greater perfection then is a line but here in the definitioÌ„ of a solide or body Euclide attributeth vnto it all the three dimensioÌ„s leÌ„gth breadth and thicknes Wherfore a solide is the most perfectest quantitie which wanteth no dimension at all passing a lyne by two dimensions and passing a superâ—icies by one This definition of a solide is without any designation of â—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 dimensionâ— 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 â— cube For in that respect and consideration onely that it is long broade and thicke it beareth the name of a solide or body â—nd hath the nature and properties therof There is added to the endâ— of the definition of a solide that the terme and limite of a solide â—s a superficies Of thinges infinitie there iâ— no Arte or Scienâ—e All quantities therfore in this Arte entreated of are imagined to be finite 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 thaâ— the endes limites or borders of a solideâ— are superficieces As the side of any â—quare piece of timber or of a table or die or any other likâ— are the termes and limites of them 2 A right line is then erected perpendicularly to a pl 〈…〉 erficies wheÌ„ the right line maketh right angles with all the lines 〈…〉 it and are drawen vpon the ground plaine superficies Suppose that vpon the grounde playne superficies CDEF from the pointe B be erected a right line namely â—A so that let the point A be a loâ—e in the ayre Drawe also from the poynte â— in the playne superficies CDBF as many right lines as ye list as the lines BC BD â—â— 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 â—ngles Aâ—â— Aâ—D Aâ—E ABFâ— Aâ—G Aâ—K ABH ABL and so of others be right angles then by this definition the line AB iâ— a line â—â—â—cted vpon the superficies CDEF it is also called commonly a perpendicular line or a plumb line vnto or vpon a superficies 3 A plaine superficies is then vpright or 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 superficieces making 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 perpendicularly vpon a plaine superficies 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 superâ—icieces ABCD and CDEF Of which let the superficies CDEF be a ground plaine superficies and let the superficies ABCD be erected vnto it and let the line CD be a common terme or intersection to them both that is let it be the end or bound of either of them 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 superâ—icies ABCD which let be GL and HK which falling 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 â—G drawen in the superficies erected and vnder the GM or GN drawen in the ground superficies CDEF lying 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 For the second part of this definition which is of the inclination of a right line vnto a plaine superficies 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 the point of the falling of the perpendicular line vpon the same superficies 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 perpendicular 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◠〈…〉 â—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 4 Inclination of a plaine superficies to a plaine superficies is an acute angle contayned vnder the right lines which being drawen in either of the plaine superficieces to one the self same point of the coÌ„mon section make with the section right angles Suppose that there be two superficieces ABCD EFGH 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 coÌ„mon terme or section of which two superficieces let be the line CD From some one point aâ— 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 superficies ABCD draw the line ML Now
â—ignifieth Last of all a Dodecahedron for that it is made of Pâ—ntagoâ— whose angles are more ample and large then the angles of the other bodies and by that â—eaâ—â—â— draw more â—â— rounâ—nes 〈◊〉 to the forme and nature of a sphere they assigned to a sphere namely 〈…〉 Who so will 〈…〉 in his Tineus shall â—ead of these figures and of their mutuall proportionâ—â—â—raunge maâ—terâ— which hâ—re are not to be entreated of this which is sayd shall be sufficient for the 〈◊〉 of them and for thâ— 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 quadrangle figures of which those which are opposite are parallels Because these fiue regular bodies here defined are not by these figures here set so fully and liuely expressed that the studious beholder can throughly 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 aptly ye may do in fine pasted paper such as pastwiues make womeÌ„s pastes of theÌ„ with a knife cut euery line finely not through but halfe way only if theÌ„ 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 hadâ—â—tore of that pasted paper by you for so shal yoâ— vpon it 〈…〉 the formes of other bodies as Prismes and Parallelipopedons 〈…〉 set forth in these fiue bookes following and see the very 〈◊〉 of thâ—se bodies there meÌ„cioned 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 Describe thiâ— figurâ— which consistâ—th of twâ—luâ—â—quilâ—â—â—â— and â—quianglâ— Pâ—ntâ—â—â—â—â— vpoâ— the foresaid mattâ—r and finely cut as before was â—â—ught tâ—â—â—lâ—uâ—n lines containâ—d within thâ— figurâ— and bow and folde the Penâ—â—gonâ— accordingly And they will so close toâ—ethâ—â— thaâ— thâ—y will â—â—kâ— thâ— very forme of a Dodecahedron If ye describe this figure which consisteth of twentie equilater and equiangle triangles vpon the foresaid matter and finely cut as before was shewed the ninâ—tâ—ne lines which are contayned within the figure and then bowe and folde them accordingly they will in such sort close together that therâ— will be made a perfecte forme of an Icosahedron 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 â— Pentagonâ— 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 â—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. 