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THE POPULAR SCIENCE MONTHLY. 



stick. At the end of a fortnight or a month the tally-notches were 

 reckoned up and the account was settled. The number of notches 

 represented the number of loaves of bread bought, and this number, 

 multiplied by the price per loaf, gave the amount of money I had to 

 take to the baker. 



Although in our present article we shall make use of systems of 

 numeration, and particularly of the decimal system, it is proper to ob- 

 serve that the most important properties of numbers are independent 

 of such systems, and that they are used by the arithmetician in his 

 calculations only for aids, as the chemist uses bottles and retorts. We 

 give two specimens of properties of numbers, which we see illustrated 

 in the problems called the flight of cranes and the square of the cabbages. 

 Cranes in their flight dispose themselves regularly in triangles. We 

 wish to get a rule for finding the number of the birds when we know the 

 number of files ; or, supposing that we have arranged the files with 

 increasing numbers from unity to a determined limit, we seek to find 

 the total of the unities contained in the collection. To make the mat- 

 ter plainer, let us seek the sum of the first six numbers, or the number 

 of units represented to the left of the broken line in Fig. 1, by the black 

 pawns. We will represent the same numbers, in an inverse order, by 

 white pawns, to the right of the same line. We shall see at once that 

 each horizontal line contains six units plus one ; and, since there are 

 six lines, the number of units in the whole square is six times seven. 

 The number we are seeking, then, or the number in the half -square, is 

 half of forty-two, or twenty-one. The same reasoning may be applied 



PIQ. 1. 



FiQ, 8. 



to any number ; so the sum of the first hundred numbers is half of a 

 hundred times a hundred and one, or 5,050. Therefore, to obtain the 

 sura of all the numbers of any series beginning with unity, all that is 

 necessary is to take half the product of the last number of the series 

 by the next one. An analogue of this mode of reasoning appears in 

 the geometrical demonstration of the proposition that the area of the 

 triangle ABC (Fig. 2) is half that of the parallelogram A B C D, of 

 the same base and height ; and reflection will show us that the arith- 

 metical and the geometrical theorems are really one. The numbers we 

 have just learned to calculate, which represent all collections of objects 

 regularly disposed in triangles, are called triangular numbers. The 



