These six books contain collections of problems, with their solutions, relating chiefly to properties of numbers, such as square and cube numbers, proportional numbers, &c. Ἐκ δὲ γάμων πέμπτω παιδ ̓ ἐπένευσεν ἔτει. 1 Diophanti Alexandrini Rerum Arithmeticarum libri sex, quorum primi duo adjecta habent Scholia Maximi (ut conjectura est) Planudis. Item liber de numeris polygonis seu multangulis. Opus incomparabile, veræ Arithmetica Logistica perfectionem continens, paucis adhuc visum. A. Guil. Xylandro Augustano incredibili labore Latine redditum et commentariis explanatum, inque lucem editum ad Illustriss. Principem Ludovicum Vuirtembergensem. Basiliæ per Eusebium Episcopium et Nicolai Fr. hæredes, 1571. (fol.) Diophanti Alexandrini Arithmeticorum libri sex, et de numeris multangulis liber unus. Nunc primum Græce et Latine editi, atque absolutissimis Commentariis illustrati. Auctore Claudio Gaspare Bacheto Meziriaco Sebusiano. Lutetiæ Parisiorum, sumptibus Sebastiani Cramoisy, 1621, fol. A second edition was published, with the following title:-Diophanti Alexandrini Arithmeticorum libri sex, et de numeris multangulis liber unus. Cum Com.mentariis C. G. Bacheti, V.C., et Observationibus D.P. de Fermat Senatoris Tolosani. Accessit doctrinæ Analytica inventum novum, collectum ex variis ejusdem D. P. de Fermat Epistolis. Obloquitur numeris Septem discrimina vocuni. Tolosa, excudebat Bernardus Bosc, e regione Collegii Societatis Jesu. M. DC. LXX. The Greek arithmetical notation is employed, but no account is given of the first principles of the science, or of the methods employed in the elementary operations of addition or subtraction, multiplication or division; nor even any explanation of these processes. In the first book, which contains forty-three problems with their solutions, are prefixed eleven definitions or explanations of the terms employed in the solution of the problems. The first three definitions explain the method of forming the successive powers of numbers, and give the initial letters of the names by which they are denoted. The second power of a number is named δύναμις ; the third power κύβος. The powers above the square and cube are named from the sums, not from the products of the lower powers. Thus, the fourth power is named dvrapodvrapus, which is formed by the product of two equal square numbers; the fifth power is named ivvaμókvßoc, the product of a square by a cube number; the sixth power by kʊßókʊßoç, which last, in modern language, is more correctly designated by the square of the cube, or the cube of the square. In the fourth book is shown how to form the third power of a binomial, and in the sixth book the fourth power. Diophantus calls a positive quantity raps (substance) and a negative quantity Meus (defect or want), and employs a decurtated inverted and prefixed to denote minus, but employs no corresponding mark to denote plus. The ninth definition states that minus multiplied by minus produces plus, and that minus multiplied by plus produces minus, but offers no reason nor explanation.1 The unknown number is named apoμòs, and is denoted by s, the final letter; a lineal quantity is also designated by the same word. The monad or general unit is denoted by the letters μ. In The eleventh definition contains some general remarks for forming the equations. No general rules are laid down for the solution of them, but different assumptions are made, and ingenious artifices are employed, whereby the equations are very much simplified. general, a direct assumption is made for the number required. Sometimes if the sum of two numbers be given in the problem, the difference is assumed; but if the difference be given, the sum is assumed for the unknown number. Sometimes when two quantities are required, their ratio is assumed, and other artifices are employed whereby almost all the solutions are effected by means of equations of the first degree. 