•i44 PHILOSOPHICAL TRANSACTIONS. [aNNO l6gQ. 



known, it is often convenient to use the 7th and 8th proposition of the intro- 

 duction by means of which the difference of the terms of an additive ratio, or 

 the sum of the terms of a subtractive one, may be expressed, whence you may 

 argue by division or composition. Now the 7th proposition of the introduction 

 is this ; if a right line is divided into two equal parts, and into two unequal 

 parts, the middle part is the half diflerence of the unequal parts. The 8th pro- 

 position 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 sum of the line that is added, and of that which is compounded of the 

 whole and the line added. 



Second Definition. — That ratio is called common, which is common to two 

 proportions, whether it be direct or reciprocal. Let there be two proportions, 

 a, b : : d, e, and b, c :: e, 1, having the same terms b and e, and constituting 

 a direct ratio; this ratio is called common, becau«e it is common to both pro- 

 portions : in like manner let there be two proportions a, b : : e, 1 and b, c :: d, e, 

 each having the same terms b and e which constitute a reciprocal ratio, this ratio 

 is called common, because it is common to both proportions. 



5. Therefore if two proportions have a common ratio, we may argue by 

 equality ; but if a common ratio is wanting, it must be introduced, that we 

 may proceed farther, which will be done by the reduction of some ratio into 

 another equal to it. Likewise if a proportion lies in a triangle, or any other 

 figure, we must use a new proportion, by repeating some angle, that is, by 

 changing its position, that so we may have two equal terms in two different pro- 

 portions, and so may argue by equality : hence it is evident, that that angle ought 

 to be transposed, which together with the other angles and sides of the figure, 

 shows the most convenient similitude of triangles, 



6. Now what is sought being assumed as granted, all our endeavours must 

 be to retain in arguing those magnitudes which are already known, and to 

 extinguish as much as we can the unknown point ; and the analyst under- 

 standing where to use additive or subtractive ratio in one proportion, and how 

 to introduce a common ratio in two proportions, if it be wanting, will come to 

 the end of this resolution by necessary consequences : now this end is obtained 

 when the unknown magnitude is found equal to some known one, or the un- 

 known point is in one term, which is a 4th proportional, or in two terms either 

 means or extremes whose sum or difference is known ; for a 4th proportional, or 

 two reciprocals will do it. 



7. The analysis being ended, the order of the construction and demon- 

 stration is evident, for nothing else is required for the construction, but what 

 has, or is supposed to have been done in the analysis ; and for the demonstra- 



