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 eleuenth booke of Eu∣clides Elementes. (Book 11)

* 1.1HITHERTO HATH ••VCLID•• IN TH••S•• former bookes with a wonderfull Methode and order entreated of such kindes of figures superficial, which are or may be described in a superficies or plaine. And hath taught and set forth their properties, natures, generati∣ons, and productions euen from the first roote, ground, and beginning of them: namely, from a point, which al∣though it be indiuisible, yet is it the beginning of all quantitie,* 1.2 and of it and of the motion and slowing ther∣of is produced a line, and consequently all quantitie cō∣tinuall, as all figures playne and solide what so euer. Eu∣clide therefore in his first booke began with it,* 1.3 and from thence went he to a line, as to a thing most simple next vnto a point, then to a superficies, and to angles, and so through the whole first booke,* 1.4 he intreated of these most simple and plaine groundes. In the second booke he entrea∣ted further,* 1.5 and went vnto more harder matter, and taught of diuisions of lines, and of the multiplication of lines, and of their partes, and of their passions and properties. And for that rightlined ••igures are far distant in nature and propertie from round and circular figures, in the third booke he instructeth the reader of the nature and conditiō of circles.* 1.6 In the fourth booke he compareth figures of right lines and circles together,* 1.7 and teacheth how to describe a figure of right lines with in or about a circle: and con∣tra••iwi••e a circle with in or about a rectiline figure. In the fifth booke he searcheth out the nature of proportion (a matter of wonderfull vse and deepe consideration),* 1.8 for that otherwise he could not compare ••igure with figure, or the sides of figures together. For whatsoeuer is compared to any other thing, is compared vnto it vndoubtedly vn∣der some kinde of proportion. Wherefore in the sixth booke he compareth figures to∣gether,* 1.9 one to an other, likewise their sides. And for that the nature of proportion, can not be fully and clearely sene without the knowledge of number, wherein it is first and chiefely found: in the seuenth,* 1.10 eight,* 1.11 and ninth bookes,* 1.12 he entreat••th of number, & of the kindes and properties thereof. And because that the sides of solide bodyes, for the most part are of such sort, that compared together, they haue such proportion the one to the other,* 1.13 which can not be expres••ed by any number certayne, and therefore are cal∣led irrational lines, he in the tēth boke hath writtē & taught which line•• are cōmēsura∣ble or incōmēsurable the one to the other, and of the diuersitie of kindes of irrationall lines, with all the conditions & proprieties of them. And thus hath Euclide in these ten foresayd bokes, fully & most plēteously in a meruelous order taught, whatsoeuer semed necessary, and requisite to the knowledge of all superficiall figures, of what sort & forme so euer they be. Now in these bookes following he entreateth of figures of an other kinde, namely, of bodely figures:* 1.14 as of Cubes, Piramids, Cones, Columnes, Cilinders, Parallelipipedons. Spheres and such others•• and sheweth the diuersitie of thē, the gene∣ration, and production of them, and demonstrateth with great and wonderfull art, their proprieties and passions, with all their natures and conditions. He also compareth one o•• them to an other, whereby to know the reason and proportion of the one to the o∣ther, chiefely of the fiue bodyes which are called regular bodyes.* 1.15 And these are the thinges of all other entreated of in Geometrie, most worthy and of greatest dignitie, and as it were the end and finall entent of the whole are of Geometrie, and for whose cause hath bene written, and spoken whatsoeuer hath hitherto in the former bookes bene sayd or written. As the first booke was a ground, and a necessary entrye to all the r••st ••ollowing, so is this eleuenth booke a necessary entrie and ground to the rest which follow.* 1.16 And as that contayned the declaration of wordes, and definitions of thinge••

requisite to the knowledge of superficiall figures, and entreated of lines (and of their di¦uisions and sections) which are the termes and limites of superficiall figures: so in this booke is set forth the declaration of wordes and definitions of thinges pertayning to solide and corporall figures: and also of superficieces which are the termes & limites of solides: moreouer of the diuision and intersection of them, and diuers other thinges, without which the knowledge of bodely and solide formes can not be attayned vnto. And first is set the definitions as follow••th.

Definitions. A solide or body is that which hath length, breadth, and thicknes,* 1.17 and the terme or limite of a solide is a superficies.

There are three kindes of continuall quantitie, a line, a superficies, and a solide or body: the begin∣ning of all which (as before hath bene sayd) is a poynt, which is indiuisible. Two of these quantities, namely, a line and a superficies, were defined of Euclide before in his first booke. But the third kinde, namely, a solide or body he there defined not, as a thing which pertayned not then to his purpose: but here in this place he setteth the definitiō therof, as that which chiefely now pertayneth to his purpose, and without which nothing in these thinges can profitably be taught. A solide (sayth he) is that which hath lēgth, breadth, and thicknes, or depth. There are (as before hath bene taught) three reasons or meanes of measuring, which are called cōmonly dimensions, namely, l••ngth, breadth, and thicknes. These di∣mensions are ascribed vnto quantities onely. By these are all kindes of quantitie de••ined, •••• are counted perfect or imperfect, according as they are pertaker of fewer or more of them. As Euclide defined a line, ascribing vnto it onely one of these dimensions, namely, length: Wherefore a line is the imperfectest kinde of quantitie. In defining of a superficies, he ascribed vnto it two dimensions, namely, length, and breadth: whereby a superficies is a quantitie of greater perfection then is a line, but here in the defini∣tiō of a solide or body. Euclide attributeth vnto it all the three dimensiōs, lēgth, breadth, and thicknes. Wherfore a solide is the most perfectest quantitie,* 1.18 which wanteth no dimension at all, passing a lyne by two dimensions, and passing a super••icies by one. This definition of a solide is without any designation of ••orme or figure easily vnderstanded, onely conceiuing in minde, or beholding with the eye a piece of timber or stone, or what matter so euer els, whose dimension•• let be equall or vnequall. For example let the length therof be 5. inches, the breadth 4. and the thicknes 2. if the dimensions were equall, the reason is like, and all one, as it is in a Sphere and in •• cube. For in that respect and consideration onely, that it is long, broade, and thicke, it beareth the name of a solide or body, ••nd hath the nature and pro∣perties therof. There is added to the end•• of the definition of a solide, that the terme and limite of a so∣lide ••s a superficies. Of thinges infinitie there i•• no Arte or Scien••e. All quantities therfore in this Arte entreated of, are imagined to be finite,* 1.19 and to haue their endes and borders as hath bene shewed in the first booke, that the limites and endes of a line are pointes, and the limites or borders of a superficies are lines, so now he saith tha•• the endes, limites, or borders of a solide•• are superficieces. As the side of any ••quare piece of timber, or of a table, or die, or any other lik••, are the termes and limites of them.

2 A right line is then erected perpendicularly to a pl〈…〉〈…〉erficies, whē the right line maketh right angles with all the lines 〈…〉〈…〉 it,* 1.20 and are drawen vpon the ground plaine superficies.

Suppose that vpon the grounde playne superfi∣cies,

[illustration]

CDEF from the pointe B be erected a right line, namely, ••A, so that let the point A be a lo••e in the ayre. Drawe also from the poynte •• in the playne superficies C∣DBF, as many right lines as ye list, as the lines BC, BD, ••••, BF, BG, HK, BH, and BL. If the erected line BA with all these lines drawen in the superficies CDEF make a right angle, so that all those ••ngles A••••, A••D, A••E, ABF•• A••G, A••K, ABH, ABL, and so of others, be right angles, then by this definition, the line AB, i•• a line ••••••cted vpon the superficies CDEF: it is also called com∣monly a perpendicular line or a plumb line, vnto or vpon a superficies.

* 1.213 A plaine superficies is then vpright or erected perpendicularly to a plaine superficies, when all the right lines drawen in one of the plaine su∣perficieces vnto the common section of those two plaine superficieces, ma∣king therwith right angles, do also make right angles to the other plaine superficies. Inclination or leaning of a right line, to a plaine superficies, is an acute angle, contained vnder a right line falling from a point aboue to the plaine superficies, and vnder an other right line, from the lower end of the sayd line (let downe) drawen in the same plaine superficies, by a certaine point assigned, where a right line from the first point aboue, to the same plaine superficies falling perpendicularly, toucheth.

In this third definition are included two definitions: the first is of a plaine superficies erected per∣pendicularly vpon a plaine superficies.* 1.22 The second is of the inclination or leaning of a right line vnto a superficies: of the first take this example. Suppose ye haue two super••icieces ABCD and CDEF. Of which let the superficies CDEF be a ground plaine superficies, and let the superficies ABCD be e∣rected vnto it, and let the line CD be a common terme or in∣tersection

[illustration]

to them both, that is, let it be the end or bound of either of them,* 1.23 & be drawen in either of them: in which line note at pleasure certaine pointes, as the point G, H. From which pointes vnto the line CD, draw perpendicular lines in the super••icies ABCD, which let be GL and HK, which fal∣ling vpon the superficies CDEF, if they cause right angles with it, that is, with lines drawen in it from the same pointes G and H, as if the angle LGM or the angle LGN contayned vnder the line ••G drawen in the superficies erected, and vnder the GM or GN drawen in the ground superficies CDEF ly∣ing flat, be a right angle, then by this definition, the superficies ABCD is vpright or erected vpon the superficies CDEF. It is also commonly called a superficies perpendicular vpon or vnto a superficies.

* 1.24For the second part of this definition, which is of the inclination of a right line vnto a plaine su∣perficies, take this example. Let ABCD be a ground plaine superficies, vpon which from a point being a loft, namely, the point E, suppose a right line to fall, which let be the line EG, touching the plaine superficies ABCD at the poynt G. Againe, from the point E, being the toppe or higher limite and end of the inclining line EG, let a perpendicular line fall vnto the plaine superficies ABCD, which let be the line EF, and let F be the point where EF toucheth the plaine superficies ABCD. Then from the point of the fall of the line inclining vpon the superficies vnto

[illustration]

the point of the falling of the perpendicular line vpon the same super∣ficies, that is, from the point G to the point F, draw a right line GF. Now by this definition, the acute angle EGF is the inclination of the line EG vnto the superficies ABCD. Because it is contayned of the inclining line, and of the right line drawen in the superficies, from the point of the fall of the line inclining to the point of the fall of the per∣pendicular line: which angle must of necessitie be an acute angle. For the angle EFG is by construction a right angle, and three angles in a triangle are equ••〈…〉〈…〉 ••ight angles. Wherefore the other two angles, namely, the angles EGF, and GEF, are equ••〈…〉〈…〉 right angle. Wherfore either of them is lesse then a right angle. Wherfore the angle EGF is an 〈…〉〈…〉gle.

* 1.254 Inclination of a plaine superficies to a plaine superficies, is an acute an∣gle contayned vnder the right lines, which being drawen in either of the plaine superficieces to one & the self same point of the cōmon section, make with the section right angles.

Suppose that there be two superficieces ABCD & EFGH,

[illustration]

and let the superficies ABCD be supposed to be erected not perpendicularly, but somewhat leaning and inclining vnto the plaine superficies EFGH, as much or as litle as ye will: the cōmon terme or section of which two superficieces let be the line CD. From some one point, a•• from the point M assigned in the common section of the two superficieces, namely, in the line CD, draw a perpendicular line in either superficies. In the ground superficies EFGH draw the line MK, and in the su∣perficies ABCD draw the line ML. Now if the angle LMK be an acute angle, then is that angle the inclination of the su∣perficies ABCD vnto the superficies EFGH, by this defini∣tion, because it is contained of perpendicular lines drawen in either of the superficieces to one and the self same point being the common section of them both.

5 Plaine superficieces are in like sort inclined the on•• 〈…〉〈…〉her,* 1.26 when the sayd angles of inclination are equall the one to the o〈…〉〈…〉

This definition needeth no declaration at all, but is most manifest by the definition last going before. For in considering the inclinations of diuers superficieces to others, if the acute angles contayned vn∣der the perpendicular lines drawen in them from one point assigned in ech of their common sections be equall, as if to the angle LMK in the former example be geuen an other angle in the inclination of two other superficieces equall, then is the inclination of these superficieces like, and are by this defini∣tion sayd in like sort to incline the one to the other.

Theodosius geueth an other definition of like inclination of plaine superficieces the one to the other, after this maner.

One plaine superficies is like inclined to an other, as an other superficies is to an other, when in either of the plaine superficieces right lines being drawen, and making right angles with their common section, containe in the same pointes equall angles. This definition is in substance the same with that geuen of Euclide, and is an elucidation of it.

For example let ABCD be a ground plaine superficies, vnto which let the superficies EFIK incline and leane. And let the common section of these two superficieces be the line EF. Then drawe in eche of thes•• superficieces right lines to some one point of the common section EF, which let be the point G: with which section let them make right angles. As in the super∣ficies

[illustration]

ABCD draw the line ••G, which in the point G let it make with the common sectiō a right angle HGF or HGE. Also in the superficies EFIK draw the line LG, which in the point G together with the common section EF let make also a right angle LGF, or the right angle LG••.

Now also let there be an other ground plaine superficies, namely, the superficies MNOP, vnto whom also let leane and incline the superficies Q••••T, and let the common section or segment of them be the line QR. And draw in the superficies MNOP to some one point of the cōmon section as to the point X the line VX, making with the common section right angles, namely, the angle VXR, or the angle VXQ: also in the superficies STQR draw the right line YX to the same point X in the common secti∣on, making therwith right angles, as the angle YX••, or the angle YXQ. Now (as sayth the definition) if the angles contayned vnder the right lines drawen in these superficieces & making right angles with the common section, be in the pointes, that is, in the pointes of their meting in the common section, e∣quall: then is the inclination of the superficieces equall. As in this example, if the angle LGH con∣tayned vnder the line LG being in the inclining superficies ••KEF and vnder the line HG being in the ground superficies ABCD, bē equall, to the angle YXV contayned vnder the line VX being in the

ground superficies MNOP and vnder the line YX being in the inclining superficies STQR: then is the inclination of the super••icies IKEF vnto the superficies ABCD, like vnto the inclination of the superficies STQR vnto the superficies MNOP. And so by this definition these two superficieces are sayd to be in like sort inclined.

* 1.276 Parallell plaine superficieces are those, which being produced or exten∣ded any way neuer touch or concurre together.

Neither needeth this definition any declaration, but is very easie to be vnderstanded by the defini∣tion of parallell lines: ••or as they being drawen on any part, neuer touch or come together: so parallel plaine super••icieces are such, which admitte no touch, that is, being produced any way infinitely neuer meete or come together.

* 1.287 Like solide or bodily figures are such, which are contained vnder like plaine superficieces, and equall in multitude.

What plaine super••icieces are called like, hath in the beginning of the sixth booke, bene sufficiently declared. Now when solide figures or bodies be contained vnder such like plaine superficieces as there are defined, and equall in number, that is, that the one solide haue as many in number as the other, in their sides and limites: they are called like solide figures, or like bodies.

* 1.298 Equall and like solide (or bodely) figures are those which are contained vnder like superficieces, and equall both in multitude and in magnitude.

In like solide figures it is su••ficient, that the superficieces which containe them be like and equall in number onely, but in like solide figures and equall, it is necessary that the like superficieces contaynyng them, be also equal in magnitude. So that besides the likenes betwene them, they be (eche being com∣pared to his correspondent super••icies) o•• one greatnes, and that their areas or fieldes be equal. When such super••icieces contayne bodies or solides, then are such bodies equall and like solides or bodies.

* 1.309 A solide or bodily angle, is an inclination of moe then two lines to all the lines which touch themselues mutually, and are not in one and the selfe same super••icies.

Or els thus: A solide or bodily angle is that which is contayned vnder mo then two playne angles, not being in one and the selfe same plaine superfi∣cies, but consisting all at one point.

Of a solide angle doth Euclide here geue two seue••all definitiōs. The first is geuen by the concurse and touch of many lines. The second by the touch & concurse of many superficiall angles. And both these definitions tende to one, and are not much different, for that lynes are the limittes and termes of superficieces. But the second geuen by super••iciall angles is the more naturall definition, because that supe••ficieces a••e the next and immediate limites of bodies, and so are not lines. An example of a solide angle cannot wel and at ••ully be geuē or described in a pla••••e superficies. But touchyng this first defini∣tiō, lay before you a cube or a die, and cōsider any of the corners or angles therof so shal ye see that at eue••y angle there concurre thre lines (for two lines cōcurring cannot make a solide angle) namely, the line or edge of his breadth, of his lēgth, and of his thicknes, which their so inclining & cōcurring toue∣ther, make a solide angle, and so of others. And now cōc••rning the second definitiō, what super••icial or plaine angles be hath bene taught before in the first bok••, namely, that it is the touch of two right lines. And as a super••iciall or playne angle is caused & cōtained of right lines, so si a solide angle caused & cō∣tayned of plaine superficiall angles. Two right lines touching together, make a plaine angle, but two plaine angles ioyned together can not make a solide angle, but according to the definitiō, they must be moe thē two, as three, ••oure, ••iue, or mo••: which also must not be in one & the selfe same superfici••s, but must be in diuers superficieces, ••eeting at one point. This definition is not hard, but may easily be cōceiued in a cube or a die, where ye see three angles of any three superficieces or sides of the die con∣curre and meete together in one point, which three playne angles so ioyned together, make a solide angle. Likewise in a Pyrami•• or a spi••e of a steple or any other such thing, all the sides therof tēding vp∣ward

narower and narower, at length ende their angles (at the heig••••

[illustration]

or toppe therof) in one point. So all their angles there ioyned toge∣ther, make a solide angle. And for the better ••ig••t thereof, I haue set here a figure wherby ye shall more easily conceiue •••• the base of the figure is a triangle, namely, ABC, if on euery side of the triangle ABC, ye rayse vp a triangle, as vpon the side AB, ye raise vp the triangle AFB, and vpon the side AC the triangle AFC, and vpon the side BC, the triangle BFC, and so bowing the triangles raised vp, that their toppes, namely, the pointes F meete and ioyne together in one point, ye shal easily and plainly see how these three superficiall angles AFBBFC, CFA, ioyne and close together, touching the one the other in the point F, and so make a solide angle.

10 A Pyramis is a solide figure contained vnder many playne superficieces set vpon one playne superficies, and gathered together to one point.* 1.31

Two superficieces raysed vpon any ground can not make a Pyramis, for that two superficiall angles ioyned together in the toppe, cannot (as before is sayd) make a solide angle. Wherfore whē thre, foure, fiue, or moe (how many soeuer) superficieces are raised vp frō one superficies being the ground, or base, and euer ascēding diminish their breadth, till at the lēgth all their angles cōcurre in one point, making there a solide angle: the solide inclosed, bounded, and terminated by these superficieces is called a Py∣ramis, as ye see in a taper of foure sides, and in a spire of a towre which containeth many sides, either of which is a Pyramis.

And because that all the superficieces of euery Pyramis ascend from one playne superficies as from the base, and tende to one poynt, it must of necessitie come to passe; that all the superficieces of a Pyra∣mis are trianguler, except the base, which may be of any forme or figure except a circle. For if the base be a circle, then it ascendeth not with sides, or diuers superficieces, but with one round superficies, and hath not the name of a Pyramis, but is called (as hereafter shall appeare) a Cone.

Of Pyramid, there are diuers kindes. For according to the varietie of the base is brought forth the varietie and diuersitie of kindes of Pyramids. If the base of a Pyramis be a triangle, then is it called a triangled Pyramis. If the base be a figure of fower angles, it is called a quadrangled Pyramis. If the base be a Pentagon, then is it a Pentagonall or fiue angled Pyramis. And so forth according to the increase of the angles of the base infinitely. Although the fi∣gure

[illustration]

of a Pyramis can not be well expressed in a playne superficies, yet may ye sufficiently conceaue of it both by the figure before set in the de••inition of a solide angle, and by the figure here set, if ye ima∣gine the point A together with the lines AB, AC, and AD, to be eleuated on high. And yet that the reader may more clerely see the forme of a Pyramis, I haue h••re set two sundry Pyramids which will appeare bodilike, if ye erecte the papers wherin are drawen the trian∣gular sides of eche Pyramis, in such sort that the pointes of the angles F of ech triangle may in euery Pyramis concurre in one point, and make a solide angle: one of which hath to his base a fower sided fi∣gure, and the other a fiue sided figure. T〈◊〉〈◊〉me of a triangled Pyra∣mis ye may before beholde in the examp〈◊〉〈◊〉 solide angle. And by these may ye conceaue of all other kindes o•• ••yramids.

* 1.3211 A prisme is a solide or a bodily figure contained vnder many plaine superficieces, of which the two superficieces which are opposite, are equall and like, and parallells, & all the other superficieces are parallelogrāmes.

Although you may in a plaing superfi∣cies

[illustration]

by this figure here set,* 1.33 without any hardne•• conceaue what a prisme is, name∣ly, if ye imagine the super••icies ABDC to be the ground & base of the solid••, and the two sup••••ficieces, namely, the superficies AEFB, and the superficies CEFD to be erected vpon the sides of the base, the one on the one side, namely; on the line AB, and the other on the other side, namely, on the line DC, not perpendicularly, but in∣clining and bending the one to the other, till they meete in the toppe, namely, on the line EF. For so ye see that this solide figure is contained vnder many plaine superficieces, of which two, namely, the superficies AEC, and the su∣perficies BFD, which are the endes of the solide, and opposite the one to the other, are equall like and parallels, and all the other superficieces, namely, the base ABCD, & the two erected superficieces, that is, the super••icies AEFB, and the superficies CEED are parallelogrammes. Yet notwithstan∣ding, to make the thing more clere vnto the reader, I haue here set a Prisme which will appeare bodi∣like,

[illustration]

if you erecte bending wise the••

[illustration]

papers wherein are drawen the pa∣rallelogrāmes ABEF, & CDE•• that they may concurr•••• 〈…〉〈…〉 EF in the toppe, and 〈…〉〈…〉 pers wherein are drawen th•• 〈…〉〈…〉 gles AC•• and BDE, that the 〈◊〉〈◊〉 AE of the one triangle may exactly agree with the side AE of the one parallelogramme ABEF, and the side CE of the same triangle, with the side CE of the parallelo∣gramme CDEF: and moreouer, the side BF of the other triangle, with the side DF of the parallelogramme CDEF: and finally, the side BF of the same triangle, with the side BF of the parallelo∣gramme ABEF. And so shall you most easilie see the forme of a Prisme: that it consisteth of two equall, like, and parallell triangular superficieces, and of three parallelogrammes: wherof the one is the base, and the other two are erected bending wise. Here also be∣holde the forme thereof as it is by arte described in a plaine to ap∣peare bodilike.

Flussas here noteth that Theon and Campane disagree in defi∣ning a Prisme, and he preferreth the definition geuen of Campane before the de••inition geuen of Euclide (which because he may seme with out lesse offence to reiect, he calleth it Theons definition) and following Campane he geueth an other definition, which is this.

* 1.34A Prisme is a solide figure, which is contayned vnder fiue playne superficieces, of which two are triangles, like, equall, and parallels, and the rest are parallelogrammes.

The example before set agreeth likewise with this definition, and manifestly declareth the same. For in it were ••iue superficieces, the base, the two erected superficieces, and the two endes: of which the two endes are triangles like, equall and parallels, and all the other are parallelogrammes as this definiti∣on requireth. The cause why he preferreth the difinion of Campane before the difinition of Theon (as he calleth it, but in very deede it is Euclides definition, as certainely, as are all those which are geuen of him in the former bookes, neither is there any cause at all, why it should be doubted in this one defini∣tion more then in any of the other) as he him selfe alledgeth, is, for that it is (as he sayth) to large, and comprehendeth many mo kindes of solide figures besides Prismes, as Columnes hauing sides, and all Parallelipipedons, which a definition should not doo: but should be conuertible with the thing defi∣ned, and declare the nature of it onely, and stretch no farther.

Me ••hinketh Flussas ought not to haue made so much a doo in this matter, nor to haue bene so sharpe in sight and so quicke as to see and espy out such faultes, which can of no man that will see right∣ly withou•• affection be espyed for such great faultes. For it may well be aunswered that these faultes which he noteth (if yet they be faultes) are not to be found in this definion. It may be sayd that it exten∣deth it selfe not ••arther then it should, but declareth onely the thing defined, namely, a Prisme. Neither

doth it agree (as ••lussas cauilleth) with all Parallelipipedons and Columnes hauing sides. All Paralleli∣pipedons what so euer right angled, or not right angled which are described of equidistant sides or su∣perficieces, haue their sides opposit. So that in any of them there is no one side, but it hath a side oppo∣sit vnto it. So likewise is it of euē sided Columnes, eche hath his opposite side directly agaynst it, which agreeth not with this definition of Euclide. Here it is euidently sayd, that of all the superficieces, the two which are opposite are equall, like, and parallels, meaning vndoubtedly onely two & no moe. Which is manifest by that which followeth. The other (sayth he) are parallelogrammes, signifiing most euidently that none of the rest besides the two aforesayd, which are equall, like, and parallels, are opposite: but two of necessitie are raysed vp, and concurre in one common line, and the other is the base. So that it contayneth not vnder it the figures aforesayd, that is sided Columnes, & al Parallelipipedons, as Flussas hath not so aduisedly noted.

Agayne where Flussas setteth in his definition, as an essentiall part thereof, that of the fiue superfi∣cieces, of which a Prisme is contayned, two of them must be triangles, that vndoubtedly is not of ne∣cessitie, they may be of some other figure. Suppose that in the figure before geuen that in the place of the two opposite figures, which there were two triangl••s, were placed two pentagōs: yet should the fi∣gure remayne a Prisme still, and agree with the definition of Euclide, and ••alleth not vnder the definiti∣on of Flussas. So that his definitiō semeth to be to narrow and stretcheth not so farre as it ought to do, nor declareth the whole nature of the thing defined. Wherefore it is not to be preferrd before Euclides definition, as he woulde haue it. This figure of Euclide called a Prisme, is called of Campane and certayne others Figura Serr••tilis, for that it repres••teth in some maner the forme of a Sawe.* 1.35 And of some others it is called Cuneus, that is, a Wedge, because it beareth the figure of a wedge.