1 The words of the definition are: Λειψις ἐπὶ λέιψιν πολλαπλασιασθείσα, ποιει ὕπαρξιν. λειψις δὲ ἐπὶ ὕπαρξιν, ποιει λειψιν. "Penuria in penuriam ducta, copiam ; in copiam ducta, penuriam procreat.” The second book contains thirty-six problems with solutions; the third, twenty-four problems on squares; the fourth book, forty-six on cubes and squares; the fifth, twenty-three on square and cube numbers, and some involving numbers in geometrical progression. To these the editor has added upwards of forty questions from the Greek Anthology. The sixth book contains twenty-six problems relating to properties of right-angled triangles. The fragment on triangular and polygonal numbers contains only the ten problems that are known to be extant. of The school of Alexandria continued to flourish after the age Diophantus. Pappus, an eminent mathematician, lived in the latter part of the fourth century. His mathematical collections consist of eight books; of these, the first book and the early part of the second are not known to be extant. The last twelve propositions of the second book were printed by Dr. Wallis in 1688, at the end of his work entitled, "Aristarchus Samius;" from which it would appear that Arithmetic was the subject of the second book of the collections. The work of Pappus is especially valuable as containing an account of the state of the mathematical sciences at that period. Theon was contemporary with Pappus, and became president of the school at Alexandria. As a philosopher and a mathematician, he appears from the remains of his writings to have been a student of the laws of nature, as well as of the sciences. He understood the higher use of the mathematical sciences as an intellectual discipline, as will appear from the following extract from his writings: "The science of number is the mover and the guide to truth. It is not to be studied with gross and vulgar views, but in such a manner as may enable the student to attain to the contemplation of the nature of numbers; not learning it merely for the purpose of selling or of dealing with merchants or retailers, but for the improvement of the mind, considering it as the path which leads to the knowledge of truth and reality. The study of this science exalts the mind, and compels it to think and reason on numbers themselves, as abstract. There are not to be admitted into its reasonings, as necessary, any objects, either visible or capable of being numbered. And besides this study renders a student competent for all branches of the mathematics, and enables the slow in intellect to become more acute." 1 Proclus, in the fifth century, was a student at Alexandria, but after 1 Theon. Smyrn. c. i. 2 Proclus wrote the following epitaph to the memory of his instructor Syrianus :- Πρόκλος ἐγὼ γενόμην Λύκιος γένος, ὃν Συριανὸς Ενθάδ ̓ ἀμοιβὸν ἑῆς θρέψε διδασκαλίης Ξυνὺς δ ̓ ἀμφοτέρων ὅδε σώματα δέξατο τύμβος wards removed to the Platonic school at Athens. At this period, the study of the mathematical sciences at Alexandria was not pursued in the manner of Euclid and others of his school, but had degenerated into the dreams and fancies of the later Platonists. They imagined that they had discovered mysteries in numbers and in their properties. They assumed imaginary analogies as elementary truths, and constructed from them the most strange and absurd theories. It is very singular that these absurdities of the Platonists have not been without an advocate in modern times; as the author of "Theoretic Arithmetic demands: "Whether it is possible that these philosophers could have spoken thus sublimely of number, unless they had considered it as possessing an essence separate from sensibles, and a transcendency fabricative, and at the same time paradigmatic." After the end of the fourth century, there appear no names of great eminence who advanced the knowledge of the sciences. The dreams and subtleties of the later Platonists appear to have absorbed the attention both of the philosophers and their disciples; and Alexandria continued to be the chief seat of learning until that city was besieged, in A.D. 640, and captured by the Saracens, and the great library destroyed. This magnificent library, from its foundation, rapidly increased, and became the most extensive depository of writings on science and learning in the world, at that period. Even in the time of Ptolemy Philadelphus, the son of the founder, it is reported to have contained no less than 10,000 volumes, and it continued to increase, until, at the time of its destruction, it contained between 700,000 and 800,000 volumes, among which were the original writings of Eschylus, Sophocles, and Euripides. It has been maintained that the astronomical and other mathematical sciences of the Hindus were borrowed from the Greeks;1 and 1 Professor Dugald Stewart, in his "Philosophy of the Human Mind," has adduced considerations to show that the Sanscrit language is derived from the Greek. He considers that the Sanscrit language was formed on the model of the Greek, by the Brahmins, after the invasion of India by Alexander the Great, and that it grew rapidly and attained its perfection as a new language a century before the Christian era. The names of the