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 29. Theoreme. The 34. Proposition. In equall Parallelipipedons the bases are reciprokall to their altitudes. And Parallelipipedons whose bases are reciprokall to their altitudes, are equall the one to the other.

SVppose that these Parallelipipedons AB & CD be equall the one to the other.* 1.1 Then I say, that the bases of the Parallelipipedons AB and CD are reciprokall to their altitudes, that is, as the base EH is to the base NP, so is the altitude of the solide CD to the altitude of the solide AB. First let the standing lines AG, EF, LB, HK, of the solide AB, & the stāding lines CM, NX, OD, and PR, of the solide CD, be erectedperpē∣dicularly to the bases EH & NP.* 1.2 Thē I say, that as the base EH is to the base NP, so is the line CM to the line AG.* 1.3 Now if

[illustration]

the base EH be equal to the base NP, and the solide AB is equall to the solide CD, wherefore the line CM is equall to the line AG* 1.4. For if the bases EH and NP being equall, the altitudes AG and CM be not equall, nei∣ther also shall the solide AB be equall to the solide CD, but they are supposed to be equall. Wherefore the altitude CM is not vnequall to the altitude AG. Wherefore it is equall. And therefore as the base EH is to the base PN, so is the altitude CM to the altitude AG. Wher∣fore it is manifest, that the bases of the Parallelipipedons AB and CD are reciprokall to their altitudes.

But now suppose that the base EH be not equall to the base NP.* 1.5 But let the base EH be the greater. Now the solide AB is equall to the solide CD. Wherefore also the altitude CM is greater then the altitude AG * 1.6. For if not, then againe are not the solides AB and CD equall: but they are by supposition equall. Wherefore (by the 2. of the first) put vnto the line AG an equall line CT. And vpon the base NP and the altitude being CT, make perfecte a solide contained vnder parallel plaine superficieces, and let the same be CZ. And forasmuch

as the solide AB is equall to the solide CD, and there is a certaine other solide, namely, CZ, but vnto one and the selfe same magnitude equall magnitudes haue one and the selfe same proportion (by the 7. of the fift). Wherefore as the solide AB is to the solide CZ, so is the so∣lide CD to the solide CZ. But

[illustration]

as the solide AB is to the solide CZ, so is the base EH to the base NP (by the 32. of the ele∣uenth)•• for the solides AB and CZ are vnder equall altitudes. And as the solide CD is to the solide CZ, so is the base MP to the base PT, and the line MC to the line CT. Wherefore (by the 11. of the fift) as the base EH is to the base NP, so is the line CM to the line CT. But the line CT is equall to the line AG. Wherefore (by the 7. of the fift) as the base EH is to the base NP, so is the altitude CM to the altitude AG. Wherfore in these Parallelipipedons AB and CD the bases are reciprokall to th••ir altitudes.

But now againe suppose that the bases of the Parallelipipedons AB and CD be recipro∣kall to their altitudes,* 1.7 that is, as the base EH is to the base NP, so let the altitude of the so∣lide CD be to the altitude of the solide AB. Then I say, that the solide AB is equall to the solide CD. For againe let the standing lines be erected perpendicularly to their bases.

And now if the base EH be equall to the base NP: but as the base EH is to the base [ 1] NP, so is the altitude of the solide CD to the altitude of the solide AB. Wherefore the al∣titude o•• the solide CD is equall to the altitude of the solide AB. But Parallelipipedons con∣sisting vpon equall bases and vnder one and the selfe same altitude, are (by the 31. of the eleuenth) equall the one to the other. Wherefore the solide AB is equall to the solide CD.

[ 2] But now suppose that the base EH be not equall to the base NP: but let the base EH be the greater. Wherefore also the altitude of the solide CD, that is, the line CM is greater then the altitude of the solide AB, that is, then the line AG. Put againe (by the 3. of the first) the line CT equall to the line AG, and make perfecte the solide CZ. Now for that as the base EH is to the base NP, so is the line MC to the line AG. But the line AG is e∣quall to the line CT. Wherefore as the base EH is to the base NP, so is the line CM to the line CT. But as the base EH is to the base NP, so (by the 32. of the eleuenth) is the solide AB to the solide CZ, ••or the solides AB and CZ are vnder equall altitudes. And as the line CM is to the line CT, so (by the 1. of the sixt) is the base MP to the base P••T, and (by the 32. of the eleuenth) the solide CD to the solide CZ. Wherefore also (by the 11. and 9. of the fift) as the solide AB is to the solide CZ, so is the solide CD to the solide CZ. Wher∣fore either of these solides AB and CD haue to the solide CZ one and the same proportion. Wherefore (by the 7. of the fift) the solide AB is equall to the solide CD: which was requi∣red to be demonstrated.

