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 31. Theoreme. The 36. Proposition. If there be three right lines proportionall: a Parallelipipedon described of those three right lines, is equall to the Parallelipipedon described of the middle line, so that it consiste of equall sides, and also be equiangle to the foresayd Parallelipipedon.

SVppose that these three lines A, B, C, be proportionall, as A is to B, so let B be to C. Then I say, that the Parallelipipedon made of the lines A, B, C, is equall to the Pa∣rallelipipedon made of the line B, so that the solide made of the line B consist of e∣quall

sides, and be also equiangle to the solide made of the lines A, B, C. Describe (by the 23. of the eleuenth) a solide angle E contained vnder three superficiall angles,* 1.1 that is, DEG, GEF, and FED: and (by the 3. of the first) put vnto the line B euery one of these lines DE, GE, & EF, equall:

[illustration]

and make perfecte the so∣lide EK. And vnto the line A let the line LM be equall. And (by the 26. of the eleuēth) vnto the right line LM, and at the point in it L, describe vnto the solide angle E an equall so∣lide angle, cōtained vnder these plaine superficiall an∣gles NLX, XLM, and NLM, and vnto the line B put the line LX equall, & the line LN to the line C. Now for that as the line A is to the line B,* 1.2 so is the line B to the line C: but the line A is equall to the line LM, and the line B to eue∣ry one of these lines LX, EF, EG, and ED, and the line C to the line LN. Wherefore as LM is to EF, so is DE to LN: So then the sides about the e∣qual angles MLN, & D∣EF, are reciprokall: Wher∣fore (by the 14. of the sixt) the parallelogrāme MN is equall to the parallelogramme DF. And forasmuch as two plaine superficiall angles, name∣ly, DEF and NLM are equall the one to the other, and vpon them are erected vpward e∣quall right lines, LX and EG; comprehending with the right lines put at the beginning e∣quall angles the one to the other. Wherefore * 1.3 perpendicular lines drawen from the pointes X and G to the plaine super••icieces wherin are the angles NLM, and DEF, are (by the Co∣rollary of the former Proposition) equall the one to the other: and those perpendiculars are the altitudes of the Parallelipipedons LH and EK, by the 4. definition of the sixt. Wherfore the solides LH and EK, are vnder one and the selfe same altitude. But Parallelipipedons consisting vpon equall bases, and being vnder one and the selfe altitude, are (by the 31. of the eleuenth) equall the one to the other. Wherefore the solide LH is equall to the solide EK. But the solide LH is described of the lines A, B, C, and the solide EK is described of the line B. Wherefore the Parallelipipedon described of the lines A, B, C, is equall to the Pa∣rallelipipedon made of the line B, which consisteth of equall sides, and is also equiangle to the foresaid Parallelipipedon. If therfore there be three right lines proportionall, a Parallelipipe∣dō described of those three lines is equall to the Parallelipipedō described of the middle line, so that is consist of equall sides, and also be equiangle to the foresaid Parallelipipedon: which was required to be proued.

The construction and demonstration of this Proposition, and of the next Proposition following, may easily be conceaued and vnderstanded by the figures described in the plaine belonging to them. But ye may for the more full sight of them, describe such bodies of pasted paper, hauing their sides pro∣portionall, as is required in the Propositions.

¶New inuentions (coincident) added by Master Iohn Dee.

A Corollary. 1.

Hereby it is euident, that if three right lines be proportionall: the Cube produced of the middle line, is equall to the rectangle Parallelipipedon made of those three lines.

For a Cube is a Parallelipipedon of equall sides: and also rectangled: as we suppose the Parallelipi∣pedon, made of the three lines to be likewise rectangled.

¶A Probleme. 1.

A Cube being geuen, to finde three right lines proportionall, in any proportion geuen betwene two right lines: of which three lines, the rectangle Parallelipipedon produced, shall be equall to the Cube geuen.

Suppose AC to be the

[illustration]

Cube geuen: whose roote, suppose to be AB. Let the proportion geuen, be that which in betwene the two right lines D and E, I say now, three right lines are to be found, proportionall, in the proportion of D to E, of which, the rectangle Pa∣rallelipipedon produced, shall be equall to AC. By the 12. of the sixt let a line be found, which to AB haue that proportion that D hath to E. Let that line be F: and by the same 12. of the sixth, let an other line be found, to which, AB, hath that proportion that D hath to E: and let that line found be H. Let a rectangle Paral∣lelipipedon mathematically be produced of the three right lines F, AB, and H, which suppose to be K: I say now, that F, AB, and H, are three right lines found pro∣portionall in the proporti∣on of D to E, of which, the rectangle Parallelipipedon K, produced, is equall to AC the Cube geuen. First it is euident that F, AB, and H, are proportionall in the proportion of D to E. For, by construction, as D is to E, so is F to AB: and by construction likewise, as D is to E, so is AB to H. Wherefore F is to AB, and AB is to H, as D is to E. So then it is manifest, F, AB, and H, to be proportionall in the proportion of D to E, and AB to be the middle line. By my former Corollary, therefore, the rectangle parallelipipedon made of F, AB, and H, is equall to the Cnbe made of AB. But AC, is (by supposi••ion) the Cube made of AB•• and of the three lines F, AB, and H, the rectangle parallelipipedon produced, is K, by construction: Wher∣fore, K, is equall to AC: A Cube being geuen, therefore, three right lines are found, proportionall in

••ny prop••••••ion geuē be••wene two right lines, of which three right lines the rectangle parallelipipedon p••oduced, is equall to the Cube geuen. Which ought to be done.

A Probleme. 2.

A re••••angle Parallelipipedon being geuen, to finde three right lines proportionall: of the which, the rectangle Parallelipipedon produced, is equall to the rectangle Parallelipipedon geuen.

* 1.4Listen to this new deuise, you couragious Mathematiciens: consider, how nere this crepeth to the famous Probleme of doubling the Cube. What hope may (in maner) any young beginner cōceiue, by one meanes or other, at one time or other, to execute this Probleme? * 1.5 Seing to a Cube may in∣finitely in••inite Parallelipipedons be found equall: all which Parallelipipedons shall be produced of three right lines proportionall, by the fo••me•• P••obleme: but to any ••ectangle Parallelipipedon geuen, some one Cube is equall •• as is ••asie to demonst••at••: We can not doubt, but vnto our rectangle Paral∣lelipipedon geuen, many other ••ectangle Parallelipipedons are also equall, hauing their three lines of p••oduction, proportionall.* 1.6 In th•• former Probleme•• in••ini••••ly in••inite Parallelipipedons may be found of three proportionall lines p••oduced, equall to the Cube g••uen: it is to wete, the three lines to be of all proportions, that a man can deuise betwene two right lines: and here any one will serue: where also i•• infinite varietie: though all of one quantitie: as be••ore in the Cube. I leaue as now, with thys marke here set vp to shoote at. Hit it who can.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  It is euidēt that those per∣pendiculars are all one with the sta••∣ding lines of the solides, if their solide an¦gles be made of super••i∣c••••il right an∣gles onely.
• * 1.4

  D••u••ling of the 〈◊〉〈◊〉. &c.
• * 1.5

  D••m••••••••rati∣o•• ••f p••s••ibilitie in the ••••oblem.
• * 1.6

  An other argu∣me••t to com••o••t the studious.