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 47. Proposition. A line contayning in power two medials, is in one point onely deuided into his names.

SVppose that AB being a line containing in power two medialls, be deuided into his names in the point C, so that let the lines AC and CB be incommensurable in power, hauing that which is composed of the squares of the lines AC & CB mediall, and that also which is contained vnder the lines AC and CB mediall, and moreouer incommensurable ••o that which is composed of the squares of the lines AC and CB. Then I say,* 1.1 that the line AB can in no other point

[illustration]

be deuided into his names but onely in the point C. For if it be possible, let it be deuided into his names in the point D, so that let not the line AC be one and the same, that is, equall with the line DB: but by supposition let the line AC be the greater. And take a rationall line EF. And (by the 43. of the first) vpon the line EF apply a rectangle paralle∣logrāme EG equall to that which is cōposed of the squares of the lines AC and CB: and likewise vpon the line HC, which is equall to the line EF, apply the parallelogramme HK equall to that which is contained vnder the lines AC and CB twise. Wherefore the whole parallelogramme EK is equall to the square of the line AB.* 1.2 Againe vpon the same line EF describe the paralle∣logramme EL equall to the squares of the lines AD and DB. Wherefore the residue, name∣ly, that which is contayned vnder the lines AD and DB twise, is equall to the parallelo∣gramme remaining, namely, to MK. And forasmuch as that which is cōposed of the squares of the lines AC and CB, is (by supposition) mediall, therefore the parallelogrāme EG which is equall vnto it, is also mediall: and it is applied vpon the rationall line EF. Wherefore (by the 22. of the tenth) the line HE is rationall and incommensurable in length vnto the line EF. And by the same reason also the line HN is rationall and incommensurable in length to the same line EF. And forasmuch as that which is composed of the squares of the lines AC and CB is incommensurable to that which is contained vnder the lines AC and CB twise (for it is supposed to be incommensurable to that which is cōtained vnder the lines AC and CB once): therefore the parallelogramme EG is incommensurable to the paralle∣logramme H ••. Wherefore the line EH also is incommensurable in length to the line HN, and they are rationall lines: wherfore the lines EH and HN are rationall commensurable in power onely. Wherefore the whole line EN is a binomiall line, and is deuided into his names in the point H. And in like sort may we proue, that the same binomiall line EN is de∣uided into his names in the point M, and that the line EH is not one and the same that is e∣quall with the line MN, as it was proued in the end of the demonstration of the 44. of this booke. Wherefore a binomiall line is deuided into his names in two sundry pointes: which is impossible (by the 42. of the tenth). Wherefore a line containing in power two medials, is not in sundry pointes deuided into his names. Wherefore it is deuided in one point onely: which was required to be demonstrated.

¶Second Definitions.

IT was shewed before that of binomiall lines there were sixe kindes,* 1.3 the definitions of all which are here now set, and are called second definitiōs. All binomiall lines, as all other kindes of irrationall lines, are cōceaued, cōsidered, and perfectly vnderstanded onely in respecte of a rationall line (whose partes as before is taught, are certayne, and knowen, and may be distinctly expressed by number) vnto which line they are compared. Thys rational•• line must ye euer haue before your eyes, in all these definitions, so shall they all be ••asie inough.

* 1.4A binomiall line (ye know) is made of two partes or names, wherof the one is greater then the other. Wherfore the power or square also of the one is greater then the power or square o•• the other. The three first kindes of binomiall lines, namely, the first, the se∣con••, & the third, are produced, when the square of the greater name or part of a bino∣m••all e••cedeth the square of the lesse name or part, by the square of a line which is com∣m••nsurable in length to it, namely, to the greater. The three last kindes, namely, the fourth, the ••i••t, and the sixt, are produced, when the square of the greater name or part ••••••••edeth the square of the lesse name or part, by the square of a line incommensurable in length vnto it, that is, to the greater part.

* 1.5A first binomiall line is, whose square of the greater part exceedeth the square of t••e lesse part ••y the square of a line commensurable in length to the greater part, and the greater part is also commensurable in length to t••e rationall line first set.

As l••t the ra••ion••ll line first set be AB: whose partes are distinctly knowen: suppose also that the line CE be a binomiall line, whose names or partes let be CD and DE. And let the square of the line CD the greater part excede the square of the line DE the lesse part by the square of the line FG: which line FG, let b••e commen∣su••able

[illustration]

in length to the line CD, which is the greater part of the binomiall line. And moreouer let the line CD the greater pa••t be com¦mensurble in length to the rationall line first set, name∣ly, to AB. So by this d••••ini∣tion the binomiall line CE is a first binomiall line.

