TEEOKEMAWM GEOMETRICORVM 133 



DA == y ( M= -I- Qf - 2 MQ. X) 



FGzzV ( tf=-f- ^^ -2 tf ^. X) 

 Hinc itiiqiie , fubftituds hi(ce valoribus , deprehendetur , 

 efle AB^ -\- BC^ -{- CD= -f- DA^ =r M^ H- N^ -f- 2 M 



N . X -I- N^-i- P' - 2NP.x-+-p^-j-q^-i-2Pq.x-i- 



M^ H- 0; - 2 MQ. X ~ 2 ( M^ -I- N^ -I- F^ H- Q_') 



H- 2 X (MN - NP -h PQ.- MQ_) = 2 ( M^ -f- N^ -I- P 



Q_^)H- 2X (M- PxN-Q_) = 2 (A^-+- 2.Aa-i-a* 

 -i- B» — 2 B^ -I- ^' -h A= — 2 A^ -}- rt^H- B^ -f- 2B^ 

 H- ^^)- 8 abX, — 4(A^-j-B^-h«'-|-^*- 2^^X) 

 =1 4 (A^ -f- B' -I- FG^) — 4 A^ -i- 4 B^ -h 4- FG^ z= . 

 AC^ H- DB^ H- 4 ¥G\ 



I. Q. E. D. 

 Per {q itaque patct , fi quadrilaterum fiierit parallelogram 

 mum : tum FG in nihilum abire , adeaque futurum efle 

 AB- -i- BC^ H- CD^-i- DA= = AC^ -+- DB-\ 



Alterum Theorema non oftendit minus vfum Lemmatis 

 iam explicati , fed et multo difficilius demouflratur , ob 

 liimmam , in qua pofitum eft , atque extenfiflimam vni- 

 ver(alitatem. Inuentorem habet hoc altemm Theorcma 

 Celeberr. Cotefmm , in cuius opufculis poftremis editiim il- 

 iiid eft: a Rob. Smitb , led fine demonftratione ; quam 

 deinde alii Geometiiie addidenint ; omnibus autem in hac 

 reperiuuda palmam praeripuit , vti alias femper ,. loh. Ber- 

 noitUius ,, quem nuper admodum viuis ereptum luget ad. 

 hnc , diuque lugebit ,, ciuitas omnis Geometrica. Hanc vi- 

 rii , pofl: fata etiam illuftris. ,, demonftrationera videre li- 

 cet in Eiusdem Operibus , Tomo IV. png. 6j , perfedn 

 Indudione ,, atque euidenti ferierum coafecutione ekbora- 



