316 MATHEMATICS. 



We proceed to speak more particularly of the individual branches 

 of Mathematics ; commencing with Arithmetic, taking next Algebra, 

 Geometry, and Analytic Geometry, or Ancylometry, and concluding 

 with the Calculus, or Rheometry. 



CHAPTER I. 



ARITHMETIC. 



ARITHMETIC is that branch of Mathematics which treats of calcula- 

 tion by means of the Arabic numerals, or ordinary characters repre- 

 senting numbers. Its name is derived from the Greek apt^^oj, a 

 number; and it is regarded as a science, when we study its theory 

 or principles ; but as an art, when we apply it in practice. From 

 its constant application to other sciences, and to the common pursuits 

 of life, it is one of the most useful branches of knowledge, among 

 those which are necessary to complete an elementary education. It 

 is much to be regretted that in teaching its rules, attention is not more 

 generally paid to the theory or reasons on which those rules are 

 founded ; both as rendering them more intelligible, and as serving to 

 discipline the mind. 



The invention of Arithmetic, is attributed by Josephus to the He- 

 brews ; by Strabo, to the Phcenicians ; and by others to the Egyp- 

 tians, Chaldeans, or Indians. Its first principles, were evidently 

 known, at a very early period, by the Chaldeans and Egyptians, 

 from whom the Hebrews and Phoenicians doubtless received them. 

 They were introduced into Greece, by Thales, and Pythagoras ; both 

 of whom travelled among the nations just named ; doubtless acquiring, 

 as well as communicating knowledge. Pythagoras either invented 

 or borrowed the Multiplication Table, about 520 B. C. : and, in 

 some books, it is still called by his name. Much of the Arithmetic 

 of Pythagoras, and the other philosophers, related to imaginary mys- 

 tical properties of numbers ; such as the Tetrachys, or most perfect 

 number, (36 or 40), to which they attributed wonderful virtues. The 

 Sieve of Eratosthenes, was a contrivance for finding the series of 

 Prime numbers, by successively cancelling all those which admit of 

 exact division : and the properties of Square numbers, were the 

 subject of many problems invented by Diophantus. 



We have already remarked that although the ancients reckoned by 

 tens, they did not use our modern Decimal notation. The Greeks 

 used the first letters of their alphabet for the successive numbers as 

 far as ten ; but the next letter stood for 20, and the next for 30, thus 

 proceeding as far as 100 ; after which the next letter stood for 200, 

 and so on to 1000, which was represented by the first letter of the 

 alphabet, with a dash beneath it. Three additional characters, how- 

 ever, were used in this scheme, as already explained, (p. 56.) They 

 also used numerals, similar to the Roman, though not the same. The 

 Roman numerals probably originated as follows. They expressed 

 the numbers from one to five, by straight marks, which afterwards 

 took the form of the letter I. Five was expressed by two straight 



