Elements of Algebra

CHAP. V. OF ALGEBRA. 145 look first in the Table for the xxv-gonal number, whose side is $n$, and it is found to be $\frac{23n^2-21n}{2}$. Then making $n=36$, we find 14526 for the number sought. 439. Question. A person bought a house, and he is asked how much he paid for it. He answers that the 365th gonal number of 12 is the number of crowns which it cost him. In order to find this number, we make $m=365$, and $n=12$; and substituting these values in the general formula, we find for the price of the house 23970 crowns*. * This chapter is entitled “Of Figurate or Polygonal Numbers.” It is not however without foundation that some algebraists make a distinction between figurate numbers and polygonal numbers. For the numbers commonly called figurate are all derived from a single arithmetical progression, and each series of numbers is formed from it by adding together the terms of the series which goes before. On the other hand, every series of polygonal numbers is produced from a different arithmetical progression. Hence, in strictness, we cannot speak of a single series of figurate numbers, as being at the same time a series of polygonal numbers. This will be made more evident by the following Tables. TABLE OF FIGURATE NUMBERS. Constant numbers - - 1. 1. 1. 1. 1. 1. \ldots Natural - - - - - 1. 2. 3. 4. 5. 6. \ldots Triangular - - - - 1. 3. 6. 10. 15. 21. \ldots Pyramidal - - - - 1. 4. 10. 20. 35. 56. \ldots Triangular-pyramidal - 1. 5. 15. 35. 70. 126. \ldots TABLE OF POLYGONAL NUMBERS. Diff. of the progr | Numbers 1 | triangular - 1. 3. 6. 10. 15. \ldots 2 | square - - 1. 4. 9. 16. 25. \ldots 3 | pentagon - 1. 5. 12. 22. 35. \ldots 4 | hexagon - 1. 6. 15. 28. 45. \ldots Powers likewise form particular series of numbers. The first two are to be found among the figurate numbers, and the third among the polygonal; which will appear by successively substituting for $a$ the numbers $1, 2, 3, \ldots$. TABLE OF POWERS. $a^0$ - - - - - 1. 1. 1. 1. 1. \ldots $a^1$ - - - - - 1. 2. 3. 4. 5. \ldots $a^2$ - - - - - 1. 4. 9. 16. 25. \ldots $a^3$ - - - - - 1. 8. 27. 64. 125. \ldots $a^4$ - - - - - 1. 16. 81. 256. 625. \ldots