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

1. ¶An Assumpt.

Suppose that there be a rectangle triangle ABC, hauing the angle BAC a right angle. And (••y the 12. of the first) from the poynt A to the right line BC, a perpendicular line being drawen AD: then I say first, that the parallelogramme contayned vnder the lines [ 1] C•• and BD, is equall to the square of the line BA. Secondly I say, that the parallelo∣gramme [ 2] contay••ed vnder the lines BC and CD, is equall to the square of the line CA. Thirdly I say, that the parallelogramme contayned vnder the lines BD and DC, is equall to [ 3] the square of the line AD. And fourthly I say, that the parallelogramme contayned vnder [ 4] the lines BC & AD, is equall to the parallelogramme cōtayned vnder the lines BA & AC.

As tou••hing ••he first, that the parallelogramme contayned vnder the lines CB and BD, is equall to the square of the line AB, is thus proued.

[illustration]

For foras••uch as in the rectangle triangle BAC, ••••om the right angle vnto the base is drawen a per∣pendicular l••ne AD, ••he••••ore (by the 8. of the sixt) the triangles ABD and ADC, are like to the whole ••ria••gle ABC, and are also like the one to ••he other. A••d for that the triangle ABC is like to the triangle ADE, therefore both the triangles are equiangle by the d••finitiō of like figures. Wher∣fore (by the 4. of the sixt) as the line CB is to the line BA, so is the line AB to the line BD. * 1.1 Wherefore (by the 17. of the sixt) the parallelogramme con∣tained vnder the lines BC & BD, is equall to the, square of the line AB.* 1.2

As touching the second, that the parellelogramme con∣tained vnder the lines B•• and CD, is equall to the square of the line AC is by the selfe same reason proued. For the triangle ABC is like to the triangle ADC. Wherefore as the line BC is to the line AC,* 1.3 so is the line, AC to the line DC. * 1.4 Wherefore the parallelogramme contained vnder the lines BC and CD, is equall to the square of the line AC.* 1.5 As tou∣ching the third, that the parallelogramme contained vnder the lines BD and DC, is equall to the square of the line DA, is thus proued. For, forasmuch as if in a rectangle triangle be drawn frō the right angle to the base a perpendicular line, the perpendicular so drawen is the meane proportional betwene the segmēts of the base (by the corollary of the 8. of the sixt): ther¦••ore as the line BD is to the line DA, so is th•• line AD to the line DC. Wherefore (by the 1••. of the sixt) the parallelogramme contayned vnder the lines BD and DC, is equall to the square of the line DA. As touching the fourth, that the parallelogramme contained vnder the lines BC and AD, is equall to the parallelogramme contained vnder the lines BA and AC, is thus proued. For forasmuch as (as we haue already declared) the triangle ABC is like, and therefore equiangle, to the triangle ABD, therefore as the line BC is to

the line AC,* 1.6 so is the line BA to the line AD (by the 4. of the sixt).* 1.7 But if there be foure right lines proportionall, that which is contained vnder the first and the last, is equall to that which is contained vnder the two meanes (by the 16. of the sixt). Wherefore that which is contained vnder the lines BC and AD, is equall to that which is contayned vnder the lines BA and AC.* 1.8

I say moreouer, that if there be made a parallelogramme complete,* 1.9 contained vnder the lines BC and AD, which let be EC: and if likewise be made complete the parallelogramme contained vnder the lines BA and AC, which let be AF, it may by an other way be proued that the parallelogramme EC is equall to the parallelogramme AF. For, forasmuch as ei∣ther of them is double to the triangle ACB (by the 41. of the first): and thinges which are double to one and the selfe same thing, are equall the one to the other. Wherefore that which is contained vnder the lines BC and AD, is equall to that which is contained vnder the lines BA and AC.

Notes

• * 1.1

  A Corollary added by I. Dee.
• * 1.2

  Therefore if you deuide the square of the side AB by the side BC: the quotient will be BD. Which maketh DC also knowen: by ei∣the•• of which (by the 47. of the first) the per∣pendicular AD is knowen: which considera∣tion in practise, is very profitable, where the three sides AB, AC, and BC, are knowen or geuen.
• * 1.3

  I. Dee
• * 1.4

  The second Corollary.
• * 1.5

  Therefore if you deuide the square of the side AC, by the side BC, the portion DC, will be the product. &c. as in the former Corol∣l••ry.
• * 1.6

  I. D••e
• * 1.7

  The thirde Corollary.
• * 1.8

  Therfore if the parallelogramme of BA, and AC, be deuided by BC, the product will geue the p••••p••ndicular D A. These three Co∣rollaryes in practise Logisticall and Geometri∣call are profitable.
• * 1.9

  An other de∣monstration of this fourth part of the determinatiō.