NE WTON 5 



**For the difference of the inscribed and circum- 

 scribed figures is the sum of the parallelograms K/, 

 \^rn, M«, D(?, that is (from the equality of ali their 

 bases), the rectangle under one of their bases K^ 

 and the sum of their altitudes ha, that is, the 

 rectangle AB/<a:. But this rectangle, because its 

 breadth AB is supposed diminished in infinitum, 

 becomes less than any given space. And therefore 

 (by Lem. I) the figures inscribed and circumscribed 

 become ultimately equal one to the other ; and 

 much more will the intermediate curvilinear figure 

 be ultimately equal to either. O.E.D." 



Principia, Book /, Section /, Lemma XI, 

 Scholiuin (first part omitted) 



IO. ". . . Quai de curvis lineis deque super- 

 ficiebus comprehensis demonstrata sunt, facile appli- 

 cantur.ad solidorum superficies curvas & contenta. 

 Praemisi vero haec Lemmata, ut effugerem tasdium 

 deducendi perplexas^ demonstrationes, more veterum 

 Geometrarum, ad absurdum. Contractiores enim 

 redduntur demonstrationes per methodum Indivisi- 

 bilium. Sed quoniam durior est Indivisibilium 

 hypothesis, & propterea methodus illa minus Geo- 

 metrica censetur ; malui demonstrationes rerum 

 sequentium ad ultimas quantitatum evanescentium 

 summas & rationes, primasque nascentium, id est, 

 ad limites summarum & rationum deducere ; & 

 propterea limitum illorum demonstrationes qua potui 

 brevitate praemittere. His enim idem prsestatur 



^ In the third edition " longas" takes the place of " perplexas." 



