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 35. Theoreme. The 35. Proposition. If there be numbers in continuall proportion how many soeuer, and if from the second and last be taken away numbers equall vnto the first, as the ex∣cesse of the second is to the first, so is the excesse of the last to all the nūbers going before the last.

SVppose that these numbers A, BC, D, and EF, be in continuall proportion be∣ginning at A the least. And from BC, which is the second, take away CG equall vnto the first, namely, to A, and likewise from EF the last take away FH e∣quall also vnto the first, namely, to A. Then I say, that as the excesse BG is to A the first, so is HE the excesse, to all the numbers D, BC, and A, which go before the last number, namely, EF.* 1.1 Forasmuch as EF is the greater (for the second is supposed greater then the first) put the number FL equall to the number D, and likewise the number FK equall to the number BC. And forasmuch as FK is equall vnto CB, of which FH is equall vnto GC, therefore the residue HK is equall vnto the residue GB. And for that as the whole F••, is to the whole FL, so is the part taken away FL, to the part taken away FK, therefore the residue LE is to

[illustration]

the residue KL, as the whole ••E is to the whole FL (by the 11. of the seuenth). So likewise for that FL is to FK, as FK is to FH, KL shall be to HK, as the whole FL is to the whole FK (by the same Proposition). But as FE is to FL, and as FL is to FK, and FK to FH, so were FE to D, and D to BC, and BC 〈◊〉〈◊〉 A. Wherefore as LE is to KL, and as KL is to HK, so is D to BC. Wherefore alter∣nately (by the 23. of the seuenth) as LE is to D, so is KL to be BC, and as KL is to BC, so is HK to A. Wherefore also as one of the antecedentes is to one of the consequentes; so are

all the antecedentes to all the consequentes. Wherefore as KH is to A, so are HK, KL, and LE, to D, BC, and A (by the 12. of the seuenth). But it is proued, that KH is e∣quall vnto BG. Wherefore as BG, which is the excesse of the second, is to A, so is EH the excesse of the last vnto the numbers going before D, BC, and A. Wherefore as the excesse of the second is vnto the first, so is the excesse of the last to all the numbers going before the last: which was required to be proued.

Notes

• * 1.1

  Demonstra∣tion.