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

A parallelipipedon is a solide figure comprehended vnder foure playne qua∣drangle figures, of which those which are opposite are parallels.

As in playne superficieces a parallelogramme is that which is contained

[illustration]

vnder foure sides beyng lines, and whose opposite sides are equidistant and parallel lynes, so in solide figures a Parallelipipedon is that solide which is contayned vnder foure quadrangle superficieces, whose opposite sides are al∣so parallels, as it is easily to be sene and conceaued in a cube or die, all whose opposite sides are parallel superficieces, & so of others like, ye may also some∣what conceiue therof by the example in the margent.

There is also in these bookes following, mencion made of solides, whose two bases are Poligonon figures, lyke, equall, equilater, and parallels, and the sides set vpon the bases are parallelogrammes: which kynde of solides Cam∣pane calleth sided Columnes (and which as was before noted, may be cōpre∣ded vnder the definition of a Prisme) a forme wherof although grosely•• be∣hold in this example, whose bases are two like equall, equilater, equiangle, and parallel hexagons, and the sides set vppon those bases are sixe parallelo∣grāmes: ye may better cōceiue the forme therof by

[illustration]

the figure put vnder the figure of the parallelipipe∣don, which apeareth more bodilike. There may of these be infinite formes according to the diuersitie of their bases.

Because these fiue regular bodies here defined are not by these figures here set, so fully and liuely expressed, that the studious beholder can through∣ly according to their definitions conceyue them. I haue here geuen of them other descriptions drawn in a playne, by which ye may easily attayne to the knowledge of them. For if ye draw the like formes in matter that wil bow and geue place, as most apt∣ly ye may do in fine pasted paper, such as pastwiues make womēs pastes of, & thē with a knife cut eue∣ry line finely, not through, but halfe way only, if thē ye bow and bende them accordingly, ye shall most plainly and manifestly see the formes and shapes of these bodies, euen as their definitions shew. And it shall be very necessary for you to had•• ••tore of that pasted paper by you, for so shal yo•• vpon it 〈…〉〈…〉 the formes of other bodies, as Prismes and Parallelipopedons, 〈…〉〈…〉 set forth in these fiue bookes following, and see the very 〈◊〉〈◊〉 of th••se bodies there mēcioned: which will make these bokes concerning bodies, as easy vnto you as were the other bookes, whose figures you might plainly see vpon a playne superficies.

If ye draw this figure con••i••••ing a••

[illustration]

ye s••e of ••ower ••quila••er and equian∣gle triangles vpō pasted paper, or vp∣pon ••ny other such like matter that will bowe and geue place, and then cut not through the paper, but as it were halfe the thicknes of the p••per, the three lines contained within the figure, and bowe & folde in the fower triangles accordingly•• they will close together in such sort, that they will make the perfecte forme of a T••tra∣hedron.

This figure (consisting of

[illustration]

sixe equall squares) drawen vp∣on pasted paper, and the fiue lines contained within the fi∣gure being cut finely halfe the thicknes of the paper, or not through, if their ye bowe and folde accordingly the sixe e∣quall squares, they will so close together, that they will caus•• the perfecte forme of a Cube.

This figure (which consisteth of eight ••∣quilater

[illustration]

and equiangle triangle••) drawen vp∣on the foresayd matter, and the s••u••n lin•••• contained within the figure being 〈◊〉〈◊〉 as b••∣fore was taught, and the triangles bowed and folded accordingly, they will clos•• to••ether in such sort, that they will mak•• th•• per••••c•••• forme of an Octohedron.

Describe thi•• figur••, which consist••th of tw••lu•• ••quil•••••••• and ••quiangl•• P••nt••••••••••, vpo•• the fore∣said matt••r, and finely cut as before was ••••ught t•••• ••l••u••n lines contain••d within th•• figur••, and bow and folde the Pen••••gon•• accordingly. And they will so close to••eth••••, tha•• th••y will ••••k•• th•• very forme of a Dodecahedron.

[illustration]

If ye describe this figure which consisteth of twentie equilater and equiangle triangles vpon the foresaid matter, and finely cut as before was shewed the nin••t••ne lines which are contayned within the figure, and then bowe and folde them accordingly, they will in such sort close together, that ther•• will be made a perfecte forme of an Icosahedron.

[illustration]

Because in these fiue bookes there are sometimes required other bodies besides the foresaid fiue regular bodies, as Pyramises of diuers formes, Prismes, and others, I haue here set forth three figures of three sundry Pyramises, one hauing to his base a triangle, an other a quadrangle figure, the other •• Pentagon•• which if ye describe vpon the foresaid matter & finely cut as it was before taught the lines contained within ech figure, namely, in the first, three lines, in the second, fower lines, and in the third, fiue lines, and so bend and folde them accordingly, they will so close together at the toppes, that they will ••ake Pyramids of that forme that their bases are of. And if ye conceaue well the describing of these, ye may most easily describe the body of a Pyramis of what forme so euer ye will.

[illustration]

[illustration]

[illustration]

Likewise if ye describe this figure

[illustration]

vpon the foresaid matter, and finely cutte the fower lines cōtained within the figure, and bowe and folde them together accordingly, the three paral∣lelogrammes and the two triangles will so close together, that they will cause the perfecte forme of a Prisme cōtained vnder three parallelogrāmes and two equedistant triangles. And conceauing this description well, it shall not be hard to describe any o∣ther Prisme of any other forme.

Touching the descrip∣tion

[illustration]

of Parallelipipedons I shall not neede to speake. For if ye consider well the description of a Cube, it shall not be hard to de∣scribe a Parallelipipedon of what forme ye will. Onely where as in a Cube all the parallelogrāmes in the description of that fi∣gure are squares, in the de∣scribing of a Parallelipipe∣don, the sayd parallelo∣gramme may be of what forme ye will. So that ye take heede that the oppo∣site parallelogrammes be equal & equiangle. Which opposite parallelogrāmes in the figure as it lieth in a plaine, is any two paralle∣logrames leauing one pa∣rallelogramme betwene them. An example wher∣of beholde in this figure.

Because these fiue bookes following are somewhat hard for young beginners, by reason they must in the figures described in a plaine imagine lines and superficieces to be eleuated and erected, the one to the other, and also conceaue solides or bodies, which, for that they haue not hitherto bene acquain∣ted with, will at the first sight be somwhat s••raunge vnto thē, I haue for their more ••ase, in this eleuenth booke, at the end of the demonstration of euery Proposition either set new figures, if they concerne the eleuating or erecting of lines or superficieces, or els if they concerne bodies, I haue shewed how they shall describe bodies to be compared with the constructions and demonstrations of the Proposi∣tions to them belonging. And if they diligently weigh the maner obserued in this eleuenth booke tou∣ching the description of new figures agreing with the figures described in the plaine, it shall not be hard for them of them selues to do the like in the other bookes following, when they come to a Propositi∣on which concerneth either the eleuating or erecting of lines and superficieces, or any kindes of bodies to be imagined.

Notes

• Diffinition of a parallelipipe∣don.
• A sided Co∣lumne.
• A Tetrahe∣dron.
• A c••be.
• A•• octo••e∣d••o••.
• A D••d••••••••e∣dron.
• An Icosa••e∣dron.
• A triangled Pyramis.
• The forme of a quadrang∣led Pyramis.
• The forme of a fiue angled Pyramis.
• The forme of a prisme.
• The forme of a parallelip••∣pedon.