Moreouer although it were so, that the definitiō of a Prisme should be so large, that it should cōtaine all these figures noted of Flussas as sided Columnes, & all Parallelipipedons: yet should not Flussas haue so great a cause to finde so notably a fault, so vtterly to reiect it. It is no rare thing in all learninges, chiefely in the Mathematicalls, to haue one thing more generall then an other. Is it not true that euery Isosceles is a triangle, but not euery triangle is an Isosceles? And why may not likewise a Prisme be more generall, then a Parallelepipedon, or a Columne hauing sides (and contayne them vnder it as a triangle cōtayneth vnder it an Isosceles and other kinds of triangles). So that euery Prallelipipedon, or euery si∣ded Columne be a Prisme, but not euery Prisme a Parallelipipedō or a sided Columne. This ought not to be so much offensiue. And indeede it semeth manifestly of many, yea & of the learned so to be takē, as clearely appeareth by the wordes of Psellus in his Epitome of Geometrie, where he entreateth of the production and constitution of these bodyes. His wordes are these.* 1.36 All r••ctili••e figures being erected vpon their playnes or bases by right angles, make Prismes. Who perceaueth not but that a Pentagon erected vpō his base of ••iue sides maketh by his motion a sided Columne of fiue sides? Likewise an Hexagon erected at right angles produceth a Columne hauing sixe sides: and so of all other rectillne figures. All which solides or bodyes so produced, whether they be sided Columnes or Parallelipipedons, be here in most plaine words (of this excellēt and auncient Greke author Psellus) called Prismes. Wherfore if the defini∣tiō of a Prisme geuē of Euclide should extend it selfe so largely as Flussas imagineth, and should enclude such figures or bodyes, as he noted: he ought not yet for all that so much to be offended, and so na∣rowly to haue sought faultes. For Euclide in so defining mought haue that meaning & sense of a Prisme which Psellus had. So ye see that Euclide may be defended either of these two wayes, either by that that the definition extendeth not to these figures, and so not to be ouer generall nor stretch farther then it ought: or ells by that that if it should stretch so far it is not so haynous. For that as ye se many haue tak•• it in that sense. In deede cōmonly a Prisme is taken in that significatiō and meaning in which Campa•••••• Flussas and others take it. In which sense it semeth also that in diuers propositions in these bookes fol∣lowing it ought of necessitie to be taken.

12 A Sphere is a figure which is made,* 1.37 when the diameter of a semicircle abiding fixed, the semicircle is turned round about, vntill it returne vnto the selfe same place from whence it began to be moued.

To the end we may fully and perfectly vnderstand this definiti∣on,

[illustration]

how a Sphere is produced of the motion of a semicircle, it shall be expedient to cōsider how quantities Mathematically are by ima∣gination conceaued to be produced, by flowing and motion, as was somewhat touched in the beginning of the first booke. Euer the lesse quantitie by his motion bringeth for••h the quātitie next aboue it. As a point mouing, flowing, or gliding, bringeth forth a line, which is the first quantitie, and next to a point. A line mouing pro∣duceth a superficies, which is the second quantitie, and next vnto a line. And last of all, a superficies mouing bringeth forth a solide or body, which is the third & last quantitie. These thinges well mar∣ked, it shall not be very hard to attaine to the right vnderstanding of this definition. Vpon the line AB being the diameter, describe a

[illustration]

semicircle ACB, whose

[illustration]

centre let be D: the dia∣meter AB being sixed on his endes or•• pointes, imagine the whole su∣perficies of the semicir∣cle to moue round from some one point assigned, till it returne to the same point againe. So shall it produce a perfect Sphere or Globe, the forme whereof you see in a ball or bowle. And it is fully round and solide, for that it is described of a semicircle which is per∣fectly round, as our countrey man Iohannes de Sacro Busco in his booke of the Sphere,* 1.38 of this definition which he taketh out of Euclide, doth well collecte. But it is to be noted and taken heede of, that none be deceaued by the definition of a Sphere geuen by Iohannes de Sacro Busco: A Sphere (sayth he) is the passage or mouing of the circumference of a semicircle, till it returne vnto the place where it beganne, which agreeth not with Euclide. Euclide plain∣ly sayth, that a Sphere is the passage or motion of a semicircle, and not the passage or motion of the cir∣cumference of a semicircle: neither can it be true that the circumference of a semicircle, which is a line, should describe a body. It was before noted that euery quantitie moued, describeth and produceth the quantitie next vnto it. Wherefore a line moued can not bring forth a body, but a superficies onely. As if ye imagine a right line fastened at one of his endes to moue about from some one point till it returne to the same againe, it shall describe a plaine superficies, namely, a circle. So also if ye likewise conceaue of a crooked line, such as is the circumference of a semicircle, that his diameter fastened on both the endes it should moue from a point assigned till it returne to the same againe, it should describe & pro∣duce a ••ound superficies onely, which is the superficies and limite of the Sphere, and should not pro∣duce the body and soliditie of the Sphere. But the whole semicircle, which is a superficies, by his moti∣on, as is before said, produceth a body, that is, a perfect Sphere. So see you the errour of this definiti∣on of the author of the Sphere: which whether it happened by the author him selfe, which I thinke not: or that that particle was thrust in by some one after him, which is more likely, it it not certaine. But it is certaine, that it is vnaptly put in, and maketh an vntrue definition: which thing is not here spoken, any thing to derogate the author of the booke, which assuredly was a man of excellent know∣ledge•• neither to the hindrance or diminishing of the worthines of the booke, which vndoubtedly is a very necessary booke, then which I know none more meere to be taught and red in scholes touching the groundes and principles of Astronomie and Geographie: but onely to admonishe the young and vnskil••ull reader of not falling into errour.* 1.39 Theodosius in his booke De Sphericis (a booke very necessa∣ry for all those which will see the groundes and principles of Geometrie and Astronomie, which also I haue translated into our vulgare tounge, ready to the presse) defineth a Sphere after thys maner: A Sphere is a solide or body contained vnder one superficies, in the midle wherof there is a point, frō which all lines drawen to the circumference are equall. This definition of Theodosius is more essentiall and naturall, then is the other geuen by Euclide. The other did not so much declare the inward nature and substance of a Sphere, as it shewed the industry and knowledge of the producing of a Sphere, and therfore is a causall definition geuen by the cause efficient, or rather a description then a definition. But this definition is very es••entiall, declaring the natu••e and substance of a Sphere. As if a circle should be thus defined, as it well may: A circle is the passage or mouing of a line from a point till it returne to the same point againe•• it is a causall definition, shewing the efficient cause wherof a circle is produced, namely, of the motion of a line. And it is a very good description fully shewing what a circle is. Such like description is the de••inition of a Sphere geuen o•• Euclide•• by the motion of a semicircle. But when a circle is defined to be a plaine superficies, in the middest wherof is a point, from which all lines drawen to the circumfe∣rence therof, are equall: this definition is essentiall and formall, and declareth the very nature of a cir∣cle. And vnto this definition of a circle, is correspondent the de••inition of a Sphere geuē by Theodosius, saying: that it is a solide o•• body, in the middest, whereof there is a point, from which all the lines drawen to the circumference are equall. So see you the affinitie betwene a circle and a Sphere. For what a circle is in a plaine, that is a Sphere in a Solide. The fulnes and content of a circle is described by the motion of a line moued about: but the circumference therof, which is the limite and border thereof, is described of the end and point of the same line moued about. So the fulnes, content, and body of a Sphere or Globe is described of a semicircle moued about. But the Sphericall superficies, which is the limite and border of a Sphere,* 1.40 is described of the circumference of the same semicircle moued about. And this is the superficies ment in the definition, when it is sayd, that it is contained vnder one super∣ficies, which superficies is called of Iohannes de ••acro Busco & others, the circumference of the Sphere.

* 1.41Galene in his booke de diffinitionibus medici•••• geueth yet an other definitiō of a Sphere, by his proper∣tie or cōmon accidēce of mouing, which is thus. A Sphere is a figure most apt to all motion, as hauing no base whereon th stay. This is a very plaine and witty de••inition, declaring the dignitie thereof aboue all figures generally.* 1.42 All other bodyes or solides, as Cubes, Pyramids, and others haue sides, bases, and angles, all which are stayes to rest vpon, or impedimentes and lets to motion. But the Sphere hauing no side or

base to stay one, nor angle to let the course thereof, but onely in a poynt touching the playne wherein 〈◊〉〈◊〉 standeth, moueth freely and fully with out let. And for the dignity and worthines thereof, this circular and Sphericall motion is attributed to the heauens, which are the most worthy bodyes. Wherefore there is ascribed vnto them this chiefe kinde of motion. This solide or bodely figure is also commonly called a Globe.* 1.43

13 The axe of a Sphere is that right line which abideth fixed, about which the semicircle was moued.* 1.44

As in the example before geuen in the definition of a Sphere, the line AB, about which his endes being fixed, the semicircle was moued (which line also yet remayneth after the motion ended) is the axe of the Sphere described of that semicircle. Theodosius defineth the axe of a Sphere after this maner.* 1.45 The axe of a Sphere is a certayne right line drawen by the centre, ending on either side in the superficies of the Sphere, about which being fixed the Sphere is turned. As the line AB in the former example. There nedeth to this definition no other declaration, but onely to consider, that the whole Sphere turneth vpon that line AB, which passeth by the centre D, and is extended one either side to the superficies of the Sphere, wherefore by this definition of Theodosius it is the axe of the Sphere.

14 The centre of a Sphere is that poynt which is also the centre of the se∣micircle.* 1.46

This definition of the centre of a Sphere is geuen as was the other definition of the axe, namely, hauing a relation to the definition of a Sphere here geuen of Euclide: where it was sayd that a Sphere is made by the reuolution of a semicircle, whose diameter abideth fixed. The diameter of a circle and of a semicrcle is all one. And in the diameter either of a circle or of a semicircle is contayned the center of either of them, for that they diameter of eche euer passeth by the centre. Now (sayth Euclide) the poynt which is the center of the semicircle, by whose motion the Sphere was described, is also the centre of the Sphere. As in the example there geuen, the poynt D is the centre both of the semicircle & also of the Sphere. Theodosius geueth as other definition of the centre of a Sphere which is thus.* 1.47 The centre of a Sphere is a poynt with in the Sphere, from which all lines drawen to the superficies of the Sphere are equall. As in a circle being a playne figure there is a poynt in the middest, from which all lines drawen to the circum∣frence are equall, which is the centre of the circle: so in like maner with in a Sphere which is a solide and bodely figure, there must be conceaued a poynt in the middest thereof, from which all lines drawen to the superficies thereof are equall. And this poynt is the centre of the Sphere by this definition of Theodosius. Flussas in defining the centre of a Sphere comprehendeth both those definitions in one, after this sort. The centre of a Sphere is a poynt assigned in a Sphere, from which all the lines drawen to the superfi∣cies are equall, and it is the same which was also the centre of the semicircle which described the Sphere.* 1.48 This defi∣nition is superfluous and contayneth more thē nedeth. For either part thereof is a full and sufficient dif∣finition, as before hath bene shewed. Or ells had Euclide bene insufficient for leauing out the one part, or Theodosius for leauing out the other. Paraduenture Flussas did it for the more explication of either, that the one part might open the other.

15 The diameter of a Sphere is a certayne right line drawen by the cētre, and one eche side ending at the superficies of the same Sphere.* 1.49

This definitiō also is not hard, but may easely be couceaued by the definitiō of the diameter of a cir∣cle. For as the diameter of a circle is a right line drawne frō one side of the circūfrence of a circle to the other, passing by the centre of the circle: so imagine you a right line to be drawen from one side of the superficies of a Sphere to the other, passing by the center of the Sphere,* 1.50 and that line is the diameter of the Sphere. So it is not all one to say, the axe of a Sphere, and the diameter of a Sphere. Any line in a Sphere drawen from side to side by the centre is a diameter. But not euery line so drawen by the centre is the axe of the Sphere, but onely one right line about which the Sphere is imagined to be moued•• So that the name of a diameter of a Sphere is more general, then is the name of an axe. For euery axe in a Sphere is a diameter of the same: but not euery diameter of a Sphere is an axe of the same. And there∣fore Flussas setteth a diameter in the definition of an axe as a more generall word ••n this maner. The axe

of a Sphere, is that fixed diameter aboue which the Sphere is moued. A Sphere (as also a circle) may haue infi∣nite diameters, but it can haue but onely one axe.

* 1.5116 A cone is a solide or bodely figure which is made, when one of the sides of a rectangle triangle, namely, one of the sides which contayne the right angle, abiding fixed, the triangle is moued about, vntill it returne vnto the selfe same place from whence it began first to be moued. Now if the right line which abideth fixed be equall to the other side which is moued about and containeth the right angle: then the cone is a rectangle cone. But if it be lesse, then is it an obtuse angle cone. And if it be greater, thē is it an a cute∣angle cone,

This definition of a Cone is of the nature and condition that the definition of a Sphere was, for either is geuen by the motion of a superficies. There, as to the production of a Sphere was imagined a semicircle to moue round, from some one point till it returned to the same point againe: so here must ye imagine a rectangle triangle to moue about till it come againe to the place where it beganne. Let ABC be a rectangle triangle, hauing

[illustration]

the angle ABC a right angle, which let be contained vnder the lines AB and BC.

[illustration]

Now suppose the side AB, namely, one of the lines which cōtaine the right angle ABC to be fastened, and about it suppose the triangle ABC to be moued from some one poynt assigned till it re∣turne to the same agayne (as vppon the diameter in the definition of a Sphere ye imagined a se∣micircle to moue about): so shall the solide or body thus described be a perfect Cone. As you may imagine by this figure here set. And the forme of a Cone you may sufficiently conceaue by the figure set in the margent. There are of Cone three kindes, namely, a rectangle Cone, an obtuseangle Cone, and an acute angle Cone, all which were before in the former definitiō defined: Name∣ly, the first kinde after this maner.

If the right line which abideth fixed, be equall to the other side which moueth ro••••d about, and containeth the right angle, then the Cone is a rectangle Cone.

* 1.52As suppose in the former example, that the line AB which is fixed, and about which the triangle was moued, and after the motion yet remayneth, be equall to the line BC, which is the other line con∣tayning the right angle, which also is moued about together with the whole triangle•• then is the Cone described, as the Cone ADC in this example, a right angled Cone: so called for that the angle at the toppe of the Cone is a right angle. For forasmuch as the lines AB and BC of the triangle ABC are e∣quall, the angle BAC is equall to the angle BCA (by the 5. of the first). And eche of them is the halfe of the right angle ABC (by the 32. of the first). In like sort may it be shewed in the triangle ABD, that the angle ••DA is equall to the angle ••AD, and that eche of them is the halfe of a right angle. Wherefore the whole angle CAD, which is composed of the two halfe right angles, namely, DA•• and CA•• is a right angle. And so haue ye what is a right angled Cone.

But if it be lesse, then is it an obtuseangle Cone. As in this ex∣ample,

[illustration]

the line AB fixed is lesse then the line BC moued a∣bout. Wherefore the Cone described of the circumuoluti∣on of the triangle ABC about the line A••, is an obtusean∣gle Cone, for that the angle at the toppe DAC is greater then a right angle. Wherefore it is an obtuseangle. And therefore the Cone is called an obtuse angle Cone.

And if it be greater, then i•• i•• an acuteangle Cone. As in

[illustration]

this figure, the line AB fastened, is greater then the line BC moued about. Wherefore the Cone de∣scribed by the motion and turning of the triangle ABC about AB is an acuteangle Cone, hauing the angle at the toppe BAC an acute angle. Of whome the Cone is called an acuteangle Cone. For the ea∣sier sight & cōsideration of all these kindes of Cones, and also for the plainer demonstration of the varie∣ties of their angles in their toppes, I haue described them all three in one playne figure, of which the Cone ACB is a right angled Cone, hauyng his fix∣ed side CF equall to the line FB, and hys angle ACB a right angle: the Cone AEB is an obtuse angle Cone, and ADB an acuteangle Cone. By which figure ye may easily demonstrate (by the 21. of the first) that the angle ADB of the Cone ADB, whose fixed line DF is greater then the side FB, is lesse then the right angle ACB, and so is an acute angle. And also (by the same 21. of the first) ye shall with like facilitie perceaue how the angle AEB of the Cone AEB whose fix∣ed line EF is lesse then the side FB, is grea∣ter then the right angle ACB: and there∣fore is an obtuse angle.

This figure of a Cone is of Campane, of Vitellio, and of others which haue written in these latter times, called a round Pyramis,* 1.53 which is not so aptly. For a Pyramis, and a Cone, are farre distant, & of sundry natures. A Cone is a regular body produced of one circumuolution of a rectangle triangle, and limited and bordered with one onely round super∣ficies. But a Pyramis is terminated and bordered with diuers superficieces. Therefore can not a Cone by any iust reason beare the name of a Pyramis. This solide of many is called Turbo, which to our purpose may be Englished a Top or Ghyg: and moreouer, peculiarly Campane calleth a Cone the Py∣ramis of a round Columne, namely, of that Columne which is produced of the motion of a parallelo∣gramme (contained of the lines AB and BC) moued about, the line AB being fixed. Of which Co∣lumnes shall be shewed hereafter.

17 The axe of a Cone is that line, which abideth fixed, about which the triangle is moued.* 1.54 And the base of the Cone is the circle which is described by the right line which is moued about.

As in the example the line AB is sup∣posed

[illustration]

to be the line about which the right angled triangle ABC (to the pro∣duction of the Cone) was moued: and that line is here of Euclide called the axe of the Cone described. The base of the Cone is the circle which is described by the right line which is moued about. As the line AB was fixed and slayed, so was the line BC (together with the whole triangle ABC) moued and turned a∣bout. A line moued, as hath bene sayd before, produceth a superficies: and be∣cause the line BC is moued about a point, namely, the point B, being the end of the axe of the Cone AB, it produceth by his motion, and reuolution a circle, which circle is the base of the Cone: as in this example, the circle CDE.

The line which produceth the base of the Cone, is the line of the triangle which together with the axe of the Cone contayneth the right angle. The other side also of the triangle, namely, the line AC, is moued about also with the motion of the triangle, which with his reuolution describeth also a super∣ficies,* 1.55 which is a round superficies, & is erected vpon the base of the Cone, & endeth in a point, name∣ly, in the higher part or toppe of the Cone. And it is commonly called a Conicall superficies.

* 1.5618 A cylinder is a solide or bodely figure which is made, when one of the sides of a rectangle parallelogramme, abiding fixed, the parallelogramme is moued about, vntill it returne to the selfe same place from whence it be∣gan to be moued.

This definition also is of the same sort and condition, that the two definition•• before geu•• were, namely, the definition of a Sphere and the definition of a Cone. For all are geuen by mouing of a su∣perficies about a right line fixed, the one of a semicircle about his diameter, the other of a rectangle tri∣angle about one of his sides And this solide or body here de••ined is caused of the motion of a rectangle

[illustration]

parallelogrāme hauing one of his sides contayning

[illustration]

the right angle fixed from some one poynt till it re∣turne to the same agayne where it began. As suppose ABCD to be a rectangle parallelogramme, hauing his side AB fastned, about which imagine the whole parallelogramme to be turned, till it returne to the poynt where it began, then is that solide or body, by this motion described, a Cylinder: which because of his roundnes can not at full be described in a playne superficies, yet haue you for an example thereof a suf∣ficient designation therof in the margent 〈◊〉〈◊〉 as in a plaine may be. If you wil perfectly behold the forme of a cilinder. Consider a round piller that is perfect∣ly round.

* 1.5719 The axe of a cilinder is that right line which abydeth fixed, about which the parallelogramme is moued. And the bases of the cilinder are the circles described of the two opposite sides which are moued about.

Euen as in the description of a Sphere the line fastened was the axe of the Sphere pro••uced: and in the description of a cone, the line fastened was the axe of the cone brought forth: so in this descripti∣on of a cilinder the line abiding, which was fixed, about which the rectangle parallelogramme was mo∣ued is the axe of that cilinder. As in this example is the line AB. The bases of the cilinder ••••c. In the reuo∣lution of a parallelogramme onely one side is fixed, therefore the three other sides are moued about: of which the two sides which with the axe make right angles, and which also are opposite sides, in their motion describe eche of them a circle, which two circles are called the bases of the cilinder. As ye see in the figure before put two circles described of the motiō of the two opposit lines AD and BC, which are the bases of the Cilinder.

* 1.58The other line of the rectangle parallelogramme moued, by his motion describeth the round su∣perficies about the Cilinder. As the third line or side of a rectangle triangle by his motion described the round Conical superficies about the Cone. And as the circūferēce of the semicircle described the round sphericall superficies about the Sphere. In this example it is the superficies described of the line DC.

* 1.59By this definition it is playne that the two circles, or bases of a cilinder are euer equall and paral∣lels: for that the lines moued which produced them remayned alwayes equall and parallels. Also the axe of a cilinder is euer an erected line vnto either of the bases. For with all the lines described in the bases, and touching it, it maketh right angles,

* 1.60Campane, Vitell••o, with other later writers, call this solide or body a round Column•• or piller. And Campane addeth vnto this definition this, as a corrollary. That of a round Columne, of a Sphere, and

of a circle the cētre is one and the selfe same.* 1.61 That is (as he him selfe declareth it & proueth the same) where the Columne, the Sphere, and the circle haue one diameter.

20 Like cones and cilinders are those,* 1.62 whose axes and diameters of their bases are proportionall

The similitude of cones and cilin∣ders

[illustration]

standeth in the proportion of those right lines, of which they haue their ori∣ginall and spring. For by the diameters of their bases is had their length and breadth, and by their axe is had their heigth or deepenes. Wherefore to see whether they be like or vnlike, ye must compare their axes together, which is their depth, and also their diameters to∣gether, which is thier length & breadth. As if the axe ••G of the cone ABC be to to the axe EI of the cone DEF, as the diameter AC of the cone ABC is to the diameter DF of the cone DEF, then a••e the cones ABC and DEF like cones. Likewise in the cilinders. If the axe LN of the cilinder LHMN haue that pro∣portion to the axe OQ of the cilinder ROPQ, which the diameter HM hath to the diameter RP: then are the cilinders HLMN and ROPQ like cilinders, and so of all others.

21 A Cube is a solide or bodely figure contayned vnder sixe e∣quall squares.* 1.63

As is a dye which hath sixe sides, and eche of

[illustration]

them is a full and perfect square, as limites or bor∣ders vnder which it is contayned. And as ye may conceiue in a piece of timber contayning a foote square euery way, or in any such like. So that a Cube is such a solide whose three dimensions are equall, the length is equall to the breadth thereof, and eche of them equall to the depth. Here is as it may be in a playne superficies set an image therof, in these two figures wherof the first is as it is com∣monly described in a playne, the second (which is in the beginning of the other side of this leafe) is drawn as it is described by arte vpō a playne su∣perficies to shew somwhat bodilike. And in deede the latter descriptiō is for the sight better thē the first. But the first for the demōstrations of Euclides propositions in the fiue bookes following is of more vse, for that in it may be considered and sene

all the fixe sides of the Cube. And so any lines or sections drawen

[illustration]

in any one of the sixe sides. Which can not be so wel sene in the o∣ther figure described vpon a playnd. And as touching the first figure (which is set at the ende of the other side of this leafe) ye see that there are sixe parallelogrammes which ye must conceyue to be both equilater and rectangle, although in dede there can be in this description onely two of them rectangle, they may in dede be described al equilater. Now if ye imagine one of the sixe paral∣lelogrammes, as in this example, the parallelogramme ABCD to be the base lieng vpon a ground playne superfices. And so con∣ceiue the parallelogramme EFGH to be in the toppe ouer it, in such sort, that the lines AE, CG, DH, & BF may be erected per∣pendicularly from the pointes A, C, B, D, to the ground playne su∣perficies or square ABCD. For by this imagination this figure wil shew vnto you bodilike. And this imagination perfectly had, wil make many of the propositions in these fiue bookes following, in which are required to be descri∣bed such like solides (although not all cubes) to be more plainly and easily conceiued.

In many examples of the Greeke and also of the Latin, there is in this place set the diffinition of a Te∣trahedron, which is thus.

* 1.6422 A Tetrahedron is a solide which is contained vnder fower triangles equall and equilater.

A forme of this solide ye may see in these two examples here set,

[illustration]

whereof one is as it is commonly described in a playne. Neither is it hard to conceaue. For (as we before taught in a Pyramis) if ye imagine the triangle BCD to lie vpon a ground plaine superficies, and the point A to be pulled vp together with the lines AB, AC, and AD, ye shall perceaue the forme of the Tetrahedron to be contayned vnder 4. triangles, which ye must imagine to be al fower equilater and equiangle, though they can not so be drawen in a plaine. And a Te∣trahedron thus described, is of more vse in these fiue bookes follow∣ing, then is the other, although the other appeare in forme to the eye more bodilike.* 1.65

Why this definition is here left out both of Campane and of Flussas,

[illustration]

I can not but maruell, considering that a Tetrahedron, is of all Philo∣sophers counted one of the fiue chiefe solides which are here de∣fined of Euclide, which are called cōmonly regular bodies, with∣out mencion of which, the entreatie of these should seeme much maimed: vnlesse they thought it sufficiently defined vnder the definition of a Pyramis,* 1.66 which plainly and generally taken, inclu∣deth in deede a Tetrahedron, although a Tetrahedron properly much differe••h from a Pyramis, as a thing speciall or a particular, from a more generall. For so taking it, euery Tetrahedron is a Py∣ramis, but not euery Pyramis is a Tetrahedron. By the generall definition of a Pyramis, the superficieces of the sides may be as many in number as ye list, as 3.4. 5.6. or moe, according to the forme of the base, whereon it is set, whereof before in the defi∣nition of a Pyramis were examples geuen. But in a Tetrahedron the superficieces erected can be but three in number according to the base therof, which is euer a triangle. Againe, by the generall definition of a Pyrami••, the superfi∣cieces erected may ascend as high as ye list, but in a Tetrahedron they must all be equall to the base. Wherefore a Pyramis may seeme to be more generall then a Tetrahedron, as before a Prisme seemed to be more generall then a Parallelipipedon, or a sided Columne: so that euery Parallelipipedon is a Prisme, but not euery Prisme is a Parallelipipedon. And euery axe in a Sphere is a diameter: but not euery diameter of a Sphere is the axe therof. So also noting well the definition of a Pyramis, euery Te∣trahedron may be called a Pyramis, but not euery Pyramis a Tetrahedron. And in dede Psellus in num∣bring of these fiue solides or bodies, calleth a Tetrahedron a Pyramis in manifest wordes.* 1.67 This I say might make Flussas & others (as I thinke it did) to omitte the definition of a Tetrahedron in this place, as sufficiently comprehended within the definition of a Pyramis geuen before. But why then did he not count that de••inition of a Pyramis faultie, for that it extendeth it selfe to large, and comprehendeth vnder it a Tetrahedron (which differeth from a Pyramis by that it is contayned of equall triangles) as he not so aduisedly did before the definition of a Prisme.

