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 41. Theoreme. The 59. Proposition. If a superficies be contayned vnder a rationall line, and a sixt binomiall line, the lyne which contayneth in power that superficies, is irrational, & is called a line contayning in power two medials.

SVppose that the superficies ABCD be contained vnder the rationall line AB, and vnder a sixt binomiall line AD, and let the line AD be supposed to be deuided ••••∣to his names in the point E, so that let the line AE be the greater name. Then I say that the line that containeth in power the superficies AC is irrationall, and is a line contay∣ning in power two medials.* 1.1 Let the selfe same constructiōs be in this, that were in the former propositions. Now it is manifest that the line MX containeth in power the superficies AC, [ 1] and that the line MN is incommensurable in power to the line NX. And forasmuch as the [ 2] line AE is incommensurable in length to the line AB, therfore the lines AE and AB are

rationall commensurable in power only. Wherfore (by the 〈◊〉〈◊〉 of the tenth) the parallelogrām•• AK that is, that which is composed of the squares of the lines MN and NX added together [ 3] is mediall. Againe forasmuch as the line ED is incommensurable in length to the line AB, therefore also the line EF is incōmēsurable in lēgth to the line EK. Wherfore the lines EF and EK are rationall commensurable in power onely. Wherfore the parallelogramme EL, that is, the parallelogramme MR which is contained vnder the lines MN and NX is me∣diall.

[illustration]

And forasmuch as the line AE is incommensurable in length to the line EF, therfore [ 4] the parallelogramme AK is also incommensurable to the parallelogramme EL (by the first of the sixt, and 10•• of the tenth.) But the parallelogramme AK is equal to that which is com∣posed of the squares of the lines MN and NX added together. And the parallelogramme EL is equall to that which is cōtai••••d vnd••r the lines MN and NX. Wherfore that which [ 5] is c••••posed of the squares of the lines MN and NY added together, is incommensurable to that which is contained vnder the l••nes MN and NX•• and e••••her of them, nam••ly, that which is composed of the squares of the lines MN and NX added tog••••her, and that which is contained v••der the lines MN and N••, is proued mediall; and the lines MN and NX are proued incommensurable in power. Wherfore (by the 41. of the tenth) the whole line MX is a line contayning in power two medials, and it containeth in power the superfices AC: which was required to be d••••onstr••ted.

An A••••umpt.

If a right line be deuided into two vnequall partes, the squares which are made of the vnequall partes, are greater then the rectangle parallelogramme c••••tayned vnder the vnequall partes, twise.

Suppose that AB be a right line, and let it be 〈…〉〈…〉 point C. And let the line AC be the greater part. 〈…〉〈…〉 and ••B, are greater thē that which is contained vnder the lines A•• and CB twise. D••••id•• (by the 10. of the first) the line AB into two e∣quall

[illustration]

partes, in the point D. Now forasmuch as the right line AB is deuided into two equall parte•• in the point D, and into two vnequall parte•• in the point C, therfore (by the 5. of the second) that which is contained vnder the lines 〈…〉〈…〉 line CD, is equall to the square of the line AD. 〈…〉〈…〉

the lines AC and CB, (omitting the square of the line CD) is lesse then the square of the AD (by the 9. common sentence, and the seuenth of the fifth:) Wherefore that which is con∣tained vnder the lines AC and CB, twise, is lesse then the double of the square of the line AD (that is, thē twise the square of the line AD)

[illustration]

by * 1.2 alternate proportiō, and the 14. of the fift. But the squares of the lines AC and CB are double to the squares of the lines AD and DC (by the 9. of the secōd). Therfore the squares of AC and CB are more then double to the square of AD alone, (leauing out the square of DC) by the 8. of the fift. But the parallelogramme contained vnder the lines AC and ••B twise, is proued lesse thē the double of the square of the line AD. Therfore the same parallelogramme contained vnder the lines AC and CB twise, is much lesse then the squares of the lines AC and CB. If a right line therfore be deuided into two vnequall partes, the squares which are made of the vnequall partes, are greater thē the rectangle parallelogramme contained vnder the vnequall partes, twise: which was required to be demonstrated.

In numbers I neede not to haue so alleaged, for the 17. of the seuenth had confirmed the doubles to be one to the other, as their singles were, but in our magnitudes, it likewise is true and euident by alternate proportion, thus. As the parallelogramme of the lines AC and CB is to his double, so is the square of the line AD to his double (eche being halfe). Wherfore, alternately, as the parallelogramme is to the square, so is the parallelograme his double to the double of the square. But the parallelograme was proued lesse then th•• square: wherfore his double is lesse then the square his double, by the 14. of the fifth.

This Assumpt is in some bookes not read, for that in maner it semeth to be all one with that which was put after the 39. of this booke: but for the diners maner of demon∣strating, it is necessary.* 1.3 For the feare of inuentiō is therby furthered. And though Zam∣bert did in the demonstration hereof, omitte that which P. Montaureus could not sup∣ply•• but plainly doubted of the sufficiencie of this proofe, yet M. Dee, by onely allega∣tion of the due places of credite, whose pithe & force•• Theon his wordes do containe, hath restored to the demonstration sufficiently, both light and authoritie, as you may perceiue, and chiefly such may iudge, who can compare this demonstration here (thus furnished) with the Greeke of Theon, or latine translation of Zambert.

Notes

• * 1.1

  Demonstra∣tion.
• * 1.2

  Looke after the Assumpt concluded at this marke: for plainer ope∣ning of this place.
• * 1.3

  The vse of this Assumpt is in the next proposition, & other follow∣ing.