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 acquainted with will at the first sight be somwhat sâ—raunge vnto theÌ„ I haue for their more â—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 Propositions to them belonging And if they diligently weigh the maner obserued in this eleuenth booke touching 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 Proposition 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 namely the part AB be in a ground playne superficies and the other part therof namely BC be eleuated vpwarde And produce directly vpoÌ„ the ground playne superficies the right line AB beyond the point B vnto the point D. Wherfore vnto two right lines geuen ABC and ABD the line AB is a common section or part which is impossible For a right line can not touche a right line in 〈◊〉 pointes then one vâ—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 demonstratioÌ„ if ye eleuate the superficies wheriâ— the line BC. An other demonstration after Flâ—sâ—â—s If it be possible let there be a right line ABG whose part AB let be in the ground playne superficies AED 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 draâ— vnto the point G a right lyne CG Wherefore in the triangle 〈…〉 the outward angâ—â— ABâ— is eqâ—â—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 â— right line Whâ—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 playâ—er declaration of Euclides demonstration iâ— will not be hard for you to coâ—â—â—â—e this figure which â—lussâ—s putteth for his demonstâ—â—tion â— wherein
is no difference but onely the draught of the lyne GC ¶ The 2. Theoreme The 2. Proposition If two right line cut the ouâ— to the other they are â—â—â—ne and the selfe same playne superficies euery triangle is in one the selfe same superficieâ— SVppose that these two right lines AB and CD doo 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 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 For if part of the triangle EBC namely the triangle FCH or the triangle GBK be in the ground superficies and the residuâ— 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 superficies 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 also 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 demonstration of the former proposition where 〈◊〉 may eleâ—â—â—â— 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 cutte the one the other and let their common sectiâ—â—â—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 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 â—aue the selfe saâ—â— eâ—deâ— and therefore doo include a superficies which by the last common sentence is impossibleâ— 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 This figure here set sheweth most playnely not onely this third proposition but also the demonstration 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 perpendicularly erected from the playne superficies by the sayd two lines passing SVppose that there be two right lines AB 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 is also erected perpendicular to the plaine superficies which passeth by the lines A B 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 And forasmuch as these two right lines AE ED are equall to these two lines CE and EB and they comprehend 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 forasmuch 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 equall 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 Wherefore 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
perpendicularly 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 superficies by the 2. definition of the eleuenth Wherefore the right line FE is erected perpendicularly to the ground playne superficies And the ground plaine superficies is that which passeth by these right lines AB and CD Wherefore the right line FE is erected perpendicularly 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 proposition and demonstration if ye erect perpendicularly vnto the ground playne superficies ACBD theâ— triangle AFB and eleuate the triangles AFD CFB in such sort that the line AF of the triangle AFB may ioyne make one line with the line AF of the triangle 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 perpendicular 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 erected 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 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 superficies namely in the superficies wherein the lines AB and BC are And forasmuch as the right line AB is erected perpendicularly to either of these lines BD and BE therefore the line AB is also by the 4. of the eleuenth erected perpendicularly 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 perpendicularly 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 superficies 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 superficies 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 demonstrated This figure here set more playnely declareth the demonstration of the former proposition if ye erect perpendicularly vnto the ground superficies the sâ—â—perficies wherein is drawne the line 〈◊〉 and so compare it with the sayd deâ—â—â—â—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. 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 perpendicular line DE. And by the 2. of the first put the line DE equall to the line AB And draw these right lines BE AE and AD. And forasmuch as the line AB is erected perpendicularly to the ground superficies therfore by the 2. definition of the eleuenth the line AB maketh right angles with all the lines which are drawne vpon the ground playne superâ—icies and touch it But either of tâ—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 angleâ— 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 perpeÌ„dicularly to the line DA and it is also erected perpeÌ„dicularly 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