* 1.6823 An Octohedron is a solide or bodily figure cōtained vnder eight equall and equilater triangles.

As a Cube is a solide figure contayned vnder sixe superficiall fi∣gures

[illustration]

of foure sides or squares which are equilater, equiangle, and equall the one to the other: so is an Octohedron a solide figure contained vnder eight triangles which are equilater and equall the one to the other. As ye may in these two figures here set beholde. Whereof the first is drawen according as this solide is commonly described vpon a plaine superficies. The second is drawen as it is described by arte vpon a plaine, to shewe bodilike. And in deede although the second appeare to the eye more bodilike, yet as I be∣fore noted in a Cube, for the vnderstanding of diuers Propositions in these fiue bookes following, is the first description of more vse yea & of necessitie. For without it, ye can not cōceaue the draught of lines and sections in any one of the eight sides which are some∣times in the descriptions of some of those Propositions required. Wherefore to the consideration of this first description, imagine first that vppon the vpper face of the superficies of the parallelo∣gramme ABCD, be described a Pyramis, hauing his fower trian∣gles AFB, AFC, CFD, and DFB, equilater and equiangle, and con∣curring in the point F. Thē cōceaue that on the lower face of the super••icies of the former parallelogramme be described an other Pyramis, hauing his fower triangles AEB, AEC, CED, & DEB, equilater and equiangle, and concurring in the point E. For so al∣though somewhat grosly by reason the triangles can not be de∣scribed equilater, you may in a plaine perceaue the forme of this solide, and by that meanes conceaue any lines or sections requi∣red to be drawen in any of the sayd eight triangles which are the sides of that body.

24 A Dodecahedron is a solide or bodily fi∣gure cōtained vnder twelue equall, equilater,* 1.69 and equiangle Pentagons.

As a Cube, a Tetrahedron, and an Octohe∣dron,

[illustration]

are contayned vnder equall plaine figures, a Cube vnder squares, the other two vnder tri∣angles: so is this solide figure contained vnder twelue equilater, equiangle, and equall Penta∣gons, or figures of fiue sides. As in these two fi∣gures here set you may perceaue. Of which the first (which thinge also was before noted of a Cube, a Tetrahedron, and an Octohedron) is the common description of it in a plaine, the other is the description of it by arte vppon a plaine to make it to appeare somwhat bodilike. The first description in deede is very obscure to conceaue, but yet of necessitie it must so, ney∣ther can it otherwise, be in a plaine described to vnderstād those Propositions of Euclide in these fiue bokes a following which concerne the same. For in it although rudely, may you see all the twelue Pentagons, which should in deede be all equall, equilater, and equiangle. And now how you may somewhat conceaue the first figure de∣scribed in the plaine to be a body. Imagine first the Pentagon ABCDE ••o be vpon a ground plaine superficies, then imagine the Pentagon FGHKL to be on high opposite vnto the Penta∣gon ABCDE. And betwene those two Pentagons there will be ten Pentagons pulled vp, fiue frō the fiue sides of the ground Pen∣tagon, namely, from the side AB the Pentagon ABONM, from the side BC the Pentagon BCQPO, from the side CD the Pentagon CDSRQ, from the side DE, the Pentagon DEVTS, from the side EA the Pentagon EAMXV, the other fiue Penta∣gons haue eche one of their sides common with one of the sides of the Pentagon FGHKL, which is opposite vnto the Pentagon in the ground superficies: namely, these are the other fiue Penta∣gons FGNMX, GHPON, HKRQP, KLRST, LFXVT.

So here you may behold twelue Pentagons, which if you imagine to be equall, equilater, & equiangle, and to be lifted vp, ye shall (although somewhat rudely) conceaue the bodily forme of a Pentagon. And some light it will geue to the vnderstanding of certaine Propositions of the fiue bookes following con∣cerning the same.

* 1.7025 An Icosahedron is a solide or bodily figure contained vnder twentie equall and equilater triangles.

As the solides before last mentioned are all

[illustration]

described by the number and forme of the su∣perficieces which containe them: so this body likewise is de••ined by that that it is contayned of twentie triangles equall, equilater, and e∣quiangle. And although this solide also be ve∣ry hard to conceaue, as it is commonly descri∣bed vpon a plaine (an example wherof you haue in the first figure here set): yet is it of necessitie that in that forme it be described, if we will vnderstand such descriptions as are set forth of Euclide touching that body in the fiue bookes following. Howbeit you may by it (although somewhat rudely) see the 20. triangles, which are imagined to be equall, equilater, and equian∣gle, if you consider ••iu: angles of fiue triangles to concurre together at a point. And forasmuch as there are in this solide 20. triangles, and euery t••iangle hath three angles, the concurse of the said triangles will be in twelue pointes. As in this example the pointes of the concurse are A, B, C, D, E, F, G, H, K, L, M, & N. Where note that in this plaine the two poyntes M and N are but one point, yet must ye imagine one of those pointes to be erected vpward, and the other down∣ward. Now the ••iue triangles which concurre in the point M, are these, BMD, DMF, FMH, HML, and LMB: the fiue triangles which concurre in the point N, and are imagined to be erected downward, are these, ANC, CNE, ENG, GNK, and KNA: the other ten triangles which include this body, are these, ABC, BCD, CDE, DEF, EFG, FGH, GHK, HKL, KLA, LAB. The second figure here appeareth more bodilike vnto the eye.

These ••iue solides now last defined, namely, a Cube, a Tetrahedrō, an Octohedron, a Dodecahedron and an Icosahedrō are called regular bodies.* 1.71 As in plaine superficieces, those are called regular figures, whose sides and angles are equal, as are equilater triangles, equilater pentagons, hexagons, & such lyke, so in solides such only are counted and called regular, which are cōprehēded vnder equal playne super∣ficieces, which haue equal sides and equal angles, as all these fiue foresayd haue, as manifestly appeareth by their definitions, which were all geuen by this proprietie of equalitie of their superficieces, which haue also their sides and angles equall. And in all the course of nature there are no other bodies of this condition and perfection, but onely these fiue. Wherfore they haue euer of the auncient Philosophers bene had in great estimation and admiration, and haue bene thought worthy of much contemplacion, about which they haue bestowed most diligent study and endeuour to searche out the natures & pro∣perties of them. They are as it were the ende and perfection of all Geometry, for whose sake is written whatsoeuer is written in Geometry. They were (as men say) first inuented by the most witty Pithago∣ras then afterward set forth by the diuine Plato, and last of all meruelously taught and declared by the most excellent Philosopher Euclide in these bookes following, and euer since wonderfully embraced of all learned Philosophers.* 1.72 The knowledge of them containeth infinite secretes of nature. Pithag••ras, Ti∣meus and Plato, by them searched out the cōposition of the world, with the harmony and preseruation therof, and applied these ••iue solides to the simple partes therof, the Pyramis, or Tetrahedrō they ascri∣bed to the ••ire,* 1.73 for that it ascendeth vpward according to the figure of the Pyramis. To the ayre they ascribed the Octohedron,* 1.74 for that through the subtle moisture which it hath, it extendeth it selfe euery way to the one side, and to the other, accordyng as that figure doth. Vnto the water they assigned the

Ikosahedron, for that it is continually flowing and mouing,* 1.75 and as it were makyng angle•• 〈…〉〈…〉 ••ide according to that figure. And to the earth they attributed a Cube,* 1.76 as to a thing stable•• 〈◊〉〈◊〉 and sure as the figure ••ignifieth. Last of all a Dodecahedron,* 1.77 for that it is made of P••ntago••, whose angles are more ample and large then the angles of the other bodies, and by that ••ea•••••• draw more •••• roun••nes, 〈◊〉〈◊〉 & to the forme and nature of a sphere, they assigned to a sphere, namely, 〈…〉〈…〉. Who so will 〈…〉〈…〉 in his Tineus, shall ••ead of these figures, and of their mutuall proportion•• ••••raunge ma••ter••, which h••re are not to be entreated of, this which is sayd, shall be sufficient for the 〈◊〉〈◊〉 of them and for th•• declaration of their diffinitions.

After all these diffinitions here set of Euclide, Flussas hath added an other diffinition, which 〈◊〉〈◊〉 of a Parallelipipedon, which bicause it hath not hitherto of Euclide in any place bene defined, and because it is very good and necessary to be had, I thought good not to omitte it, thus it is.

A parallelipipedon is a solide figure comprehended vnder foure playne qua∣drangle figures,* 1.78 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••* 1.79 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,* 1.80 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,* 1.81 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,* 1.82 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.* 1.83

[illustration]

* 1.84If 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.

* 1.85

[illustration]

[illustration]

* 1.86

[illustration]

* 1.87

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.* 1.88 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.* 1.89 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.

¶The 1. Theoreme. The 1. Proposition. That part of a right line should be in a ground playne superficies, & part eleuated vpward is impossible.

FOr if it be possible, let part of the right line ABC, name∣ly, the part AB be in a ground playne superficies, and the other part therof, namely, BC be eleuated vpwarde. And produce directly

[illustration]

vpō the ground playne superfi∣cies the right line AB beyond the point B vn∣to the point D. Wherfore vnto two right lines geuen ABC, and ABD, the line AB is a common section or part, which is impossi∣ble.* 1.90 For a right line can not touche a right line in 〈◊〉〈◊〉 pointes then one, v••lesse those right be exactly agreing and laid the one vpon the other. Wherfore that part of a right line should be in a ground plaine superficies, and part eleuated vpward is impossible: which was required to be proued.

This figure more plainly setteth forth the foresaid de∣monstratiō,

[illustration]

if ye eleuate the superficies wheri•• the line BC.

An other demonstration after Fl••s••••s.

If it be possible let there be a right line ABG,

[illustration]

whose part AB let be in the ground playne super∣ficies AED;* 1.91 and let the rest therof BG be eleuated on high, that is, without the playne AED. Then I say that ABG is not one right line. For forasmuch as AED is a plaine superficies, produce directly & equally vpon the sayd playne AED the right lyne AB towardes D, which by the 4. definition of the first shall be a right line. And from some one point of the right line ABD, namely, from C, dra•• vnto the point G a right lyne CG. Wherefore in the triangle 〈…〉〈…〉 the outward ang•••• AB•• is eq••••ll to the two inward and opposite angles (by the 32. of the first) and therfore it is lesse then two right angles (by the 17. of the same) Wherfore the lyne ABG forasmuch as it maketh an angle, is not •• right line. Wh••refore that part of a right line should be in a ground playne superficies, and part eleuated vpward is impossible.

If ye marke well the figure before added for the play••er declaration of Euclides demonstration, i•• will not be hard for you to co••••••••e this figure which ••luss••s putteth for his demonst••••tion •• wherein is no difference but onely the draught of the lyne GC.

¶The 2. Theoreme. The 2. Proposition. If two right line cut the ou•• to the other, they are •••• ••ne and the selfe same playne superficies: & euery triangle is in one & the selfe same superficie••.

SVppose that these two right lines AB and CD doo

[illustration]

cutte the one the other in the point E. Then I say that these lines AB and CD are in one and the selfe same superficies, and that euery triangle is in one & selfe same playne superficies.* 1.92 Take in the lines EC and EB points at all auentures, and let the same be F and G, and draw a right line from the poynt B to the point C, and an other from the point F to the point G. And draw the lines FH and GK. First I say that the triangle EBC is in one and the same ground superficies.* 1.93 For if part of the triangle EBC, namely the triangle FCH, or the triangle GBK be in the ground superficies, and the residu•• be in an other, then also part of one of the right lines EC or EB shall be in the ground superficies, and part in an other. So also if part of the triangle EBC, namely, the part EFG be in the ground superficies and the residue be in an other, then also one part of eche of the right lines EC and EB shall be in the ground superficies, & an other part in an other superficies, which (by the first of the eleuenth) is proued to be impossible. Wherfore the triangle EBC is in one and the selfe same playne su∣perficies. For in what superficies the triangle BCE is, in the same also is either of the lines EC and EB, and in what superficies either of the lines EC and EB is, in the selfe same al∣so are the lines AB and CD. Wherfore the right lines lines AB and CD are in one & the selfe same playne superficies, and euery triangle is in one & the selfe same playne superficies: which was required to be proued.

In this figure here set may ye more playnely conceaue the demon∣stration

[illustration]

of the former proposition where 〈◊〉〈◊〉 may ele•••••••• what part of the triangle ECB ye will, namely the part FCH or the part GBK, or finally the part FCGB as is required in the demonstration.

¶The 3. Theoreme. The 3. Proposition. If two playne superficieces cutte the one the other: their common section is a right line.

SVppose that these two superficieces AB & BC do

[illustration]

cutte the one the other, and let their common secti∣•••• ••e the line DB. Then I say that DB is a right line. For if not, draw from the poynt D to the point B a right line DFB in the playne superficies AB,* 1.94 and likewise from the same poyntes draw an other right line DEB in the playne superficies BC. Now therfore two right lines DEB and DFB shall ••aue the selfe sa•••• e••de••, and therefore doo include a superficies which (by the last common sentence) is impossible•• Wherefore the lines DEB and DFB are not right lines. In

like sort also may we proue that no other right line can be drawne from the poynt D to the point B besides the line DB which is the common section of the two superficieces AB and BC. If therefore two playne superficieces cutte the one the other, their common section is a right line: which was required to be demonstrated.

[illustration]

This figure here set, sheweth most playnely not onely this third proposition, but also the demonstra∣tion thereof, if ye eleuate the superficies AB, and so compare it with the demonstration.

¶The 4. Theoreme. The 4. Proposition. If from two right lines, cutting the one the other, at their common section, a right line be perpendicularly erected: the same shall also be perpendicular∣ly erected from the playne superficies by the sayd two lines passing.

SVppose that there be two right lines AB

[illustration]

and CD cutting the one the other in the poynt E. And from the poynt E let there be erected a right line EF perpendicularly to the sayd two right lines AB and CD: then I say that the right line EF,* 1.95 is also erected perpendicular to the plaine superficies which passeth by the lines AB and CD. Let these lines AE, EB, EC, and ED be put equall the one to the other. And by the poynt E extend a right line at all auentures, and let the same be GEH. And drawe these right lines AD, CB, FA, FG, FD, FC, FH, and FB.* 1.96 And forasmuch as these two right lines AE & ED are equall to these two lines CE and EB, and they com∣prehend equall angles (by the 15. of the first): therefore (by the 4. of the first) the base AD is equall to the base CB, and the triangle AED is equall to the triangle CEB. Wherefore also the angle DAE, is equall to the angle EBC. But the angle AEG is equall to the angle BEH (by the 15 of the first). Wherefore there are two triangles AGE, and BEH hauing two angles of the one equall to two angles of the other, eche to his correspondent angle, and one side of the one equall to one side of the other, namely one of the sides which lye betwene the equall angles, namely, the side AE is equall to the side EB. Wherefore (by the 26. of the first) the sides remayning are equall to the sides remayning. Wherefore the side GE is equall to the side EH, and the side AG to the side BH. And for∣asmuch as the line AE is equall to the line EB, and the line FE is common to them both,

and maketh with them right angles, wherefore (by the fourth of the first) the base FA it e∣quall to the base FB. And (by the same reason) the base FC is equall to the base FD. And forasmuch as the line AD is equall to the line BC, and the line FA is equall to the line FB as it hath bene proued. Therefore these two lines FA and AD are equall to these two lines FB & BC, the one to the other, & the base FD is equall to the base FC. Wherfore also the angle FAD is equall to the angle FBC. And againe forasmuch as it hath bene proued, that the line AG is equall to the line BH, but the line FA is equall to the line FB. Where∣fore there are two lines FA and AG equall to two lines FB and BH and it is proued that the angle FAG is equall to the angle FBH: wherefore (by the 4. of the first) the base FG is equal to the base FH. Agayne forasmuch as it hath bene proued that the line GE is equal to the line EH, and the line EF is common to them both: wherefore these two lines GE and EF are equall to these two lines HE and EF, and the base FH is equall to the base FG: wherefore the angle GEF is equall to the angle HEF. Wherefore either of the angles GEF, and HEF is a right angle. Wherefore the line EF is erected, from the point E, perpendi∣cularly to the line GH. In like sort may we proue, that the same line FE maketh right angles with all the right lines which are drawne vpon the ground playne superficies and touch the point B. But a right line is then erected perpendicularly to a plaine superficies, when it maketh right angles with all the lines which touch it, and are drawne vpon the ground playne super∣ficies (by the 2. definition of the eleuenth). Wherefore the right line FE is erected perpendi∣cularly to the ground playne superficies. And the ground plaine superficies is that which pas∣seth by these right lines AB and CD. Wherefore the right line FE is erected perpendicu∣larly to the playne superficies which passeth by the right lines AB and CD. If therefore from two right lines cutting the one the other and at their common section a right line be perpendicularly erected: it shall also be erected perpendicularly to the plaine superficies by the sayd two lines passing: which was required to be proued.

In this figure you may most euidently conceaue the former

[illustration]

proposition and demonstration, if ye erect perpendicularly vnto the ground playne superficies ACBD the•• triangle AFB: and eleuate the triangles AFD, & CFB in such sort, that the line AF of the tri∣angle AFB may ioyne & make one line with the line AF of the tri∣angle AFD: and likewise that the line BF of the triangle AFB may ioyne & make one right line with the line BF of the triangle BFC.

¶The 5. Theoreme. The 5. Proposition. If vnto three right lines which touch the one the other, be erected a per∣pendicular line from the common point where those three lines touch: those three right lines are in one and the selfe same plaine superficies.

SVppose that vnto these three right lines BC, BD, and BE, touching the one the other in the poynt B, be erected perpendicularly from the poynt B, the line AB. Then I say, that those thre right lines BC, BD and BE, are in one & the selfe same plaine superficies. For if not, then if it be possible, let the lines BD &

BE be in the ground superficies, and let the line BC be e∣rected

[illustration]

vpward (now the lines AB and BC are in one and the same playne superficies (by the 2. of the eleuenth) for they touch the one the other in the point B). Extend the plaine superficies wherein the lines AB and BC are,* 1.97 and it shall make at the length a common section with the ground superficies, which common section shall be a right line (by the 3. of the eleuenth): let that common section be the line BF. Wherefore the three right lines AB, BC, and BF are in one and the selfe same su∣perficies, namely, in the superficies wherein the lines AB and BC are. And forasmuch as the right line AB is erected per∣pendicularly to either of these lines BD and BE, therefore the line AB is also (by the 4. of the eleuenth) erected perpendicu∣larly to the plaine superficies, wherein the lines BD and BE are. But the superficies wherein the lines BD and BE are is the ground superficies. Wherefore the line AB is erected per∣pendicularly to the ground plaine superficies. Wherefore (by the 2. definition of the eleuenth) the line AB maketh right angles with all the lines which are drawne vpon the ground super∣ficies and touch it. But the line BF which is in the ground superficies doth touch it. Wherfore the angle ABF is a right angle. And it is supposed that the angle ABC is a right angle. Wherefore the angle ABF is equall to the angle ABC, and they are in one and the selfe same plaine superficies which is impossible. Wherefore the right line BC is not in an higher superficies. Wherefore the right lines BC, BD, BE are in one and the selfe same plaine su∣perficies. If therefore vnto three right lines touching the one the one the other, be erected a perpendicular line from the common point where those three lines touch: those three right lines are in one and the selfe same plaine superficies: which was required to be demon∣strated.

This figure here set more playnely

[illustration]

declareth the demonstration of the for∣mer proposition, if ye erect perpendicu∣larly vnto the ground superficies, the s••••perficies wherein is drawne the line 〈◊〉〈◊〉 and so compare it with the sayd de••••••••stration.

The 6. Theoreme. The 6. Proposition. If two right lines be erected perpendicularly to one & the selfe same plaine superficies: those right lines are parallels the one to the other.

SVppose that these two right lines AB and CD be erected perpendicularly to a ground plaine superficies. Then I say that the line AB is a parallel to the line CD. Let the pointes which those two right lines touch in the plaine superficies be B and D.* 1.98 And draw a right line from the point B to the point D. And (by the 11. of the first) from the point D, draw vnto the line BD in the ground superficies a perpen∣dicular line DE. And (by the 2. of the first)* 1.99 put the line DE equall to the line AB. And draw these right lines BE, AE, and AD. And forasmuch as the line

[illustration]

AB is erected perpendicularly to the ground superficies, ther∣fore (by the 2. definition of the eleuenth) the line AB maketh right angles with all the lines which are drawne vpon the ground playne super••icies and touch it. But either of t••ese lines BD and BE which are in the ground superficies, touch the line AB, wherefore either of these angles ABD and ABE is a right angle•• and by the same reason also either of the angles CDB, & CDE is a right angle. And forasmuch as the line AB is equall to the line DE, and the line BD is common to them both, therfore these two lines AB and BD, are equall to these two lines ED and DB, and they contayne right angles: wherefore (by the 4. of the first) the base AD is equall to the base BE. And forasmuch as the line AB is equall to the line DE, and the line AD to the line BE, therefore these two lines AB and BE are equall to these two lines ED and DA, and the line AE is a common base to them both. Wherefore the angle ABE is (by the 8. of the first) equal to the angle EDA. But the angle ABE is a right angle: whefore also the angle EDA is a right angle: wherfore the line ED is erected perpēdicularly to the line DA: and it is also erected perpēdicularly to either of these lines BD and DC, wherefore the line ED is vnto these three right lines BD, DA, and DC erected perpendicularly from the poynt where these three right lines touch the one the other: wherefore (by the 5. of the eleuenth) these three right lines BD, DA, and DC are in one and the selfe same superficies. And in what superficies the lines BD and DA are, in the selfe same also is the line BA: for euery triangle is (by the 2. of the eleuenth) in one and the selfe same superficies. Wherefore these right lines AB, BD, and DC are in one and the selfe same superficies, and either of these angles ABD and BDC is a right angle (by sup∣position). Wherefore (by the 28. of the first) the line AB is a parallel to the line CD. If there∣fore two right lines be erected perpendicularly to one and the selfe same playne superficies, those right lines are parallels the one to the other: which was required to be proued.

Here for the better vnderstanding of this 6. proposition I

[illustration]

haue described an other figure: as touching which if ye erect the superficies ABD perpendicularly to the superficies BDE, and imagine only a line to be drawne from the poynt A to the poynt E (if ye will ye may extend a thred from the saide poynt A to the poynt E) and so compare it with the demonstration, it will make both the proposition, and also the demonstration most cleare vnto you.

¶An other demonstration of the sixth proposition by M. Dee.

Suppose that the two right lines AB & CD be perpendicularly erected to one & the same playne su∣perficies,

namely the playne superficies OP. Then I say that •••• and CD are parallels. Let the end points of the right lines AB and CD, which touch the plaine sup••••••••cies O••, be the poyntes •• and D, frō •• to D let a straight line be drawne (by the first petition) and (by the second petition) let the straight line ••D be extēded, as to the poynts M & N. Now forasmuch

[illustration]

as the right line AB, from the poynt •• produced, doth cutte the line MN (by construction). There∣fore (by the second proposition of this eleuenth booke) the right lines AB & MN are in one plain•• superficies. Which let be QR, cutting the superfi∣cies OP in the right line MN. By the same meanes may we conclude the right line CD to be in one playne superficies with the right line MN. But the right line MN (by supposition) is in the plaine su∣perficies QR: wherefore CD is in the plaine su∣perficies QR. And A•• the right line was proued to be in the same plaine superficies QR. Therfore AB and CD are in one playne superficie••, namely QR. And forasmuch as the lines A•• and CD (by supposition) are perpendicular vpon the playne superficies OP, therefore (by the second definition of this booke) with all the right lines drawne in the superficies OP and touching AB and CD, the same perpēdiculars A•• and CD, do make right angles. But (by construction) MN, being drawne in the plaine superficies OP toucheth the perpendiculars AB and CD at the poyntes •• and D. Therefore the perpen∣diculars A•• and CD, make with the right line MN two right angles namely ABN, and CDM: and MN the right line is proued to be in the one and the same playne superficies, with the right lines AB & CD: namely in the playne superficies QR. Wh••refore by the second part of the 28. proposition of the first booke, the right line•• AB and CD are parallel••. If therefore two right lines be erected perpendicularly to one and the selfe same playne superficies those right lines are parallels the one to the other: which was required to be demonstrated.

A Corollary added by M. Dee.

Hereby it is euident that any two right lines perpendicularly erected to one and the selfe same [ 1] playne superficies, are also them selues in one and the same playne superficies, which is likewis•• perpen∣dicularly [ 2] erected to the same playne superficies, vnto which the two right lines are perpendicular.

The first part hereof is proued by the former construction and demonstration, that the right lines [ 1] AB and CD are in one and the same playne superficies Q••. The second part is also manifest (that is, that [ 2] the playne superficies QR is perpendicularly erected vpon the playne superficies OP) for that A•• and CD being in the playne superficies QR, are by supposition perpendicular to the playne superficies OP: wherefore by the third definition of this booke QR is perpendicularly erected to, or vpon OP: which was required to be proued.

Io. Dee his aduise vpon the Assumpt of the 6.

As concerning the making of the line DE, equall to the right line AB, verely the second of the first, without some farther consideration, is not properly enough alledged. And no wonder it is, for that in the former booke••, whatsoe•••••• ••a••h of lines

[illustration]

bene spoken, the same hath alway••s bene imagined to be in one onely playne superficies considered or execu∣ted. But here the perpendicular line AB, is not in the same playn•• superficies, that the right line DB is. Ther∣fore some other helpe must be put into the handes of young beginners, how to bring this probleme to exe∣cution: which is this, most playne and briefe. Vnder∣stand that BD the right line, is the common section of the playne superficies, wherein the perpendiculars AB and CD are, & of the other playne superficies, to which they are perpendiculars. The first of these (in my former demonstration of the 6)•• I noted by the playne superfi∣cies QR: and the other, I noted by the plaine superficies OP. Wherfore BD being a right line common to both the playne sup••rficieces QR & OP, therby the ponits B and D are cōmon to the playnes QR and OP. Now

from BD (sufficiently extended) cutte a right line equall to AB, (which suppose to be BF) by the third of the first, and orderly to BF make DE equall, by the 3. o•• the first, if DE be greater then BF. (Which alwayes you may cause so to be, by producing of DE sufficiently). Now forasmuch as BF by constructi∣on is cutte equall to AB, and DE also, by construction, put equ••ll to BF, therefore by the 1. common sentence, DE is put equall to AB: which was required to be done.

