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 9. Theoreme. The 9. Proposition. If from•• vnitie be numbers in continuall proportion how many soeuer: and if th•••• number which followeth next after vnitie be a square number, then all the rest following also be square numbers. And if that number which followeth next after vnitie be a cube number, then all the rest following shall be cube numbers.

SVppose that from vnitie there be these numbers in continuall proportion A, B, C, D, E, F. And let A which followeth next vnto vnitie be a square number. Then I say, that all the rest following also are square numbers.* 1.1 That the third number, namely, B, is a square number, & so all forward leauing one betwene, it is plaine by the Proposition next going before. I say also that all the rest are square numbers. For, forasmuch as A, B, C, are in continuall proportion, and A is a square number, therfore (by the 22. of the eight) C also is a square number. Againe forasmuch as B, C, D, are in con∣tinuall

proportion, and B is a square num∣ber,

[illustration]

therfore D also (by the 22. of the eight) is a square number. In like sort may we proue, that all the rest are square numbers.

* 1.2But now suppose that A be a cube num∣ber. Then I say, that all the rest following are cube numbers. That the fourth from v∣nitie, that is, C is a cube number, and so all forward leauing two betwene, it is plaine (by the Proposition going before). Now I say, that all the rest also are cube numbers. For, for that as vnitie is to A, so is A to B: therefore how many times vnitie measu∣reth A, so many times A measureth B. But vnitie measureth A by those vnities which are in A. Wherefore A also measureth B by those vnities which are in A. Wherefore A multiplying him selfe produceth B. But A is a cube number. But if a cube number mutiply∣ing him selfe produce any number, the number produced, is (by the 3. of the ninth) a cube number. Wherefore B is a cube number. And forasmuch as there are foure numbers in con∣tinuall proportion A, B, C, D, and A is a cube number, therefore D also (by the 23. of the eight) is a cube number. And by the same reason E also is a cube number, and in like sort are all the r••st following: which was required to be proued.

Notes

• * 1.1

  Demostra∣tion of the first part of this proposition.
• * 1.2

  The second p••rt demonstrated.