^ 1 
a b C d 
Forifwe defign to change the Additive Ratio of 
to bd, iato Subtradive 5 let fobe made equal to 4^, and 
thus the Ratio of h to td^ that is, of to l^d^ will be 
Subtraftive^ and likewifejif the Subtraftive Ratio of l^d to 
was to be mad^ Additive 5 it is but making al^ equal 
to l^c, 
4^. This is always to be obferved^ when the Terms 
of theUatio which is to be reducM, ate known 5 but if 
they are unknown 5 and their Sum or Difference is 
known , it is often convenient to ufe the jth. and Sfh. 
Propofition of the Introdudion by means of which the 
difference of the Terms of an Additive Ratio , or the 
fum of'theTerms ofa Subtradive one, may be expreft, 
whence you may argue by Divifion or Compofition» 
Nowtheyf^. Propofition of the Introduftion is this 5 
If a Right Line is Divided into two equal Parts, and into 
two unequal Parts , the middle part is the half difference 
of the unequal parts. The 8th. Propofition is this ; If a 
Right Line is Divided into two equal parts, and a Right 
Line is added to it, that which is compounded of the 
half and of the Line added, is the half fum of the Line 
that is added, and of that which is compounded of the 
whole and the Line added. 
Second Defnition, 
That Ratio w€ call Common which is Common to 
two Proportions whether it be Diredi or Reciprocal ; 
Let there be two Proportions a b:: dy ^ndb^c : 
having the fame Terms b and and conftituting a Di~ 
rc£k Ratio ^ this Ratio we call Common, becaufe ic is 
Common to both Proportions : In like manner let thers 
be two Proportions a^ bw ej and b^ c : : d^^ each having 
the fame Terms b and e which conftitute a Reciprocal 
Ratio , this Ratio Vv'e call Common , becaufe it is Com- 
mon to both Proportions.. 
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