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 3. Theoreme. The 3. Proposition. Euery Pyramis hauing a triangle to his base: may be deuided into two Py∣ramids equall and like the one to the other, and also like to the whole, ha∣uing also triangles to their bases, and into two equall prismes: and those two prismes are greater then the halfe of the whole Pyramis.

SVppose that there be a Pyramis, whose base let be the triangle ABC, and his toppe the point D. Then I say, that the Pyramis ABCD may be deuided into two Pyramids equall and like the one to the other, and also like to the whole, ha∣uing also triangles to their bases, and into two equall prismes, and those two

prismes are greater then the halfe of the whole Pyramis•• Deuide (by the 10. of the first) the lines AB, BC, CA, AD, BD, & CD, into two equall partes in the pointes E, F, G, H, K, and L.* 1.1 And drawe these right lines EH, EG, GH, HK, KL, LH, EK, KF, and FG. Now forasmuch as the line AE is equall to the line

[illustration]

EB, and the line AH to the line HD:* 1.2 therfore (by the 2. of the sixt) the line EH is a parallel to the line DB. And by the same reason, the line HK is a parallel to the line AB: Wherefore HEKB is a parallelogramme. Wherefore the line HK is equall to the line EB. But the line EB is equall to the line AE: Wherefore the line AE is equall to the line HK. And the line AH is equall to the line HD. Now therfore there are two lines AE and AH, equall to two lines KH and HD, the one to the other, and the angle EAH is (by the 29. of the first) equall to the an∣gle KHD: Wherefore (by the 4. of the first) the ••ase EH is equall to the base KD. Wherefore the triangle AEH, is equall and like to the triangle HKD. And by the same reason also the triangle AHG, is equall and like to the triangle HLD. And forasmuch as two right lines EH & HG touching the one the other, are parallels to two right lines KD and DL touching also the one the other, and not being in one and the selfe same plaine superficies with the two first: those lines (by the 10. of the ele∣uenth) containe equall angles. Wherefore the angle EHG is equall to the angle KDL. And forasmuch as two right lines EH and HG, are equall to two right lines KD & DL, the one to the other, and the angle EHG is (by the 10. of the eleuenth) equall to the angle ••DL, therefore (by the 4. of the first) the base EG is equall to the base LK. Wherefore the trian∣gle EHG is equall and like to the triangle KDL. And by the same reason also the triangle AEG is equall and like to the triangle HKL. Wherefore the Pyramis, whose ba••e is the tri∣angle AEG, and toppe the point H, is equall and like to the Pyramis, whose base is the trian∣gle HKL, and toppe the point D.

And forasmuch as to one of the sides of the triangle ADB, namely, to the side AB, is drawen •• parallel line HK, therefore the whole triangle A••DB is equiangle to the triangle DHK, and their sides are proportionall (by the Corollary of the ••. of the sixt). Wherefore the triangle ADB is like to the triangle DHK. And by the same reason also the triangle DBC is like to the triangle DKL, and the triangle ADC to the triangle DHL. And forasmuch as two right lines BA and AC touching the one the other, are parallels to two right lines KH and HL, touching also the one the other, but not being in one and the selfe same super••icies with the two first lines, therefore (by the 10. of the eleuenth) they containe e∣quall angles. Wherefore the angle BAC is equall to the angle KHL. And as the line BA is to the line AC, so is the line KH to the line HE. Wherefore the triangle ABC is like to the triangle KHL. Wherfore the whole Pyramis whose base is the triangle ABC & top the point D, is like to the pyramis whose base is the triangle HKL, and toppe the point D. But the pryamis whose base is the triangle HKL and toppe the point D, is proued to be like to the pyramis whose base is the triangle AEG and toppe the poynt H. Wherefore also the pyramis whose base is the triangle ABC, and toppe the poynt D, is like to the pyramis whose base is the triangle AEG and toppe the poynt H (by the 21. of the sixth).* 1.3 Wherefore either of these py∣ramids AEGH and HKLD is like to the whole pyramis ABCD.