In like sort, if DE were a line geuen to whome AB

[illustration]

were to be cutte and made equall, first out of the line DB (su••••iciently produced) cutting of DG, equall to DE by the third of the first: and by the same 3. cutting from BA (sufficiently produced) BA, equall to DG: then is it euidēt, that to the right line DE, the perpēdicular line AB is put equall. And though this be easy to conceaue, yet I haue designed the figure accordingly, wherby you may instruct your imagination. Many such helpes are in this booke requisite, as well to enforme the young stu∣dentes therewith, as also to master the froward gayne∣sayer of our conclusion, or interrupter of our demon∣strations course.

¶ The 7. Theoreme. The 7. Proposition. If there be two parallel right lines, and in either of them be taken a point at all aduentures: a right line drawen by the said pointes is in the self same superficies with the parallel right lines.

SVppose that these two right lines AB and CD be parallels, and in either of thē take a point at all aduentures, namely, E and F. Then I say, that a right line drawen from the point E to the point F, is in the selfe same plaine superficies that the pa∣rallel lines are. For if not, then if it be possible,

[illustration]

let it be in an higher superficies,* 1.100 as the line EGF is, and draw the superficies wherin the line EGF is, & extend it, and it shall make a common secti∣on with the ground superficies, which section shall (by the 3. of the eleuenth) be a right line: let that section be the right line EF. Wherefore two right lines EGF and EF include a superficies: which (by the last common sentence) is impossible. Wherfore a right line drawen from the point E to the point F, is not in an higher superficies. Wherfore a right line drawen from the point E to the point F, is in the selfe same superficies wherein are the parallel right lines AB and CD. If therefore there be two parallel right lines, and in either of them be taken a point at all aduentures, a right line drawen by th••se pointes is in the selfe same plaine superficies with the parallel right lines: which was requi∣red to be demonstrated.

By this figure it is easie to see

[illustration]

the former demonstration, if ye eleuate the superficies wherin is drawen the line EGF.

The 8. Theoreme. The 8. Proposition. If there be two parallel right lines, of which one is erected perpendicularly to a round playne superficies: the other also is erected perpendicularly to the selfe same ground playne superficies.

SVppose that there be two parallel right lines AB and CD, and let one of them, namely, AB be erected perpendiculerly to a ground superficies.* 1.101 Then I say that the line CD is also erected perpendiculerly, to the selfe same ground superficies. Let the lines AB and CD fall vpon the ground superficies in t••e pointes B and D, and (by the first peticion) draw a righ•• line from the point B to the point D. And drawe (by the 11. of the first) in the ground superficies from the point D vnto the line BD a per∣pendiculer line DE, and (by the 2. of the first) put the line

[illustration]

DE equall to the line AB, and draw a right line from the point B to the point E, and an other from the point A to the point E, and an other from the ••oint A to the point D.* 1.102 And forasmuch as the line AB is erested perpendicularly to the ground superficieces, therfore (by the 2. definition of the e∣leuenth) the line AB is erected perpendicularly to all the right lines that are in the ground superficies and touche it. Wherfore either of these angles ABD & ABE is a right angle. And forasmuch as vpon these parallel lines AB and CD falleth a certaine right line BD, therefore (by the 29. of the first) the angles ABD and CDB are equal to two right angles. But the angle ABD is a right angle, wherfore also the angle CDB is a right angle. Wherfore the line CD is e∣rected perpendic••larly to the line BD. And forasmuch as the line AB is equall to the line DE, and the line ••D is common to them both, therfore these two lines AB and BD are equal to these two lines ED and DB, and the angle ABD is equall to the angle EDB for either of them is a right angle. Wherfore (by the 4. of the first) the base AD is equall to the base BE. And forasmuch as the line AB is equall to the line DE, and the line BE to the lin•• AD, therfore thes•• two lines AB and BE are equall to these two lines AD & DE, the on•• to the other, and the line AE is a common base to them both. Wherfore (by the 8. of the first) the angle ABE is equall to the angle ADE: but the angle A••E is a right angle, wherfore th•• ••ngle EDA, is also a right angle. Wherefore the line ED is erected perpendicularly to the line AD, and it is also erected perpendicularly to th•• line DB. Wherfore the line ED is erect••d perpendicularly to the plaine superficies wherin th•• l••n••s BD and BA are (by the 4. of ••his booke) Wherfore (by the 2. definition of the eleuenth) the line ED is erected perpen∣dicularly to all the right lines that touche it and are in the s••perficies wherein the lines BD and AD are. But in what superficies the lines BD and DA are, in the selfe same superficies is the line DC. For the line AD being drawen from two pointes taken in the parallel lines AB and CD is by the former proposition in the selfe same superficies with them. Now f••ras∣much as the lines AB and BD ar•• in the superficies wherin the lines BD and DA are, but in what superficies the lines AB & BD are, in the same is the line DC. Wherfore the line ED is erected perpendicularly to the line DC. Wherfore also the line CD is erected perpendi∣cularly to the line DE. And the line CD is erected perpendicularly to the line DB. For by the 29. of the first, the angle CDB being equall to the angle ABD is a right angle. Where∣fore the line CD is from the point D erected perpendicularly to two right lines DE and DB cutting the one the other in the point D. Wherfore by the 4. of the eleuenth, the line CD is erected perpendiculaaly to the plaine superficies, wherein are the lines DE and DB. But

the ground plaine superficies is that wherin are the lines DE and DB, to which superficies also the line AB is supposed to be erected perpendiculerly. Wherefore the line CD is erected perpendicularly to the ground plaine superficies, wherunto the line AB is erected perpendicu∣larly. If therfore there be two parallel right lines, of which one is erected perpendicularly to a ground plaine superficies, the other also is erected perpendicularly to the selfe same ground plaine superficies: which was required to be demonstrated.

This figure will more clearely set forth the former de∣monstration,

[illustration]

if ye erect perpendicularly the superficies ABD to the superficies BDE, and imagine a lyne to be drawen from the point A to the point D, in stede wher∣of, as in the 6. proposition ye may extende a threede.

¶ The 9. Theoreme. The 9. Pro〈◊〉〈◊〉 Right lines which are parallels to one and the selfe same right line, and are not in the selfe same superficies that it is in: are also parallels the one to the other.

SVppose that either of these right lines AB and CD be a parallel to the line EF not being in the selfe same superficies with it. Then I say that the line AB is a parallel to the line CD. Take in the line EF a point at all aduentures, and let the same be G.* 1.103 And from the point G raise vp in the superficies wherin are the lines EF and AB, vnto the line EF a perpendiculer line GH, and againe in the superficies wherin are the lines EF and CD, raise vp from the same point G to the line EF a perpen∣diculer line GK.* 1.104 And forasmuch as the line

[illustration]

EF is erected perpendiculerly to either of the lines GH and GK, therfore (by the 4. of the eleuenth) the line EF is erected perpendicu∣larly to the superficies wherein the lines GH and GK are, but the line EF is a parallel line to the line AB. Wherfore (by t••e 8. of the eleuenth) the line AB is erected perpendicu∣larly to the plaine superficies, wherin are the lines GH and GK. And by the same reason al∣so the line CD is erected perpendicularly to the plaine superficies wherin are the lines GH & GK. Wherefore either of these lines AB and CD is erected perpendicularly to the plaine superficies, wherin the lines GH and GK are. But if two right lines be erected perpendicu∣larly to one and the selfe same plaine superficies, those right lines are parallels the one to the other (by the 6. of the eleuenth) Wherfore the line AB is a parallel to the line CD. Wherfore right lines which are parallels to one & the selfe same right line, and are not in the self same superficies with it are also parallels the one to the other: which was required to be proued.

This figure more clearely manifesteth the former propo∣sition

[illustration]

and demonstration, if ye eleuate the superficieces ABEF and CDEF that they may incline and concurre in the lyne EF.

¶ The 10. Theoreme. The 1〈…〉〈…〉 If two right lines touching the one the othe•• 〈…〉〈…〉her right lines touching the one the other, and no〈…〉〈…〉lfe same superficies with the two first: those right lines cōtaine equall angles.

SVppose that these two right lines AB and BC touching the one the other, be parallells to these two lines DE and EF touching also the one the other, and not being in the selfe same superficies that the lines AB and BC are. Thē I say, that the angle ABC is equall to the angle DEF.* 1.105 For let the lines BA, BC, ED, EF, be put equall the one to the other: and draw these

[illustration]

right lines AD, CF, BE, AC, and DF.* 1.106 And forasmuch as the line BA is equall to the line ED, and also parallell vnto it, therefore (by the 33. of the first) the line AD is e∣quall and parallell to the line BE: and by the same reason also the line CF is equall & parallell to the line BE. Wher∣fore either of these lines AD and CF is equall & parallell to the line EB. But right lines which are parallells to one and the selfe same right line, and are not in the selfe same su∣perficies with it, are also (by the 9. of the eleuenth) parallells the one to the other. Wherefore the line AD is a parallell line to the line CF. And the lines AC and DF ioyne them together. Wherefore (by the 33. of the first) the line AC is equall and parallell to the line DF. And forasmuch as these two right lines AB & BC are equall to these two right lines DE and EF, and the base AC also is equall to the base DF: therefore (by the 8. of the first) the angle ABC is equall to the angle DEF. If therfore two right lines touching the one the other be parallells to two other right lines touching the one the other, and not being in one and the selfe same superficies with the two first: those righ•• lines containe equall angles: which was required to be demonstrated.

This figure here set more

[illustration]

plainly declareth the former Pro∣position and demonstration, if ye eleuate the superficieces DABE, and FCBE, till they concurre in the line FE.

¶ The 1. Probleme. The 11. Proposition. From a point geuen on high, to drawe vnto a ground plaine superficies a perpendicular right line.

* 1.107SVppose that the point geuen on high be A, and suppose a ground plaine superficies, namely, BCGH. It is required from the point A to draw vnto the ground super∣ficies a perpendicular line. Drawe in the ground superficies a right line at aduen∣tures, and let the same be BC.* 1.108 And (by the 12. of the first) from the point A draw vnto the line BC a perpendicular line AD. * 1.109 Now if AD be a perpendicular line to the ground superficies, then is that done which was sought for. But if not, then (by the 11. of the first) from the point D raise vp in the ground super∣ficies vnto the line BC a perpendicular line DE. And (by the 12. of the first) from the point A draw vnto the line DE a perpendicular line AF. And by the point F draw (by the 31. of the ••irst) vnto the

[illustration]

line BC a parallell line FH: And extend the line FH from the point F to the point G.* 1.110 And forasmuch as the line BC is erected perpendicularly to either of these lines DE and DA, therefore (by the 4. of the eleuenth) the line BC is erected perpēdicularly to the superficies wher∣in the lines ED and AD are: and to the line BC the line GH is a parallell. But i•• there be two parallell right lines, of which one is erected perpendicularly to a certaine plaine superficies, the other also (by the 8. of the eleuenth) is erected perpendicularly to the selfe same superficies. Wherefore the line GH is erected perpendicularly to the plain•• superficies wherein the lines ED and DA are. Wherfore also (by the 2. definition of the eleuenth) the line GH is erected perpendicularly to all the right lines which touch it, and are in the plaine superficies wherein the lines ED and AD are. But the line AF toucheth it being in the superficies wherein the lines ED and AD are (by the ••. of this booke). Wherefore the line GH is erected perpen∣dicularly to the line FA. Wherefore also the line FA is erected perpendicularly to the line GH: and the line AF is also erected perpendicularly to the line DE. Wherefore AF is e∣rected perpendicularly to either of these lines HG and DE. But if a right line be erected per∣pendicularly from the common section of two right lines cutting the one the other, it shall also be erected perpendicularly to the plaine superficies of the said two lines (by the 4. of the ele∣uenth). Wherefore the line AF is erected perpendicularly to that superficies wherin the lines ED and GH are. But the superficies wherein the lines ED and GH are, is the ground su∣perficies. Wherefore the line AF is erected perpendicularly to the ground superficies. Wher∣fore from a point geuen on high, namely, frō the point A, is drawen to the ground superficies a perpendicular line: which was required to be done.

In this figure shall ye much more plainely see both

[illustration]

the cases of this former demonstratiō. For as touching the first case, ye must erecte perpendicularly to the ground superficies, the superficies wherein is drawen the line AD, and compare it with the demonstration, and it will be clere vnto you. For the second case ye must erecte perpendicularly vnto the ground superfi∣cies the superficies wherein is drawen the line AF, and vnto it let the other superficies wherein is drawen the line AD, incline, so that the point A of the one may concurre with the point A of the other: and so with your figure thus ordered, compare it with the demon∣stration, and there will be in it no hardnes at all.

¶The 2. Probleme. The 12. Proposition. Vnto a playne superficies geuen, and from a poynt in it geuen, to rayse vp a perpendicular line.

SVppose that there be a ground playne superficies geuen,

[illustration]

and let the poynt in it geuen be A. It is required from the point A to raise vp vnto the ground plaine superficies a perpendicular line. Vnderstand some certayne poynt on high, and let the same be B.* 1.111 And from the poynt B draw (by the 11. of the eleuenth) a perpendicular line to the ground superficies, and let the same be BC. And (by the 31. of the first) by the poynt A drawe vnto the line BC a parallel line DA. Now forasmuch as there are two parallel right lines AD and CB, the one of them, namely,* 1.112 CB is erected perpendicularly to the 〈◊〉〈◊〉 superficies: wherefore the other line also, namely, AD, is 〈◊〉〈◊〉 perpendicularly to the same ground superficies (by the eight of 〈…〉〈…〉leuenth). Wherefore vnto a playne superficies geuen, and 〈◊〉〈◊〉 poynt in it geuen, namely, A, is raysed vp a perpendicular lyn•• ••required to be doone.

In this second figure ye may consider playnely the demonstration of the former proposition if ye erect per∣pendicularly the superficies wherein are drawne the lines AD and CB.

¶ The 11. Theoreme. The 13. Pr••position. From one and the selfe poynt, and to one and the selfe same playne superfi∣cies, can not be erected two perpendicular right lines on one and the selfe same side.

FOr if it be possible from the poynt A let there be erected perpendicularly to one and the selfe same playne superficies two righ•• lines AB and AC on one and the selfe same side.* 1.113 And extende

[illustration]

the superficies wherein are the lines AB and AC:* 1.114 and it shall make at length a common section in the ground super••icies which common section shall be a right line, and shall passe by the poynt A: let that common section be the line DAE. Wherefore (by the 3. of the ele∣uenth) the lines AB, AC, and DAE are in one and the selfe same playne superficies. And foras∣much as the line CA is erected perpendicularly to the ground superficies, therfore (by the

2. definition of the eleuenth) it maketh right angles with all the right lines that touch it, and are in the ground superficies. But the line DAE toucheth it, being in the ground superficies. Wherefore the angle CAE is a right angle, and by the same reason also the angle BAE is a right angle. Wherefore (by the 4 petition) the angle CAE is equall to the angle BAE the lesse to the more, both angles being in one & the selfe same playne superficies: which is im∣possi••le. Wherefore from one and the selfe same poynt, and to one and the selfe same playne su∣perficies can not be ••rected two perpendicular right lines on one & the selfe same side: which was required to be demonstrated.

In this figure if ye erect perpendicularly the su∣perficies

[illustration]

wherein are drawne the lines ••A and CA to the ground super••icies wherein is drawn the line DAE, and so compare it with the the demonstratiō of the for¦mer proposition it will be cleare vnto you.

M. Dee his annotation.

Euclides wordes in this 13. proposition admit two cases: one, if th•• 〈…〉〈…〉 in the playne superficies, (as cōmonly the demonstrations suppose) the other, if the poynt assigned be any where with¦out the sayd playne superficies, to which, the perpendiculars fall, is considered. Contrary to either of which, if the aduersarie affirme, admitting from one poynt two right lines, perpendiculars to one and the selfe same playne superficies, and on one and the same side thereof, by the 6. of the eleuenth he may be bridled: which will ••ore him to confesse his two perpendiculars to be also parallels. But by supposi∣tion agreed one, they concurre at one and the same poyn••, which (by the definition of parallels) i•• im∣possible. Therefore our aduersary must recant a••d yelde to out proposition.

¶ The 12. Theoreme. The 14. Proposition. To whatsoeuer plaine superficieces one and the selfe same right line is e∣rected perpendicularly: those superficieces are parallels the one to the other.

SVppose that a right line AB be

[illustration]

erected perpēdicularly to either of these plaine superficieces CD and EF. Then I say, that these superficieces CD and EF are paral∣lels the one to the other.* 1.115 For if not, then if they be extended they will at the length meete. Let them meete, if it be possible. Now then their common section shall (by the 3. of the eleuenth) be a right line. Let that common section be GH. And in the line GH take a point at all aduen∣tures and let the same be K. And drawe a right line from the point A to the point K, and an other from the point B to the point K. And forasmuch as the line AB is e∣rected

perpendicularly to the plaine superficies EF, therefore the line AB is also erected perpendicularly to the line BK which is in the extended superficies EF. Wherfore the angle ABK is a right angle. And by the same reason also the angle BAK is a right angle. Wher∣fore in the triangle ABK, these two angles ABK & BAK, are equall to two right angles: which (by the 17. of the first) is impossible. Wherefore these superficieces CD and EF being extended meete not together. Wherefore the superficieces CD and EF are parallells. Wher∣fore to what soeuer plaine superficieces one and the selfe same right line i•• erected perpendicu∣larly: those superficies are parallells the one to the other: which was required to be proued.

In this figure may ye plainly see the former demonstra∣tion

[illustration]

if ye erecte the three superficieces, GD, GE, and KLM perpēdiculary to the ground plaine super••icies: but yet in such sort that the two superfici•••• 〈…〉〈…〉 may concurre in the common line G 〈…〉〈…〉 the demonstration.

A corollary added by Campane.

If a right line be erected perpendicularly to one of those superficies, it 〈…〉〈…〉 erected perpen∣dicularly to the other.

For if it should not be erected perpendicularly to the other, then it falling vpon that other shall make with some one line thereof an angle lesse then a right angle: which line should (by the 5. petition of the first) concurre with some one line of that super••icies whereunto it is perpendicular. So that those superficieces should not be parallels: which is contrary to the supposition. For they are suppsed to be parallels.

¶ The 13. Theoreme. The 15. Proposition. If two right lines touching the one the other be parallels to two other right lines touching also the one the other and not being in the selfe same plaine superficies with the two first: the plaine superficieces extended by those right lines, are also parallells the one to the other.

SVppose that these two right lines AB and BC touching the one the other be pa∣rallells to these two right lines DE & EF touching also the one the other, and not being in the selfe same plaine super••icies with the right lines AB and BC. Then I say, that the plaine superficieces by the lines AB and BC, and the lines DE and EF being extended, shall not meete together, that

[illustration]

is, they are equedistant and parallels:* 1.116 From the point B draw (by the 11. of the eleuenth) a perpendicular line to the su∣per••icies wherein are the lines DE and EF, and let that per∣pendicular line be BG. And by the point G in the plaine su∣perficies passing by DE, and EF, draw (by the 31. of the first) vnto the line ED a parallell line GH: and likewise by that point G drawe in the same superficies vnto the line EF a pa∣rallell line GK.* 1.117 And forasmuch as the line BG is erected per∣pendicularly to the superficies wherein are the lines DE and EF, there••ore (by the 2. definition of the eleuenth) it is also erected perpendicularly to all the right lines which touch it, and are in the selfe same superficies wherein are the lines DE

and EF. But either of these lines GH and GK touch it, and

[illustration]

are also in the superficies wherein are the lines DE and EF, therefore either of these angles BGH, and BGK, is a right angle. And forasmuch as the line BA is a parallell to the line GH (that the lines GH and GK are parallells vnto the lines AB and BC it is manifest by the 9. of this booke): there∣fore (by the 29. of the first) the angles GBA and BGH are equall to two right angles. But the angle BGH is (by con∣structiō) a right angle, therfore also the angle GBA is a right angle: therefore the line GB is erected perpendicularly to the line BA. And by the same reason also may it be proued, that the line BG is erected perpendicularly to the line BC. Now forasmuch as the right line BG is erected perpendicularly to these two right lines BA and BC touching the one the other, therefore (by the 4. of the eleuenth) the line BG is erected perpendicularly to the superficies wherein are the lines BA and BC•• And it is also erected perpendicularly to the superficies wherein are the lines GH and GK. But the superficies wherein are the lines GH and GK, is that superficies wherein are the lines DE and EF: wherefore the line BG is erected perpendicularly to the superficies wherein are the lines DE and EF. Wherefore the line BG is erected perpendicularly to the superficies wherein are the lines DE and EF, and to the superficies wherein are the lines AB and BC. But if one and the selfe same right line be erected perpendicularly to plaine superficieces, those superfi∣cieces are (by the 14. of the eleuenth) parallels the one to the other. Wherefore the superficies wherin are the lines AB and BC is a par••llel to the superficies wherin are the lines DE and EF. If therefore two right lines touching the one the other be parallels to two other right lines touching also the one the other, and not being in the selfe same plaine superficies with the two first, the plaine superficieces extended by those right lines are also parallels the one to the other which was required to be demonstrated.

By this figure here put, ye may more clerely see both

[illustration]

the former 15. Proposition and also the demonstration therof: if ye erecte perpendicularly vnto the ground superficies, the three superficieces ABC, KHE, and LHBM, and so compare it with the demonstration.

¶ A Corollary added by Flussas.

Vnto a plaine superficies being geuen, to drawe by a point geuen without it, a parallel plaine superfi••ie••. Suppose as in the former description that the superficies geuē be ABC, & let the point geuē without it be G. Now then by the point G drawe (by the 31. of the first) vnto the lines AB and BC parallel lines GH and HK. And the superficies extended by the lines GH and GK shall be parallel vnto the superficies ABC, by this 15. Proposition.

The 14. Theoreme. The 16. Proposition. If two parallel playne superficieces be cut by some one playne superficies: their common sections are parallel lines.

SVppose that these two plaine superficieces AB and CD be cut by this plaine su∣perficies EFGH, ••nd let their common sections be the right lines EF and GH. Then I say that the line EF is a parallel to the line GH. For if not, then the lines EF and GH being produced, shall at the length meete together either on

the side that the pointes FH, are, or on

[illustration]

the side that the pointes E, G are. First let them be produced on that side that the pointes F, H are, and let them mete in the point K. And forasmuch as the line EFK is in the superficies AB, therfore all the points which are in the line EF are in the superficies AB (by the first of this booke) But one of the pointes which are in the right line EFK is the point K,* 1.118 therfore the point K is in the superficies AB. And by the same reason also the point K is in the superficies CD. Wherfore the two su∣perficieces AB and CD being produ∣ced do mete together, but by sup••ositiō they mete not together, for they are sup¦posed to be parallels. Wherfore the right lines EF and GH produced shall not meete together on that side that the pointes F, H are. In like sort also may we proue that the right lines EF and GH produced meete not together on that side that the pointes E, G are. But right lines which being produced on no side mete together, are parallels (by the last definicion of the first). Wherfore the line EF is a parallel to the line GH. If therfore two parallel plaine su∣perficieces be cut by some one plaine superficies their common sections are parallel lines: which was required to be proued.

This figure here set more plainl•• 〈…〉〈…〉

[illustration]

demonstration, if ye erect perpendicu•••••• 〈…〉〈…〉 superficies the three superficieces A•• 〈…〉〈…〉 and so compare it with the demonstr•• 〈…〉〈…〉

A Corollary added by Flussas.

If two plaine superficieces be parallels to one and the s••lfe same playne 〈…〉〈…〉 also be parallels the one to the other, or they shall make one and the selfe same plaine sup〈…〉〈…〉

For if the plaine superficieces DG and GH

[illustration]

being parallels to one and the selfe same superfi∣cies, namely, to AB be not also parallels the one to the other, then being produced they shall con∣curre (by the conuerse of the sixt definition of the eleuenth) Let them concurre in the right line GE. Then I say that the superficieces GD and GH are in one and the selfe same playne superficies. Draw in the playne superficies AB a right line at all aduentures AC. And by that right lyne & the point E extende a playne superficies, cutting the two superficieces DG and GH by the right lines ED and EI. Wherfore the right lines AC and DE, also AC and EI are parallels by this propositi∣on. But the lines DE and EI forasmuch as they concurre in the point E are not parallels the one to the other. Wherefore the right lines DE and EI make directly one right line (by that which is added after the 30. propositon of the first.) And therfore the plaine superficieces DG and GH are in one and the selfe same playne superficies. For

i•• ••hey be not, then part of the right line DI, namely, the part DE is in the playne superficies DG, and an other part therof, namely, EI is on high in an other superficies GH, which by the first of the eleuēth i•• impossible. Wherfore the superficieces DG and GH are in one and the selfe same playne superficies. But i•• the superficieces DG and GH neuer concurre, then are they parallels by the 6. definition of the ••leuenth.

In this figure here set, ye may more plainely see the

[illustration]

form••r demonstration, if ye eleuate to the ground super∣••icieces ACDI, the three super••icieces AB, DG, & GI, and ••o compare it with the demonstration.

The 15. Theoreme. The 17. Proposition. I•• two right lines be cut by playne superficieces being parallels: the partes o•• the lines deuided shall be proportionall.

* 1.119S••ppose that these two right lines AB and CD be deuided by these plaine super∣fi••i••ces being parallels, namely, GH, KL, MN in the points A, E, B, C, F, D. Thē I say that as the right line AE is to the right line EB, so is the right line CF to the right line FD. Draw these right lines AC, BD and AD. And let the line AD and the super••icies KL concurre in the point X. And

[illustration]

draw a right line from the point E to the point X and an o∣ther from the point X to the point F. And forasmuch as these two parallel superficieces KL and MN are cut by the super••icies EBDX, ther••ore their common sections which are the lines EX and BD, are (by the 16. of the eleuenth) parallels the one to the other. And by the same reason also ••orasmuch as the two parallel superficies GH and KL be cut by the super••icies AXFC, their common sections AC and XF are (by the 16. of the eleuenth) parallels. And ••orasmuch as to one of the sides of the triangle ABD•• namely, to the side BD is drawne a parallel line EX, therfore (by the 2. of the sixt) proportionally as the line AE is to the line EB, so is the line AX to the line XD. Againe forasmuch as to one of the sides of the triangle ADC, namely, to the side AC is drawen a parallel line XF, therfore by the 2. of the sixt, pro∣portionally as the line AX is to the line XD, so is the line CF to the line FD. And it was proued that as the line AX is to the line XD, so is the line AE to the line EB, therefore also (by the 11. of the fift) as the line AE is to the line EB, so is the line CF to the line FD. If therfore two right lines ••e deuided by plaine super••icieces being parallels, the parts of the lines deuided shal be propor∣tionall: which was required to be demonstrated.

