: INVESTIGATION of PORISMS. 183 
of the sepals of DE and DF to the fquare of DG, which 
is the fame with that of AB to N. 
Hence this conftruétion: Divide A B in L, fo that A L may 
be to LB as the fquare of AH to the fquare of BK, and 
make as the fquare of A H to the fquare of AB, fo AL toN; 
and, laftly, having drawn from L upon A C and CB the per- 
pendiculars LO and L M, make LG perpendicular to AB, and 
fuch, that as AB to N, fo the fum of the fquares of LO and 
L M to the fquare of LG; G will be the point required, and 
the given ratio, which the fquares on D F and DE have to the 
fquare on DG, will be that of AB to N. 
Tuts is the conftruction which follows moft diredtly from 
the analyfis; but it may be rendered more fimple. For fince 
A H?: AB?::AL:N, and BK’: AB?::BL:N, therefore A H? 
+BK?:AB*::AB:N.  Likewife, if AG, BG be joined, 
AB:N::AH?:AG*, and AB:N::BK?:BG*; wherefore 
AB:N::AH?+BK::AG*+BG’, that is, AH?+5 K?: 
A B}::AH?+BK?:AG?+BG’, and AG?+GBh = 
AB The angle AGB is therefore a right angle, and 
AL:LG:LB. If therefore AB be divided in L, as in the 
preceding conftruction ; and if LG, a mean proportional be- 
tween A L and LB, be placed at right angles to AB, G will be 
the point required, ; 
Cor. Ir is evident from the conftrution, that if at the points 
A and B we fuppofe weights to be placed that are as the fquares 
of the fines of the angles CAB, CBA, L will be the centre of 
” gravity of thefe weights. For AL is to LB as \C* to CB’, 
or inverfely as the fquares of the fines of the angles at A and B.. 
25. Now, the ftep in this analyfis by which a fecond intro- 
duGtion of the general hypothefis is avoided, is that in which 
_the angle GL D is concluded to be a right angle. This con- 
_clufion follows from the excefs of the fquare of DG above 
- the fquare of G L, having a given ratio to the fquare of LD, 
at the fame time that L D is of no determinate magnitude. For, 
if 