parallel lines are For if not then if it be possible let it be in an higher superficies as the line EGF is and draw the superficies wherin the line EGF is extend it and it shall make a common section 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 thâ—se pointes is in the selfe same plaine superficies with the parallel right lines which was required to be demonstrated By this figure it is easie to see 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 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 tâ—e pointes B and D and by the first peticion draw a righâ— 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 perpendiculer line DE and by the 2. of the first put the line 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 â—oint A to the point D. And forasmuch as the line AB is erested perpendicularly to the ground superficieces therfore by the 2. definition of the eleuenth 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 erected perpendicâ—larly to the line BD. And forasmuch as the line AB is equall to the line DE and the line â—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 linâ— AD therfore thesâ— two lines AB and BE are equall to these two lines AD DE the onâ— 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 Aâ—E is a right angle wherfore thâ—â—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 thâ— line DB. Wherfore the line ED is erectâ—d perpendicularly to the plaine superficies wherin thâ— lâ—nâ—s BD and BA are by the 4. of â—his booke Wherfore by the 2. definition of the eleuenth the line ED is erected perpendicularly to all the right lines that touche it and are in the sâ—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 fâ—rasmuch as the lines AB and BD arâ— 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 perpendicularly 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 Wherefore 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 perpendicularly 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 demonstration 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 wherof 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. 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 perpendiculer line GK And forasmuch as the line EF is erected perpendiculerly to either of the lines GH and GK therfore by the 4. of the eleuenth
the line EF is erected perpendicularly to the superficies wherein the lines GH and GK are but the line EF is a parallel line to the line AB Wherfore by tâ—e 8. of the eleuenth the line AB is erected perpendicularly to the plaine superficies wherin are the lines GH and GK And by the same reason also 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 perpendicularly 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 proposition 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 coÌ„taine 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 TheÌ„ I say that the angle ABC is equall to the angle DEF For let the lines BA BC ED EF be put equall the one to the other and draw these right lines AD CF BE AC and DF. 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 equall and parallell to the line BE and by the same reason also the line CF is equall parallell to the line BE. Wherfore 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 superficies 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 righâ— lines containe equall angles which was required to be demonstrated This figure here set more plainly declareth the former Proposition 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 SVppose 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 superficies a perpendicular line Drawe in the ground superficies a right line at aduentures and let the same be BC. And by the 12. of the first from the point A draw vnto the line BC a perpendicular line AD. 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 superficies 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 â—irst vnto the line BC a parallell line FH And extend the line FH from the point F to the point G. 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 perpeÌ„dicularly to the superficies wherin the lines ED and AD are and to the line BC the line GH is a parallell But iâ— 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 plainâ— 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 â— of this booke Wherefore the line GH is erected perpendicularly 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 erected perpendicularly to either of these lines HG and DE. But if a right line be erected perpendicularly 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 eleuenth 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 superficies Wherefore the line AF is erected perpendicularly to the ground superficies Wherfore from a point geuen on high namely froÌ„ 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 the cases of this former demonstratioÌ„ 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 superficies 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 demonstration 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 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. 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 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 lynâ—â—required to be doone In this second figure ye may consider playnely the demonstration of the former proposition if ye erect perpendicularly the superficies wherein are drawne the lines AD and CB. ¶ The 11. Theoreme The 13. Prâ—position From one and the selfe poynt and to one and the selfe same playne superficies 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 righâ— lines AB and AC on one and the selfe same side And extende the superficies wherein are the lines AB and AC and it shall make at length a common section in the ground superâ—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 eleuenth the lines AB AC and DAE are in one and the selfe same playne superficies And forasmuch 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 impossiâ—le Wherefore from one and the selfe same poynt and to one and the selfe same playne superficies can not be â—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 superficies wherein are drawne the lines â—A and CA to the ground superâ—icies wherein is drawn the line DAE and so compare it with the the demonstratioÌ„ of the former proposition it will be cleare vnto you M. d ee his annotation Euclides wordes in this 13. proposition admit two cases one if th◠〈…〉 in the playne superficies as coÌ„monly the demonstrations suppose the other if the poynt assigned be any where without 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 â—ore him to confesse his two perpendiculars to be also parallels But by supposition agreed one they concurre at one and the same poynâ— which by the definition of