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 28. Theoreme. The 33. Proposition. Like parallelipipedons are in treble proportion the one to the other of that in which their sides of like proportion are.

SVppose that these parallelipipedons AB and CD be like, & let the sides AE and CF be sides of like proportion. Then I say, the parallelipipedon AB is vnto the parallelipipedon CD in treble proportion of that in which the side AE is to the side CF. Extend the right lines AE, GE and HE to the pointes K, L, M.* 1.1 And (by the 2. of the first) vnto the

[illustration]

line CF put the line EK equal, and vnto the line FN put the line EL equall, and moreouer vnto the line FR put the line EM equall, and make perfect the parallelogramme KL, and the parallelipipedon KO.* 1.2 Now for∣asmuch as these two lines EK and EL are equall to these two lines CF and FN, but the angle KEL is equall to the angle CFN (for the angle AEG is equall to the angle CFM by reason that the solides AB and CD are like). Wher∣fore the parallelogramme KL is equall and like to the pa∣rallelogramme CN, and by the same reason also the parallelogramme KM is equall and

like to the parallelogramme C∣R,

[illustration]

and moreouer the parallelo∣gramme OE to the parallelo∣gramme FD. Wherefore three parallelogrammes of the paralle∣lipipedon KO are like and equall to three parallelogrammes of the parallelipipedon CD: but those three sides are equall and like to the three opposite sides: wherfore the whole parallelipipedon KO is equal and like to the whole pa∣rallelipipedon CD. Make perfect the parallelogramme GK. And vpon the bases GK and KL make perfect to the altitude of the parallelipipedon AB, the pa∣rallelipipedons EX & LP. And forasmuch as by reason that the parallelipipedons AB & CD are like, as the line AE is to the line CF, so is the line EG, to the line FN, and the line EH to the line FR. But the line CF is equall to the line EK, and the line FN to the line EL, and the line FR to the line EM, therefore as the line AE is to the line EK, so is the line GE to the line EL, and the line HE to the line EM (by construc∣tion). But as the line AE is to the line EK, so is the parallelogramme AG to the parallelo∣gramme GK (by the first of the sixt). And as the line GE is to the line EL, so is the paralle∣logramme GK to the parallelogrāme KL. And moreouer as the line HE is to the line EM, so is the parallelogramm•• PE to the parallelogramme KM. Wherefore (by the 11. of the fift) as the parallelogramme AG is to the parallelogramme GK, so is the parallelogramme GK to the parallelogramme KL, and the parallelograme PE to the parallelogramme KM. But as the parallelogramme AG is to the parallelogramme GK, so is the parallelipipedon AB to the parallellpipedon EX, by the former proposition, and as the parallelogramme GK is to the parallelogramme KL, so by the same is the parallelipipedon XE to the parallelipipe∣don PL: and agayne by the same, as the parallelogramme PE is to the parallelogramme K∣M, so is the parallelipipedon PL to the parallelipipedon KO. Wherfore as the parallelipipedō AB is to the parallelipipedon EX, so is the parallelipipedon EX to the parallelipipedon PL, and the parallelipipedon PL to the parallelipipedon KO. But if there be fower magnitudes in* continuall proportion, the first shalbe vnto the fourth in treble proportion that it is to the second (by the 10. definition of the fift). Wherefore the parallelipipedon AB is vnto the pa∣rallelipedon KO in treble proportion that the parallelipipedon AB is to the parallelipipedon EX. But as the parallelipipedon AB is to the parallelipipedō EX, so is the parallelogramme AG to the parallelogramme GK, and the right line AE to the right line EK. Wherefore also the parallelipipedon AB is to the parallelipipedon KO in treble proportiō of that which the line AE hath to the line EK. But the parallelipipedon KO is equall to the parallelipi∣pedon CD, and the right line EK to the right line CF. Wherefore the parallelipipedon AB is to the parallelipipedon CD in treble proportion that the side of like proportion, namely, A∣E is to the side of like proportion, namely, to CF. Wherefore like parallelipipedons are in tre∣ble proportion the one to the other of that in which their sides of like proportion are: which was required to be demonstrated.

¶ Corellary.

Hereby it is manifest, that if there be fower right lines in * 1.3 continuall pro∣portion, as the first is to the fourth, so shall the Parallelipipedon described of the first line, be to the Parallelipipedon described of the second, both the Parallelipipedons being like and in like sort described. For the first line is to the fourth in t••eble proportion that it is to the second: and it hath before bene

[illustration]

proued that the Parallelipipedon described of the first, is to the Parallelipipedon described of the second, in the same proportion that the first line is to the fourth.

Because the one of the figures before, described in a plaine, pertayning to the demonstration of this 33. Proposition, is not altogether so easie to a younge begin∣ner to conceaue, I haue here for the same described an other figure, which if ye first drawe vpon pasted paper, and afterward cut the lines & folde the sides accordingly, will agree with the construction & demon∣stration of the sayd Proposition. Howbeit this ye must note that ye must cut the lines OQ & MR on the contrary side ••o that which ye cut the other lines. For the so∣lides which haue to their base the paralle∣logramme LK are set on vpward and the other downward: Ye may if ye thinke good describe after the s••me maner of pasted pa∣per a body equall to the solide CD: though that be easie inough to conceaue by the fi∣gure thereof described in the plaine.

¶ Certaine most profitable Corollaries, Annotations, Theo∣remes, and Problemes, with other practises, Logisticall, and Mechanicall, partly vpon this 33. and partly vpon the 34. 36. and other following, added by Master Iohn Dee. ¶ A Corollary. 1.