* 1.4[And forasmuch as the lines BE, KH, are parallel lines and equall, as it hath bene proued, there∣fore the right lines BK and EH, which ioyne them together, are equall and parallels, by the 33. of the first. Againe fo••asmuch as the lines BE and FG are parallel lines and equall, therefore lines EG

and BF, which ioyne them together, are also equall and parallels: and by the same reason forasmuch as the lines FG and KH are equall parallels, the lines FK and GH, which ioyne them together, are also equall parallels. Wherefore BEHK, DEGF, and KHGF, are parallelogrammes. And forasmuch as their opposite sides are equall, by the 34. of the first•• therefore the triangles FHG, & BKF, are equi∣angle, by the 8. of the first: and therefore, by the 4. of the same, they are equall: and moreouer, by the 15. of the eleuenth, their superficiec••s are parallels. Wherefore the solide BKFEHG is a Prisme, by the 11. definition of the eleuenth. Likewise forasmuch as the sides of the triangle HKL are equall and parallels to the sides of the triangle GFC, as it hath before bene proued: It is manifest, that CFKL, FKHG, and CLHG, are parallelogrammes, by the 33. of the first. Wherefor•• the whole solide KL∣HFCG, is a Prisme, by the 11. de••inition of the eleuenth, and is contayned vnder the sayd parallelo∣grammes CFKL, FKHG, and CLHG, and the two triangles HKL and GFC, which are opposite and parallels.]

And forasmuch as the line BF is equall to the line FC, therefore (by the 41. of the first) the parallelogramme EBFG is double to the triangle GFC. And forasmuch as if there be two Prismes of equall altitudes, and 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 (by the 40. of the eleuenth) equall the one to the other: therefore the Prisme contained vn∣der the two triangles BKF and EHG, and vnder the three parallelogrammes EBFG, EBKH, and KHFG,* 1.5 is equall to the Prisme contained vnder the two triangles GFC, and HKL, and vnder the three parallelogrammes KFCL, LCGH, and HKFG.

And it is manifest, that both these Prismes, of which the base of one is the parallelo∣gramme EBFG,* 1.6 and the opposi••e vnto it the line KH, and the base of the other is the tri∣angle GFC, and the opposite side vnto it the triangle KLH, are greater then both these Pyramids, whose bases are the triangles AGE, and HKL, and toppes the pointes H & D. For if we drawe these right lines EF and EK, the Prisme whose base is the parallelogramme EBFG, and the opposite vnto it the right line HK, is greater then the Pyramis whose base is the triangle EBF, & toppe the point K. But the Pyramis whose base is the triangle EBF, and toppe the point K, is equall to the Pyramis whose base is the triangle AEG and toppe the point H, for they are contained vnder equall and like plaine superficieces. Wherefore also the Prisme whose base is the parallelogramme EBFG and the opposite vnto it the right line HK, is greater then the Pyramis, whose base is the triangle AEG, and toppe the point H. But the prisme whose base is the parallelogramme EBFG, and the opposite vnto it the right line HK, is equall to the prisme, whose base is the triangle GFC, and the opposite side vnto it the triangle HKL: And the Pyramis whose base is the triangle AEG, and toppe the point H, is equall to the Pyramis, whose base is the triangle HKL, and toppe the point D.* 1.7 Wherefore the foresaid two prismes are greater then the foresaid two Pyramids, whose bases are the tri∣angles AEG, HKL, and toppes the pointes H and D. Wherefore the whole Pyramis whose base is the triangle ABC,* 1.8 and toppe the point D, is deuided into two Pyramids equall and like the one to the other, and like also vnto the whole Pyramis, hauing also triangles to their bases, and into two equall prismes, and the two prismes are greater then halfe of the whole Pyramis: which was required to be demonstrated.

If ye will with diligence reade these fower bookes following of Euclide, which concerne bodyes, and clearely see the demonstrations in them conteyned, it shall be requisite for you when you come to any proposition, which concerneth a body or bodies, whether they be regular or not, first to describe of p••s••ed paper (according as I taught you in the end of the definitions of the eleuenth booke) such a body or bodyes, as are there required, and hauing your body, or bodyes thus described, when you haue no∣ted it with letters according to the figure set forth vpō a plaine in the propositiō, follow the constructi∣on required in the proposition. As for example, in this third propositiō it is sayd that, Euery pyramis ha∣uing a triangle to ••is base, may be deuided into two pyramids. &c. Here first describe a pyramis of pasted paper ha••ing his base triangled (it skilleth not whether it be equilater, or equiangled, or not, only in this pro∣position is required that the base be a triangle. Then for that the proposition supposeth the base of the pyramis to be the triangle ABC, note the base of your pyramis which you haue described with the let∣ters ABC, and the toppe of your pyramis with the letter D: For so is required in the proposition. And thus haue you your body ordered ready to the construction. Now in the construction it is required that ye deuide the lines, AB, BC, CA. &c, namely, the sixe lines which are the sids of the fower triangles con∣tayning