* 1.8But now suppose that the standing lines, namely, FE, BL, GA, KH: XN, DO, MC, and RP, be not erected perpendicularly to their bases. And (by the 11. of the eleuenth) from the pointes F, G, B, K•• X, M, D, R,* 1.9 draw vnto the plaine superficies of the bases EH and NP perpendicular lines, and let those perpendicular lines light vpō the pointes S, T, V, Z: W, Y, d, and Q, and make perfecte the Parallelipipedons FZ, and XQI say that euen in this case also, if the solides AB and CD be equall, their bases are reciprokall to their altitudes, that is, as the base EH is to the base NP, so is the altitude of the solide CD to the altitude of the solide AB. For forasmuch as the solide AB is equall to the solide

CD, but the solide AB is equall to

[illustration]

the solide BT (by the 20. of the e∣leuenth) for they are vpon one and the selfe same base,* 1.10 namely, the pa∣rallelogramme KF, and vnder one and the selfe same altitude, whose standing lines are in the selfe same right lines, namely, HZAT, and LVES: and the solide CD is by the same reason equall to the solide DY, for they both consist vpon one and the selfe same base, namely, the parallelogramme XR, are vn∣der one and the selfe same altitude, whose standing lines are in the selfe same right lines, namely, PQCY, and OhNW. Wherefore the solide

[illustration]

BT is equall to the solide DY. But in equall Parallelipipedons, whose altitudes are erected perpendicularly to their bases, their bases are re∣ciprokall to their altitudes (by the first part of this Proposition). Wherefore as the base FK is to the base XR, so is the altitude of the solide DY to the altitude of the solide BT. But the base FK is equall to the base EH, and the base XR to the base NP. Wherefore as the base EH is to the base NP, so is the altitude of the solide DY to the altitude of the solide BT. But the altitudes of the solides DY & BT, and of the solides DC & BA are one and the selfe same. Wherefore as the base EH is to the base NP, so is the altitude of the solide CD to the altitude of the solide AB. Wherfore the bases of the Parallelipipedons AB and CD are reciprokall to their altitudes.

Againe suppose that the bases of the Paralleli∣pipedons AB and CD be reciprokall to their al∣titudes,* 1.11 that is, as the base EH is to the base NP, so let the altitude of the solide CD be to the altitude of the solide AB. Then I say, that the solide AB is equall to the solide CD. For the same order of construction remayning, for that as the base EH is to the base NP, so is the altitude of the solide CD to the altitude of the solide AB: but the base EH is equall to the parallelogramme FK, and the base NP to the parallelogramme XR: wherefore as the base FK is to the base XR, so is the altitude of the solide CD to the altitude of the solide AB But the altitudes of the solides AB and BT are equall, and so also are the altitudes of the solides DC and DY. Wherefore as the base FK is to the base XR, so is the altitude of the solide DY to the altitude of the solide BT. Wherefore the bases of the Parallelipipedons BT and DY are reciprokall to their altitudes. But Parallelipipedons whose altitudes are erected per∣pendicularly to their bases, and whose bases are reciprokall to their altitudes, are equall the one to the other (by this Proposition). Wherefore the solide BT is equall to the solide DY. But the solide BT is equall to the solide BA (by the 29. of the eleuenth) for they consist vpon one and the selfe same base, namely, FK, and are vnder one and the self same altitude, whose

standing lines are in the selfe same right lines. And the solide DY is equall to the solide DC, for they consiste vpon one and the selfe same base, namely, XR, and are vnder one and the selfe same altitude, whose standing lines are in the selfe same right lines. Wherefore also the solide AB is equall to the solide CD.* 1.12 Wherefore in equall Parallelipipedons the bases are reciprokall to their altitudes. And Parallelipipedons whose bases are reciprokall with their altitudes, are equall the one to the other: which was required to be proued.

The demonstration of the first case of this Proposition is easie to conceaue by the figure as it is descri∣bed in the plaine. But ye may for your more full sight describe Parallelipipedons of pasted paper, ac∣cording as the construction teacheth you.

And for the second case, if ye remēber well the forme of the figure•• which you made for the second case of the 31. Proposition: and describe the like for this, taking ••eede to the letters that ye place them like as the cōstruction in this case requireth, ye shall most easily by them come to the full vnderstanding of the construction and demonstration of the said case.