In this figure it is more easy to see the former demonstration, if ye e∣rect

[illustration]

perpendicularly vnto the ground superficies ACBD, the thre su∣perficieces, GH, KL, and MN, or if ye so ••r••ct them that th••y be equedi∣stant one to the other.

¶ The 16. Theoreme. The 18. Proposition. If a right line be erected perpēdicularly to a plaine superficies: all the superficieces extended by that right line, are erected perpendicularly to the selfe same plaine superficies.

SVppose that a right line AB be erected perpendicularly to a ground superficies. Thē I say, that all the superficieces passing by the line AB, are erected perpendicularly to the ground superficies. Extend a superficies by the line AB, and let the same be ED, & let the cōmon section of the plaine

[illustration]

superficies and of the ground superficies be the right line CE. And take in the line CE a point at all aduentures,* 1.120 and let the same be F: and (by the 11. of the first) from the point F drawe vnto the line CE a perpendicular line in the superficies DE, and let the same be FG. And forasmuch as the line AB is erected perpendicularly to the ground super∣ficies, therefore (by the 2. definition of the e∣leuenth) the line AB is erected perpendicu∣larly to all the right lines that are in the ground plaine superficies,* 1.121 and which touch it. Wher∣fore it is erected perpendicularly to the line CE. Wherefore the angle ABE is a right angle. And the angle GFB is also a right angle (by construction). Wherefore (by the ••8. of the first) the line AB is a parallel to the line FG. But the line AB is erected perpendicularly to the ground superficies: wherefore (by the 8. of the eleuenth) the line FG is also erected per∣pendicularly to the ground superficies. And forasmuch as (by the 3. definition of the eleuenth) a plaine superficies is then erected perpendicularly to a plaine superficies, when all the right lines drawen in one of the plaine superficieces vnto the common section of those two plaine su∣perficieces making therwith right angles, do also make right angles with the other plaine su∣perficies and it is proued that the line FG drawen in one of the plaine superficieces, namely, in DE, perpendicularly to the common section of the plaine superficieces, namely, to the line CE, is erected perpendicularly to the ground superficies: wherefore the plaine superficies DE is erected perpendicularly to the ground superficies. In like sort also may we proue, that all the plaine superficieces which passe by the line AB, are erected perpendicularly to the ground su∣perficies. If therefore a right line be erected perpendicularly to a plaine superficies all the su∣perficieces passing by the right line, are erected perpendicularly to the selfe same plaine super∣ficies: which was required to be demonstrated.

In this figure here set ye may erect per∣pēdicularly

[illustration]

at your pleasure the superficies wherin are drawen the lines DC, GF, AB, and HE, to the ground superficies wherin is drawen the line CFBE, and so plainly compare it with the demonstration before put.

¶ The 17. Theoreme. The 19. Proposition. If two plaine superficieces cutting the one the other be erected perpendicu∣larly to any plaine superficies: their common section is also erected perpen∣dicularly to the selfe same plaine superficies.

SVppose that these two plaine super••icieces AB & BC cutting the one the other be e∣rected p••rp••ndicularly to a ground superficies, and let their common section be the line BD. Then I say, that the line BD is erected perpendicularly to the ground super••icies.* 1.122 For if not, then (by the 11. of the first)

[illustration]

from the point D draw in the superficies AB vnto the right line DA a perpendicular line DE. And in the superficies CB draw vnto the line DC a per∣pendicular line DF. And forasmuch as the super∣ficies AB is erected perpendicularly to the ground superficies, and in the plaine superficies AB vnto the common section of the plaine superficies and of the ground superficies, namely, to the line DA is erected a perpendicular line DE, therefore (by the conuerse of the 3. de••inition of this booke) the line DE is e∣rected perpendicularly to the ground super••icies. And in like sort may we proue, that the line DF is erected perpendicularly to the ground superficies. Wherefore from one and the selfe same point, namely, from D, are erected perpendicularly to the ground superficies two right lines both on one and the self same side: which is (by the 15. of the eleuenth) impossible. Wher∣fore from the point D can not be erected perpendicularly to the ground superficies any other right lines besides BD, which is the common section of the two superficieces AB and BC. If therefore two plaine super••icieces cutting the one the other be erected perpendicularly to any plaine super••icies, their common section is also erected perpendicularly to the selfe same plaine super••icies: which was required to be proued.

Here haue I set an other figure which

[illustration]

will more plainly shewe vnto you the for∣mer demonstration, if ye erecte perpendi∣cularly to the ground superficies AC the two superficieces AB and BC which cut the one the other in the line BD.

The 18. Theoreme. The 20. ••roposition. If a solide angle be contayned vnder three playne superficiall angles: euery

two of those three angles, which two so euer be taken, are greater then the third.

SVppose that the solide angle A be contayned vnder three playne superficiall an∣gles, that is, vnder BAC, CAD, and DAB. Then I say that two of these superficiall angles how so euer they be taken, are greater then the third. If the angles BAC, CAD, & DAB

[illustration]

be equall the one to the other, then is it manifest that two of them which two so euer be taken are greater then the third. But if not, let the angle BAC be the greater of the three angles. And vnto the right line AB and from the poynt A make in the playne superficies BAC vnto the angle DAB an equall angle BAE. And (by the 2. of the first) make the line AE equall to the line AD. Now a right line BEC drawne by the poynt E, shall cut the right lines AB and AC in the poyntes B and C: draw a right line from D to B, and an other from D to C. And forasmuch as the line DA is equall to the line AE,* 1.123 and the line AB is common to thē both, therefore these two lines DA and AB are equall to these two lines AB and AE and the angle DAB is equall to the angle BAE. Wherefore (by the 4. of the first) the base DB is equall to the base BE. And forasmuch as these two lines DB and DC are greater then the line BC, of which the line DB is proued to be equall to the line BE. Wherefore the re∣sidue, namely, the line DC is greater then the residue, namely, then the line EC. And foras∣much as the line DA is equall to the line AE, and the line AC is common to them both, and the base DC is greater then the base EC, therefore the angle DAC is greater then the angle EAC. And it is proued that the angle DAB is equall to the angle BAE: wher¦fore the angles DAB and DAC are greater then the angle BAC. If therefore a solide angle be contayned vnder three playne superficiall angles euery two of those three angles, which two so euer be taken are greater then the third: which was required to be proued.

In this figure ye may playnely behold the

[illustration]

former demonstration, if ye eleuate the three triangles ABD, A••C and ACD in such ••or••that they may all meete together in the poynt A.

The 19. Theoreme. The 21. Proposition. Euery solide angle is comprehended vnder playne angles lesse then fower right angles.

SVppose that A be a solide angle contayned vnder these superficiall angles BAC, DAC and DAB. Then I say that the angles BAC, DAC and DAB are lesse then fower right angles.* 1.124 Take in euery one of these right lines ACAB and

AD a poynt at all aduentures and let the same be B, C, D. And draw these right lines BC, CD and DB.* 1.125 And forasmuch as the angle B is a solide angle, for it is contayned vnder three superficiall angles, that is, vnder CBA, ABD and CBD, therefore (by the 20. of the eleuenth) two of them which two so euer be taken are greater then the third. Wherefore the angles CBA and ABD are greater

[illustration]

then the angle CBD: and by the same rea∣son the angles BCA and ACD are grea∣ter then the angle BCD•• and moreouer the angles CDA and ADB are greater then the angle CDB. Wherefore these sixe angles CBA, ABD, BCA, ACD, CDA, and ADB are greater thē these thre angles, namely, CBD, BCD, & CDB. But the three angles CBD, BDC, and BCD are equall to two right angles. Wherefore the sixe angles CBA, ABD, BCA, ACD, CDA, and ADB are greater thē two right an∣gles. And forasmuch as in euery one of these triangles ABC, and ABD and ACD three angles are equall two right angles (by the 32. of the first). Wherefore the nine angles of the thre triangles, that is, the angles CBA, ACB, BAC, ACD, DAC, CDA, ADB, DBA and BAD are equall to sixe right angles. Of which angles the sixe angles ABC, BCA, ACD, CDA, ADB and DBA are greater then two right angles. Wherefore the angles remayning, namely, the angles BAC, CAD and DAB which contayne the solide angle are lesse then sower right angles. Wherefore euery solide angle is comprehended vnder playne angles lesse then fower right angles: which was required to be proued.

If ye will more fully see this demonstration compare it with the figure which I put for the better sight of the demonstration of the proposition next going before. Onely here is not required the draught of the line AE.

Although this demonstration of Euclide be here put for solide angles contayned vnder three super∣ficiall angles, yet after the like maner may you proceede if the solide angle be contayned vnder superfi∣ciall angles how many so euer. As for example if it be contayned vnder fower superficiall angles, if ye follow the former construction, the base will be a quadrangled figure, whose fower angles are equall to fower right angles: but the 8. angles at the bases of the 4. triangles set vpon this quadrangled figure may by the 20. proposition of this booke be proued to be greater then those 4. angles of the quadrangled fi∣gure: As we sawe by the discourse of the former demonstration. Wherefore those 8. angles are greater then fower right angles: but the 12. angles of those fower triangles are equall to 8. right angles. Where∣fore the fower angles remayning at the toppe which make the solide angle are lesse then fower right angles. And obseruing this course ye may proceede infinitely.

¶ The 20. Theoreme. The 22. Proposition. If there be three superficiall plaine angles of which two how soeuer they be taken, be greater then the third, and if the right lines also which con∣tayne those angles be equall: then of the lines coupling those equall right lines together, it is possible to make a triangle.

SVppose that there be thre superficial angles ABC, DEF, and GHK, of which let two, which two soeuer be taken, be greater then the third, that is, let the an∣gles ABC, and DEF be greater then the angle GHK, and let the angles DEF and GHK be greater then the angle ABC: and moreouer let the angles GHK and ABC be greater then the angle DEF. And let the right lines AB, BC, DE, EF GH, and HK be equall the one to the other, and draw a right line from the point A to the

point C, and an other from the point D to the point F, and moreouer an other from the point G to the point K.* 1.126 Then I say that it is possible of three right lines equall to the lines AC, DF

[illustration]

and GK, to make a triangle, that is, that two of the right lynes AC, DF, and GK, which two soeuer be taken, are greater then the third. Now if the angles ABC, DEF,* 1.127 and GHK be equall the one to the other, it is manifest that these right lines AC, DF, and GK being also (by the 4. of the first) equall the one to the other, it is possible of three right lines equall to the lines AC, DF, and GK to make a triangle.* 1.128 But if they be not equall, let them be vnequall. And (by the 23. of the first) vnto the right line HK,* 1.129 and at the point in it H, make vnto the angle ABC an equall angle KHL. And by the ••. of the first) to one of the lines AB, BG, DE, EF, GH,* 1.130 or HK make the line HL equal, & draw these right lines KL and GL. And forasmuch as these two lines AB and DC, are equall to these two lines KH and HL, and the angle B is equall to the angle KHL, ther•••••••• (by the 4. of the first) the base AC is equall to the base KL. And forasmuch as the angles ABC, and GHK are greater then the angle DEF, but the angle GHL is equall to the angles ABC, & GHK•• therfore the angle GHL is greater then the angle DEF. And forasmuch as these two lines GH and HL are equall to these two lines DE and EF, and the angle GHL is grea••er then the angle DEF, therfore (by the 25. of the first) the base GD is greater thē the base DF. But the lines GK and KL are greater then the line GL. Wherfore then lines GK & KL are much greater then the line DF. But the line KL is equall to the line AC. Wher∣fore the lines AC and GK are greater then the line DF. In like sort also may we pro•••••• tha•• the lines AC and DF are greater then the line GK, and that the lines GK and DF are greater then the lyne AC. Wherfore it is possible to make a triangle of three lynes equall to the lines AC, DF, and GK: which was required to be demonstrated.

An other demonstration.

Suppose that the three superficiall angles be ABC, DEF, and GHK, of which angles,* 1.131 two howsoeuer they be taken, are greater then the third. And let them be contained vnder these equall right lines AB, BC, DE, EF, GH, HK, which equall right lines let th••se lines AC, DF, and GK

[illustration]

ioyne together. Then I say that it is possible of three right lines equal to the lines AC, DF, and GK to make a tri∣angle, whi••h againe is as much to say, as that two of those lines which two soeuer be ta∣ken, are greater then the third. Now againe if the angles B, E, H, be equall, the lines also AC

DF, and GK are equall, and so two of them shall be greater then the third. But if not, l••t the angles B, E, H, be vnequall, and let the angle B be greater then ••ither of the angles E and H. Therfore (by the 24. of the first) the right line AC is greater then either of the lines DF & GK. And it is manifest that the line AC with either of the lines DF or GK is greater then the third. I say also that the lines DF and GK are greater then the line AC. Vnto the right line AB,* 1.132 and to the point in it B, make (by the 22. of the first) vnto the angle GHK an equall angle AB

[illustration]

L, and vnto one of the lines AB, BC, DE, EF, GH or HK, make by the 2. of the first) an equall line BL. And draw a right line from the point A to y^e point L, and an other frō the point L to the point C.* 1.133 And forasmuch as these two lines AB & BL are equal to these two lines GH & HK the one to the other, and they containe equal an∣gles: therfore (by the 4. of the first) the base AL is equall to the base GK. And forasmuch as the angles E and H be greater then the angle ABC, of which the angle GHK is equal to the angle ABL, therfore the angle remayning, namely, the angle E is greater then the angle LBC. And forasmuch as these two lines LB and BC, are equall to these two lines DE and EF the one to the other, and the angle DEF is greater then the angle LBC, therfore (by the 25. of the first) the base DF is greater then the base LC: and it is proued that the line GK is e∣quall to the line AL. Wherfore the lines DF & GK are greater then the lin••s AL & LC. B••t the li••es AL and LC are greater then the line AC. Wherfore the lines DF & GK are much greater thē th•• line AC. Wherfore two of these right lines AC, DF & GK which two soeuer be taken are greater then th•• third. Wherfore it is possible of three right lines equall to the lines AC, DF and GK to make a triangle: which was required to be demonstrated.

¶ The 3. Probleme. The 23. Proposition. Of three plaine superficiall angles, two of which how soeuer they be taken, are greater then the third, to make a solide angle: Now it is necessary that those three superficiall angles be lesse then fower right angles.

SVppose that the superficiall angles geuen be ABC, DEF, GHK: of which let two how soeuer they be taken, be greater then the third: and moreouer, let those three angles be l••sse then fower right angles. It is required of three super∣ficiall angles equall to the angles ABC, DEF, and GHK, to make a solide

[illustration]

or bodily angle. Let the lines AB, BC, DE, EF, GH, and HK, be made equall:* 1.134 and drawe a right line from the point A to the point C, & an other from the point D to the point F, and an other from the point G to the point K. Now (by the 22. of the eleuenth) it is pos∣sible of three right lines equall to the right lines AC, DF, and GK, to make a triangle. Make such a triangle, and let the same be LMN,

[illustration]

so that let the line AC be equall to the line LM, and the line DF to the line MN, and the line GK to the line LN. And (by the 5. of the fourth) about the triangle LMN describe a circle LMN, and take (by the 1. of the third) the centre of the same circle,* 1.135 which centre shall either be within the tri∣angle LMN, or in one of the sides therof, or with∣out it. First let it be within the triangle, and let the same be the point X, & drawe these right lines LX, MX, and NX. Now I say, that the line AB is greater then the line LX.* 1.136 For if not, then the line AB is either equall to the line LX, or els it is lesse then it. First let it be equall. And forasmuch as the line AB is equall to the line LX, but the line AB is equall to the line BC, therefore the line LX is equall to the line BC: and vnto the line LX the line XM is (by the 15. definition of the first) equall: wherefore these two lines AB and BC, are equall to these two lines LX and XM the one to the other: and the base AC is supposed to be equall to the base LM. Wherefore (by the 8. of the first) the angle ABC is equall to the angle LXM. And by the same reason also the angle DEF is equall to the angle MXN, and moreouer the angle GHK to the angle NXL. Wherefore these three angles ABC, DEF, and GHK, are e∣quall to these three angles LXM, MXN, & NXL. But the three angles LXM, MXN, and NXL, are equall to fower right angles (as it is manifest to see by the 13. of the first, if a∣ny one of these lines MX, LX, or NX, be extended on the side that the point X is). Wher∣fore the three angles ABC, DEF, and GHK, are also equall to fower right angles. But they are supposed to be lesse then fower right angles: which is impossible. Wherefore the line AB is not equall to the line LX. I say also that the line AB is not lesse then the line LX. For if it be possible, let it be lesse: and (by the 2. of the first) vnto the line AB put an equall line XO: and to the line BC put an equall line XP, and draw a right line from the point O to the point P. And forasmuch as the line AB is equall to the line BC, therefore also the line XO is equall to the line XP. Wherefore the residue OL is equall to the residue MP. Wher∣fore (by the 2. of the sixt) the line LM is a parallel to the line OP•• and the triangle LMX is equiangle to the triangle OPX. Wherefore as the line XL is to the line LM, so is the line XO to the line OP. Wherefore alternately (by the 16. of the fift) as the line LX is to the line XO, so is the line LM to the line OP. But the line LX is greater then the line XO. Wher∣fore also the line LM is greater then the line OP. But the line LM is put to be equall to the line AC: wherefore also the line AC is greater then the line OP. Now forasmuch as these two right lines AB and BC are equall to these two right lines OX and XP, and the base AC is greater then the base OP, therefore (by the 25. of the first) the angle ABC is greater then the angle OXP. In like sort also may we proue, that the angle DEF is greater then the an∣gle MXN, and that the angle GHK is greater then the angle NXL. Wherefore the three angles ABC, DEF, and GHK, are greater then the three angles LXM, MXN, and NXL. But the angles ABC, DEF, and GHK, are supposed to be lesse then fower right angles: wherefore much more are the angles LXM, MXN, & NXL lesse then fower right angles. But they are also equall to fower right angles: which is impossible. Wherefore the line AB is not lesse then the line LX: and it is also proued that it is not equall vnto it. Wherfore

the line AB is greater then the line LX.

Now from the point X raise vp vnto the plaine superficies of the circle LMN a perpendi∣cular line XR (by the 12. of the eleuenth). And vnto that which the square of the line AB excedeth the square of the line XL * 1.137 let the

[illustration]

square of the line XR be equall. And draw a right line ••rom the point R to the point L, and an other from the point R to the point M, and an other from the point R to the point N. And forasmuch as the line RX is erected perpendicularly to the plaine su∣perficies of the circle LMN, therefore (by conuer∣sion of the second definition of the eleuenth) the line RX is erected perpendicularly to euery one of these lines LX, MX, and NX.* 1.138 And forasmuch as the line LX is equall to the line XM, & the line XR i•• common to them both, and is also erected per∣pendicularly to them both, therefore (by the 4. of the first) the base RL is equall to the base RM. And by the same reason also the line RN is equall to either of these lines RL and RM. Where∣fore these three lines RL, RM, and RN, are equall the one to the other. And forasmuch as vnto that which the square of the line AB excedeth the square of the line LX, the square of the line RX is supposed to be equall, therefore the square of the line AB is equall to the squares of the lines LX and RX. But vnto the squares of the lines LX and XR, the square of the line LR is (by the 47. of the first) equall, for the angle LXR is a right angle. Where∣fore the square of the line AB is equall to the square of the line RL. Wherefore also the line AB is equall to the line RL. But vnto the line AB is equall euery one of these lines BC, DE, EF, GH, and HK, and vnto the line RL is equall either of these lines RM and RN. Wher∣fore euery one of these lines AB, BC, DE, EF, GH, and HK, is equall to euery one of these lines RL, RM, and RN. And forasmuch as these two lines RL and RM are equall to these two lines AB and BC, and the base LM is supposed to be equall to the base AC, ther∣fore (by the 8. of the first) the angle LRM is equall to the angle ABC. And by the same reason also the MRN is equall to the angle DEF, and the angle LRN to the angle GHK. Wherefore of three superficiall angles LRM, MRN, and LRN, which are e∣quall to three superficiall angles geuē, namely, to the angles ABC, DEF, & GHK, is made a solide angle R, comprehended vnder the superficiall angles LRM, MRN, and LRN: which was required to be done.

* 1.139But now let the centre of the circle be in one of the sides of the triangle, let it be in th•• side MN, and let the centre be X. And draw a right line from the point L to the point X. I say againe, that the line AB is greater then the line LX. For if not, then AB is either e∣quall to

[illustration]

LX, or els it is lesse then it. First let it be equall. Now thē these two lines AB and

BC, that is, DE & EF are equall to these two lines

[illustration]

MX and XL, that is, to the line MN. But the line MN is supposed to be equall to the line DF. Where∣fore also the lines DE and EF are equall to the line DF: which (by the 20. of the first) is impossible. Wher∣fore the line AB is not equall to the line LX. In like sort also may we proue, that it is not lesse. Wherefore the line AB is greater then the line LX. And now if as before vnto the plaine superficies of the circle be e∣rected frō the point X a perpendicular line RX whose square let be equall vnto that which the square of the line AB excedeth the square of the line LX, and if the rest of the construction and demonstration be ob∣serued in this that was in the forme•• case, then shall the Probleme be finished.

But now let the centre of the circle be without the triangle LMN, and let it be in th•• point X. And draw these right lines LX, MX, and NX.* 1.140 I say that in this case also the line AB is greater then the line LX. For if not, then is it either equall or lesse. First let it be e∣quall.

[illustration]

Wherefore these two lines AB and BC are

[illustration]

equall to these two lines MX and XL the one to th•• other, and the base AC is equall to the base ML. Wherefore (by the 8. of the first) the angle ABC is equall to the angle MXL: and by the same reason also the angle GHK is equall to the angle LXN. Wherefore the whole angle MXN is equall to these two angles ABC and GHK. But the angles ABC and GHK are greater then the angle DEF. Wherefore the angle MXN is greater then the an∣gle DEF. And forasmuch as these two lines DE and EF are equall to these two lines MX and XN, and the base DF is equall to the base MN, there∣fore (by the 8. of the first) the angle MXN is equall to the angle DEF. And it is proued that it is also greater: which is impossible. Wherefore the line AB is not equall to the line LX. In like sort also may we proue, that it is not lesse. Wherfore the line AB is greater then the line LX. And againe if vnto the plaine superficies of the circle be erected perpendicularly from the point X a line XR, whose square is equall to that which the square of the line AB exceedeth the square of the line LX, and the rest of the construction be done in this that was in the for∣mer cases, then shall the Probleme be finished.

I say moreouer, that the line AB is not lesse then the line LX.* 1.141 For if it be possibl••, let it be lesse. And vnto the line AB, put (by the 2. of the first) the line XO equall: and vnto the line BC put the line XP equall: And draw a right line from the point O to the point P.

And forasmuch as the line AB is equall to the line BC, therefore the line XO is equall to the line XP. Wherefore the residue OL is equall to the residue MP. Wherefore (by the 2. of the sixt) the line LM, is a parallel to the line PO. And the triangle LXM is equiangle to the triangle PXO. Wherefore (by the 6. of the sixt) as

[illustration]

the line LX is to the line LM, so is the line XO to the line PO. Wherefore alternately (by the 16. of the fift) as the line LX is to the line XO, so is the line LM to the line OP. But the line LX is greater then the line XO. Wherefore also the line LM is greater then the line OP. But the line LM is equall to the line AC. Wherefore also the line AC is greater then the line OP. Now forasmuch as these two lines AB and BC are equall to these two lines OX & XP the one to the other, and the base AC is greater then the base OP, therefore (by the 25. of the first) the angle ABC is greater then the angle OXP. And in like sort if we put the line XR equall to either of these lines XO or XP, and draw a right line frō the point O to the point R, we may proue that the angle GHK is greater then the angle OXR. Vnto the right line LX, and vn∣to the point in it X, make (by the 23. of the first) vnto the angle ABC an equall angle LXS: and vnto the angle GHK make an equall angle LXT. And (by the second of the first) let either of these lines SX, and XT, be equall to the line OX. And drawe these lines OS, OT, and ST. And forasmuch as these two lines AB and BC are e∣quall to these two lines OX and XS, and the angle ABC is equall to the angle OXS, there∣fore (by the 4. of the first) the base AC, that is, LM, is equall to the base OS. And by the same reason also the line LN is equall to the line OT. And forasmuch as these two lines LM and LN are equall to these two lines SO and OT, and the angle MLN is greater then the angle SOT, therefore (by the 25. of the first) the base MN is greater then the base ST. But the line MN is equall to the line DF. Wherefore the line DF is greater then the line ST. Now forasmuch as these two lines DE and EF are equall to these two lines SX and XT, and the base DF is greater then the base ST, therefore (by the 25. of the first) the angle DEF is greater then the angle SXT. But the angle SXT is equall to the angles ABC and GHK. Wher••fore also the angle DEF is greater then the angles ABC and GHK. But it is also lesse: which is impossible.

* 1.142But now let vs declare how to finde out the line XR, whose square shall be equall ••o that which the square of the line AB exceedeth the square of the line LX. Take the two right lines AB and LX, and let AB be the greater, and vpon AB

[illustration]

describe a semicircle ACB, and from the poynt A apply into the semicircle a right line AC equall to the right line LX: and draw a right line from the poynt C to the poynt B. And forasmuch as in the semicircle ACB is an angle ACB, therefore (by the 31. of the third) the angle ACB is a right angle. Wherefore (by the 47. of the first) the square of the line AB is equall to the squares of the lines AC and CB: wherefore the square of the line AB is in power more then the square of the line AC by the square of the line CB: but the line AC is equall to the line LX, wherefore the square of the line AB is in power more then the square of the line LX by the square of the line CB. If therefore vnto the line CB we make the line XR equall, then is the square of the line AB greater then the square of the line LX by the square of the line XR: which was required to be doone.

In this figure may ye more fully s•••• the

[illustration]

construction and demonstration of the ••••rst case of the former 23. Propositiō, if ye erect perpendicularly the triangle •• RN, and vn∣to it bend the triangle LMR, that the an∣gles R of eche may ioyne together in the point R. And so fully vnderstanding this case, the other cases will not be hard to conceaue.

¶ The 21. Theoreme. The 24. Proposition. If a solide or body be contayned vnder* 1.143 sixe parallel playne superficieces, the opposite plaine superficieces of the same bo∣dy are equall and parallelogrammes.