parallels iâ— impossible Therefore our aduersary must recant aâ—d yelde to out proposition ¶ The 12. Theoreme The 14. Proposition To whatsoeuer plaine superficieces one and the selfe same right line is erected perpendicularly those superficieces are parallels the one to the other SVppose that a right line AB be erected perpeÌ„dicularly to either of these plaine superficieces CD and EF. Then I say that these superficieces CD and EF are parallels the one to the other 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 aduentures 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 erected 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 Wherfore 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 Wherfore to what soeuer plaine superficieces one and the selfe same right line iâ— erected perpendicularly those superficies are parallells the one to the other which was required to be proued In this figure may ye plainly see the former demonstration if ye erecte the three superficieces GD GE and KLM perpeÌ„diculary to the ground plaine superâ—icies but yet in such sort that the two superficiâ—◠〈…〉 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 perpendicularly 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 superâ—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 parallells to these two right lines DE EF touching also the one the other and not being in the selfe same plaine superâ—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 is they are equedistant and parallels From the point B draw by the 11. of the eleuenth a perpendicular line to the superâ—icies wherein are the lines DE and EF and let that perpendicular line be BG And by the point G in the plaine superficies 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 parallell line GK And forasmuch as the line BG is erected perpendicularly to the superficies wherein are the lines DE and EF thereâ—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 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 therefore by the 29. of the first the angles GBA and BGH are equall to two right angles But the angle BGH is by constructioÌ„ 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 BCâ— 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 superficieces 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 parâ—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 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 superfiâ—ieâ— Suppose as in the former description that the superficies geueÌ„ be ABC let the point geueÌ„ 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 superficies EFGH â—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 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 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 superficieces AB and CD being produced do mete together but by supâ—ositioÌ„ they mete not together for they are supposed 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 superficieces 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◠〈…〉 demonstration if ye erect perpendicuâ—â—◠〈…〉 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 sâ—lfe same playne 〈…〉 also be parallels the one to the other or they shall make one and the selfe same plaine sup 〈…〉 In this figure here set ye may more plainely see the formâ—r demonstration if ye eleuate to the ground superâ—icieces ACDI the three superâ—icieces AB DG GI and â—o compare it with the demonstration The 15. Theoreme The 17. Proposition Iâ— two right lines be cut by playne superficieces being parallels the partes oâ— the lines deuided shall be proportionall Sâ—ppose that these two right lines AB and CD be deuided by these plaine superfiâ—iâ—ces being parallels namely GH KL MN in the points A E B C F D. TheÌ„ 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 superâ—icies KL concurre in the point X. And draw a right line from the point E to the point X and
an other from the point X to the point F. And forasmuch as these two parallel superficieces KL and MN are cut by the superâ—icies EBDX therâ—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 â—orasmuch as the two parallel superficies GH and KL be cut by the superâ—icies AXFC their common sections AC and XF are by the 16. of the eleuenth parallels And â—orasmuch as to one of the sides of the triangle ABDâ— 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 proportionally 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 â—e deuided by plaine superâ—icieces being parallels the parts of the lines deuided shal be proportionall which was required to be demonstrated In this figure it is more easy to see the former demonstration if ye erect perpendicularly vnto the ground superficies ACBD the thre superficieces GH KL and MN or if ye so â—râ—ct them that thâ—y be equedistant one to the other ¶ The 16. Theoreme The 18. Proposition If a right line be erected perpeÌ„dicularly 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 TheÌ„ 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 coÌ„mon section of the plaine superficies and of the ground superficies be the right line CE. And take in the line CE a point at all aduentures 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 superficies therefore by the 2. definition of the eleuenth the line AB is erected perpendicularly to all the right lines that are in the ground plaine superficies and which touch it Wherfore 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 â—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 perpendicularly 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 superficieces making therwith right angles do also make right angles with the other plaine superficies 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 superficies If therefore a right line be erected perpendicularly to a plaine superficies all the superficieces passing by the right line are erected perpendicularly to the selfe same plaine superficies which was required to be demonstrated In this figure here set ye may erect perpeÌ„dicularly 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 perpendicularly to any plaine superficies their common section is also erected perpendicularly to the selfe same plaine superficies SVppose