1. Hereby it is manifest, that two right lines may be found, which shall haue that proportion, the one to the other, that any two like Parallelipi∣pedons, and in like sort described, geuen, haue the one to the other.

Suppose Q and X to be two like Parallelipipedons, and in like sort described. Of Q take any of the three lines, of which it is produced: as namely, RG. Of X, take that right line of his production, which line ••s aunswerable to R G in proportion (which most aptly, after the Greke name, may be

called Omologall to RG) &

[illustration]

let that be TV. By the 11. of the sixth, to RG and TV, let the third line in propor∣tion with them be founde, and let that be Y. By the same 11. of the sixth, to TV and Y, let the thirde right line be foūd, in the sayd pro¦portion of TV to Y: & let that be Z. I say now that RG hath that proportion to Z, which Q hath to X. For by construction, we haue fower right lines in continuall pro∣piotion, namely, RG, TV, Y, and Z. Wherfore by Euclides Corollary, here before, RG is to Z, as Q is to X. Where∣fore we haue foūd two right lines hauing that proportion the one to the other, which any two like Parallelipipedons of like descrip∣tion, geuen, haue the one to the other: which was required to be done.

¶A Corollary.

As a Conuerse, of my ••ormer Corollary, doth it followe: To finde two like Parallelipipedons of like description, which shall haue that proportion the one to the other, that any two right lines, geuen, haue the one to the other.

Suppose the two right lines geuen to be A and B: Imagine of foure right lines in continuall pro∣portion, A. to be the first, and B to be the fourth: (or contrariwise, B to be first, & A to be the fourth). The second and third are to be found, which may, betwene A & B, be two meanes in continuall pro∣po••tion: as now,* 1.4 suppose such two lines, found: and let them be C and D. Wherefore by Euclides Corollary, as A is to B (if A were taken as first) so shall the Parallelipipedon described of A, be to the like Parallelipipedon and in like sort described of C: being the second of the fower lines in continuall proportion: it is to we••e, A, C, D, and B. Or, if B shall be taken as first, (and that thus they are orderly in continuall proportion, B, D, C, A,) then, by the sayd Corollary, as B is to A, so shall the Parallelipi∣pedon described of B, be vnto the like Parallelipipedon and in like sort described o•• D. And vnto a Pa∣rallelipipedon of A or B, at pleasure described, may an other of C or D be made like, and in like sort si∣tuated or described, by the 27. of this eleuenth booke. Wherefore any two right lines being geuē, &c: which was required to be done.

Thus haue I most briefly brought to your vnderstanding, if (first) B were double to A, then what Parallelipipedon soeuer, were described of A, the like Parallelipipedon and in like sort described of C, shall be double to the Parallelipipedon described of A. And so likewise (secondly) if A were double to B, the Parallelipipedon of D, shoulde be double to the like, of B described, both being like situated. Wherefore if of A or B, were Cubes made, the Cubes of C and D are proued double to them:* 1.5 as that of C, to the Cube of A: and the Cube of D to the Cube of B: in the second case. * 1.6 And so of any proportion els betwene A and B.

Now also do you most clerely perceaue the Mathematicall occasion, whereby (first of all men) Hippocrates, to double any Cube geuen, was led to the former Lemma: Betwene any two right lines geuen, to finde two other right lines, which shall be with the two first lines, in conti∣nuall proportion: After whose time (many yeares) diuine Plato, Heron, Philo, Appollonius, Di••••l••••, Pappus, Sporus, Menech••us, Archytas Tarentinus (who made the wodden doue to slye) Erato••••hene, Nicomedes, with many other (to their immortall fame and renowme) published, * 1.7 diuers their witty deuises, methods, and engines (which yet are extant) whereby to execute thys Problematicall Lemma. But not withstanding all the trauailes of the ••oresayd Philosophers and Mathematiciens, yea and all others doinges and con∣triuinges (vnto this day) about the sayd Lemma, yet there remaineth sufficient matter, Mathematically so to demonstrate the same, that most exactly & readily, it may also be Mechanically practis••d: that who soeuer shall achieue that feate, shall not be counted a second Archi∣medes,

but rather a per••les Mathematicien, and Mathematicorum Princeps. I will sundry wayes (in my briefe additions and annotations vpon Euclide) excite you thereto,* 1.8 yea and bring before your eyes sundry new wayes, by meinuented: and in this booke so placed, as matter thereof, to my inuentions appertayning, may geue occasion: Leauing the farther, full, & absolute my concluding of the Lemma, to an other place and time: which will, now, more cōpendiously be done: so great a part therof, being before hand in thys booke published.

¶A Corollary added by Flussas.

Parallelipipedons consisting vpon equall bases, are in proportion the one to the other as their alti∣tudes are. For if those altitudes be cut by a plaine superficies parallel to the bases: the sections shall be in proportion the one to the other as the sections of the bases cut, by the 25. of this booke. Which sections of the bases are the one to the other in that proportion that their sides or the altitudes of the solides are, by the ••irst of the sixt. Wherefore the solides are the one to the other as their altitudes are. But if the bases be vnlike, the selfe same thing may be proued by the Corollary of the 25. of this booke, which by the 25. Proposition was proued in like bases.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  *
• * 1.4

  Note this famous L••m∣ma.
• * 1.5

  The doubling o•• ••he Cube.
• * 1.6

  Note.
• * 1.7

  〈…〉〈…〉 Lemma.
• * 1.8

  Note what i•• yet lacking re∣quisite to the doubling of the Cube.