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 92. Theoreme. The 116. Proposition. Now let vs proue that in square figures, the diameter is incommensurable in length to the side.

SVppose that ABCD be a square, and let the diameter therof be AC. Then I say that the diameter AC is incommensurable in length to the side AB. For if it be possible,* 1.1 let it be cōmensurable in lēgth. I say that thē this will follow, that one and the selfe same nūber shall be both an euen number & an odde number. It is manifest (by the 47. of the first) that the square of the line AC is double to the square of the line AB. And for that the line AC is commensurable in length to the line AB (by sup∣position), therfore the lyne AC hath vnto the line AB

[illustration]

that proportion that a number hath to a number (by the 5. of the tenth). Let the lyne AC haue vnto the line AB that proportion that the number EF hath to the num∣ber G. And let EF and G be the least numbers that haue one and the same proportion with them. Wherfore EF is not vnitie. For if EF be vnitie, and it hath to the num∣ber G that proportion that the line AC hath to the lyne AB, and the line AC is greater then the lyne AB. Wher¦fore vnitie EF is greater then the number G, which is impossible. Wherfore FE is not vnitie, wherfore it is a number. And for that as the square of the line AC is to the square of the lyne AB, so is the square number of the number EF, to the square number

of the number G, for in eche is the proportion of their sides doubled (by the corollary of the 20 of the sixt and 11. of the eight): and the proportion of the line AC to the line AB doubled, is equal to the proportiō of the nūber EF to the number G, doubled, for as the line AC is to the line AB, so is the nūber EF to the number G. But the square of the line AC is double to the square of the line AB. Wherfore the square number produced of the number EF is double to the square number produced of the number G. Wherefore the square number produced of EF is an euen number. Wherfore EF is also an euen number. For if EF were an odde num∣ber, the square number also produced of it, should (by the 23. and 29. of the ninth) be an odde number. For if odde numbers how many soeuer be added together, and if the multitude of thē be odde, the whole also shal be odde. Wherfore EF is an euen number. Deuide the number EF into two equall partes in H. And forasmuch as the numbers EF and G are the lest num∣bers in that proportion, therfore (by the 24. of the seuenth) they are prime numbers the one to the other. And EF is an euen number. Wherfore G is an odde number. For if G were an e∣uen number, the number two should measure both the number EF and the number G (for e∣uery euen nūber hath an halfe part by the definition) but these numbers EF & G are prime the one to the other. Wherfore it is impossible that they should be measured by two or by any other number besides vnitie. Wherfore G is an odde number. And forasmuch as the number EF is double to the number EH, therfore the square number produced of EF is quadruple to the square number produced of EH. And the square number produced of EF is double to the square number produced of G. Wherfore the square number produced of G is double to the square number produced of EH. Wherfore the square number produced of G is an euen number. Wherfore also by those thinges which haue bene before spoken, the number G is an euen number, but it is proued that it is an odde number, which is impossible. Wherefore the line AC is not commensurable in length to the line AB, wherfore it is incommensurable.

An other demonstration.

We may by an other demonstration proue, that the diameter of a square is incommensu∣rable to the side thereof.* 1.2 Suppose that there be a square, whose diameter let be A and let the side thereof be B. Then I say that the line A is incommensurable in length to the line B. For if it be possible let it be commensurable in length. And agayne as the line A is to the line B so let the number EF be to the number G: and let them be the least that haue one and the same proportion with them: wherefore the numbers EF and

[illustration]

G, are prime the one to the other. First I say that G is not v∣nitie. For if it be possible let it be vnitie. And for that the square of the line A is to the square of the line B as the square number produced of EF is to the square number produced of G (as it was proued in the ••ormer demonstration) but the square of the line A is double to the square of the line B. Wher¦fore the square nūber produced of EF is double to the square number produced of G. And by your supposition G is vnitie. Wherefore the square number produced of EF is the number two which is impossible. Wherefore G is not vnitie. Wherefore it is a number. And for that as the square of the line A is to the square of the line B, so is the square number produced of EF to the square number produced of G. Wherefore the square number produced of EF is double to the square number produced of G. Wherefore the square number produced of G. measureth the square number produced of EF. Wherefore also (by the 14. of the eight) the number G measureth the number EF: and the number G also measureth it selfe. Wherefore the number G measureth these numbers EF and G, when yet they are prime the one to the

other: which is impossible. Wherefore the diameter A is not commensurable in length to the side B. Wherefore it is incommensurable: which was required to be demonstrated.

An other demonstration after Flussas.

Suppose that vppon the line AB be described a square

[illustration]

whose diameter let be the line AC. Then I say that the side AB is incommensurable in length vnto the diameter AC. For∣asmuch as the lines AB and BC are equall, therefore the square of the line AC is double to the square of the line AB by the 47. of the first. Take by the 2. of the eight nūbers how many soeuer in continuall proportion frō vnitie, and in the proporti∣on of the squares of the lines AB and AC. Which let be the numbers D, E, F, G. And forasmuch as the first from vnitie name¦ly E is no square number, for that it is a prime number, neither is also any other of the sayd numbers a square number except the third from vnitie and so all the rest leuing one betwene, by the 10. of the ninth. Wherefore D is to E, or E to F, or F to G, in that proportion that a square number is to a number not square. Wherefore by the corrollary of the 25. of the eight, they are not in that proportion the one to the other that a square number is to a square number. Wherefore neither also haue the squares of the lines AB and AC (which are in the same pro∣portion) that porportion that a square number hath to a square number. Wherefore by the 9. of this booke their sides, namely, the side AB and the diameter AC are incommensurable in length the one to the other which was required to be proued.