that these two plaine superâ—icieces AB BC cutting the one the other be erected pâ—rpâ—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 superâ—icies For if not then by the 11. of the first 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 perpendicular line DF. And forasmuch as the superficies 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. deâ—inition of this booke the line DE is erected perpendicularly to the ground superâ—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 Wherfore 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 superâ—icieces cutting the one the other be erected perpendicularly to any plaine superâ—icies their common section is also erected perpendicularly to the selfe same plaine superâ—icies which was required to be proued Here haue I set an other figure which will more plainly shewe vnto you the former demonstration if ye erecte perpendicularly 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. â—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 angles 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
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 parallelogrammes Draw a right line from the point A to the point H and an other from the point D to the point F. 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 comprehend equall angles Wherfore the angle ABH is equall to the angle DCF And forasmuch as these two lines AB and BH are equall to these two lines DC and CF and the angle ABH is proued equall to the angle DCFâ— therfore by the 4. of the first the base AH is equall to the base DF and the triangle ABH is equall to the triangle DCF And forasmuch as by the 41. of the first the parallelogramme BG is double to the triangle ABH and the parallelogramme CE is also double to the triangle DCF therfore the parallelogramme BG is equall to the parallelogramme CE. In like sort 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 superficieces 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 pasted 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 ParallelipipedoÌ„ be cutte of a playne superficies 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 I haue for the better sight of the coÌ„structioÌ„ demoÌ„stration of the former 25. propositioÌ„ here set another figure whose forme if ye describe vppon 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 demoÌ„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 the seâ—â—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. definitioÌ„ 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 superâ—icieces and shal also be equiangle Wherfore if vnto the bases by producing the right lines be added like and equall bases as was before shewed in a parallelipipedon of those sections shal be made as many like Prismes as ye will. And so by the same reason namely by the commoÌ„ excesse equalitie or want of the multiplices of the bases of the sections by the 5. definitioÌ„ 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 whos 's two opposite superficieâ—es are poligonon figures like equall and parallels the other superficies which of necessitie are parallelogrammes 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 thâ— one to the other Which thing is manifest for such solides are deuided into Prismes which haue one coÌ„mon side namely the axe or right line which is drawne by the centers of the opposite bases Wherefore how many paâ—allelograÌ„mes or bases are set vpon the opposite poligonon figures of so many Prismes shal the whole solide be coÌ„posed 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 sectioÌ„s of the bases shal by the former Corollary be proportional with the sectioÌ„s of the Prismes Wherefore by the â—â— 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 poligonon figures which poligonon figures doo alter the angles of the parallelogrammes set vpon them according to the diuersitie oâ— 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 vpoÌ„ the opposite like equal poligonon and parallel superficieces are alwayes parallelogrammes Third Corollary Tâ—e foresayd solides â—omposed of Prismes being cutte by a playne superficies parallel to the oppositâ— 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 parâ—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 In thesâ— two 〈…〉 here put you may in 〈◊〉 clearely concerne the â—â—â—â—mer construction and dâ—â—monstratioÌ„ if ye erect peâ—â—pendicularly vnto the ground superficies the triangles ALB and DCE eleuate the triangles ALH and DCF that the lynes LA and CD of them may exactly agree with the lineâ— LA and CD of the â—riangles erecâ—edâ— For so ordering them if ye compare the former construction and demonstration with them they will be playnâ— vnto you Although Euclides demoÌ„stration 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
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 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 parallelogramme RY and the opposite side vnto the base the parallelogramme VZ For they consiste vpon one and the selfe fame base namely RY and are vnder one and the selfe same altitude 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 equall 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 therefore the parallelogramme QT is equall to the parallelogramme AB and the parallelogramme CD is equall to the parallelogramme AB by supposition Wherefore the parallelogramme CD is equall to the parallelogramme QT And there is a certaine other superficies 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 superfiâ—â—es RF which is a parallel to either of the opposite plaine superficieces therfore 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 Wherfore 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 staÌ„ding lines namely AG HK BE LM CX OP DF and RS be not erected perpendicularly 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 eleueÌ„th vnto the ground plaine superficieces 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 proued in this 31. Proposition the solide KZ is equall to the solide PI for they consist vpon equall bases namely KM and PS and are vnder one and the selfe same altitude whose standing lines also are erected perpendicularly 