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 2. Theoreme. The 2. Proposition. If two right line cut the ou•• to the other, they are •••• ••ne and the selfe same playne superficies: & euery triangle is in one & the selfe same superficie••.

SVppose that these two right lines AB and CD doo

[illustration]

cutte the one the other in the point E. Then I say that these lines AB and CD are in one and the selfe same superficies, and that euery triangle is in one & selfe same playne superficies.* 1.1 Take in the lines EC and EB points at all auentures, and let the same be F and G, and draw a right line from the poynt B to the point C, and an other from the point F to the point G. And draw the lines FH and GK. First I say that the triangle EBC is in one and the same ground superficies.* 1.2 For if part of the triangle EBC, namely the triangle FCH, or the triangle GBK be in the ground superficies, and the residu•• be in an other, then also part of one of the right lines EC or EB shall be in the ground superficies, and part in an other. So also if part of the triangle EBC, namely, the part EFG be in the ground superficies and the residue be in an other, then also one part of eche of the right lines EC and EB shall be in the ground superficies, & an other part in an other superficies, which (by the first of the eleuenth) is proued to be impossible. Wherfore the triangle EBC is in one and the selfe same playne su∣perficies. For in what superficies the triangle BCE is, in the same also is either of the lines EC and EB, and in what superficies either of the lines EC and EB is, in the selfe same al∣so are the lines AB and CD. Wherfore the right lines lines AB and CD are in one & the selfe same playne superficies, and euery triangle is in one & the selfe same playne superficies: which was required to be proued.

In this figure here set may ye more playnely conceaue the demon∣stration

[illustration]

of the former proposition where 〈◊〉〈◊〉 may ele•••••••• what part of the triangle ECB ye will, namely the part FCH or the part GBK, or finally the part FCGB as is required in the demonstration.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion leading to an impossibi∣litie.