M. Iohn Dee, his sundry Inuentions and Annotacions, very necessary, here to be added and considered.

A The••reme.

If fower right lines be in continuall proportion, and vpon the squar•• of the first, as a base, be erec∣ted a rectangle parallelipipedon, whose heith is the fourth line: that rectangle parallelipipedon is equal to the Cube made of the second line. And if vpon the square of the fourth line, as a base, be erected a rectangle parallelipipedon, whose heith is the first line, that parallelipipedon is equall to the Cube made of the third line.

Suppose AB,

[illustration]

CD, E••, and ••H to be fower right lines in cōtinuall proportion:* 1.13 and vpō the square of AB (which let be AI) as a base, l••t be erected a rec∣tāgle parallelipi∣pedō, hauing his heith IK, equall to GH, the fourth line. And let that parallelipipedon be AK. Of the se∣cond line CD, let a Cube be made: whose square base, let be noted wi••h CQ: and let his heith b•• no∣ted by QL: & let the whole Cube be signified by C∣L. I say that AK is equall to CL. Let the like cōstructi∣on be for the cube of the third line: that is, vpon the square of ••H (which suppose to be GN) let a rectangle paralle∣lipipedon be e∣rected,

hauyng his heith NO, equall to A••, the first line: which parallelipipedon let be noted with GO. And suppose the cube of the third line (••F) to be ••M•• whose square base, let be noted by •••••• and hys heith by RM. I say now (secondly) that GO is equal to ••M. For the first part consider,* 1.14 that AI (the square base of AK) is to CQ, the square base of CL, as A•• is to the third line EF, by the ••. Corollary of the 20. of the sixth. But as A••, is to EF, so (by alternate proportion) is CD to GH, to CD. The cubes roote, is QL, the same cubes heith equall: and to GH is IK (by construction) equall: wherefore, as AI is to CQ, so is QL to IK. The bases therefore and heithes of AK and CL, are reciprocally in proporti∣on: wherefore by the second part of this 34. proposition, AK and CL are equall.* 1.15 For proofe of the se∣cond part of my theoreme, I say, that as AB, CD, ••F, and GH, are in continuall proportion forward, so are they backward in continuall proportion, as by the fourth of the fift may be proued. Wherefore now considering GH to be as first, and so A•• to be the fourth: the square base GN, is to the square base ••K, as GH is to CD, by the 2. corollary of the 20. of the sixth: But as GH is to CD, so is •••• to A••, by alternate proportion: to the Cubik roote ••F, is RM (the heith of the same Cube ••M) equall. And to AB, is the heith NO equall, by construction: wherfore as GN is to ••R, so is RM to NO. Therfore by the second part of this 34. proposition, GO is equall to EN. If fowre right lines (therefore) be in continu∣all proportion &c. as in the proposition: which was required to be demonstrated.

A Corollary logisticall.

Of my former Theoreme it followeth: Any two numbers being geuen, betwene which two we would haue two other numbers, middle, in continuall proportion:* 1.16 That if we multiply the square of the first number geuen, by the other geuen number (as if it were the fourth): the roote Cubik of that of come or product, shall be the second number sought: And farther, if we multiply the square of the other number geuen, by the first geuen number, the roote Cubike of that of come shall be the thirde number sought.

For (by my Theoreme) those rectangle parallelipipedōs made of the squares of the first & fourth, multiplied by the fourth & the first, accordingly, are equall to the Cubes made of the second & third numbers: which we make our two ••iddl•• proportionals•• Wherefore of those parallelipipedo••s (as Cubes) the Cubik rootes, by good and vsuall arte sought and found, geue the very two middle num∣bers desired. And where those numbers, are not by logisticall consideration accounted Cubik num∣bers, ye may vse the logistical secret of approching nere to the precise verytye:* 1.17 so that therof most easily you shall p••rc••aue, that your fayle is of the s••nce neuer to be perceaued: it is to wete, as in a lyne of an inch long, not to want or exceede the thousand thousand part: or farther you may (infinitely approche at pleasure. O Mechanicall frend, be of good comfort, put to thy hand: Labor improbus, om∣nia vincit.

A Probleme. 1.

Vppon a right lined playne superficies geuen, to apply a rectangle parallelipipedon geuen.

Or we may thus expresse the same thing.

Vppon a right lined playne superficies geuen, to erect a rectangle parallelipipedon, equall to a rec∣tangle parallelipipedon geuen.