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 equall 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 described for it in a playne And yet before M. d ee 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 impossible 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 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 parallelipipedoâ— 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 correspondent side and to his correspondent angle But concerning the other solide When ye haue described these three figures vpon pasted paper Where note for the proportion of eche line to make your figure of pasted paper equall with the figure before described vpon the playne let your lines OP CX RS DF c. namely the rest of the standing 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 coÌ„tayne the bases of these figures be equal to the lines OC CR RD DO c. namely to the sids which coÌ„tayne the bases of that figure Moreouer let the lines PX Xâ— SF FP c. namely the rest of the lines which coÌ„taine the vpper superficieces of these figures be equal to the lines PX XS SF Fâ— c. namely to the rest of the lines which coÌ„taine 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 â—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 As touching the second case ye neede no new figures for it is playne to see by the figures now
superficies geuen to be â— and the rectangle parallelipipedon geuen to be AM. Vppon â— as a base must AM be applyed that is a rectangle parâ—llipipedon must be erected vppon â— as a base whiche shall be equall to AM. By the laste of the second to the right lyned figure â— let an equall square be made which suppose to be FRX produce one side of the base of the parallelipipedoÌ„ AM which let be AC extended to the point P. Let the other side of the sayde base concurring 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 â— 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 perpendicularly erected vppon the base â— hauinge the heith of the line O is equall to the parallelipipedon AM. For CD is to AG as CP is to Aâ— by the firste of the sixth and â— 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 â— and AG are reciprocally in proportion with the heithes AN and O. By this 34 therefore a solide erected perpendicularly vppon â— 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 â— Now must we make a rectangle parallelipipedon equal to A Whose heith must be equall to â— According to the manner before vsed we must frame our coÌ„struction to a reciprokall proportioÌ„ 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 geuen 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 geueÌ„ to describe a parallepipedoÌ„ like likewise situated to a parallelipipedoÌ„ geueÌ„ 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 proposition 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 angles be eleuated on high right lines comprehending together with those right lines which containe the superficiall angles equall angles eche to his corespoÌ„dent angle and if in eche of the eleuated lines be takeÌ„ 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 beginning 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 eleuated on high shall contayne equall angles SVppose that these two rectiline superficiall angles BAC and EDF be equall the one to the other 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 aueÌ„tures and let the same be G and M. And by the 11. of the eleueÌ„th from the pointes G and M draw vnto the ground playne superficieces wherein are the angles BAC and E DF perpendicular lines GL and MN and let them fall in the sayd playne superâ—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. FroÌ„ 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 perpeÌ„dicularly to the same grounde plaine superficies BAC Drawe by the 12. of the first froÌ„ the pointes K and N vnto the right lines AB AC DF DE perpeÌ„dicular right lines and let the same be KC NF KB NE. And drawe these right lines HC CB MF FE Now forasmuch aâ— by the 47. of the first the square of the line HA is equall to the squares of the lines HK and KA 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 suppositioÌ„ equal to the angle MDF Wherfore there are two triangles MDF and HAC hauing two angles 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 perpeÌ„dicularly 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 reason 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 supposition 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 angle remayning namely to EFN And by the same reasoÌ„ also the angle CBK is equal to the angle FEN Wherfore 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 other namely that side that lieth betwene the equall angles that is the side BC is equall to the side EF Wherefore 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 froÌ„ 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 perpendicular lines to the plaine superficieces in which are the angles geuen at the beginning and frâ—â— 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 together 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 other figures wherein you must erecte perpendicularly to the ground superficieces the two triangles BHK and EMN and then eleuate 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 other 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 superficies compare it with the former construction demonstration and it will make it very easye to conceaue ¶ Corollary By this it is manifest that if there be two rectiline superficiall angles equall and vpon those angles be eleuated on high equall right lines contayning together with the right lines put at the bâ—ginning equall angles perpendicular lines drawen from those eleuated lines to the ground plaine superficieces 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 drawen 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 Parallelipipedon made of the line B so that the solide made of the line B consist of equall sides and be also equiangle to the solide made of the lines A B C. Describe by the 23. of the
