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 35. Theoreme. The 40. Proposition. If there be two Prismes vnder equall altitudes, & the one haue to his base a parallelogramme, and the other a triangle, and if the parallelogramme be double to the triangle: those Prismes are equall the one to the other.

SVppose that these two Prismes ABCDEF, GHKMON, be vnder equall altitudes, and let the one haue to his base the parallelogramme AC, and the o∣ther the triangle GHK, and let the parallelogramme AC be double to the tri∣angle GHK. Then I say, that the Prisme ABCDEF is equall to the Prisme GHKMON.* 1.1 Make perfecte

[illustration]

the Parallelipipedons AX & GO. And forasmuch as the parallelo∣gramme AC is double to the tri∣angle GHK,* 1.2 but the parallelo∣gramme GH is also (by the 41. of the first) double to the triangle GHK, wherefore the parallelo∣gramme AC is equall to the pa∣rallelogramme GH. But Parallelipipedons consisting vpon equall bases and vnder one and the selfe same altitude, are equall the one to the other (by the 31. of the eleuenth). Wherefore the solide AX is equall to the solide GO. But the halfe of the solide AX is the Prisme AB∣CDEF, and the halfe of the solide GO is the Prisme GHKMON. Wherfore the Prisme ABCDEF is equall to the Prisme GHKMON. If therefore there be two Prismes vn∣der equall altitudes, and the one haue to his base a parallelogramme, & the other a triangle, and if the parallelogramme be double to the triangle: those Prismes are equall the one to the other: which was required to be proued.

This Proposition and the demonstration thereof are not hard to conceaue by the former figures: but ye may for your fuller vnderstanding of thē take two equall Parallelipipedons equilate•• and equi∣angle the one to the other described of pasted paper or such like matter, and in the base of the one Pa∣rallelipipedon draw a diagonall line, and draw an other diagonall line in the vpper superficies opposite vnto the said diagonall line drawen in the base. And in one of the parallelogrammes which are set vp∣on the base of the other Parallelipipedon draw a diagonall line, and drawe an other diagonall line in the parallelogramme opposite to the same. For so if ye extend plaine superficieces by those diagonall lines there will be made two Prismes in ech body. Ye must take heede that ye put for the bases of eche of these Parallelipipedons equall parallelogrāmes. And then note thē with letters according to the let∣ters of the figures before described in the plaine. And cōpare thē with the demonstration, and they will make both it and the Proposition very clere vnto you. They will also geue great light to the Corollary following added by Flussas.

A Corollary added by Flussas.

By this and the former propo••itions it is manifest, that Prismes and solides * 1.3 contayned vnder two poligo••on figures equall, like, and parallels, and the rest parallelogrammes: may be compared the one to the other after the selfe same maner that parallelipipedons are.

For forasmuch as (by this proposition and by the second Corollary of the 2••. of this booke) it is manifest, that euery parallelipipedon may be resolued into two like, and equal Prismes, of one and the same altitude, whose base shalbe one and the selfe same with the base of the parallelipipedon or the halfe thereof, which Pris••es also shalbe cont••yned vnder the selfe same side•• with the parallelipipedō, the sayde side•• beyng also sides of like proportion: I say that Prisme•• may be compared together after the like maner that their Parallelipipedon•• are•• For if we would deuide a Prisme like vnto his foli••e by the 25. of this booke, ye shall finde in the Corollaryes of the 25. propo••••tiō, that that which is set forth touching a parallelipipedon, followeth not onely in a Prisme, but also in any sided columne whose op∣posite bases are equall, and like, and his sides parallelogrammes.

If it be required by the 27. proposition vpon a right line geuen to describe a Prisme like and in like sorte situate to a Prisme geuen: describe ••••••st the whole parallelipipedon whereof the prisme geuen is the halfe (which thing ye see by this 40. proposition may be done). And vnto that parallelipipedō de∣scribe vpon the right line geuen by the sayd 27. proposition an other parallelipipedon like: and the halfe thereof shalbe the prisme which ye seeke for, namely, shalbe a prisme described vpon the right line geuen, and like vnto the prisme geuen.

In deede Prismes can not be cut according to the 28. proposition. For that in their opposite sides can be drawen no diagonall lines: howbeit by that 28. proposition those Prismes are manifestly con∣firmed to be equall and like, which are the halues of one and the selfe same parallelipipedon.