SVppose that this solide body CDHG be contained vnder these 6. parallel plaine superficieces, namely, AC, GF, BG, CE, FB, and AE. Then I say that the opposite superficieces of the same body, are equal and parallelogrāmes, it is to wete, the two opposites AC and GF, and the two opposites BG

[illustration]

and CE, and the two opposites FB and AE to be equall, and al to be parallelogrammes.* 1.144 For forasmuch as two pa∣rallel plaine superficieces, that is, BG, and CE are deui∣ded by the plaine superficies AC, their common sections are (by the 16. of the eleuenth) parallels. Wherfore the line AB is a parallel to the line CD. Again•• forasmuch as two parallel plaine superficieces FB and AE are deui∣ded by the plaine superficies AC their common sections are by the same proposition, parallels. Wherfore the lyne AD is a parallel to the line BC. And it is also proued, that the line AB is a parallel to the line DC. Wherfore the superficies AC is a parallelogramme. In like sort also may we proue, that euery one of these superficices CE, GF, BG, FB, and AE are parallelo∣grammes. Draw a right line from the point A, to the point H, and an other from the point D to the point F.* 1.145 Aud forasmuch as the line AB is proued a parallel to the line CD, and the lyne BH to the line CF, therfore these two right lines AB and BH touching the one the other, are parallels to these two right lines DC and CF touching also the one the other, and not being in one and the selfe same plaine superficies. Wherfore (by the 10. of the eleuenth) they compre∣hend equall angles. Wherfore the angle ABH is equall to the angle DCF. And forasmuch as these two lines AB and BH are* 1.146 equall to these two lines DC and CF, and the angle ABH is proued equall to the angle DCF•• therfore (by the 4. of the first) the base AH is equall to the base DF, and the triangle ABH is e∣quall to the triangle DCF. And forasmuch as (by the 41. of the first) the parallelogramme BG is double to the triangle ABH, and the parallelo∣gramme CE is also double to the triangle DCF, therfore the parallelo∣gramme

BG is equall to the parallelogramme CE. In like sort

[illustration]

also may we proue that the parallelogramme AC is equall to the parallelogramme GF, and the parallelograme AE to the parallelogramme FB. If therfore a solide or body be contained vnder sixe parallel plaine superficieces, the opposite plaine super¦ficieces of the same body are equal & parallelogrammes which was required to be demonstrated.

I haue for the better helpe of young beginners, described here an other figure whose forme if it be described vpon pa∣sted paper with the letters placed in the same order that it is here, and then if ye cut finely these lines AG, DE and CF not through the paper, and folde it accordingly, compare it with the demonstration, and it will shake of all hardenes from it.

The 22. Theoreme. The 25. Proposition. If a Parallelipipedō be cutte of a playne superfi∣cies beyng a parallel to the two opposite playne superficieces of the same body: then, as the base is to the base, so is the one solide to the other solide.

SVppose that this solide ABCD being contained

[illustration]

vnder parallel plaine superficieces (and therfore called a parallelipipedō) be cut of the plaine super∣ficies VE, being a parallel to the two opposite superficieces of the same body, namely, to the superfici••ces AR & DH. Thē I say that as the base AEFW is to the base EHCF, so is the solide ABFV to the solide EGGD.* 1.147 Extēd the line AH on either side, & put vnto y^e line EH as many equal lines as you wil, namely, HM, & MN: & likewise vnto the line AE, put as many equal lines as you will, namely, AK & KL, & make perfect these parallelogrāmes LO, KW, HZ, & MS, and likewise make perfect these solides or bodies LP, KR, DM, and MT.* 1.148 And forasmuch as these right lines, LK, KA, and AE are equall the one to the other, therfore these parallelogrammes LO, KW, and AF, are also (by the first of the sixt) equall the one to the other: and so also (by the same) are these parallelogrammes KX, KB, and AG equall the one to the other. And likewise (by the 24. of the eleuenth) are the parallelogrammes LY, KP, and AR, e∣qual, for they are opposite the one to the other. And by the same reason also the parallelogrammes EC, HZ, and MS are equall the one to the other, and the parallelogrammes HG, HI, and IN are equal the one to the other. And more¦uer the parallelogrammes DH, MQ, and NT are (by the 24. of the eleuenth) equall the one to the other, for they

are opposite: wherefore three plaine superficies of the solides LP, RK, and AV are equall to three plaine superficies: but vnto eche of these three superficieces are equall the three opposite superficieces (by the 24. of the eleuēth) Wherfore these three solides or bodies LP, KR, and AV, are equal the one to the other, by the 8. definition of the eleuēth. And by the same reason also the three solides, ED, DM & MT are equal the one to the other. Wherfore how multiple x the base LF is, to the base AF, so multiplex is the solide LV to the solide AV. And by the same reason also how multiplex the base NF is to the base FH, so multiplex is the solide NV to the solide HV: so that if the base LF be equall to the base NF, the solide also LV shall be equall to the solide VN, and if the base LF exceede the base NF, the solide also LV shall exceede the solide VN, and if the base LF be lesse then the base NF, the solide also LV shall be lesse then the solide VN (by the 1. and 14. of the fift.) Now then there are foure magnitudes, namely, the two bases AF and FH, and the two solides or bodies AV and VH, of which are takē their equemultiplices, namely, the equemultiplices of the base AF, & of the solide AV, or the base LF, & the solide LV, & the equemultiplices of the base HF, & of the solide HV are the base NF, and the solide NV. And it is proued that if the base LF excede the base NF, the solide also LV excedeth the solide NV, & if it be equal, it is equal, and if it be lesse, it is lesse. Wherfore (by the 6. definitiō of the fift) as the base AF is to the base FH, so is the solide AV to the solide HV. If therfore a parallelipipedon be cut of a playne su∣perficies being a parallel to the two opposite playne superficieces of the same body, then, as the base is to the base, so is the one solide to the other solide: which was required to be proued.

I haue for the better

[illustration]

sight of the cōstructiō & demōstration of the for∣mer 25. propositiō, here set another figure, whose forme if ye describe vp∣pon pasted paper, and finely cut the three lines XI, BS, and TY, not through the paper but halfe way, and then fold it accordingly, and compare it with the construction and demō∣stration, you shall see that it will geue great light therunto.

Here Flussas addeth three Corollaries.

First Corollary.

If a Prisme be cutte of a playne superficies parallel to the opposite superficieces,* 1.149 the se••••ions of the Prisme shall be the one to the other in that proportion, that the sections of the base are the one to the other.

For the sections of the bases, which bases (by the 11. definitiō of this booke) are parallelogrammes, shall likewise be parallelogrammes, by the 16. of this booke (when as the superficies which cutteth is parallelel to the opposite super••icieces) and shal also be equiangle. Wherfore if vnto the bases (by produ∣cing the right lines) be added like and equall bases, as was before shewed in a parallelipipedon, of those sections shalbe made as many like Prismes as ye will. And so by the same reason, namely, by the commō excesse, equalitie, or want of the multiplices of the bases & of the sections by the 5. definitiō of the fifth may be proued, that euery section of the Prisme multiplyed by any multiplycation whatsoeuer, shall haue to any other section that proportion that the sections of the bases haue.

Second Corollary.

Solides whose two opposite superficie••es are poligonon figures like equall and parallels, the other su∣perficies, which of necessitie are parallelogrammes,* 1.150 being cutte of a playne superficies parallel to the two opposite superficies: the sections of the base are the one to the other, as the sections of the solide are th•• one to the other.

Which thing is manifest, for such solides are deuided into Prismes, which haue one cōmon side, namely, the axe or right line, which is drawne by the centers of the opposite bases. Wherefore how many pa••allelogrāmes or bases are set vpon the opposite poligonon figures, of so many Prismes shal the whole solide be cōposed. For those poligonon figures are deuided into so many like triangles by the 20. of the sixth, which describe Prismes. Which Prismes being cut by a superficies parallel to the opposite superficieces, the sectiōs of the bases shal (by the former Corollary) be proportional with the sectiōs of the Prismes. Wherefore by the •••• of the fifth, as the sections of the one are the one to the other, so are the sections of the whole the one to the other.

Of these solides there are infinite kindes, according to the varietie of the opposite and parallel po∣ligonon figures, which poligonon figures doo alter the angles of the parallelogrammes set vpon them according to the diuersitie o•• their situation. But this is no let at all to this corollary, for that which we haue proued will alwayes follow. When as the superficieces which are set vpō the opposite like & equal poligonon and parallel superficieces are alwayes parallelogrammes.

Third Corollary.

* 1.151T••e foresayd solides ••omposed of Prismes, being cutte by a playne superficies parallel to the opposit•• superficieces, are the one to the other as the heades or higher parts cutte are.

For it is proued that they are the one to the other as the bases are. Which bases forasmuch as they are par••llelogrammes, are the one to the other as the right lines are vpon which they are set by, the first of the sixth, which right lines are the heddes or higher parts of the Prismes.

The 4. Probleme. The 26. Proposition. Vpon a right lyne geuen, and at a point in it geuen, to make a solide angle equall to a solide angle geuen.

SVppose that the right line geuen be AB, and let the point in it geuen be A, and let the solide or bodily angle geuen be D being contained vnder these superfici∣all angles EDC, EDF and FDC. It is required vpon the right line AB, & at the point in it geuen A to make a solide angle equall to the solide angle D. Take in the line DF a point at all aduentures, and let the same be F.* 1.152 And (by the 11. of the eleuenth) frō the point F. Draw vnto the superficies wherin are the lines ED & DC a per∣pendicular line FG, and let it fall vpon the plaine superficies in the point G, & draw a right line from the point D to the point G. And (by the 23. of the first) vnto the line AB, and at the point A make vnto the angle EDC an equall angle BAL, and vnto the angle EDG put the angle BAK equall: and by the 2. of the first, put the line AK equall to the line DG, and (by the 12. of the eleuenth) from the point K raise vp vnto the plaine superficies BAL a perpendicular line KH, and put the line KH equall to the line GF, and draw a right lyne

from the point H to the point A. Now I say that the so••ide angle A contained vnder the su∣perficiall angles BAL, BAH, and HAL is equall to the solide angle D contained vnder the superficiall angles EDCEDF, and FD••. Le•• the the li••es AB and DE be put e∣quall, and draw these right lines. HB, BK, FE, and EG. And forasmuch as the line FG is erected perpendicularly to the ground superficies,* 1.153 therfore by the 2. definition of the eleuenth, the lin•• FG is also erected perpendicularly to all the right lines that are in the ground super∣ficies and touche it. Wherfore either of these angles FGD and FGE is a right angle, and by the same reason also either of the angles HKA and HKB is a right angle. And foras∣much as these two lines KA & AB are equall to these two lines GD & DE, the one to the other, and they containe equall

[illustration]

angles (by construction). Wher¦fore (by the 4. of the first) the base KB is equall to the base EG, and the line KH is equall to the line GF, and they cōprehēd right angles. Wherfore the line BH is equall to the line FE. Agayne, forasmuch as these two lines AK and KH are equal to these two lines DG and GF, and they containe right angles. Wherfore y^e base AH is (by the 4. of the first) equall to the base DF. And the line AB is equall to the line DE. Wherfore these two lines AB and AH are equall to these two lines FD and DE, and the base BH is equall to the base FE. Wherfore (by the 8. of the first) the angle BAH is equall to the angle EDF. And by the same reason also the angle HKL is equall to the angle FGC. Wherfore if we put these lines AL and DC equall, and draw these right lines KL, HL, GC, and FC: for∣asmuch as the whole angle BAL is equall to the whole angle EDC, of which the angle BAK is supposed to be equall to the angle EDG, therfore the angle remayning, namely, KAL is equall to the angle remayning GDC. And forasmuch as these two lines KA and AL are equall to these two lines GD and DC, and they containe equall angles, therefore by the 4. of the first, the base KL is equall to the base GC, and the line KH is equall to the line GF, wherfore thes•• ••wo lines LK and KH are equall to these two lines CG and GF, and they cō∣taine right angles. Wherfore the base HL is (by the 4. of the first) equal to the base FC. And forasmuch as these two lines HA and AL are equall to these two FD and DC, and the base HL is equall to the base FC, therfore (by the 8. of the first) the angle HAL is equall to the angle FDC, and by construction, the angle BAL is equall to the angle EDC. Wherefore vnto the right line geuen, and at the point in it geuen, namely, A, is made a solide angle equal to the solide angle geuen D: which was required to be done.

In thes•• two 〈…〉〈…〉

[illustration]

here put, you may in 〈◊〉〈◊〉 clearely concerne the ••••••••mer construction and d••••monstratiō, if ye erect pe••••pendicularly vnto the ground superficies the tri∣angles ALB and DCE, & eleuate the triangles ALH and DCF that the lynes

LA and CD of them may exactly agree with the line•• LA and CD of the ••riangles erec••ed•• For so or∣dering them, if ye compare the former construction and demonstration with them, they will be playn•• vnto you.

Although Euclides demōstration be touching solide angles which are contained only vnder three superficiall angles, that is, whose bases are triangles: yet by it may ye performe the Probleme touching solide angles contained vnder superficiall angles how many soeuer, that is, hauing to their bases any kinde of Poligonon figures. For euery Poligonon figure may by the 20. of the sixt, be resolued into like tringles. And so also shall the solide angle be deuided into so many solide angles as there be triangles in the base. Vnto euery one of which solide angles you may by this proposition describe 〈◊〉〈◊〉 equall solide angle vpon a line geuen, and at a point in it geuen. And so at the length the whole solide angle after this maner described shall be equall to the solide angle geuen.

The 5. Theoreme. The 27. Proposition. Vpon a right line geuen to describe a parallelipipedon like and in like sort si∣tuate to a parallelipipedon geuen.

••Vppose that the right line geuen be AB, and let the parallelipipedon geuen be CD. It is required vpon the right line geuen AB to describe a parallelipipe∣don like and in like sort situate to the parallelipipedon geuen, namely, to CD.* 1.154 Vnto the right line AB and at the poynt in it A describe (by the 26. of the ele∣••enth

[illustration]

(a solide angle equall to the solide angle C, and let it be contayned vnder these superfi∣ciall angles BAH, HAK, and KAB, so that let the angle BAH be equall to the angle ECF, and the angle BAK to the angle ECG, and moreouer the angle KAH to the an∣gle GCF. And as the line EC is to the line CG, so let the AB be to the line AK (by the 12. of the sixth) and as the line GC is to the line CF, so let the KA be 〈◊〉〈◊〉 the line AH. Wherefore of equalitie (by the 22. of the fift) as the line EC is to the line CF, so is the line BA to the line AH.* 1.155 Make perfect the parallelogramme BH, and also the solide AL. Now for that as the line EC is to the line CG, so is the line BA to the line AK, therefore the sides which contayne the equall angles, namely, the angles ECG and BAK are proportio∣nall: wherefore (by the first definition of the sixth) the parallelogramme GE is like to the pa∣rallelogramme KB. And by the same reason the parallelogramme KH is like to the paralle∣logramme GF, and moreouer the parallelogramme FE to the parallelogramme HB•• Wher∣fore there are three parallelogrammes of the solide CD like to the three parallelogrammes of the solide AL. But the three other sides in eche of these solides are equall and like to their opposite sides. Wherefore the whole solide CD is like to the whole solide AL. Wherfore vpon the right line geuen AB is described the solide AL contayned vnder parallel playne superfi∣cieces like and in like sort situate to the solide geuē CD contayned also vnder parall••l playne superficieces: which was required to be doone.

This demonstration is not hard to conceaue by the former figure as it is described in a playne, if ye

that imagination of parallelipipedons described in a playne which we before taught in the diffinition of a cube. Howbeit I haue here for the more ease of such as are not yet acquainted with solides, de••∣cribed

[illustration]

two figures, whose formes first describe vpon pasted paper with the like letters noted in them, and then finely cutte the three midle lines of eche figure, and so fold them accordingly, and that doone compare them with the construction and demonstration of this 27. proposition, and they will be very easy to conceaue.

The 23. Theoreme. The 28. Proposition. If a parallelipipedō be cutte by a plaine superficies drawne by the diagonall lines of those playne superficieces which are opposite: that solide is by this playne superficies cutte into two equall partes.

SVppose that the parallelipipedon

[illustration]

AB be cutte by the playne super∣ficies CDEF drawne by the dia∣gonal lines of y^e plaine superficieces which are opposite, namely, of the superficieces CF and DE. Then I say that the parallelipi∣pedon AB is cutte into two equall partes by the superficies CDEF. For forasmuch as (by the 34. of the first) the triangle CGF is equall to the triangle CBF,* 1.156 and the triangle ADE to the triangle DEH, and the parallelograme CA is equall to the parallelogramme BE, for they are opposite, and the parallelogramme GE is also equall to the parallelogramme CH, and the parallelogramme CE, is the common section by supposition: Wherfore the prisme contai∣ned vnder the two triangles CGF, and DAE, and vnder the three parallelogrammes GE, AC, and CE is (by the 8. definition of the eleuenth) equall to the prisme contayned vnder the two triangles CFB and DEH and vnder the three parallelogrammes CH, BE, and CE. For they are cōtayned vnder playne superficieces equall both in multitude and in magnitude. Wherefore the whole parallelipipedon AB is cutte into two equall partes by the playne superficies CDEF: which was required to be demonstrated.

A diagonall line is a right line which in angular figures is drawne from one angle and extended to his contrary angle as you see in the figure AB.

Describe for the better sight of this demonstration a figure vpon pasted

[illustration]

paper like vnto that which you described for the 24. proposition onely altering the letters: namely, in steade of the letter A put the letter F, and in steade of the letter H the letter C: moreouer in steade of the letter C put the letter H, and the letter E for the letter D, and the letter A for the letter E, and finally put the letter D for the letter F. And your figure thus ordered compare it with the de∣monstratiō, only imagining a superficies to passe through the body by the dia∣gonall lines CF and DE.

¶ The 24. Theoreme. The 29. Proposition. Parallelipipedons consisting vpon one and the selfe same base, and vn∣der one and the selfe same altitude, whose * 1.157 standing lines are in the selfe same right lines, are equall the one to the other.

SVppose that that these parallelipipedons CM and CN doo consist vpon one and the selfe same base, namely AB, and let them be vnder one and the selfe same altitude, whose standing lines, that is, the fower sides of eche solide which fall vpon the base, as the lines AF, CD, BH, LM of the solide CM, and the lines CE, BK, AG, and LN

[illustration]

of the solide CN, let be in the selfe same right lines or paral∣lel lines FN, DK.* 1.158 Then I say that the solide CM is equall to the solide CN. For forasmuch as either of these superficieces CBDH, CBEK is a paralle∣logramme, therefore (by the 34. of the first) the line CB is e∣quall to either of these lines DH and EK. Wherefore also the line DH is equall to the line EK. Take away EH which is common to them both, where∣fore the residue namely DE is equall to the residue HK. Wherfore also the triangle DCE is equall to the triangle HKB. And the parallelogramme DG is equall to the parallelo∣gramme HN. And by the same reason the triangle AGF is equall to the triangle MLN, and the parallelogramme CF is equall to the parallelogramme BM. But the parallelogrāme CG is equall to the parallelogramme BN, by the 24. of the tenth for they are opposite the one to the other. Wherefore the prisme contayned vnder the two triangles FAG and DCE and vnder the three parrallelogrāmes AD, DG, and CG is equall to the prisme cōtayned vnder the two triangles MLN and HBK, and vnder the three parallelogrāmes, that is, BM, NH, and BN. Put that solide common to them both, whose base is the parallelogrāme AB, and the opposite side vnto the base is the superficies GEHM. Wherefore the whole pa∣rallelipipedon CM is equall to the whole parallelipipedon CN: Wherfore parallelipipedons consisting vpon one and the selfe same base, and vnder one and the selfe same altitude, whose standing lines are in the selfe same right lines, are equall the one to the other: which was re∣qui••ed to be demonstrated.

Although this demonstration

[illustration]

be not hard to a good imaginati∣on to conceaue by the former fi∣gure (which yet by M. Dee•• refor∣ming is much better then the figure of this proposition commonly des∣cribed in other copyes both greake and lattin): yet for the ease of those which are young beginners in thys matter of solides, I haue here set an other figure whose forme if it be described vpon pasted paper, with the like letters to euery line as they be here put, and then if ye finely cut not thorough but as it were halfe way the three lines LA, NMGF, and KHED, & so folde it accordingly, & compare it with the demonstratiō, it will geue great light thereunto.

Stāding lines are called those fower right lines of euery parallelipipedon which ioyne together the angles of the vpper and nether bases of the same body.* 1.159 Which according to the diuersitie of the angles of the solides, may either be perpendicular vpon the base, or fall obliquely. And forasmuch as in thys proposition and in the next proposition following, the solides compared together are supposed to haue one and the selfe same base, it is manifest that the standing lines are in respect of the lower base in the selfe same parallel lines, namely, in the two sides of the lower base. But because there are put two solides vpon one and the selfe same base, and vnder one and the selfe same altitude, the two vpper bases of the solides may be diuersly placed. For forasmuch as they are equall and like (by the 24. of this booke) either they may be placed betwene the selfe same parallel lines: and thē the standing lines are in either solide sayd to be in the selfe same parallel lines, or right lines: namely, when the two sides of eche of the vpper bases are contayned in the selfe same parallel lines: but contrariwise if those two sides of the vp∣per bases be not contayned in the selfe same parallel or right lines, neither shal the standing lines which are ioyned to those sides be sayd to be in the selfe same parallel or right lines. And therefore the stan∣ding lines are sayd to be in the selfe same right lines, when the sides of the vpper bases, at the least two of the sides are contayned in the selfe same right lines: which thing is required in the supposition of this 29, proposition. But the standing lines are sayd not to be in the selfe same right lines, when none of the two sides of the vpper bases are contayned in the selfe same right lines, which thing the next propositi∣on following supposeth.

¶The 25. Theoreme. The 30. Proposition. Parallelipipedons consisting vpon one and the selfe same base, and vnder the selfe same altitude, whose standing lines are not in the selfe same right lines, are equall the one to the other.

SVppose that these Parallelipipedons CM and CN, do consist vpon one and the selfe same base, namely, AB, and vnder one and the selfe same altitude, whose standing lines, namely, the lines AF, CD, BH, and LM, of the Parallelipipedon CM, and the standing lines AG, CH, BK, and LN, of the Parallelipipedon CN, let not be in the selfe same right lines. Then I say, that the Parallelipipedon CM, is equall to the Parallelipipedon CN. Forasmuch as the vpper superficieces FH & ••K, of the former Parallelipipedons, are in one and the selfe same superficies (by reason they are supposed to be of one and the selfe same alti∣tude):* 1.160 Extend the lines FD and MH, till they concurre with the lines N•• and KE (suffi∣ciently both waies extended: for in diuers cases their concurse is diuers). Let ••D extended, meete with NG, and cut it in the point X: and with KE in the point P. Let likewise MH extended, meete with NG (sufficiently produced) in the point O, and with KE in the point R. And drawe these right lines AX, LO, CP, and BR. Now (by the 29. of the eleuenth) the solide CM, whose base is the parallelogramme ACBL, and the parallelogramme opposite vn∣to

it is FDHM, is equall to the solide CO, whose base is ACBL and the opposite side the parallelogramme XPRO,* 1.161 for they consiste vpon one and the selfe same base, namely, vpon the parallelogrāme ACBL, whose standing lines AF, AX, LM, LO, CD, CP, BH, and ••K,

[illustration]

* 1.162 are in the selfe same right lines FP and MR. But the solide CO, whose base is the paralle∣logramme ACBL, and the opposite superficies vnto it is XPRO, is equall to the solide CN, whose base is the parallelogramme ACBL, and the opposite superficies vnto it is the superfi∣cies G••KN, for they are vpon one and the selfe same base, namely, ACBL, and their stan∣ding lines AG, AX, CF, CP, LN, LO, BK, and BR, are in the selfe same right lines NX, and PK. Wherefore also the solide CM, is equall to the solide CN. Wherefore Parallelipi∣pedons consisting vpon one and the selfe same base, and vnder the selfe same altitude, whose standing lines are not in the selfe same right lines, are equall the one to the other: which was required to be proued.

This demonstra∣tion

[illustration]

is somwhat har∣der then the former to conceaue by the fi∣gure described in the plaine (and yet before M. Dee inuented that figure which is placed for it, it was much harder) by reason one solide is contained in an other. And there∣fore for the clerer light therof, describe vpō pasted paper this figure here put with the like letters and finely cut the lines AC, CB, EG, BL, EPK, ROHM, and folde it accordingly that euery line may exactly agree with his correspondent lyne (which obseruing the letters of euery line

ye shall easily do) and so cōpare your figure with the demonstration, and it will make it very plaine vnto you.

The 26. Theoreme. The 31. Proposit. Parallelipipedons consisting vp∣on equall bases, and being vnder one and the selfe same altitude, are equall the one to the other.

SVppose that vppon these equall

[illustration]

bases AB and CD do consiste these parallelipipedons AE and CF, being vnder one & the self same altitude.* 1.163 Thē I say, that the solide AE is equall to the solide CF. First let the stan∣ding lines, namely, HK, BE, AG, LM, OP, DF, CX, and RS, be erected perpendicular∣ly to the bases AB and CD, and let the an∣gle ALB not be equall to the angle CRD. Extend the line CR to the point T. And (by the 23. of the first) vpon the right line RT, and at the point in it R, describe vn∣to* 1.164

the angle ALB an equall angle TRV.