Suppose the right lined playne superficies geuen to be ••: and the rectangle parallelipipedon ge∣uen to be AM. Vppon ••, as a base must AM be applyed: that is, a rectangle par••llipipedon must be ere∣cted vppon ••,

[illustration]

as a base, whi∣che shall be e∣quall to AM. By the laste of the second, to the right ly∣ned figure ••, let an equall square be made: which suppose to be FRX.* 1.18 produce one side of the base of the pa∣rallelipipedō

AM, which

[illustration]

let be AC, extended to the point P. Let the o∣ther side of the sayde base, con∣curring with AC, be CG. As CG is to FR (the side of the square FRX) so let the same FR be to a line cut of from CP sufficiently extended: by the 11. of the sixth: and let that third proportionall line be CP. Let the rectangle parallelogramme be made perfect, as CD. It is euident, that CD, is equall to the square FRX by the 17. of the sixth: and by construction FRX, is equall to B. Wherfore CD, is equall to ••. By the 12. of the sixth, as CP, is to AC, so let AN (the heith of AM) be to the right line O. I say that a solide perpen∣dicularly erected vppon the base ••, hauinge the heith of the line O, is equall to the parallelipipedon AM. For CD is to AG, as CP is to A•• by the firste of the sixth, and •• is proued equall to CD: Wherfore by the 7. of the fifth, B is to AG as CP is to AC: But as CP is to AC, so is AN to O, by construction: Wherefore B is to AG as AN is to O. So than the bases •• and AG are reciprocally in proportion with the heithes AN and O. By this 34 therefore, a solide erected perpendicularly vppon •• as a base, hauing the height O, is equall to AM. Wherefore vppon a right lyned playn superficies geuen, we haue applied a rectangle parallelipipedon geuen: Which was requisite to be donne.

A Probleme 2.

A rectangle parallelipipedon being geuen to make an other equall to it of any heith assigned.

Suppose the rectangle parallelipipedon geuen to be A, and the heith assigned to be the right line ••: Now must we make a rectangle parallelipipedon, equal to A: Whose heith must be equall to ••. Ac∣cording to the manner before vsed, we must frame our cōstruction to a reciprokall proportiō betwene the bases and heithes. Which will be done if, as the heith assigned beareth it selfe in proportion to the heith of the parallelipipedon giuen: so, one of the sides of the base of the parallelipipedon giuen, be to a fourth line, by the 12. of the sixth found. For that line founde, and the other side of the base of the ge∣uen parallelipipedon, contayne a parallelogramme, which doth serue for the base, (which onely, we wanted) to vse with our giuen heith: and so is the Probleme to be executed.

Note.

Euclide in the 27. of this eleuenth hath taught, how, of a right line geuē, to describe a parallepipedō, like, & likewise situated, to a parallelipipedō geuē: I haue also added, How, to a parallepipedon geuen, an other may be made equall, vppon any right lined base geuen, or of any heith assigned: But if either Euclide, or any other before our time (answerably to the 25. of the sixth, in playns) had among solids inuented this proposi∣tion:* 1.19

Two vnequall and vnlike parallelipipedons being geuen, to describe a parallelipipedon equall to the one, and like to the other, we would haue geuen them their deserued praise: and I would also haue ben right glad to haue ben eased of my great trauayles and discourses about the inuenting thereof.

Here ende I. Dee his additions vppon this 34. Proposition.

Notes

• * 1.1

  Two cases in the first part of this propo∣tion.
• * 1.2

  First case, which also may be two wayes.
• * 1.3

  First way.
• * 1.4

  This follow∣eth also of the Corollary ad∣ded of F••us••as after the 3••. proposition of this booke.
• * 1.5

  Second way.
• * 1.6

  This also followeth of the former Corollary.
• * 1.7

  The conuerse of both the partes of the first case.
• * 1.8

  Second case.
• * 1.9

  Construction.
• * 1.10

  Demonstra∣tion.
• * 1.11

  The conuerse of the second case.
• * 1.12

  The generall conclusion.
• * 1.13

  Construction.
• * 1.14

  Demonstratiō of the first part.
• * 1.15

  Demonstratiō of the second part.
• * 1.16

  To finde two midd••e propor¦tionals be∣twene two numbers geuē.
• * 1.17

  Note the prac¦tise of app••o∣ching to pre∣cie••es in Cu∣bik rootes.
• * 1.18

  This is the way to apply any square geuen, to a line also geuē, sufficienty ex∣tended.
• * 1.19

  A probleme worth the searching for.