C 354 
5 Therefore if two Proportions have a Common 
Ratio , we may argue by Equality • but if a Common 
Ratio is wanting, it muft be introduced, that we may 
proceed farther, which will be done by the Reda^ion of 
fome Ratio into another equal to it. 
Likewife if a Proportion lies in a Triangle or any 
other Figure , you muft uie a new Proportion by re- 
peating Ibmc Angle, that is^by changing its Pofition^that 
fo you may have two equal Terms m two different Pro- 
portions, and fo may argue by Equality ; Hence it is evi- 
dent that 5 that Angle ought to be tranfpofed, which 
together with the other Angles and Sides of the Figure, 
fliews the moft convenient fimilitude of Triangles. 
So. Now what is fought being affumM as granted, 
all our endeavours muft be to retain in arguing thofe 
magnitudes which are already known, and to extin- 
guifti as much as we can the unknown Point, and the 
Analyft underftanding where to ufe Additive or Sub- 
tra(3[ive Ratio in one Proportion , and how to Introduce 
a Common Ratio in two Proportions, if it be wanting, 
will come to the end of thisRefoiution by neceffary confe- 
quences : Now this cad is obtain'd when the unknown 
Magnitude is found equal to fome known Magfikudc, 
or the unknown Point is in one Term, which is a 
Proportional, or in two Terms either Means or Extreams 
whofe fum or difference is known, for a 4^^. Propor- 
tional, or two Reciprocals will do it. 
JO. The Analyfis being ended , the order of the Con- 
ftrudionand Demonftration is evident , for nothing elfe 
is required for the Conftrudion, but what has , or is 
fuppos'd to^have been done in the Analyfis , and for the 
Demonftration ^ nothing but to begin from the end of 
the Analyfis and proceed to the beginning of it,obfer- 
ving that where the Analyfis argues by Alternate or In- 
'^e^ted Propofitions^ the Synthefis argues by the fame, 