[illustration]

And (by the third of the first) put the line RT equall to the line LB, and the line RV equall to the line AL. And (by the 31. of the first) by the point V draw vnto the line RT a parallel line VW:* 1.165 and make perfecte the base RW, and the solide YV. Now forasmuch as these two lines TR and RV are equall to these two lines BL and LA, and they containe equall angles, ther∣fore the parallelogramme RW is equall and like to the parallelogramme AB. Againe, forasmuch as the line LB is equall to the line RT, and the line LM to the line RS (for the lines LM and RS are the altitudes of the Parallelipipedons AE and CF, which altitudes are supposed to be equall) and they containe right angles by supposition, there∣fore the parallelogramme RY is equall and like to the parallelogramme BM. And by the same reason also the parallelogramme LG is equall and like to the parallelogrāme SV. Wherefore three parallelogrammes of

the solide AE are equall and like to the three parallelogrammes of the solide YV. But these three parallelogrammes are equall and like to the three opposite sides. Wherefore the whole Parallelipipedon AE is equall and like to the whole Parallelipipedon YV. Extend (by the second petition) the lines DR and WV, vntill they concurre, and let them concurre in the point Q. And (by the 31. of the first) by the point T drawe vnto the line RQ a parallel line T 4, and extend duely the lines Ta and DO vntill they concurre, and let them concurre in the point ✚. And make perfecte the solides QY and RI. Now the solide QY, whose * 1.166 base is the parallelogramme RY, and the opposite side vnto the base the parallelogramme Qb, is equall to the solide YV, whose base is the paral∣lelogramme RY, and the opposite side vnto the base the parallelogramme VZ. For they con∣siste vpon one and the selfe fame base, namely, RY, and are vnder one and the selfe same al∣titude, and their standing lines, namely, RQ, RV, Ta, TW, SN, Sd, Yb, and YZ, are in the selfe same right lines, namely, QW, and NZ. But the solide YV is proued e∣quall to the solide AE. Wherefore also the solide YQ is equall to the solide AE. Now forasmuch as the parallelogramme RVWT is equall to the parallelogramme QT (by the 35. of the first) and the parallelogramme AB is equall to the parallelogramme RW: there∣fore the parallelogramme QT is equall to the parallelogramme AB, and the parallelo∣gramme CD is equall to the parallelogramme AB (by supposition). Wherefore the paralle∣logramme CD is equall to the parallelogramme QT. And there is a certaine other super∣ficies, namely, DT. Wherefore (by the 7. of the fift) as the base CD is to the base DT, so is the base QT to the base DT. And forasmuch as the whole Parallelipipedon CI is cut by the plaine superfi••••es RF, which is a parallel to either of the opposite plaine superficieces, ther∣fore as the base CD is to the base DT, so is the solide CF to the solide RI (by the 25. of the eleuenth). And by the same reason also, forasmuch as the whole Parallelipipedon QI is cut by the plaine superficies RY, which is a parallel to either of the opposite plaine superficieces, therefore as the base QT is to the base DT, so is the solide QY to the solide RI. But as the base CD is to the base DT, so is the base QT to the base TD. Wherefore (by the 11. of the fift) as the solide CF is to the solide RI, so is the solide QY to the solide RI. Wherefore either of these solides CF and QY, haue to the solide RI one and the same proportion. Wher∣fore the solide CF is equall to the solide QY. But it is proued that the solide QY is equall to the solide AE. Wherefore also the solide CF is equall to the solide AE.

But now suppose that the

[illustration]

stāding lines, namely, AG, HK,* 1.167 BE, LM, CX, OP, DF, and RS, be not erected perpendicu∣larly to the bases AB and CD. Then also I say, that the solide AE is equall to the solide CF. Draw (by the 11. of the eleuēth) vnto the ground plaine superfi∣cieces AB and CD from these pointes K, E, C, M, P, F, X, S, these perpendicular lines KN, ET, GV, MZ, PW, FY, XQ, and SI. And draw these right lines NT, NV, ZV, ZT, WY, WQ, IQ, and IY. Now (by that which hath before bene pro∣ued in this 31. Proposition) the

solide KZ is equall to the solide

[illustration]

PI, for they consist vpon equall * 1.168 bases, namely, KM, and PS, and are vnder one and the selfe same altitude, whose standing lines also are erected perpendicu∣larly to the bases. But the solide KZ is equall to the solide AE (by the 29. of the eleuenth): and the solide PI is (by the same) e∣quall to the solide CF, for they consist vppon one and the selfe same base, and are vnder one & the selfe same altitude, whose standing lines are vpon the selfe same right lines. Wherefore also the solide AE is equall to the solide CE. Wherefore Parallelipipedons consisting vpon equall bases and being vnder one and the selfe same altitude, are equall the one to the other: which was required to be demonstrated.

The demonstration of the first case of this proposition is very hard to conceaue by the figure des∣cribed for it in a playne. And yet before M. Dee inuented that figure which we haue there placed for it, it was much harder. For both in the Greke and Lattin Euclide, it is very ill made, and it is in a maner im∣possible to conceaue by it the construction and demonstration thereto appertayning. Wherefore I haue here described other figures, which first describe vpon pasted paper, or such like matter and then order them in maner following.

[illustration]

As touching the solide AE in the first case, I neede not to make any new description. For it is playne inough to conceaue as it is there drawne. Although you may for your more ease of imagination describe of pasted paper a parallelipipedo•• hauing his sides equall with the sides of the parallelipipedon AE before described, and hauing also the sixe parallelogrammes thereof (contayned vnder those sides)

equiangle with the sixe parallelogrammes of that figure, ech side and eche angle equall to his correspon¦dent side, and to his correspondent angle.

But concerning the other solide. When ye haue described these three figures vpon pasted pa∣per: Where note for the proportion of eche line, to make your figure of pasted paper equall with the fi∣gure before described vpon the playne, let your lines OP, CX, RS, DF, &c. namely, the rest of the stan∣ding lines, of these figures, be equall to the standing lines OP, CX, RS, DF, &c. of that figure. Likewise let the lines OC, CR, RD, DO, &c. namely, the sides which cōtayne the bases of these figures be equal to the lines OC, CR, RD, DO, &c. namely, to the sids which cōtayne the bases of that figure. Moreouer let the lines PX, X••, SF, FP, &c. namely, the rest of the lines which cōtaine the vpper superficieces of these figures, be equal to the lines PX, XS, SF, F••, &c. namely, to the rest of the lines which cōtaine the vpper superficieces of that figure (to haue described all those foresaid lines of these figures equal to all the lines of that figure, would haue required much more space then here can be spared: I haue made them equall onely to the halues of those lines, but by the example of these ye may, if ye will describe the like figures hauing their lines equall to the whole lines of the figure in the playne, eche line to his correspondent line). When I say ye haue as before is taught described these three figures, cut finely the lines XC, SR, FD of the first figure, and the lines SR, YT, and I ✚ of the second figure: likewise the lines ••R, NQ, ZVV, and YT, of the third figure, and fold these figures accordingly, which ye can not chuse but doo if ye marke well the letters of euery line.

The three former figures being after this sort described, set them vpon this figure here described vpon a playne, as vpō their bases, namely, the lines OC, CR, RD, DO: RT, T ✚, ✚ D, DR: VR, RT, TW, Wa, VQ, and QR of these three figures vpon the lines correspondent vnto them in this figure. And they so standing compare them with the

[illustration]

construction and demonstration of the first case, and they will geue great light vnto it. This also ye must note,* 1.169 that if ye make the lines of the fore sayd three figures equall to the lines of the figure of the plaine described before in the demōstrati∣on of the first case: then must ye make a new base for them like vnto this, which is easy to doo, if ye draw a pallelogramme equall and like to the parallelogramme OCT ✚, and thē cut of from the same a parallelogramme DRT ✚ equall and in like sort situate to the parallelogramme DRT ✚ of that figure: & vpon the line RT describe two parallelogrammes, the one equall like, and in like sort situate to the parallelogramme RTQa of that figure, and the other equall, like and in like sort situate to the parallelogramme RTVW of the same figure. The lines of this base which I haue here put are equall onely to the halues of the lines of that figure in the demonstration.

As touching the second case ye neede no new figures, for it is playne to see by the figures (now reformed by M. Dee) described for it in the playne, especially if ye remember the forme of the figure of the 29. proposition of this booke. Only that which there ye conceaue to be the base, imagine here in both the figures of this second case to be the vpper superficies opposite to the base, and that which was there supposed to be the vpper superficies conceaue here to be the base. Ye may describe them vpon pasted paper for your better sight, taking hede ye note the letters rightly according as the construction requireth.

Flussas demonstrateth this proposition an otherway taking onely the bases of the solides, and that after this maner.

Take equall bases (which

[illustration]

yet for the surer vnderstan∣ding let be vtterly vnlike) namely, AEBF and ADCH, and let one of the sides of eche concurre in one & the same right line AED, & the bases being vpon one and the selfe same playne let there be supposed to be set vpon thē parallelipipedons vnder one & the selfe same altitude. Then I say that the solide set vpō the base AB is equal to the solide set vpon the base AH. By the poynt E draw vnto the line AC a

parallel line EG, which if it

[illustration]

fall without the base AB, produce the right line HC to the poynt I. Now foras∣much as AB and AH are pa∣rallelogrmaes, therefore by the 24. of this booke, the triangles ACI and EGL shall be equaliter the one to the other: and by the 4. of the first, they shal be equian∣gle and equall: and by the first definition of the sixth, and fourth Proposition of the same, they shall be like. Wherfore Prismes erected vppon those triangles and vnder the same altitude that the solides AB and AH a••e, shall be equall and like, by the 8. definition of this booke. For they are contayned vnder like playne superficieces equall both in multitude and mag¦nitude. Adde the solide set vpon the base ACLE common to them both. Wherefore the solide set vp∣pon the base AEGC, is equall to the solide set vpon the base AELI. And forasmuch as the superficie∣ces AEBF, and ADHC are equall (by supposition): and the part taken away AG is equall to the part taken away AL: therefore the residue BI shall be equall to the residue GD. Wherefore as AG is to GD as AL is to BI (namely, equalls to equalls). But as AG is to GD, so i•• the solide set vpon AG to the solide set vpon GD by the 25. of this booke, for it is cut by a playne superficies set vpon the line GE, which superficies is parallel to the opposite superficieces. Wherefore as AL is to BI, so is the solide set vpon AL to the solide set vpon BI. Wherefore (by the 11. of the fifth) as the solide set vpon AG (or vpon AL which is equall vnto it) is to the solide set vpon GD, so is the same solide set vpon AG or AL to the solide set vpon BI. Wherefore (by the 2. part of the 9. of the fifth) the solides set vpon GD and BI shall be equall. Vnto which solides if ye adde equall solides, namely, the solide set vpon AG to the solide set vpon GD, and the solide set vpon AL to the solide set vpon BI: the whole solides set vpon the base AH and vpon the base AB ••hall be equall. Wherefore Parallelipedons consisting vpon equall bases and being vnder one and the selfe same altitude are equall the one to the other: which was required to be proued.

¶The 27. Theoreme. The 32. Proposition. Parallelipipedons being vnder one and the selfe same altitude, are in that proportion the one to the other that their bases are.

SVppose that these parallelipipedons AB and CD be vnder one & the selfe same altitude. Then I say that those parallelipipedons AB and CD are in that pro∣portion the one to the other, that their bases are, that is, that as the base AE is to the base CF, so is the parallelipipedon AB to the parallelipipedon CD.* 1.170 Vpon the line FG describe (by the 45.

[illustration]

of the first) the parallelo∣gramme FH equall to the parallelogramme AE and e∣quiangle with the parallelo∣gramme CF. And vpon the base FH describe a paralleli¦pipedō of the selfe same alti∣tude that the parallelipipedō CD is, & let y^e same be GK.* 1.171 Now (by the 31. of the ele∣uenth) the parallelipipedon AB is equall to the parallelipipedon GK, for they consist vpon e∣quall

bases, namely, AE and FH, and are vnder one and the selfe same altitude. And foras∣much as the parallelipipedon CK is cut by a plaine superficies DG, being parallel to either of the opposite plaine super••icieces, therfore (by the 25. of the eleuenth) as the base HF is to the base FC, so is the parallelipipedon GK to parallelipipedon CD: but the base HF is equal to the base AE, and the parallelipipedon GK is proued equall to the parallelipipedon AB. Wherfore as the base A••E is to the base CF, so is the parallelipedon AB, to the parallelipipe∣don CD. Wherfore parallelipipedons being vnder one and the selfe same altitude, are in that proportion the one to the other that their bases are: which was required to be demonstrated.

I neede not to put any other figure for the declaration of this demonstration, for it is easie to see by the figure there described. Howbeit ye may for the more full sight therof, describe solides of pasted paper according to the construction there set forth, which will not be hard for you to do, if ye remem∣ber the descriptions of such bodies before taught.

A Corollary added by Flussas.

Equall parallelipipedons cōtained vnder one and the selfe same altitude, haue also their bases equal. For if the bases should be vnequall, the parallelipipedons also should be vnequal by this 32 propositiō. And equall parallelipipedons hauing equall bases, haue also one and the selfe same altitude. For if they should haue a greater altitude, they should exceede the equall parallelipipedons which haue the selfe same altitude: But if they should haue a lesse they should want so much of those selfe same equal parallelipipedons.

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.* 1.172 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.* 1.173 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 * 1.174 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,* 1.175 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:* 1.176 as that of C, to the Cube of A: and the Cube of D to the Cube of B: in the second case. * 1.177 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, * 1.178 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,* 1.179 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.

¶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.* 1.180 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.* 1.181 Thē I say, that as the base EH is to the base NP, so is the line CM to the line AG.* 1.182 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* 1.183. 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.* 1.184 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 * 1.185. 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,* 1.186 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.

* 1.187But 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,* 1.188 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,* 1.189 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,* 1.190 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.* 1.191 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:* 1.192 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,* 1.193 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.* 1.194 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:* 1.195 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:* 1.196 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.* 1.197 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:* 1.198

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.

The 30. Theoreme. The 35. Proposition. If there be two superficiall angles equall, and from the pointes of those an∣gles

be eleuated on high right lines, comprehending together with those right lines which containe the superficiall angles, equall angles, eche to his corespōdent angle, and if in eche of the eleuated lines be takē a point at all auentures, and from those pointes be drawen perpendicular lines to the ground playne superficieces in which are the angles geuen at the begin∣ning, and from the pointes which are by those perpendicular lines made in the two playne superficieces be ioyned to those angles which were put at the beginning right lines: those right lines together with the lines eleua∣ted on high shall contayne equall angles.

SVppose that these two rectiline superficiall angles BAC, and EDF be equall the one to the other:* 1.199 and from the pointes A and D let there be eleuated vpward these right lines AG and DM, comprehendinge together with the lines put at the beginninge equall angles, ech to his correspondent angle, that is, the angle MDE to the angle GAB, and the angle MDF to the angle GAC, and take in the lines AG and DM pointes at all auētures and let the same be G and M. And (by the 11. of the eleuēth) from the pointes G and M draw vnto the ground playne superficieces wherein are the an∣gles BAC and E

[illustration]

DF perpendicular lines GL and MN and let them fall in the sayd playne super••icieces in the pointes N and L, and drawe a right line from the point L to the point A and an other from the pointe N to the pointe D. Then I say that the angle GAL is equall to the angle MDN. Frō the greater of the two lines AG and DM, (which let be AG) cut of by the 3. of the first the line AH equall vnto the line DM. And (by the 31. of the first) by the point H, drawe vnto the line GL a parallel line, and let the same be HK. Now the line GL is erected perpendicularly to the grounde playne superficies BAL: Wherfore also (by the 8. of the eleuenth) the line HK is erected perpēdicularly to the same grounde plaine superficies BAC. Drawe (by the 12. of the first) frō the pointes K and N vnto the right lines AB, AC, DF, & DE perpēdicular right lines, and let the same be KC, NF, KB, NE. And drawe these right lines HC, CB, MF, FE. Now forasmuch a•• (by the 47. of the first) the square of the line HA is equall to the squares of the lines HK and KA,* 1.200 but vnto the square of the line KA are equall the squares of the lines KC and CA: Wherefore the square of the line HA is equall to the squares of the lines HK, KC and CA. But by the same vnto the squares of the lines HK and KC is equall the square of the line HC: Wherefore the square of the line HA is equall to the squares of the lines HC and CA: wherfore the angle HCA is (by the 48. of the first) a right angle. And by the same reason also the angle MFD is a right angle. Wherefore the angle HCA is equall to the angle MFD. But the angle HAC is (by suppositiō) equal to the angle MDF. Wherfore there are two triangles MDF and HAC hauing two an∣gles of the one equall to twoo angles of the other, eche to his correspondent angle, and one side of the one equall to one side of the other, namely, that side which subtendeth one of the equall

angles, that is, the side HA is equall to the side DM by construction. Wherefore the sides remayning are (by the 26. of the first) equall to the sides remayning. Wherefore the side AC is equall to the side DF. In like sort may we proue that the side AB is equall to the side DE, if ye drawe a right line from the point H to the point B, and an other from the point M to the point E. For forasmuch as the square of the line AH is (by the 47. of the firste) equall to the squares of the lines AK and KH, and (by the same) vnto the square of the line AK are equall the squares of the lines AB and BK. Wherefore the squares of the lines AB, BK, and KH are equall to the square of the line AH. But vnto the squares of the lines BK and KH is equall the square of the line BH (by the 47. of the first) for the angle HKB is a right angle, for that the line HK is erected perpēdicularly to the ground playne superficies: Wherefore the square of the line AH is equall to the squares of the lines AB and BH. Wherefore (by the 48. of the first) the angle ABH is a right angle. And by the same rea∣son the angle DEM is a right angle. Now the angle BAH is equall to the angle EDM, for it is so supposed, and the line AH is equall to the line DM. Wherefore (by the 26. of the firste) the line AB is equall to the line DE. Now forasmuch as the line AC is equall to the line DF, and the line AB to the line DE, therefore these two lines AC and AB are equall to these two lines FD and DE. But the angle also CAB is by suppositi∣on equall to the angle FDE. Wherefore (by the 4. of the firste) the base BC is equall to the base EF, and the triangle to the triangle, and the rest of the angles to the reste of the angles. Wherefore the angle ACB is equall to the angle DFE. And the right angle ACK is equal to the right angle DFN. Wherfore the angle remayning, namely, BCK, is equall to the an∣gle remayning, namely, to EFN. And by the same reasō also the angle CBK is equal to the angle FEN. Wher¦fore

[illustration]

there are two triangles BCK, & EFN, hauing two angles of the one equal to two angles of the other, eche to his correspondent angle, and one side of the one equall to one side of the o∣ther, namely, that side that lieth betwene the equall angles, that is the side BC is equall to the side EF: Where∣fore (by the 26. of the first) the sides remaininge are equall to the sides remayning. Wherfore the side CK is equall to the side FN: but the side AC is equall to the side DF. Wherefore these two sides AC and CK are equall to these two sides DF and FN, and they contayne equall angles: Wherefore (by the 4. of the first) the base AK is equall to the base DN. And forasmuch as the line AH is equall to the line DM, therefore the square of the line AH is equall to the square of the line DM. But vnto the square of the line AH are equall the squares of the lines AK and KH (by the 47. of the first) for the angle AKH is a right angle. And to the square of the line DM are equall the squares of the lines DN and NM, for the angle DNM is a right angle. Wherefore the squares of the lines AK and KH are equall to the squares of the lines DN and NM: of which two, the square of the line AK is equall to the square of the line DN (for the line AK is proued equall to the line AN). Wherefore the residue, namely, the square of the line KH is equal to the residue, namely, to the square of the line NM. Wherefore the line HK is equall to the line MN. And forasmuch as these two lines HA and AK are equall to these two lines MD and DN, the one to the other, and the

base HK is equall to the base MN: therfore (by the 8. of the first) the angle HAK is equall to the angle MDN. If therefore there be two superficiall angles equall, and frō the pointes of those angles be eleuated on high right lines, comprehending together with those right lines which were put at the beginning, equall angles, ech to his corespondent angle, and if in ech of the erected lines be taken a point at all aduentures, and from those pointes be drawen perpen∣dicular lines to the plaine superficieces in which are the angles geuen at the beginning, and fr•••• the pointes which are by the perpendicular lines made in the two plaine superficieces be ioyned right lines to those angles which were put at the beginning, those right lines shall to∣gether with the lines eleuated on high make equall angles which was required to be proued.

Because the figures of the former demonstration are somewhat hard to conceaue as they are there drawen in a plaine, by reason of the lines that are imagined to be eleuated on high, I haue here set o∣ther figures, wherein you must e∣recte

[illustration]

perpendicularly to the ground superficieces the two triangles BHK, and EMN, and then ele∣uate the triangles DFM, & ACH, in such sort that the angles M and H of these triangles, may concurre with the angles M and H of the o∣ther erected triangles. And then imagining only a line to be drawen from the point G of the line AG to the point L in the ground superfi∣cies, compare it with the former construction & demonstration, and it will make it very easye to con∣ceaue.

¶ Corollary.

By this it is manifest, that if there be two rectiline superficiall angles e∣quall, and vpon those angles be eleuated on high equall right lines contay∣ning together with the right lines put at the b••ginning equall angles: per∣pendicular lines drawen from those eleuated lines to the ground plaine su∣perficieces wherein are the angles put at the beginning, are equall the one to the other. For it is manifest, that the perpendicular lines HK, & MN, which are dra∣wen from the endes of the equall eleuated lines AH, and DM, to the ground superficieces, are equall.

¶The 31. Theoreme. The 36. Proposition. If there be three right lines proportionall: a Parallelipipedon described of those three right lines, is equall to the Parallelipipedon described of the middle line, so that it consiste of equall sides, and also be equiangle to the foresayd Parallelipipedon.

SVppose that these three lines A, B, C, be proportionall, as A is to B, so let B be to C. Then I say, that the Parallelipipedon made of the lines A, B, C, is equall to the Pa∣rallelipipedon made of the line B, so that the solide made of the line B consist of e∣quall

sides, and be also equiangle to the solide made of the lines A, B, C. Describe (by the 23. of the eleuenth) a solide angle E contained vnder three superficiall angles,* 1.201 that is, DEG, GEF, and FED: and (by the 3. of the first) put vnto the line B euery one of these lines DE, GE, & EF, equall:

[illustration]

and make perfecte the so∣lide EK. And vnto the line A let the line LM be equall. And (by the 26. of the eleuēth) vnto the right line LM, and at the point in it L, describe vnto the solide angle E an equall so∣lide angle, cōtained vnder these plaine superficiall an∣gles NLX, XLM, and NLM, and vnto the line B put the line LX equall, & the line LN to the line C. Now for that as the line A is to the line B,* 1.202 so is the line B to the line C: but the line A is equall to the line LM, and the line B to eue∣ry one of these lines LX, EF, EG, and ED, and the line C to the line LN. Wherefore as LM is to EF, so is DE to LN: So then the sides about the e∣qual angles MLN, & D∣EF, are reciprokall: Wher∣fore (by the 14. of the sixt) the parallelogrāme MN is equall to the parallelogramme DF. And forasmuch as two plaine superficiall angles, name∣ly, DEF and NLM are equall the one to the other, and vpon them are erected vpward e∣quall right lines, LX and EG; comprehending with the right lines put at the beginning e∣quall angles the one to the other. Wherefore * 1.203 perpendicular lines drawen from the pointes X and G to the plaine super••icieces wherin are the angles NLM, and DEF, are (by the Co∣rollary of the former Proposition) equall the one to the other: and those perpendiculars are the altitudes of the Parallelipipedons LH and EK, by the 4. definition of the sixt. Wherfore the solides LH and EK, are vnder one and the selfe same altitude. But Parallelipipedons consisting vpon equall bases, and being vnder one and the selfe altitude, are (by the 31. of the eleuenth) equall the one to the other. Wherefore the solide LH is equall to the solide EK. But the solide LH is described of the lines A, B, C, and the solide EK is described of the line B. Wherefore the Parallelipipedon described of the lines A, B, C, is equall to the Pa∣rallelipipedon made of the line B, which consisteth of equall sides, and is also equiangle to the foresaid Parallelipipedon. If therfore there be three right lines proportionall, a Parallelipipe∣dō described of those three lines is equall to the Parallelipipedō described of the middle line, so that is consist of equall sides, and also be equiangle to the foresaid Parallelipipedon: which was required to be proued.

The construction and demonstration of this Proposition, and of the next Proposition following, may easily be conceaued and vnderstanded by the figures described in the plaine belonging to them. But ye may for the more full sight of them, describe such bodies of pasted paper, hauing their sides pro∣portionall, as is required in the Propositions.

¶New inuentions (coincident) added by Master Iohn Dee.

A Corollary. 1.

Hereby it is euident, that if three right lines be proportionall: the Cube produced of the middle line, is equall to the rectangle Parallelipipedon made of those three lines.

For a Cube is a Parallelipipedon of equall sides: and also rectangled: as we suppose the Parallelipi∣pedon, made of the three lines to be likewise rectangled.

¶A Probleme. 1.

A Cube being geuen, to finde three right lines proportionall, in any proportion geuen betwene two right lines: of which three lines, the rectangle Parallelipipedon produced, shall be equall to the Cube geuen.

Suppose AC to be the

[illustration]

Cube geuen: whose roote, suppose to be AB. Let the proportion geuen, be that which in betwene the two right lines D and E, I say now, three right lines are to be found, proportionall, in the proportion of D to E, of which, the rectangle Pa∣rallelipipedon produced, shall be equall to AC. By the 12. of the sixt let a line be found, which to AB haue that proportion that D hath to E. Let that line be F: and by the same 12. of the sixth, let an other line be found, to which, AB, hath that proportion that D hath to E: and let that line found be H. Let a rectangle Paral∣lelipipedon mathematically be produced of the three right lines F, AB, and H, which suppose to be K: I say now, that F, AB, and H, are three right lines found pro∣portionall in the proporti∣on of D to E, of which, the rectangle Parallelipipedon K, produced, is equall to AC the Cube geuen. First it is euident that F, AB, and H, are proportionall in the proportion of D to E. For, by construction, as D is to E, so is F to AB: and by construction likewise, as D is to E, so is AB to H. Wherefore F is to AB, and AB is to H, as D is to E. So then it is manifest, F, AB, and H, to be proportionall in the proportion of D to E, and AB to be the middle line. By my former Corollary, therefore, the rectangle parallelipipedon made of F, AB, and H, is equall to the Cnbe made of AB. But AC, is (by supposi••ion) the Cube made of AB•• and of the three lines F, AB, and H, the rectangle parallelipipedon produced, is K, by construction: Wher∣fore, K, is equall to AC: A Cube being geuen, therefore, three right lines are found, proportionall in

••ny prop••••••ion geuē be••wene two right lines, of which three right lines the rectangle parallelipipedon p••oduced, is equall to the Cube geuen. Which ought to be done.

A Probleme. 2.

A re••••angle Parallelipipedon being geuen, to finde three right lines proportionall: of the which, the rectangle Parallelipipedon produced, is equall to the rectangle Parallelipipedon geuen.

