71 
sition of his third Book, which many of the modems have 
been very illustrious about, but for elegance am afraid they 
must all yield to the ancients.” Another corollary gives the 
“ property made use of by Diophantus, ^[inJQuestion 24 
Book V. ; ” and by generalising his method he deduces “ the 
Porism quoted by Diophantus in the third question of the 
fifth Booh of Arithmetic .” 
Mr. Wildbore now takes up the second of the two Lemmas 
previously noticed, and, by an elegant and somewhat 
elaborate process, he deduces u the very remarkable Porism* 
quoted by Diophantus as the 5 th question of the Fifth Book.” 
He further adds that — “ there are many other curious pro- 
perties flowing from the squares here deduced. Thus in the 
second square, if DC=H, we have question 33, Book II., of 
Diophantus ; and if there be any number which is equal to 
the product of two numbers differing by unity ; and if to 
this number unity be added, the square of the sum may be 
thus divided into three squares. Hence the sum of three 
squares may be easily divided into three other squares.” 
Towards the close of the letter Mr. Wildbore gives a 
geometrical solution to the following case of the general 
Problem of Inclinations. " between two given semicircles, 
having their bases m the same right line, to insert a right 
line of a given length which shall verge to an angle of one 
of the semicircles.” He says, “ it is as curious as any we 
know that were done by the ancients, and having some time 
since bestowed pains to solve it without Dr. Horsley’s alge- 
braical Lemmas, I will here subjoin the solution.” He con- 
cludes by stating that should Mr. Lawson “communicate 
the preceding to his lordship, as I am in hopes you will, 
please to make my most respectful compliments to him, and 
tell him that I shall wait with patience till I can be favoured 
with a sight of what Dr. Simson has done.” 
# “ Etant donnes deux nombres carres cousecutifs, on pent trouver un 
troisieme nombre egal au double de la somme de ces deux premiers plus 2, 
tel, que le produit de deux de ces nombres augmente, soit de la somme des deux 
memes, soit du troisieme nombre, fasse un carre.” — Chasles, Porismes, p. 49.) 
If x, y , z be the numbers — then xy-\-{ xJ ry) 5 xz-\-(x-\-z) ; yz(y-\-z) ; 
x-\-yz ; y-\-xz ; z-\-xy , — are all to be squares under the conditions proposed. 
Quest. III. requires that x-\~ a ; y~)r a ; and also xy-\-a, be square numbers. 
