[(^74 3 
gles ABC, and S ; which excefles are equal, for M is equal 
to ABCj the 2 fides about the right angle, being 2 fides of 
a fquare, upon AB by fuppofition equal to AC, and ti e id 
fide equal to BC, therefore the whole triangles are equal, 
after tlie fame manner S and U are proved to be equal, 
therefore the fquare of CB is equal to the Iquare ot the 
2other fides C^ED. 
But if the fides be unequal (as in Fig. ^d) let the fquare 
be defcribed, and the^-z^^/Z^/c^r^/w LQ^compleated, the 
whole Figure exceeds the fiquare upon BC, by: 3 triangles 
X,R,Z, and exceeds alio the fquare LA, AD, by the trian- 
gle ABC, and the Parallelogram P. which excefles I 
fay are. equal, for Z is equal to ABC, the fide OC=BC, 
CP=AC, the angle D=A, and OCD=BCA> which is ma- 
nifeft by taking the common angle ACO out of the 
2 right angles BCO, ACD, therefore by fuperimpofition 
the whole triangles are equal. In like manner X is prov- 
ed equal to ABC, alfoR; and the parrallelogram PQ^ 
to be double of the triangle ABC s thus the excelTes be- 
ing proved equal, the remainders alfo will be equal, 
th'e fquare of BC to the fquare of AB, AC (QJ&- i»^- 
iiifeft cerollaries^ from hence are the 3^^^ and i6th oi 
the id book, alfo the ^^th and 13^/; of 2.d. And here I 
fliall obferve that by this Method of proving the 47. i. 
EucL\\s manifeft that that propofition maybedemon- 
flratedotherwife then Euclide has done it, andyetwith- 
qutth^ help of proportions, ^^MxchPeletarius denyed as 
pofSble. 
The fijft I p propofitions of the %d book are evidently 
demonftrated,only by fubftiUiteing or letters ia- 
jftead of lines, and multiplying them according to the te- 
nor of the propofition; thus to inftance in one or two; 
in.F/^. 4 call the whole line A, and its parts B and C there- 
fore A---B 4 C and, confequently AA=BB 4 CC 4 2l BC 
which is the very fence ot the ^th of the 2^book. Thus 
alfo (ini^/ij. s) let a line be cut into equal parts F,F, 
and let another line S be added thereto, tis manifeft thajc 