This demonstration I thought good to adde, for that the former demonstrations seme not so full, and they are thought of some to be none of Theons, as also the proposi∣tion to be none of Euclides.

Here followeth an instruction by some studious and skilfull Grecian (perchance Theon) which teacheth vs of farther vse and fruite of these irrationall lines.

Seing that there are founde out right lines incommensurable in length the one to the ••∣ther, as the lines A and B, there may also be founde out many other magnitudes hauing lēgth and breadth (such as are playne superficieces) which shalbe incommēsurable the one to the o∣ther. For if (by the 13. of the sixth) betwene the lines A and B there be taken the meane pro∣portionall line, namely C, then (by the second corrollary of the 20. of the sixth) as the line A is to the line B, so is the figure described vpon the line A to the figure described vpon the line C, being both like and in like sort described, that is, whether they be squares (which are al∣wayes like the one to the other), or whether they be any other like rectiline figures, or whether they be circles aboute the diameters A and C. For cir∣cles

[illustration]

haue that proportion the one to the other, that the squares of their diameters haue (by the 2. of the twelfth). Wherfore (by the second part of the 10. of the tenth) the ••igures described vpon the lines A and C being like and in like sort described are incommensurable the one to the other. Wherfore by this meanes there are founde out superficieces incommensurable the one to the other. In like sort there may be founde out figures cōmēsurable the one to the other, if ye put the lines A and B to be cōmensurable in lēgth the one to the other. And seing that it is so, now let vs also proue that euen in soli••es also or bodyes there are some commensurable the one to the other, and other some incommensurable the one to the other. For if from eche of the squares of the lines A and B, or from any other rectiline figures equal to these squares

be ••rected solides of equall alti••ude, whether those solides be comp••sed of equidistant supersi∣cieces, or whether they be p••ramids or prismes, thos•• solides s•• er••c••ed shalbe in that proportiō the one to the other that theyr bases are (by the 32. o•• the eleuenth and 5. and 6. of the twelfth) Howbeit there is no such proposition concerning prismes, And so if the bases of the solides b•• commensurable the one to the other, the solides also shall be commensurable the one to the other, and if the bases be incommensurable the one to the other, the solides also shall be incom∣mensurable the one to the other (by the 10. of the tenth). And if there be two circles A and B: and vpon ech of the circles be erected Cones or Cilinders of equal altitude, those Cones & Ci∣linders s••all be in that proportion the one to the other that the circles are, which are their ba∣ses (by the 11. of the twelfth): and so if the circles be commensurable the one to the other, the Cones and Cilinders also shall be commensurable the one to the other. But if the circles be in∣cōmensurable the one to the other, the Cones also and Cilinders shalbe incōmensurable the one to the other, (by the 10. of the tenth). Wherefore it is manifest that not onely in lines and super••icieces, but also in solides or bodyes is found commensurabilitie or incommensurability.

An aduertisement by Iohn Dee.

Although this proposition were by Euclide to this booke alotted, (as by the auncient grecian pub∣lished vnder the name of Aristoteles 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, it would seme to be, and also the property of the same, agreable to the matter of this booke, and the proposition it selfe, so famous in Philosophy and Logicke, as it was, would in maner craue his elemētal place, in this tēth boke) yet the dignitie & perfec∣tion•• of Mathematicall Method can not allow it here: as in due order following: But most aptly after the 9. propositiō of this booke, as a Corrollary of the last part thereof. And vndoubtedly the propo••itiō hath for this 2000. yeares bene notably regarded among the greke Philosophers: and before Aristotles time was concluded with the very same inconuenience to the gaynesayer, that the first demonstration here induceth, namely, Odde number to be equall to euen: as may appear•• in Aristotles worke, named Analitica prima, the first booke and 40. chapter. But els in very many places of his workes he maketh mention of the proposition. Euident also it is that Euclide was about Aristotles time, and in that age the most excel∣lent Geometrician among the Grekes. Wherefore, seing it was so publike in his time, so famous, and so appertayning to the property of this booke: it is most likely, both to be knowne to Euclide, and also to haue bene by him in apt order placed. But of the disordring of it, can remayne no doubt, if ye consider in Zamberts translation, two other propositions going next before it, so farre misplaced, that where they are, word for word, before du••ly placed, being the 105. and 106. yet here (after the booke ended), they are repeated with the numbers of 116. and 117. proposition. Zambert therein was more faythfull to fol∣low as he found in his greke example, than he was skilfull or carefull to doe what was necessary. Yea and some greke written auncient copyes haue them not so: Though in deede they be well demonstrated, yet truth disorded, is halfe disgraced•• especially where the patterne of good order, by profession is auouched to be. But through ignoraunce, arrogan∣cy and ••emerltie of vnskilfull Methode Masters, many thinges remayne yet, in these Geometricall Elementes, vnduely tumbled in: though true, yet with disgrace: which by helpe of so many wittes and habilitie of such, as now may haue good cause to be skilfull herein, will (I hope) ere long be taken a∣way: and thinges of impor∣tance (wanting) sup∣plied.

Notes

• * 1.1

  Demonstratiō leading to an impossibili••ie.
• * 1.2

  An other de∣monstration leading to an impossibili••ie.