CHAP. I] POLYGONAL, PYRAMIDAL AND FIGURATE NUMBERS. 



The rth figurate number of order n is the binomial coefficient 



- 1\ (r + n - l)(r + n - 2)- -r 



J 1-2- -n 



Thus fl is the rth triangular number p r 3 , while f 3 is the rth pyramidal or 

 tetrahedral number P r 3 . In a comment on the Polygonal Numbers of 

 Diophantus, Fermat 41 stated a theorem which, in the present notation, is 



and called fl the rth triangulo-triangular number. 



In April, 1638, St. Croix proposed to Descartes the problem: "Trouver 

 un trigone [triangular number] qui, plus un trigone te"tragone, fasse un 

 te*tragone [square], et de rechef , et que de la somme des cotes des tetragones 

 requite le premier des trigones et de la multiplication d'elle par son milieu 

 le second. J'ai donne 15 et 120. J'attends que quelqu'un y satisfasse par 

 d'autres nombres ou qu'il montre que la chose est impossible." The 

 problem, without the example, was proposed to Fermat (Oeuvres, II, 63) 

 in 1636, who did not solve it. 



Descartes 42 understood a trigone tetragone to be the square A 2 of a 

 triangular number, and proved that 15, 120 is the only solution if the 

 problem is understood to require two triangular numbers such that, if 

 either be added to the same A 2 , the sum is a square; while if one is per- 

 mitted to add both A 2 and a new A' 2 to the second required triangular 

 number, the two triangular numbers may be taken to be 45 and 1035, since 



45 + 6 2 = 9 2 , 1035 + 6 2 + 15 2 = 36 2 , 36+9 = 45, 45-46/2 = 1035. 



St. Croix did not admit the validity of Descartes' solution, and probably 

 meant a trigone tetragone to be a number both triangular and square (like 

 1, 36). The question would then be to find two numbers of the form 

 n(n -f- l)/2 such that, if a number both triangular and square be added 

 to each, there result two squares; further, the sum of the square roots of 

 these squares must equal the first required triangular number and must 

 also be the first factor n used in forming the second triangular number. 

 If, as seems intended, the numbers to be added to the triangular numbers 

 are to be identical, the only solution is 15, 120. Cf. Gerardin. 220 



Fermat 43 proposed to Frenicle the problem to find a number which 

 shall be polygonal in a given number of ways. Neither gave a solution. 

 [Cf. Euler, 59 end.] 



John Wallis 44 derived by summation the expression for the general 

 triangular number (p. 139), pyramidal number with triangular base P r 3 

 (p. 143), the sum (called trianguli-pyramidal number) of the latter for 

 r = 1, 2, , Z (p. 145), and the sum (called pyramidi-pyramidal number) 

 of these last for Z = 1,2, . The values found are the expanded forms of 



"Oeuvres, I, 341; French transl., Ill, 273. Also, II, 70, 84-5; French transl., Ill, 291-2; 

 letters to Mersenne, Sept., 1636, and to Roberval, Nov. 4, 1636. 



42 Oeuvres, II, 1898, 158-165, letter from Descartes to Mersenne, June 3, 1638. 



43 Oeuvres, II, 225, 230, 435, June and Aug., 1641, Aug., 1659. 



44 Arithmetica Infinitorvm, Oxford, 1656. 