* 1.204Listen to this new deuise, you couragious Mathematiciens: consider, how nere this crepeth to the famous Probleme of doubling the Cube. What hope may (in maner) any young beginner cōceiue, by one meanes or other, at one time or other, to execute this Probleme? * 1.205 Seing to a Cube may in∣finitely in••inite Parallelipipedons be found equall: all which Parallelipipedons shall be produced of three right lines proportionall, by the fo••me•• P••obleme: but to any ••ectangle Parallelipipedon geuen, some one Cube is equall •• as is ••asie to demonst••at••: We can not doubt, but vnto our rectangle Paral∣lelipipedon geuen, many other ••ectangle Parallelipipedons are also equall, hauing their three lines of p••oduction, proportionall.* 1.206 In th•• former Probleme•• in••ini••••ly in••inite Parallelipipedons may be found of three proportionall lines p••oduced, equall to the Cube g••uen: it is to wete, the three lines to be of all proportions, that a man can deuise betwene two right lines: and here any one will serue: where also i•• infinite varietie: though all of one quantitie: as be••ore in the Cube. I leaue as now, with thys marke here set vp to shoote at. Hit it who can.

¶ The 32. Theoreme. The 37. Proposition. If there be fower right lines proportionall: the Parallelipipedons descri∣bed of those lines, being like and in like sort described, shall be proportio∣nall. And i•• the Parallelipipedons described of them, being like and in like sort described, be proportionall: those right lines also shall be proportionall.

SVppose that these fower right lines AB, CD, EF, and GH, be proportionall, as AB is to CD, so let EF be to GH, and vpon the lines AB, CD, EF, and GH, describe these Parallelipipedons KA, LC, ME, and NG, being like and in like sort desc••••bed. Then I say, that as the solide KA is ••o the solide LC, so is the solide ME to the solide

[illustration]

NG.* 1.207 For forasmuch as the Pa∣rallelipipedon KA is like to the Parallelipipedon LC: therfore (by the 33. of the eleuenth) the solide KA is to the solide LC in treble proportion of that which the side AB is to the side CD••: and by the same reason the Parallelipipedon ME is to the Parallelipipedon NG in treble proportion of that which the side EF is to the side GH. Wherfore (by the 11. of the fift) as the Parallelipipedon KA is to the Parallelipipedon LC, so is the Parallelipipedon ME t•• the Parallelipipedon NG.

But now suppose, that as the Parallelipipedon KA is to

the Parallelipipedon LC, so is the Parallelipipedon ME to the Parallelipipedon NG. Then I say,* 1.208 that as the right line AB is to the right line CD, so is the right li•••• EF to the right line GH. For againe forasmuch as the solide KA is to the solide LC in treble proportion of that which the side AB is to the side CD, and the solide ME also is to the solide NG in treble proportion of that which the line EF is to the line GH, and as the solide KA is to the solide LC, so is the solide ME to the solide NG. Wherefore also as the line AB is to the line CD, so is the line EF to the line GH. If therefore there be fower right lines proportionall: the Parallelipipedons described of those lines, being like & in like sort described, shall be pro∣portionall. And if the Parallelipipedons described of them, and being like and in like sort de∣scribed, be proportionall: those right lines also shall be proportionall•• which was required to be proued.

¶ The 33. Theoreme. The 38. Proposition. If a plaine superficies be erected perpendicularly to a plaine superficies, and from a point taken in one of the plaine superficieces be drawen to the other plaine superficies, a perpendicular line: that perpendicular line shall fall vpon the common section of those plaine superficieces.

SVppose that the plaine superficies CD be erected perpēdicularly to the plaine superfi∣cies AB, and let their common section be the line DA: and in the superficies CD take a point at all aduentures, and let the same be E. Then I say, that a perpendicu∣lar line drawen from the point E to the

[illustration]

plaine superficies AB, shall fall vpon the right line DA. For if not, then let it fall without the line DA, as the line EF doth, and let it fall vpon the plaine su∣perficies AB in the point F.* 1.209 And (by the 12. of the first) from the point F draw vnto the line DA, being in the superficies AB a perpendicular line F∣G, which line also is erected perpendicu∣larly to the plaine superficies CD: by the third diffinitiō: by reason we presuppose CD and AB to be perpendicularly erec∣ted ech to other. Draw a right line from the point E to the point G. And foras∣much as the line FG is erected perpendi∣cularly to the plaine superficies CD, and the line EG toucheth it being in the superficies CD. Wherefore the angle FGE is (by the 2. definition of the eleuenth) a right angle. But the line EF is also erected perpēdicularly to the superficies AB: wherefore the angle EFG is a right angle. Now therefore two angles of the triangle EFG, are equall to two right angles: which (by the 17. of the first) is impossible. Wherfore a perpendicular line drawen frō the point E to the s••••erficies AB, falleth not with∣out the line DA. Wherefore it falleth vpon the line DA: which was required to be proued.

¶ Note.

Campane maketh this as a Corollary, following vpon the 13: and very well, with small ayde of other Propositions, he proueth it•• whose demonstratiō there, Flussas hath in this place, and none other: though he sayth that Campane of such a Propositiō, as of Euclides, maketh no mention.

In this figure ye may more fully see the former Proposi∣tion

[illustration]

and demonstration if ye erecte perpendicularly vnto the ground plaine superficies AB the superficies CD, and imagine a line to be extended from the point E to the point F, instede whereof ye may extend if ye will a thred.

¶ The 34. Theoreme. The 39. Proposition. If the opposite sides of a Parallelipipedon be deuided into two equall partes, and by their common sections be extended plaine superficieces: the commō section of those plaine superficieces, and the diameter of the Paral∣lelipipedon shall deuide the one the other into two equall partes.

SVppose that AF be a Parallelipipedon, and let the opposite sides thereof CF and AH be deuided into two equall partes in the pointes K, L, M, N, and likewise let the opposite sides AD and GF be deuided into two equall partes in the point••s X, P, O, R, and by those sections extend these two plaine superficieces KN & XR, and let the common section of those plaine superficieces be the line VS, and let the diagonall line of the solide AB be the line

[illustration]

DG. Then I say, that the lines VS and DG do deuide the one the o∣ther into two equall partes,* 1.210 that is, that the line VT is equall to the line TS, and the line DT to the line TG. Drawe these right lines DV, VE, BS, and SG.* 1.211 Now for∣asmuch as the line DX is a parallel to the line OE, therfore (by the 29. of the first) the angles DXV and VOE being alternate angles, are equall the one to the other. And forasmuch as the line DX is equall to the line OE, and the line XV to the line VO, and they comprehend equall angles: Wherefore the base DV is equall to the base VE (by the 4. of the first) and the triangle DXV is equall to the triangle VOE, and the rest of the angles to the rest of the angles. Wherefore the angle XVD is equall to the angle OVE. Wherefore DVE is one right line, and by the same reason BSG is also one right line, and the line BS is equall to the line SG. And forasmuch as the line CA is equall to the line DB, and is vn∣to it a parallel, but the line CA is equall to the line GE, and is vnto it also a parallel: wher∣fore (by the firs•• common sentence) the line DB is equall to the line GE, & is also a parallel vnto it: but the right lines DE and BG do ioyne these parallel lines together: Wherefore (by the 33. of the first) the line DE is a parallel vnto the line BG. And in either of these lines

are taken pointes at all aduentures, namely, D, V, G, S, and a right line is drawen from the point D to the point G, and an other from the point V to the point S. Wherefore (by the 7. of the eleuenth) the lines DG and VS are in one and the selfe same plaine superficies. And for∣asmuch as the line DE is a parallel to the line BG, therefore (by the 24. of the first) the an∣gle EDT is equall to the angle BGT, for they are alternate angles, and likewise the angle DTV is equall to the angle GTS. Now then there are two triangles, that is, DTV and GTS, hauing two angles of the one equall to two angles of the other, and one side of the one equall to one side of the other, namely, the side which subtendeth the equall angles, that is, the side DV to the side GS, for they are the halfes of the lines DE and BG: Wherefore the sides remayning are equall to the sides remayning. Wherfore the line DT is equall to the line TG, and the line VT to the line TS. If therefore the opposite sides of a Parallelipipedon be de∣uided into two equall partes, and by their sections be extended plaine superficieces, the com∣mon section of those plaine superficieces, and the diameter of the Parallelipipedon, do deuide the one the other into two equall partes: which was required to be demonstrated.

A Corollary added by Flussas.

Euery playne superficies extended by the center of a parallelipipedon, diuideth that solide into two equall partes: and so doth not any other playne superficies not extended by the center.

For euery playne extended by the center, cutteth the diameter of the parallelipipedon in the cen∣ter into two equall partes. For it is proued, that playne superficieces which cutte the solide into two equall partes, do cut the dimetient into two equall partes in the center. Wherefore all the lines drawen by the center in that playne superficies shall make angles with the dimetient. And forasmuch as the di∣ameter falleth vpon the parallel right lines of the solide, which describe the opposite sides of the sayde solide, or vpon the parallel playne superficieces of the solide, which make angels at the endes of the diameter: the triangles contayned vnder the diameter, and the right line extended in that playne by the center, and the right line, which being drawen in the opposite superficieces of the solide, ioyneth together the endes of the foresayde right lines, namely, the ende of the diameter, and the ende of the line drawen by the center in the superficies extended by the center, shall alwayes be equall, and equi∣angle, by the 26. of the first. For the opposite right lines drawen by the opposite playne superficieces of the solide do make equall angles with the diameter, forasmuch as they are parallel lines, by the 16. of this booke. But the angles at the cēter are equall, by the 15. of the first, for they are head angles: & one side is equall to one side, namely, halfe the dimetient. Wherefore the triangles contayned vnder e∣uery right line drawen by the center of the parallelipipedon in the superficies, which is extended also by the sayd center, and the diameter thereof, whose endes are the angles of the solide, are equall, equi∣later, & equiangle (by the 26. of the first). Wherfore it followeth that the playne superficies which cut∣teth the parallelipipedon, doth make the partes of the bases on the opposite side, equall, and equiangle, and therefore like, and equall both in multitude, and in magnitude: wherefore the two solide sections of that solide, shalbe equall and like, by the 8. diffinition of this booke. And now that no other playne superficies, besides that which is extended by the center, deuideth the parallelipipedon into two equall partes, it is manifest: if vnto the playne superficies which is not extended by the center, we extend by the center a parallel playne superficies (by the Corollary of the 15. of this booke). For forasmuch as that superficies which is extended by the center, doth deuide the parallelipipedō into two equall par••••: it is manifest, that the other playne superficies (which is parallel to the superficies which deuideth the solide into two equall partes) is in one of the equall partes of the solide: wherefore seing that the whole is euer greater then his partes, it must nedes be that one of these sections is lesse then the halfe of the solide, and therefore the other is greater.

For the better vnderstanding of this former proposition, & also of this Corollary added by Flussas, it shalbe very nedefull for you to describe of pasted paper or such like matter a parallelipipedō or a Cube, and to deuide all the parallelogrāmes therof into two equall parts, by drawing by the c••̄ters of the sayd parallelogrammes (which centers are the poynts made by the cutting of diagonall lines drawen frō th•• opposite angles of the sayd parallelogrāmes) lines parallels to the sides of the parallelogrāmes: as in the former figure described in a plaine ye may see, are the sixe parallelogrāmes DE, EH, HA, AD, DH, and CG, whom these parallel lines drawen by the cēters of the sayd parallelogrāmes, namely, XO, OR, PR, and PX, do deuide into two equall parts: by which fower lines ye must imagine a playne superfi∣cies to be extended, also these parallel lynes KL, LN, NM, and M••, by which fower lines likewise y•• must imagine a playne superficies to be extended ye: may if ye will put within your body made thus of pasted paper, two superficieces made also of the sayd paper, hauing to their limites lines equall to the foresayde parallel lines: which superficieces must also be deuided into two equall partes by parallel

lines drawen by their centers, and must cut the one the other by these parallel lines. And for the dia∣meter of this body, extēd a thred from one angle in the base of the solide to his opposite angle, which shall passe by the center of the parallelipipedon, as doth the line DG in the figure before described in the playne. And draw in the base and the opposite superficies vnto it, Diagonall lines, from the angles from which is extended the diameter of the solide: as in the former description are the lines BG and DE. And when you haue thus described this body, compare it with the former demonstration, and it will make it very playne vnto you, so your letters agree with the letters of the figure described in the booke. And this description will playnely set forth vnto you the corollary following that proposition. For where as to the vnderstanding of the demonstration of the proposition the superficieces put within the body were extended by parallel lynes drawen by the cēters of the bases of the parallelipipe∣don: to the vnderstanding of the sayd Corollary, ye may extende a superficies by any other lines dra∣wen in the sayd bases, so that yet it passe through the middest of the thred, which is supposed to be the center of the parallelipipedon.

¶ 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.* 1.212 Make perfecte

[illustration]

the Parallelipipedons AX & GO. And forasmuch as the parallelo∣gramme AC is double to the tri∣angle GHK,* 1.213 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 * 1.214 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,* 1.215 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.

The end of the eleuenth booke of Euclides Elementes.

Notes

• * 1.1

  The argument of the eleuenth booke.
• * 1.2

  A point the be∣ginning of all quantitie con∣tinuall.
• * 1.3

  The methode vsed by Euclide in the ten ••••••∣mer boo••es.
• * 1.4

  ••irst bo••••e.
• * 1.5

  Second ••••o••e.
• * 1.6

  Third boo••e.
• * 1.7

  ••ourth b••o••e.
• * 1.8

  ••iueth bo••••e.
• * 1.9

  Sixth boo••e.
• * 1.10

  Seuenth book••
• * 1.11

  ••ight boo••••.
• * 1.12

  Ninth booke.
• * 1.13

  Tenth boo••e.
• * 1.14

  What is entrea¦••ea of in the fi••e boo••es fol∣l••wi••••.
• * 1.15

  〈◊〉〈◊〉 ••••••ular bo∣dies•• the ••••all ende 〈…〉〈…〉 o•• I u••••••••es ••eo∣me••••••all ••••e∣mentes.
• * 1.16

  Co••••a••is••n •••• the 〈◊〉〈◊〉 ••••o••e and 〈◊〉〈◊〉 booke 〈◊〉〈◊〉.
• * 1.17

  First dif••ini∣tion
• * 1.18

  A solide the most perfec∣test quantitie.
• * 1.19

  No science of thinges infi∣nite.
• * 1.20

  Second diffi∣nition.
• * 1.21

  Third diffini∣tion.
• * 1.22

  Two dif••ini∣tions included in this di••••i∣nition.
• * 1.23

  Declaratiō of the first part.
• * 1.24

  Declaration of the second part.
• * 1.25

  Fourth diffi∣nition.
• * 1.26

  Fifth diffini∣tion.
• * 1.27

  Sixth diffini∣tion.
• * 1.28

  Seuenth def••i∣nition.
• * 1.29

  Eighth di••fi∣nition.
• * 1.30

  Ninth di••••i∣••ition.
• * 1.31

  Tenth diffini∣tion.
• * 1.32

  Eleuenth diffi¦nition.
• * 1.33

  This diffiniti∣on agreeth to the generall name of a prisme, which also may be applied to Pa∣rallel••pedons (a•• may ap∣peare in the 12. booke fol∣lowing & 10. propo••••tiō, his demonstratiō) and to some o∣ther bodie likewise.
• * 1.34

  An other dif∣finition of a prisme, which is a speciall diffinition of a prisme: as it is commonly called and vsed.
• * 1.35

  This bodie called Figura Serratilis.
• * 1.36

  Psellus.
• * 1.37

  Twelueth dif∣finition.
• * 1.38

  What is to be ta••••n heede of in the diffini∣tion of a sphere geuen by Io∣hannes de Sa∣cro Busco.
• * 1.39

  Theodosiu•• di••••inition of a sphere.
• * 1.40

  The circumfe∣rence of a sphere.
• * 1.41

  Galens diffi∣nition 〈◊〉〈◊〉 a sph••r••.
• * 1.42

  The dig••itie of a s••here.
• * 1.43

  A sphere cal∣led a Globe.
• * 1.44

  Thirtenth dif∣finition.
• * 1.45

  Theodosius diffinition of the axe of a sphere.
• * 1.46

  Fourtenth diffinition.
• * 1.47

  Theodosius diffinition of the center of a sphere.
• * 1.48

  Flussas diffi∣nition of the center of a sphere.
• * 1.49

  Fiuetenth dif∣finition
• * 1.50

  Difference be∣twene the di∣ameter & axe of a sphere.
• * 1.51

  Seuententh diffinition.
• * 1.52

  First kinde of Cones.
• * 1.53

  A Cone called of Campane a ro••••de Pira∣mis.
• * 1.54

  Seuententh diffinition.
• * 1.55

  A conicall su∣perficies.
• * 1.56

  Eightenth diffinition.
• * 1.57

  Ninetenth diffinition.
• * 1.58

  A cillindricall superficies.
• * 1.59

  Corollary.
• * 1.60

  A round•• Co∣lumne or sphere.
• * 1.61

  A Corollary added by Cam¦pane.
• * 1.62

  Twenty diffi∣nition.
• * 1.63

  Twenty one diffinitio••.
• * 1.64

  Twenty two diffinition.
• * 1.65

  A Tetrahe∣dron one of the fiue regu∣lar bodyes.
• * 1.66

  Di••••erence betwene a Tetrahedron and a Pira∣mis.
• * 1.67

  Psellus calleth a Tetrahedron a Piramis.
• * 1.68

  Twenty three definition.
• * 1.69

  Twēty ••o••er definition.
• * 1.70

  Twenty fiue diffinition.
• * 1.71

  Fiue regular bodies.
• * 1.72

  The dignity of these bodies.
• * 1.73

  A Tetrahe∣dron ascribed vnto the fire.
• * 1.74

  An octohe∣dron ascribed vnto the ayre.
• * 1.75

  An Ikosahedron assigned vnto the water.
• * 1.76

  A cube assigned vnto the earth.
• * 1.77

  A dodecahe∣dron assigned to heauen.
• * 1.78

  Diffinition of a parallelipipe∣don.
• * 1.79

  A sided Co∣lumne.
• * 1.80

  A Tetrahe∣dron.
• * 1.81

  A c••be.
• * 1.82

  A•• octo••e∣d••o••.
• * 1.83

  A D••d••••••••e∣dron.
• * 1.84

  An Icosa••e∣dron.
• * 1.85

  A triangled Pyramis.
• * 1.86

  The forme of a quadrang∣led Pyramis.
• * 1.87

  The forme of a fiue angled Pyramis.
• * 1.88

  The forme of a prisme.
• * 1.89

  The forme of a parallelip••∣pedon.
• * 1.90

  Demonstrati∣on leading to an impossibi∣litie.
• * 1.91

  An other de∣monstration after Flussas.
• * 1.92

  Construction.
• * 1.93

  Demonstra∣tion leading to an impossibi∣litie.
• * 1.94

  Demonstra∣tion leading to an impossibi∣litie.
• * 1.95

  Construction.
• * 1.96

  Demonstra∣tion.
• * 1.97

  Demonstrati∣on leading to an impossibi∣litie.
• * 1.98

  Construction.
• * 1.99

  An Assumpt as M. Dee pr••ueth it. Demonstra∣tion.
• * 1.100

  Demonstra∣tion leading to an impossibi∣litie.
• * 1.101

  This proposi∣tion is as it were the con∣uerse of the sixth Construction.
• * 1.102

  Demonstra∣tion.
• * 1.103

  Construction.
• * 1.104

  Demonstra∣tion.
• * 1.105

  Construction.
• * 1.106

  Demonstra∣tion.
• * 1.107

  Construction.

  Two cases in this proposi∣tion.

  The first case.
• * 1.108

  Iohn Dee.
• * 1.109

  This requireth the imagination of a plaine superficies passing by the pointe A, and the straight line BC. And so helpe your selfe in the lyke cases either Mathematically imagi∣ning, or Me∣chanically practising.
• * 1.110

  Second cas••.

  Demonst••a∣t••on.
• * 1.111

  Construction.
• * 1.112

  Demonstra∣tion.
• * 1.113

  Demonstra∣tion leading to an impossibi∣litie.
• * 1.114

  Note this maner of i∣magination Mathemati∣call.
• * 1.115

  Demonstra∣tion leading to an impossibi∣litie.
• * 1.116

  Construction.
• * 1.117

  Demonstra∣tion.
• * 1.118

  Demonstra∣tion leading to an absurditie••
• * 1.119

  In t••is ••rono∣••••••o•••••• must vnd••rs••and the prop••rtio∣••••ll ••artes or s••••••ions to be th••se which are c••ntai••ed 〈◊〉〈◊〉 the parallel super∣••••cies.

  Construction.

  Dem••nstra∣tion.
• * 1.120

  Construction.
• * 1.121

  Demonstra∣tion.
• * 1.122

  Demonstrati∣on leading to an impossibi∣litie.
• * 1.123

  Demonstra∣tion.
• * 1.124

  Construction
• * 1.125

  Demonstra∣••ion.
• * 1.126

  Two cases in this proposi∣tion.
• * 1.127

  The first case.
• * 1.128

  Second case.
• * 1.129

  Constructi••••••
• * 1.130

  Demonstra∣tion.
• * 1.131

  An other de∣monstration.
• * 1.132

  Construction.
• * 1.133

  Demonstra∣••••••n.
• * 1.134

  Construction.
• * 1.135

  Three cases in this propo∣sition.

  The first case.
• * 1.136

  A necessary thing to be proued before he p••oceede any ••arther in the con∣struction of the Problem••.
• * 1.137

  Which how to finde out is taught at the end of this de¦monstration, and also was taught in the as••umpt put before the 14. proposition of the tēth boke.
• * 1.138

  Demonstra∣tion of the first case.
• * 1.139

  Second case.
• * 1.140

  Third case.
• * 1.141

  An other demō∣stratiō to proue that th•• line AB is not lesse thē the line LX.
• * 1.142

  This was before ta••ght in the tenth booke in the assumpt put before the 14. proposition.
• * 1.143

  M. Dee (to auoide cauillation) addeth to Euclides proposition this worde sixe: whome I haue follow∣ed accordingly, and not Zamberts, in this.

  This kinde of body mencioned in the proposition is called a Paral∣lelipipedō according to the di••fini∣tion before geuen thereof.
• * 1.144

  Demonstra∣tion that the opposite sides are parallelo∣grammes.
• * 1.145

  Demonstratiō that the oppo∣site superficies are equall.
• * 1.146

  AB is equall to DC, because the superficies AC is proued a pa∣rallelogrāme, and by the same rea∣son, is BH equall to CF, because the superficies FB is proued a pa∣rallelogramme: therefore the 34. of the first is our proofe.
• * 1.147

  Construction.
• * 1.148

  Demonstra∣tion.
• * 1.149

  F•••••••• Corolla∣ry.
• * 1.150

  Second Corol∣ry.

  These solides which he speaketh of in this Corollary are of some called sid••d column••s.
• * 1.151

  Third Corol∣lary.
• * 1.152

  Co••struc••ion.
• * 1.153

  Demonstra∣••ion.
• * 1.154

  Constr••ction.
• * 1.155

  Demonstr••∣tion.
• * 1.156

  Demonstra∣tion.
• * 1.157

  Looke at the end of the de∣monstratio•• what is vn∣derstanded by stāding lines.
• * 1.158

  Iohn Dee his figure.

  By this figure it app••ar••th why ••uch Pris¦mes were cal∣led ••••edges: of 〈◊〉〈◊〉 v••ry shape of a wedge, as is the solide DEFGA∣C. &c.
• * 1.159

  Stāding lines.
• * 1.160

  Construction.
• * 1.161

  Demonstra∣tion.
• * 1.162

  I. Dees figure.
• * 1.163

  Two cases in this proposi∣tion.

  Th•• first case.

  Construction.
• * 1.164

  We are behol∣ding to M. Dee for inuenting this figure, with other, which till his reforming were as much mishappen as this was, and so both in the Greeke and La∣tine copies re∣maine.
• * 1.165

  Demonstra∣tion.
• * 1.166

  Note now, how the base respec∣tiuely is taken•• for so may alte∣ratiō of respects alter the name of the bowndes eyther of solides or playnes.
• * 1.167

  Second case.
• * 1.168

  There you perceaue how the base is di∣uersly consi∣dered & cho∣sen: as before we aduertised you.
• * 1.169

  Construction.
• * 1.170

  Construction.
• * 1.171

  Demonstra∣tion.
• * 1.172

  Construction.
• * 1.173

  Demonstra∣tion.
• * 1.174

  *
• * 1.175

  Note this famous L••m∣ma.
• * 1.176

  The doubling o•• ••he Cube.
• * 1.177

  Note.
• * 1.178

  〈…〉〈…〉 Lemma.
• * 1.179

  Note what i•• yet lacking re∣quisite to the doubling of the Cube.
• * 1.180

  Two cases in the first part of this propo∣tion.
• * 1.181

  First case, which also may be two wayes.
• * 1.182

  First way.
• * 1.183

  This follow∣eth also of the Corollary ad∣ded of F••us••as after the 3••. proposition of this booke.
• * 1.184

  Second way.
• * 1.185

  This also followeth of the former Corollary.
• * 1.186

  The conuerse of both the partes of the first case.
• * 1.187

  Second case.
• * 1.188

  Construction.
• * 1.189

  Demonstra∣tion.
• * 1.190

  The conuerse of the second case.
• * 1.191

  The generall conclusion.
• * 1.192

  Construction.
• * 1.193

  Demonstratiō of the first part.
• * 1.194

  Demonstratiō of the second part.
• * 1.195

  To finde two midd••e propor¦tionals be∣twene two numbers geuē.
• * 1.196

  Note the prac¦tise of app••o∣ching to pre∣cie••es in Cu∣bik rootes.
• * 1.197

  This is the way to apply any square geuen, to a line also geuē, sufficienty ex∣tended.
• * 1.198

  A probleme worth the searching for.
• * 1.199

  Construction.
• * 1.200

  Demonstra∣tion.
• * 1.201

  Construction.
• * 1.202

  Demonstra∣tion.
• * 1.203

  It is euidēt that those per∣pendiculars are all one with the sta••∣ding lines of the solides, if their solide an¦gles be made of super••i∣c••••il right an∣gles onely.
• * 1.204

  D••u••ling of the 〈◊〉〈◊〉. &c.
• * 1.205

  D••m••••••••rati∣o•• ••f p••s••ibilitie in the ••••oblem.
• * 1.206

  An other argu∣me••t to com••o••t the studious.
• * 1.207

  Demonstrati∣on of the first part.
• * 1.208

  Demonstra∣tion of the second part, which is the conuerse of the first part.
• * 1.209

  Demonstra∣tion leading to an impossibi∣litie.
• * 1.210

  Construction.
• * 1.211

  Demonstra∣tion.
• * 1.212

  Construction.
• * 1.213

  Demonstra∣tion.
• * 1.214

  Which of some are cal∣led sided Co∣lumnes.
• * 1.215

  Sided Co∣lumnes.