set before the thing required In some propositions there are more things geueÌ„ then one and mo thinges required then one In some there is nothing geuen at all Moreouer euery Probleme Theoreme beyng perfect and absolute ought to haue all these partes namely First the Proposition to be proued Then the exposition which is the explication of the thing geuen After that followeth the determinatioÌ„ which is the declaration of the thing required Then is set the construction of suche things which are necessary ether for the doing of the propositioÌ„ or for the demoÌ„stration Afterward followeth the demonstration which is the reason and proofe of the proposition And last of all is put the conclusion which is inferred proued by the demonstration and is euer the proposition But all those partes are not of necessitie required in euery Probleme and Theoreme But the Proposition demonstration and conclusion are necessary partes can neuer be absent the other partes may sometymes be away Further in diuers propositions there happen diuers cases which are nothing els but varietie of delineatioâ— and construction or chaunge of position as when pointes lines superâ—iciesses or bodies are chaunged VVhich thinges happen in diuers propositions NOw then in this Probleme the thing geuen is the line geueÌ„ the thing required to be serched out is how vpoÌ„ that line to describe an equilater triangle The Proposition of this Probleme is Vpon a right line geuen not beyng infinite to describe an equilater triangle The exposition is Suppose that the right line geuen be AB and this declareth onely the thing geuen The determination is It is required vpon the line AB to describe an equilater triangle for therby as you see is declared onely the thing required The construction beginneth at these woâ—ds Now therfore making the ceÌ„tre the point A the space AB describe by the third peâ—icion a circle c and continueth vntil you come to these wordes And forasmuch at the point A c. For thetheâ—to are described circles and lines necessarye both for the doyng of the proposition and also for the demonstration therof VVhich demonstration beginneth at these wordes And forasmuche as the point A is the centre of the circle CBD c And so procedeth till you come to these wordes VVherfore vpon the line AB is described an equilater triangle ABC For vntill you come thether is by groundes before set and constructions had proued and made euident that the triangle made is equilater And then in these wordes wherfore vpon the line AB is described an equilater triangle ABC is put the first conclusion For there are commonly in euery proposiâ—ion two conclusions the one perticuler the other vniuersal and from the first you go to the last And this is the first and perticuler conclusion â—or that it concluâ—eth that vpon the lyne AB is described an equilater triaÌ„gle which is according to the exposition After it followeth the last and vniuersal conclusion wherfore vpon a right line geuen not being infinite is described an equilater triangle For wheâ—her the line geuen be greater or lesse then thys lyne the â—ame constructioÌ„s and demonstrations proue the same conclusion Last of all is added this clause Which is the thing which was required to be done wherby as we haue before noted is declared that this proposition is a Probleme and not a Theoreme As for varietie of cases in this proposition there is none for that the line geuen can haue no diuersitie of position As you haue in this Probleme sene plainelye set foorthe the thing geuen and the thing required moreouer the proposition exposition determination construction demonstration and conclusion which are generall also to many other both Problemes and Theoremes so may you by the example therof distinct them and searche them out in other Problemes â— and also Theoremes This also is to be noted that there are three kyndes of demonstration The one is called Demonstratio a priori or composition The other is called Demonstratiâ— a posteriori or resolution And the third is a demonstration leadyng to an impossibilitie A demonstration a priori or composition is when in reasoâ—ing from the principles and first groundes we passe discending continually till after many reasons made we come at the lengâ—h to conclude that which we first chiefly entend And this kinde of demonstration vseth Euclide in his bookeâ— for the most part A demonstration a posteriori or resolution is when contrariwise in reasoning we passe from the last conclusion made by the premisses and by the premisses of the premisses continually ascending til we come to the first principles and grounds which are indemonstrable and for theyr simplicity can suffer no farther resolution A demonstration leadyng to an impossibilitie is that argument whose coÌ„clusion is impossible that is when it concludeth directly against any principle or against any proposition before proued by principles or propositioâ—s beâ—ore proued Premisses in an argument are propositions goyng before the conclusion by which the conclusion is proued Composition passeth from the cause to the effect or from thinges simple to thinges more compounded Resolution contrariwise passeth from thinges compounded to thinges more simple or from the effect to â—he cause Composition or the first kynde of demonstration which passeth from the principles may easely be sene in this first proposition of Euclide The demonstration wherof beginneth thus And forasmuch as the point A is the centre of the circle CBD therfore the line AC is equal to the line AB This reason you see taketh his beginnyng of a principle namely of the definition of a circle And this is the first reason Agayne forasmuch as B is the centre of the circle CAE therfore the line BC is equall to the lyne BA which is the second reason And it was before proued that the lyne AC is equall to the line AB wherfore either of these lines CA CB is equal to the lyne AB And this is the third reasoÌ„ But things which are equall to one the selfe same thyng are also equall the one to the other VVherfore the line CA is equal to the line CB. And this is the fourth argument VVherfore these three lines CA AB and BC are equall the one to the other which is the conclusion and the thing to be proued You may also in the same first PropositioÌ„ easely take an exaÌ„ple of ResolutioÌ„ vsing a contrary order passyng backward froÌ„ the last conclusioÌ„ of the former demonstration til you come to the first principle or ground wheron it began For the last argument or reason in composition is the first in Resolution the first in composition is the last in resolution Thus therfore must ye procede The triangle ABC is contained of three equall right lines
