The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed

The 11. Theoreme. The 11. Proposition. Proportions which are one and the selfe same to any one pro∣portion, are also the selfe same the one to the other.

SVpppose that as A is to B, so is C to D, and as C is to D, so is E to F. Then I say that as A is to B, so is E to F.* 1.1 Take equemultiplices to A, C and E, which let be G, H, K. And likewise to B, D and F take a∣ny other equemultiplices, which let be L, M, and N. And because as A is to B, so is C to D: and to A and G are taken equemultiplices G & H; & to B and D are take certaine other equemultiplices L & M.* 1.2 If therfore G exceede L, then also H excedeth M, and if it be equall it is equall, and if it be lesse it is lesse (by the conuerse of the 6•• definition of the fifth). Agayne because that as C is to D, so is E to F: and to C and E are taken ••••••em••ltiplices H ••••d K: and likewise to D & F are takē certaine other equemultiplices M & N. If therfore H exceede M, then also K excedeth N: and if it be equall, it is equall, and if it be lesse, it is lesse (by the same conuerse) But if K exceede M, then also G exce∣deth

L: and

[illustration]

if it be equal it is equall, and if it be lesse, it is lesse (by the same con∣uerse) Wher¦fore if G ex∣cede L, then K also ex∣cedeth N, and if it be equal it is e∣quall, and if it be lesse, it is lesse. But G & K are equemultipli¦ces of A & E. And L & N are certaine other equemultiplices of B & F. Wher∣fore (by the 6. definition) as A is to B, so is E to F. Proportions therfore which are one and the selfe same to any one proportion, are also the selfe same one to the other: which was required to be proued.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion••