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 33. Theoreme. The 33. Proposition. A number whose halfe part is odde, is euenly odde onely.

SVppose that A be a number whose halfe part is odde. Then I say that A is euenly od onely. That it is euenly odde it is manifest: for his halfe being odde measureth him by an euē number, namely, by 2.* 1.1 (by the defini∣tion).

[illustration]

I say also that it is euenly odde onely. For if A be euen∣ly euen, his halfe also is euen. For (by the definition) an euen number measureth him by an euen number. Wherefore that euen number which measureth him by an euen number shall also measure the halfe thereof being an odde number by the 4. common sentence of the seuenth which is absurd. Wherfore A is a number euenly odde onely: which was required to be proued.

An other demonstration to proue the same.

Suppose that the number A haue to his halfe an od nūber, namely, B. Thē I say that A is euēly od onely. That it is euenly odde needeth no profe: forasmuch as the number 2. an euen number measu∣reth it by the halfe thereof which is an odde number.* 1.2 Let C be the number 2. by which B measureth A

[illustration]

(for that A is supposed to be double vnto B). And let an euen number, namely, D measure A (which is possible for that A is an euen number by the definition) by F. And forasmuch as that which is produced of C into B is equall to that which is produced of D into F, therefore by the 19. of the seuenth, as C is to D, so is B to F. But C the number two measureth D being an euen number: wherfore F also measureth B which is the halfe of A. Wherfore F is an odde number. For if F were an euen number then should it in the B whome it measureth an odde number also by the 21. of this booke, which is contrary to the supposition. And in like maner may we proue that all the euē nūbers which measure the number A•• do measure it by odde numbers. Wherefore A is a number euenly odde onely: which was required to be proued.

Notes

• * 1.1

  Demonstra∣tion leading to an absurditie.
• * 1.2

  An other de∣monstration.