the piramis, into two equall partes in the poyntet ••, F, G, &c. That is, ye must deuide the line AB of your pyramis into two equall partes, and note the poynt of the deuision with the letter E, and so the line BC being deuided into two equall partes, note the poynt of the deuision with the letter F. And so the rest, and this order follow ye as touching the rest of the construction there put, and when ye haue finished the construction, compare your body thus described with the demonstration: and it will make it very playne and easy to be vnderstāded. Whereas without such a body described of matter, it is hard

[illustration]

for young beginners (vnlesse they haue a very deepe imagination) fully to conceaue the demonstration by the sig••e as it is described in a plaine. Here for the better declaration of that which I haue sayd, haue I set a figure, whose forme if ye describe vpon pasted paper noted with the like letters, and cut the lines ••A, DA, DC, and folde it accordingly, it will make a Pyramis described according to the construction required in the proposition. And this order follow ye as touching all other propositions which con∣cerne bodyes.

¶An other demonstration after Campane of the 3. proposition.

Suppose that there be a Pyramis ABCD hauing to his base the triangle BCD, and let his toppe be the solide angle A: from which let there be drawne three subtended lines AB, AC, and AD to the three angles of the base, and deuide all the sides of the base into two equall partes in the three poyntes E, F, G: deuide also the three subtēded lines AB, AC, and AD in two equall partes in the three points H, K, L. And draw in the base these two lines EF and EG:

[illustration]

So shall the base of the pyramis be deuided into three su∣perficieces: whereof two are the two triangles BEF, and EGD, which are like both the one to the other, and also to the whole base, by the 2 part of the secōd of the sixth, & by the definitiō of like super••iciec••s, & they are equal the one to the other, by the 8. of the first: the third superficies is a quadrangled parallelogramme, namely, EFGC: which is double to the triangle EGD, by the 40. and 41. of the first. Now then agayne from the poynt H draw vnto the points E and F these two subtendent lines HE and HF: draw also a subtended line from the poynt K to the poynt G. And draw these lines HK, KL, and LH. Wherefore the whole pyramis ABCD is deuided into two pyramids, which are HBEF, and AHKL, and into two prismes of which the one is EHFGKC, and is set vpon the quadrangled base CFGE: the other is EGDHKL, and hath to his base the triangle EGD. Now as touching the two pyramids HBEF and AHKL, that they are equall the one to the other, and also that they are like both the one to the o∣ther and also to the whole, it is manifest by the definition of equall and like bodyes, and by the 10. of the eleuenth, and by 2. part of the second of the sixth. And that the two Prismes are equall it is manifest by the last of the eleuenth. And now that both the prismes taken together are greater then the halfe of the whole pyramis, hereby it is manifest, for that either of them may be deuided into two pyramids, of which the one is a triangular pyramis equall to one of the two pyramids into which together with the two prismes is deuided the whole pyramis, and the other is a quadrangled pyramis double to the other pyramis. Wherefore it is playne that the two prismes taken together are three quarters of the whole

pyramis deuided. But if ye are desirous to know the proportiō betwen them, reade the ••, of this booke. But now here to this purpose it shall be sufficient to know, that the two prismes taken together do ex∣ceede in quantity the two partial pyramids taken together, into which together with the two prismes the whole pyramis was deuided.

Notes

• * 1.1

  Cons••ruction.
• * 1.2

  Demonstrati∣on of the first part, namely, that the whole Pyra∣mids is de••i∣ded into two Pyramis e∣quall and like the one to the other, and also to the whole, & hauing tri¦angles to their bases.
• * 1.3

  The conclusi∣on of the first part.
• * 1.4

  Demonstratiō of the second part, namely, that it is deui∣ded moreouer into two equal Prismes.
• * 1.5

  Conclusion of the second part.
• * 1.6

  Demonstratiō of the last part that the two Prismes are greater then the halfe of the whole Py∣ramis.
• * 1.7

  Conclusion of the last part.
• * 1.8

  Conclusion of the whole proposition.