Since therefore it is certain that the sum of DM and EM is the transverse ax DE; if it be demonstrated that DM is = KN and EM = Kn, the business will be done, because the sum of KN and Kn is also equal to the transverse ax DE.
Resolve the □ KN.
It is certain that NIq+••Kq=KNq.
Substitute for NIq, by the 7. lib. 1. CIq+CNq+2NC••
Then will CIq+CNq+2NC••+IKq=KNq.
Substitute for C••q, by the 9. lib. 1. CDq+DIE; then will CDq+DIE+CNq+2NC••+Kq=KNq.
Resolve also □ DM.
It is certain that CMq+CDq+
- 2DCM
- 2NCI
Substitute for CMq its value by the Preparation, and you'l have CNq−D••E+••Kq+CDq+2NCI=DMq:
Which were before = KNq.
Therefore KN=DM; which is one.
In like manner resolve □ Kn.
It is certain that nlq+••Kq=Knq.
Substitute for nIq, by Consect. 1. Prop. 10. Lib. 1. C••q+CNq−2nCI, and you'l have
CIq+CNq−2nCI+IKq=Knq.
Substitute for CIq, by the 9. lib. 1. CDq+D••E, and you'l have 〈 math 〉〈 math 〉=Knq.