1676-] 
adding to both MQ^RQ^MP, if therefore from thefe 
equal triangles you take what is common W^^.MLQ^ the 
remainders will be equal RXLM=QLZP s to both which 
addXLZ, and the whole parallelograms will be equal, 
RZ=^QZ (Q^E.DJ that triangles alfo having a com- 
nion baj/s, ^ltq in the proportion of their altitudes does 
hence follow, becaufe they are the halfs of parallelograms 
upon the lame bajis^ this alfo is true, and the demonftra- 
tion exa£Hy the fame in prifms, Pyramids, Cylinders, and 
cones, having the fame bafis. 
To prove the i6th of the 6th I fuppofe (\n Fig* 6.) the 
4 lines A, B, C, E. to be proportional, that is, granting A 
and C to be the leSer terms, the fame way that A is con- 
tained in B,fo is C in E, and that D is the denominator of 
the ratio, 'twill follow then that B is made up of A, mul- 
tiplied by D, and E of C multiplied by D, io that AD=B, 
andCD— E, draw therefore the extremes upon one ano- 
ther, that is A upon CD and the meanes, that is, C upon 
AD, the factors being the fame, I fay the produds ACD 
and CAD are the fame and confequently equal (QJE.. D. 
I know not whether it be worth the while to add fome- 
what (tho altogether impertinent to this prcfentfub- 
jed:} concerning Mons : Cornier s probleme which he lately 
propofed with oftentation enough to all Mathematicians 
to be folved, as if it contained fomething new, whereas 
tisnomore then the old bufinefs oi doubling the Cube z 
little difguifed, this has been fliewn by feveral, but by 
none /I think) after the <^^^^r^^V^/ way, or fo briefly as 
follows in 8. 
A: 2X::X.i_Xq=P 
A 
Aq4 2AX — iXc=4^qq per 47. I. 
A Aq 
Aqq4iAcX- — 2XcA = 4Xqq 
Aqq 4 2 AcX=:4 Xqq 4 ^ Xc A refolving which aquation 
A 4 2 X : 2 A 4 4 X : : Xc : Ac : into an Anally. 
that is, Xc (the cube uj^on X) is i of Ac (the cube upon A) 
An 