by his motion described the round Conical superficies about the Cone And as the circuÌ„fereÌ„ce of the semicircle described the round sphericall superficies about the Sphere In this example it is the superficies described of the line DC By this definition it is playne that the two circles or bases of a cilinder are euer equall and parallels 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 Campane Vitellâ—o with other later writers call this solide or body a round Columnâ— 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 ceÌ„tre is one and the selfe same 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 whose axes and diameters of their bases are proportionall The similitude of cones and cilinders standeth in the proportion of those right lines of which they haue their originall 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 together which is thier length breadth As if the axe â—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 aâ—e the cones ABC and DEF like cones Likewise in the cilinders If the axe LN of the cilinder LHMN haue that proportion 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 equall squares As is a dye which hath sixe sides and eche of them is a full and perfect square as limites or borders 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 commonly 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 vpoÌ„ a playne superficies to shew somwhat bodilike And in deede the latter descriptioÌ„ is for the sight better theÌ„ the first But the first for the demoÌ„strations 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 in any one of the sixe sides Which can not be so wel sene in the other 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 parallelogrammes as in this example the parallelogramme ABCD to be the base lieng vpon a ground playne superfices And so conceiue the parallelogramme EFGH to be in the toppe ouer it in such sort that the lines AE CG DH BF may be erected perpendicularly from the pointes A C B D to the ground playne superficies 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 described 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 Tetrahedron which is thus 22 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 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 Tetrahedron thus described is of more vse in these fiue bookes following then is the other although the other appeare in forme to the eye more bodilike Why this definition is here left out both of Campane and of Flussas I can not but maruell considering that a Tetrahedron is of all Philosophers counted one of the fiue chiefe solides which are here defined of Euclide which are called coÌ„monly regular bodies without mencion of which the entreatie of these should seeme much maimed vnlesse they thought it sufficiently defined vnder the definition of a Pyramis which plainly and generally taken includeth in deede a Tetrahedron although a Tetrahedron properly much differeâ—h from a Pyramis as a thing speciall or a particular from a more generall For so taking it euery Tetrahedron is a Pyramis 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 definition 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 Pyramiâ— the superficieces 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 Tetrahedron may be called a Pyramis
shall be proportionall And iâ— 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 descâ—â—bed Then I say that as the solide KA is â—o the solide LC so is the solide ME to the solide NG For forasmuch as the Parallelipipedon 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 CDâ— 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 tâ— 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 that as the right line AB is to the right line CD so is the right liâ—â— 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 proportionall And if the Parallelipipedons described of them and being like and in like sort described be proportionall those right lines also shall be proportionallâ— 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 perpeÌ„dicularly to the plaine superficies 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 perpendicular line drawen from the point E to the 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 superficies AB in the point F. And by the 12. of the first from the point F draw vnto the line DA being in the superficies AB a perpendicular line FG which line also is erected perpendicularly to the plaine superficies CD by the third diffinitioÌ„ by reason we presuppose CD and AB to be perpendicularly erected ech to other Draw a right line from the point E to the point G. And forasmuch as the line FG is erected perpendicularly 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 perpeÌ„dicularly 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 froÌ„ the point E to the sâ—â—erficies AB falleth not without 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 itâ— whose demonstratioÌ„ there Flussas hath in this place and none other though he sayth that Campane of such a PropositioÌ„ as of Euclides maketh no mention In this figure ye may more fully see the former Proposition 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 commoÌ„ section of those plaine superficieces and the diameter of the Parallelipipedon 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 pointâ—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 DG Then I say that the lines VS and DG do deuide the one the other into two equall partes that is that the line VT is equall to the line T S and the line DT to the line TG Drawe these right lines DV VE BS and SG Now forasmuch 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 vnto it a parallel but the line CA is equall to the line GE and is vnto it also a parallel wherfore by the firsâ— 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