[ 5i7 ] 
which, with fome ufeful additions from later writers, 
have been handed down to us. 
And as the advantages of this ancient fyftem of 
analyfis cannot be too much inculcated in an age, 
wherein it has been fo little known, and almoft to- 
tally negledted, permit me, Sir, to clofe this addrefs 
to you with an example in each fpecies of problems. 
Were it propofed to draw a triangle given in fpe- 
cies, that two of its angles might touch each a right 
line given in pofition, and the third angle a given 
point. It is obvious, how difficult it would be to 
adopt a commodious algebraic calculation to this prob- 
lem notwithftanding it admits of more than one 
very concife folution, as follows. 
Let the lines given in pofition be AB, Fjo . 
A C and the given point D, the triangle 2 o,' 2 
given in fpecies being E D F. 
In the firft place fuppofe a circle to pafs thr , 
the three points A, E, D, which fhall inter- Fja 1 
fed AC in G. Then E G, D G being a ‘ 9 ' 
joined, the angle DEG will be equal to the given 
angle DAC, both infilling on the fame arch DG; 
alfo the angle E D G is the complement to two right 
of the given angle B A C : thefe angles therefore 
are given, and the whole figure EFGD given in 
fpecies. Confequently the angle E G F, and its equal 
ADE will be given together with the fide D E of 
the triangle in pofition. 
Again, fuppofe a circle to pafs through the three 
points A, E, F, cutting AD in H, and 
EH, F H joined. Here the angle E F H ' g ' 
will be equal to the given angle E A H, and the an- 
Vol. LIH. Xxx gle 