* 1.6A second binomiall line is, when the square of the greater part exceedeth the square of the lesse part by the square of a line commensurable in length vnto it, and the lesse part is commensurable in length to the rationall line first set.

As (supposing euer the rationall line) let CE be a binomiall line deuided in the poynt D. The square of whose greater part CD let exceede the square of the lesse part DE by the square of the line FG, which

[illustration]

line ••G let be cōmensurable in length vnto the line CD t••e gr••ater p•••••• o•• the binomiall line. And let also the line DE the lesse part of the binomiall line be commensu∣••able in l••ngth to the rationall line first set AB. So by this definition the binomiall line CE is a second binomiall line.

* 1.7A third binomiall line is, when the square of the greater part excedeth the

square of the lesse part, by the square of a line cōmensurable in length vnto it. And neither part is commensurable in length to the rationall line geuē.

As suppose the line CE to be a binomiall line: whose partes are ioyned together in the poynt D: and let the square of the line CD the greater part exceede the square of the lesse part DE by the square of the line FG, and

[illustration]

let the line FG be commensurable in length to the line CD the greater part of the binomiall. Moreouer, let neither the greater part CD, nor the lesse part DE, be commensurable in length to the rationall line AB, then is the line CE by this definition a third binomiall line.

A fourth binomiall line is,* 1.8 when the square of the greater part exceedeth the square of the lesse by the square of a line incommensurable in length vn∣to the greater part. And the greater is also commensurable in length to the rationall line.

As let the line CE be a binomiall line, whose partes let be CD & DE, & let the square of the line CD the greater part

[illustration]

exceede the square of the line DE the lesse, by the square of the line FG. And let the line FG be incommensu∣rable in length to the line CD the greater. Let also the line CD the greater part be commensurable in length vnto the rat••onall line AB. Then by this definition the line CE is a fourth binomiall line.

A fift binomiall line is,* 1.9 when the square of the greater part exceedeth the square of the lesse part, by the square of a line incommensurable vnto it in length. And the lesse part also is commensurable in length to the rationall line geuen.

As suppose that CE be a binomiall line, whose greater part let be CD, and let the lesse part be DE. And let the square of the line CD excede the square of the line DE by the square of the line FG, which let be incōmensurable in length vn∣to

[illustration]

the line CD the greater part of the binomiall line. And let the line DE the second part of the binomi∣all line be commensurable in length vnto the rationall line AB. So is the line CE by this definition a fift bi∣nomiall line.

A sixt binomiall line is,* 1.10 when the square of the greater part exceedeth the square of the lesse, by the square of a line incommensurable in length vnto it. And neither part is commensurable in length to the rationall line geuen.

As let the line CE be a binomiall line, deuided into his names in the point D. The square of whose greater part CD let exceede the square of the lesse part DE by the square of the line FG, and let the line FG be incommensurable in length to the line CD the greater part of the binomiall line. Let also

n••ither CD the greater part, nor DE

[illustration]

the lesse part be commensurable in length to the rationall line AB. And so by this definition the line CE is a sixt binomiall line. So ye see that by these definitions, & their examples, and declarations, all the kindes of binomiall lines are made very playne.

This is to be noted that here is nothing spoken of those lines, both whose portions a••e com••ensurable in length vnto the rationall line first set, for that such lines cannot be binomiall lines. ••or binomiall lines are composed of two rationall lines commensu∣rable in Power onely (by the 36. of this booke). But lines both whose portions are comm••nsurable in length to the rationall line first set are not binomiall lines. For that the partes of such lines should by the 12. of this booke be commensurable in length the one to the other. And so should they not be such lines as are required to the compositi∣on of a binomiall line. Moreouer such lines should not be irrationall but rationall, for that they are commensurable t•• ••ch of the parts whereof they are cōposed (by the 15. o•• this booke). And therefore they should be rationall for that the lines which compos•• them are rationall.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion leading to an absurditie.
• * 1.3

  Sixe kindes of binomiall lines.
• * 1.4

  A binomiall line co••••••ste••h of two pa••t••s.
• * 1.5

  Firs•• d••••••ini∣ti••n.
• * 1.6

  Secon•• diffi∣nition.
• * 1.7

  Third ••••••••••••i∣tion.
• * 1.8

  Fourth diffi∣nition.
• * 1.9

  Fifth dif••ini∣tion.
• * 1.10

  Sixth diffini∣tion.