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NATURE 



{Jan. 17, I i 



by Alexander is duly recorded by his biographer, Quintus 

 Curtius. What Alexandria actually became, is thus briefly 

 and graphically described : — 



"The earliest attempt to found a University, as we 

 understand the word, was made at Alexandria. Richly 

 endowed, supplied with lecture-rooms, libraries, museums, 

 laboratories, gardens, and all the plant and machinery 

 that ingenuity could suggest, it became at once the intel- 

 lectual metropolis of the Greek race, and remained so 

 for a thousand years. It was particularly fortunate in 

 producing, within the first century of its existence, three 

 of the greatest mathematicians of antiquity— Euclid, 

 Archimedes, and Apollonius. They laid down the lines 

 on which mathematics were subsequently studied, and, 

 largely owing to their influence, the history of mathe- 

 matics centres more or le-^s round that of Alexandria, 

 until the destruction of the city by the Arabs in 641 

 A.D." 



It would occupy too much space to discuss, or even to 

 enumerate, the writings of the Alexandrian mathema- 

 ticians. The most precious relics they have left behind 

 them are : the greater part of the numerous works of 

 Euclid, many of the writings of Archimedes, the " Conies'' 

 of Apollonius, the " Almagest " of Ptolemy, the " Mathe- 

 matical Collections" of Pappus, and the "Arithmetic," or, 

 rather, the " Algebra," of Diophantus. These and other 

 valuable pieces of work, which, like them, have reached 

 us in a more or less mutilated condition, are reviewed 

 in the fourth and fifth chapters of Mr. Ball's " History,'' 

 in which the best editions of these classical authors 

 are mentioned, and other sources of information con- 

 terning them are referred to. We owe the preservation 

 of most of them to the Greek refugees at Constantinople, 

 as will be seen from the following quotation : — 



! "After the capture of Alexandria by the Mohamme- 

 ,dans, the majority of the philosophers, who had previously 

 "been teaching there, migrated to Constantinople, which 

 then became the centre of Greek learning in the East, 

 and remained so for 900 years. But, though its history 

 tovers such an immense interval of time, it is utterly 

 '.barren of any scientific interest ; and its chief merit is 

 ithat it preserved for us the works of the different Greek 

 ■schools. The revelation of these works to the West in 

 the fifteenth century was one of the most important 

 sources of the stream of modern European thought, and 

 the history of the school may be summed up by saying 

 "that it played the part of a conduit-pipe in conveying to 

 ,us the results of an earlier and brighter age." 



Before the fall of Constantinople in 1453, which is alluded 

 to in the above extract, such mathematics as were known 

 in Western Europe were derived from Arabian sources. 



The history of Arab m ithematics and their introduc- 

 tion into Europe forms ihe subject-matter of the ninth 

 and tenth chapters of Mr Ball's book. The first of these 

 'excellent chapters tells us, in the beginning, how the 

 Arabs, by their intercourse with Constantinople and 

 India, in the reign of Al Mamun, the successor of the re- 

 nowned Caliph Haroun Al Raschid, acquired a knowledge 

 of the principal Greek and Hindu authors ; it then gives 

 an account of the works of the three chief Hindu mathe- 

 maticians, Arya-Bhatta. Brahmagupta, and Bhaskara ; 

 and finishes with an anaKsis of the great treatise of Alka- 

 rismi, the first Arab mathematician, and an enumeration 

 of the works of the most prominent among his successors 



from Tabit-ibn-Korra down to Alhazen and Abdel-gehl. 

 The account of Bhaskara is very much fuller than that 

 given by M. Maximilien Marie in his " Histoire des 

 Sciences Mathdmatiques et Physiques " (twelve vols. 8vo, 

 1883-88), and in other parts of the chapter some very 

 interesting facts are mentioned, which we do not find 

 noticed by M. Marie. Among these we may instance 

 the solution of the cubic by Tabit-ibn-Korra, about 650 

 years before the time of Tartaglia, and, what is even 

 more remarkable, the enunciation by Alkhodjandi of the 

 proposition that the sum of two cubes can never be a 

 cube. 



The next chapter begins with the introduction of 

 mathematics into Europe by the Moorish conquerors 

 of Spain in the eighth century ; shows how the Christians 

 gained from them some knowledge of Arab science in 

 the twelfth century, and, before the end of the thirteenth, 

 were in possession of "copies of Euclid, Archimedes, 

 Apollonius, Ptolemy, and some of the Arab works on 

 algebra " ; and brings the history of European mathe- 

 matics down to the middle of the fifteenth century. 

 During this long interval there lived only two great 

 mathematicians in all Christendom, both of whom be- 

 longed to the thirteenth century. One was the famous 

 Roger Bacon ; the other, Leonardo Fibonacci, of Pisa, 

 was the earliest European writer on algebra that we are 

 acquainted with. Their biographies, though concisely 

 written, necessarily occupy a large portion of the chapter. 



The three following chapters contain the history of 

 mathematics from the invention of printing to the year 

 1637, when the " Geomdtrie " of Descartes made its 

 appearance. In this brief space of tiqie, barely three- 

 quarters of a century, owing to the labours of Pacioli, 

 Recorde, Stifel, Tartaglia, Cardan, Ferrari, Bombelli, 

 Vieta, Harriot, Oughtred, Stevinus, and others, vast im- 

 provements in algebra had been effected ; trigonometrical 

 and logarithmic tables had been brought to a high state 

 of perfection by Regiomontanus, Rheticus, Napier, and 

 Briggs ; Desargues had invented the modern projective 

 geometry ; while, in astronomy, Copernicus, Kepler, and 

 Galileo had replaced the old Ptolemaic system by a still 

 older one (propounded by the Pythagoreans), which was 

 now, for the first time, established on a firm basis. 



Our author, as he tells us in the preface, has " usually 

 omitted all reference to practical astronomers, unless 

 there is some mathematical interest in the theories they 

 proposed," and, accordingly, the name of Tycho Brahe 

 does not figure in the above list. It would be better, in 

 our opinion, to treat Copernicus in the same manner, rather 

 than to do him the injustice of speaking of " his conjecture 

 that the earth and planets revolved round the sun." Grant- 

 ing that "he advocated it only on the ground that it gave 

 a simple explanation of natural phenomena," we would 

 ask what other, or what better, proof could he have of it ? 

 It should be borne in mind that Copernicus spent the 

 best years of his life in testing his "conjecture" by ob- 

 servations, and that nothing short of a firm conviction of 

 its truth could possibly have induced him to pubhsh it 

 in the face of the fierce opposition which he well knew 

 it would provoke. 



With this exception, the short sketches of the lives and 

 writings of all the mathematicians we have named are 

 well drawn, and convey a clear idea of the importance of 



