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

An other demonstration of the same first part after Montaureus.

Suppose that there be two lines commensurable in length A and B. Then I say that the squares described of those lin••s shalbe in proportion the one to the other as a square number is to a square nū∣ber.* 1.1 For forasmuch as the lines A and B are commēsurable in length, they shalbe in proportion the one to the other as number is to number (by the 5. of this booke). Let A be to B in duple proportion, which is in such proportion as number is to number, namely, as 4. is to 2, and 6. to 3, and so of many other. And (by the s••cond o•• the eight) take three least numbers in continuall proportion, and in duple propor¦tion, & let the same be the numbers 4.2.1: whe••fore by

[illustration]

the corrollary of the second of the eight, the numbers 4. and 1. shalbe square numbers. (For as 4. is a square number produced of 2. multiplied into him selfe, so is 1. also a square number, for it is produced of vnitie multi∣plied into him selfe.) I say moreouer that those are the square numbers, whose proportion the squares of the lines A and B haue the one to the other. For as the number 4. is to the number 2. so is the line A to the line B (••or either proportion is double by supposition): but as the line A is to the line B, so is the square of the line A to the parallelograme contained vnder the lines A and B (by the first of the sixt). Wherefore as the number 4. is to the number 2: so shall the square of the line A be to the parallelo∣grame contained vnder the lines A and B. Likewise as the number 2. is to the number 1. so is the line A to the line B (For either proportion is duple by supposition): but as the line A is to the line B, so is the parallelograme contained vnder the lines A and B to the square of the line B (by the selfe same first of the sixt). Wherefore as the number 2. is to 1. so is the parallelograme contained vnder the lines A and B to the square of the line B: wherefore of equallitie (by the 22. of the fifth) as the square of the line A is to the square of the line B, so is the number 4. to 1. which are proued to be square numbers.

* 1.2But now suppose that the square of

[illustration]

the line A, be vnto the square of the line B, as the square nūber produced of the number C is to the square number produced of the number D. Then I say that the lines A & B are cōmensurable in length. For for that as the square of the line A is to the square of the line B, so is the square number produced of the number C to the square number pro∣duced of the number D: but the propor¦tion of the square of the line A, is vnto the square of the line B double to that proportion which the line A hath vnto the line B (by the corollary of the 20. of

the sixt). And the proportion of the square number which is produced of the number C to the square number produced of the number D is (by the 11. of the eight) double to that pro∣portion which the number C hath vnto the number D. Wherefore as the line A is to the line B, so is the number G to the number D. Wherefore the line A hath vnto the line B the same proportion that the number C hath to the number D. Wherefore (by the 6. of this booke) the lines A and B are commensurable in length: which was required to be proued.

Notes

• * 1.1

  An oth••r de∣mon••••ra••ion o•• the same first part after Montaureus.
• * 1.2

  Demonstra∣tion of the se∣conde part which is the co••••erse of the former.