countries, rivers, towns, &c., recorded in the writings of Quintius Curtius, Arrian, and others who have written on Alexander's expedition to the East, clearly evince to the reader that the Sanscrit language had an existence before the age of Alexander. And further, there are no evidences to show that the Greek language was in use in India, or that any writings in that language were extant anterior to those in Sanscrit. Sanscrit words are found in all the various dialects spoken in India, and these dialects, so different from each other, and extending over very large tracts of country, may have been derived from a common primeval tongue, before the Sanscrit language became a language developed. There are very striking affinities between the Sanscrit and the Greek and Latin languages, both in the words themselves and in their inflexions. All languages are further, that the Greek language, after the conquests of Alexander in the East, gave birth to the Sanscrit language. The settlement of a part of Alexander's army in Bactriana, and their government, which lasted for 130 years, might have influenced in some degree the populations of that and the neighbouring countries. It is not very likely that there was much intercourse between the rulers of Bactriana and the people of Greece. Of the numerous facts adduced in favour of the priority of Greek science, one may be here mentioned. It is true that the twelve signs of the zodiac are identical both in the astronomy of the Greeks and the Hindus.' But it is uncertain whether the Hindus subject to growth and improvement, and it cannot be conceived that all the words and constructions of the Greek, or any other language, were coeval with the earliest use of that language. This may be understood by a comparison of the English language in its present form, with the language in the time of Chaucer, and with the earlier form of the Anglo-Saxon. The obvious affinity between Sanscrit, Greek, and Latin words, and the variety of inflected forms existing in Sanscrit, and almost the same found in Greek and Latin inflexions, afford at least a strong ground for the presumption that the tribes which founded the Greek and Latin colonies had brought with them a language, either the Sanscrit or some cognate dialect of that language. In fact, the common origin both of Sanscrit, Greek, and Latin, as well as their offshoots, must be referred to some primitive tongue. The resemblances in words, with their derivations and inflexions, notwithstanding the changes which these languages have undergone in the course of ages, supply an indication of their relation to the same family. Examples may be seen in Dr. Donaldson's New Cratylus, 3rd Ed., 1859. From p. 477 of the third volume of E. T. Colebrooke's Essays, edited by Professor Cowell, the following note of the Professor is extracted :-" Dr. Kern gives a list of thirty-six Greek words which occur in Varahamihira's Vrihat Sanhita [Great Course of Astrology]. The signs of the zodiac (except Cancer), Kriya, Tavuri, Jituma, Leya, Pathena, Dyuka or Juka, Kaurpya, Taukshika, Akokera, Hridroga, Ittham :-Heli (Atos), Himna ('Epuñs), Ara ("Apns), Jyau (Zeús), Kona (Kpóvos), Asphujit ('Appodíτn), hora, kendra, dresh kana or drekkana, lipta, anapha (ἀναφή), sunapha (συναφή), dumdhara (δορυφορία), kemadruma (χρηματισμός), vesi (φάσις), apoklima (ἀπόκλιμα), panaphara (ἐπαναφορά), hibuka (ὑπόγειον), jamitra (διάμετρος), meshurana (μεσουράνημα), dyunam or dyutam (δυτικόν ?), rilpha (ῥιφή), and harija (ópíÇwv).” 1 The Hindu division of the ecliptic is the same as that of the Greeks. Their astronomical year was sidereal (being the space of time in which the sun departing from a star returns to the same), and commenced at the instant when the sun enters the first degree of the sign Aries, or the Hindu constellation Mesha. The inclination of the ecliptic to the equator is stated in the Surya Siddhanta to be twentyfour degrees. It was the opinion of Sir William Jones that the Indian zodiac was not borrowed from the Greek; and if the solar divisions of it in India were the same as in Greece, it may reasonably be concluded, that both Greeks and Hindus had received it from an older nation which first gave names to the constellations, and from which both Greeks and Hindus, as the similarity of their language fully evinces, had a common origin. Sir William Jones received the following account from two learned Brahmins. "The Hindus divide the circle of the heavens into 360 portions, and allot 30 to each of the twelve constellations of the zodiac in this order : |