CAL C ULA TING-MA CHINES. 



443 



FiQ. 3. 



theory of these numbers was born on the Nile at a remote epoch, and 

 was developed by Diophantes, the father of arithmetic, at the school 

 of Alexandria. In his treatise occurs the proposition, giving, as an 

 essential condition of such 

 a number, that the octuple 

 of a triangular number, aug- 

 mented by unity, is a perfect 

 square. This fact is made 

 evident by the examination 

 of the diagram (Fig. 3). 



An arithmetical progres- 

 sion is a series of numbers in 

 which each member is equal 

 to the preceding member, 

 plus a constant number 

 which is denominated the 

 common difference. Thus 

 the odd numbers one, three, 

 five, seven, nine, eleven, 

 form an arithmetical pro- 

 gression, the common differ- 

 ence of which is two. We 

 can demonstrate, as was 

 done in the preceding case, 

 that the sum of the terms 

 in such a procession is equal to the product of the number of terms by 

 the half-sum of the extremes ; and, in the same way, the area of a 

 trapeze is half the area of a parallelogram of the same height, the base 

 of which is equal to the sum of the 

 bases of the trapeze (Fig. 4). 



Our second example is borrowed 

 from Plato. Fig. 5 represents a 

 square of cabbages. To get the 

 number of cabbages contained in 

 the square, we multiply by itself the 

 number on one of the sides. We 

 have marked lines bounding the suc- 

 cessive squares that contain one, two, 

 three, four, five, or six cabbages to 

 the side. Now observe the differ- 

 ence between the numbers of cab- 

 bages in one square and the next one. 

 We find that the numbers included in the successive inclosures bor- 

 dered by our lines are one, three, five, seven, nine, eleven ; and we per- 

 ceive, by reference to the short dotted lines, that the number from one 

 inclosure to another increases by two. We come at once to the propo- 



FiG. 4. 



Fig. 5.— The Square op Cabbages. 



