One of the earliest writers on arithmetic was Nico machus. The period in which he flourished is not well determined ; nor is it of much importance to be known. He refers, in his writings, to an invention of Eratos thenes, and we cannot err far in assigning him a place soon after that mathematician. His treatise on the arithmetic of the ancients, presents us with a view of the distribution of numbers, according to the doctrines of the Pythagorean and Platonic schools. Science would have had but little to regret, had this work shared the fate of another written on the same subject, by Por phyry. His Praxis Aritl2metica, which unhappily, is lost, would have been a more valuable acquisition, and thrown much light on the methods of computation em ployed by the ancients. Jamblichus, Proclus, Asclepius, and Piliponus, have written commentaries on the works of Nicomachus ; but only those of Jamblichus now exist. The work of Boethius is rather a translation than a com mentary.
About the second century of the Christian zra, a new method of notation was introduced, called the sexagesi mal arithmetic, of which Ptolemy is supposed to he the inventor. Some traces of it are still to be found in the divisions and subdivisions of the circle, hour, &c. The principal design of this notation was to avoid the incon veniences of the common method, especially in fractions. Every unit was divided into 60 parts, and each of these parts into 60 others : and, in order to render the com putation more simple, the progression in whole numbers was also made sexagesimal. From unity to 59, the numbers were represented in the common way ; and 60, which was called a sexagesima prima, was denoted by unity with a dash over it thus I'. Twice 60 was II'; thrice 60, III'; and so on to fifty-nine times 60, where the series was resumed as before, only sixty times 60 was denoted by unity with two dashes, thus I". When a number less than 60 was joined to a sexagesimal, it was annexed in its proper character, thus I'V represent ed 65 ; ten times 60 and 11, or 611, Ste. Frac tions were denoted by placing the dash at the bottom, or at the left hand of the numeral letter : thus I, or 'I expressed A.
It is not unworthy of remark, that the notation of the sexagesimal arithmetic is founded on the same princi ples with that of the Arabians, and differs from it in no respect but in the extent of the radix, or scale. The dash corresponds exactly to the cypher. It does not appear, however, that any advantage was taken of the distinguished properties which this form of notation possesses for simplifying the operations of arithmetic. The radix is divisible by 2, 5, 4, 5, 6, 10, 12, 15, 20, and 30, and indeed has the greatest number of aliquot parts of any number below 120. This property proba
bly recommended it to the attention of Ptolemy in the choice of a scale. The only difficulty under which it labours, is the great number of characters that would be necessary for a system of arithmetic, founded on so high a radix. But this objection is not insurmountable ; the characters in the Chinese language are said to amount, at least, to as many thousands. 1.
Altlmugh the sexagesimal arithmetic is commonly as cribed to Ptolemy, it is probably an Eastern invention. The Indians, to this day, employ the sexagesimal divi sion of time. They divide the day into 60 equal parts, called Buries ; each gurie, into 60 equal parts, called tolls ; and lastly, each poll, into 60 equal parts, called or twinklings of an eye. They also employ periods of 60 years as we do centuries.
After the establishment of the Alexandrian school, the various branches of mathematics were cultivated with increasing ardour and success. Arithmetic in particular, was enriched by Diophantus, the supposed inventor of algebra, with some valuable problems which still retain his name. The time in which he lived is uncertain, but it probably was shout the middle of the fourth century. According to Abulpharagius, he flourished under the emperor Julian. He could not have been much after that period, as the celebrated Hypatia, who fell a victim to religious bigotry about the beginning of the fifth cen tury, wrote a commentary on his works. Diophantus is the author of what is called the indeterminate analysis, which furnishes methods of solving many problems in pure arithmetic, as well as in algebra and the higher geometry. He wrote thirteen books on arithmetic, of which only six have escaped the destroying hand of time, if we except a seventh, de Xumeris llfultangulis, which is found in some editions of his works. His writings which have come down to us attest the greatness of his genius, and increase our regret for those which we have lost.
An edition of the works of Diophantus was found in the Vatican, about the middle of the sixteenth century, and, according to Regiomontanus, it then contained all the thirteen books. In this, however, he was probably mistaken ; at least Bombelli, in the preface to his Alge bra, written in 1572, expressly affirms, that there were only six of them in the library at that time. These six books were published at Basil by Xylander in 1575, with a kind of commentary. But Xylander was no great ma thematician, and committed several mistakes. A far more accurate edition was published at Paris in 1621, by M. Bochet de Meziriac, the value of which was af terwards much increased by some excellent notes of M. Fermat.