And as touching the 29. proposition, and the three following it, which proueth that parallelipi∣pedons vnder one and the selfe same altitude, and vpon equall bases, or the selfe same bases, are equal: or if they be vnder one and the selfe same alti••••d•• they are in proportion the one to the other, as their bases are•• to apply these comparisons vnto 〈◊〉〈◊〉 it is to 〈◊〉〈◊〉 required, that the bases of the Prismes compared together, be either all parallelogrammes, or all tria••gles. For so one and the selfe altitude remayning, the comparison of thinges equall 〈◊〉〈◊〉 one and th•••• selfe same, and the halfes of the bases are euer the one to the other in the same proportion, that their wholes are. Wherfore Prismes which are the halues of the parallelipipedons, and which haue the same proportion the one to the other that the whole parallelipipedons haue, which are vnder one and th•• self•• ••ame altitude: must needes cause that their bases being the halues of the base•• of the parallelip••p•••••••• ••••e in the same proportiō the one to the other, that their whole parallelipip••don•• are. If there••o•••• the w••ole parallelipipedons be in the proportion of the whole bases, their h••l•••••• also (which are Prismes) shalbe in the (proportion either of the wholes if their bases be parallel••gr••mm•••••• or of the hal•••••• ••f they be triangles, which is euer all one by the 15. of the fiueth.

And forasmuch as by the 33. proposition, like parallelipipedons which are the doubles of their Prismes are in treble proportion the one to the other that their sides of like proportion are, it is mani∣fest, that Prismes being their halues (which haue the one to the other the same proportion that their wholes haue, by the 15 of the fiueth) and hauing the selfe same sides that thei•• parallelipipedons haue, are the one to the other in treble proportion of that which the sides of like proportion are.

And for that Prismes are the one to the other in the same proportion that their parallelipipedons are, and the bases of the Prismes (being all either triangles or parallelogrāmes) are the one to the other in the same proportion that the bases of the parallelipipedons are, whose altitudes also are alwayes e∣quall, we may by the 34. proposition conclude, that the bases of the prismes and the bases of the paral∣lelipipedons their doubles (being ech the one to the other in one and the selfe same proportion) are to the altitudes, in the same proportion that the bases of the double solides, namely, of the parallelipi∣pedons are. For if the bases of the equall parallelipipedōs be reciprokall with their altitudes, then their halues which are Prismes shall haue their bases reciprokall with their altitudes.

By the 36. proposition we may conclude, that if there be three right lines proportionall, the an∣gle of a Prisme made of these three lines (being common with the angle of his parallelipipedon which is double) doth make a prisme, which is equall to the Prisme described of the middle line, and contay∣ning the like angle, consisting also of equall sides. For a•• in the parallelipipedon, so also in the Prisme, this one thing is required, namely, that the three dimensions of the proportionall lines do make an an∣gle like vnto the angle contayned of the middle line taken three tymes. Now then if the solide angle of the Prisme be made of those three right lines, there shall of them be made an angle like to the angle of the parallelipipedon which is double vnto it. Wherefore it followeth of necessitie, that the Prismes which are alwayes the halues of the Parallelipipedons, are equiangle the one to the other, as also are their doubles, although they be not equilater: and therefore those halues of equall solides are equall the one to the other: namely, that which is described of the middle proportionall line is equall to that which is described of the three proportionall lines.

By the 37. proposition also we may conclude the same touching Prismes which was concluded

touching Parallelipipedons. For forasmuch as Prismes, described like & in like sort of the lines geuen, are the halues of the Parallelipipedons which are like and in like sort described, it followeth that these Prismes haue the one to the other the same proportion that the solides which are their doubles haue. And therfore if the lines which describe them be porportionall, they shalbe proportionall, and so con∣uersedly according to the rule of the sayd 37. proposition.

But forasmuch as the 39. proposition supposeth the opposite superficiall sides of the solide to be parallelogrammes, and the same solide to haue one diameter, which thinges a Prisme can not haue, therefore this proposition can by ••o meanes by applyed to Prismes.

But as touching solides whose bases are two like, equall, and parallel poligonon figures, and their sides are parallelogrammes,* 1.4 forasmuch as by the second Corollary of the 25. of this booke it hath bene declared, that such solides are composed of Prismes, it may easely be proued that their nature is such, as is the nature of the Prismes, whereof they are composed. Wherefore a paralle∣lipipedon being by the 27. proposition of this booke described, there may also be de∣scribed the halfe thereof, which is a Prisme: and by the description of Prismes, there may be composed a solide like vnto a solide geuen composed of Prismes. So that it is manifest, that that which the 29. 30. 31. 32. 33. 34. and 37. propositions set forth touching parallelipipedons, may well be applyed also to these kyndes of solides.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  Which of some are cal∣led sided Co∣lumnes.
• * 1.4

  Sided Co∣lumnes.