two sides EH and EK of the triangle EHK and the angle FEG is greater then the angle HEK therfore by the 24. of the first the base FG is greater then the base HK And by the same reason may it be proued that the line AC is greater then the line AB And so is manifest the whole Proposition The 15. Theoreme The 16. Proposition If from the end of the diameter of a circle be drawen a right line making right angles it shall fall without the circle and betwene that right line and the circumference can not be drawen an other right line and the angle of the semicircle is greater then any acute angle made of right lines but the other angle is lesse then any acute angle made of right lines SVppose that there be a circle ABC whose centre let be the point D and let the diameter therof be AB Then I say that a right line drawen from the poynt A making with the diameter AB right angles shall fall without the circle For if it do not then if it be possible let it fall within the circle as the line AC doth and draw a line from the point D to the point C. And for asmuch as by the 15. definition of the first the line DA is equall to the line DC for they are drawen from the centre to the circumference therefore the angle DAC is equall to the angle ACD But the angle DAC is by supposition a right angle Wherfore also the angle ACD is a right angle Wherefore the angles DAC and ACD are equall to two right angles which by the 17. of the first is impossible Wherefore a right line drawen from the poynt A making with the diameter AB right angles shall not fall within the circle In like sort also may we proue that it falleth not in the circumference Wherefore it falleth without as the line AE doth I say also that betwene the right line AE and the circumference ACB can not be drawen an other right line For if it be possible let the line AF so be drawen And by the 12. of the first from the poynt D draw vnto the line FA a perpendicular line DG And for asmuch as AGD is a right angle but DAG is lesse then a right angle therefore by the 19. of the first the side AD is greater then the side DG But the line DA is equall to the line DH for they are drawen from the centre to the circumference Wherefore the line DH is greater then the line DG namely the lesse greater then the greater which is impossible Wherefore betwene the right line AE and the circumference ACB can not be drawen an other right line I say moreouer that the angle of the semicircle contayned vnder the right line AB and the circuÌ„ference CHA is greater then any acute angle made of right lines And the angle remayning coÌ„tayned vnder the circumference CHA and the right line AE is lesse then any acute angle made of right lines For if there be any angle made of right lines greater then that angle which is contayned vnder the right line BA and the circumference CHA or lesse then that which is contayned vnder the circumference CHA and the right line AE then betwene the circumference CHA and the right line AE there shall fall a right line which maketh the angle contayned vnder the right lines greater then that angle which is contayned vnder the right line BA and the circumference CHA and lesse then the angle which is contayned vnder the circumference CHA and the right line AE But there can fall no such line as it hath before bene proued Wherfore no acute angle contained vnder right lines is greater then the angle contayned vnder the right line BA and the circumference CHA nor also lesse then the angle contayned vnder the circumference CHA and the line AE Correlary Hereby it is manifest that a perpendicular line drawen froÌ„ the end of the diameter of a circle toucheth the circle and that a right line toucheth a circle in one poynt onely For it was proued by the 2. of the third that a right line drawen from two pointes taken in the circumference of a circle shall fall within the circle Which was required to be demonstrated The 2. Probleme The 17. Proposition From a poynt geuen to draw a right line which shall touch a circle geuen SVppose that the poynt geuen be A and let the circle geuen be BCD It is required from the poynt A to draw a right line which shall touch the circle BCD Take by the first of the third the centre of the circle and let the same be E. And by the first petition draw the right line ADE And making the centre E and the space AE describe by the third petition a circle AFG And from the poynt D raise vp by the 11. of the first vnto the line EA a perpendicular line DF. And by the first petition drawe these lines EBF and AB Then I say that from the point A is drawen to the circle BCD a touch line AB For for asmuch as the point E is the centre of the circle BCD and also of the circle AFG therfore the line EA is equall to the line EF and the line ED to the line EB for they are drawen from the centre to the circumference Wherefore to these two lines AE and EB are equall these two lines EF ED and the angle at the poynt E is common to them both Wherefore by the 4. of the first the base DF is equall to the base AB and the triangle DEF is equall to the triangle EBA and the rest of the angles remayning to the rest of the angles remayning Wherefore the angle EDF is equall to the angle EBA But the angle EDF is a right angle Wherfore also the angle EBA is a right angle and the line EB is drawen from the centre But a perpendicular line drawen from the end of the diameter of a circle toucheth the circle by the Corellary of the 16. of the third Wherefore the line AB toucheth the circle BCD Wherfore from the point geuen namely A is drawen vnto the circle geueÌ„ BCD a touch line AB which was required to be done ¶ An addition of Pelitarius Vnto a right lyne which cutteth a circle to drawe a parallel line which shall touch the circle Sâ—ppose that the right lyne AB do out the circle ABC in the poyntes A and B. It is required to drawe vnto the line AB a parallel lyne which shall touche the circle Let the centre of the circle be the point D. And deuide the lyne AB into two equall partes in the point E. And by the point E and by the centre D draw the diameter CDEF And from the point F which is the ende of the diameter rayse vp by the 11. of the first vnto the diameter CF a perpendicular line GFH Then I
the touch and cutting the circle may eyther passe by the centre or not If it passe by the centre then is it manifest by the 18. of thys booke that it falleth perpendicularly vpon the touch line and deuideth the circle into two equall partes so that all the angles in eche semicircle are by the former Proposition right angles and therfore equall to the alternate angles made by the sayd perpendicular line and the touch line If it passe not by the centre then followe the construction and demonstration before put The 5. Probleme The 33. Proposition Vppon a right lyne geuen to describe a segment of a circle which shall contayne an angle equall to a rectiline angle geueÌ„ SVppose that the right line geuen be AB and let the rectiline angle geâ—â—n be C. It is required vpon the right line geueÌ„ AB to describe a segment of a circle which shall contayne an angle equall to the angle C. Now the angle C is either an acute angle or a right angle or an obtuse angle First let it be an acute angle as in the first description And by the 23 of the first vpon the right line AB and to the point in it A describe an angle equal to the angle C and let the same be DAB Wherfore the angle DAB is an acute angle From the point A raise vp by the 11. of the first vnto the line AD a perpendiculer line AF. And by the 10. of the first deuide the line AB into two equall partes in the point F. And by the 11. of the same from the point F raise vp vnto the line AB a perpendicular lyne FG and draw a line from G to B. And forasmuch as the line AF is equall to the line FB and the line FG is common to them both therfore these two lines AF and FG are equall to these two lines FB and FG and the angle AFG is by the 4. peticion equall to the angle GFB Wherfore by the 4. of the same the base AG is equall to the base GB Wherfore making the centre G and the space GA describe by the 3. peticion a circle and it shall passe by the point B describe such a circle let the same be ABE And draw a line from E to B. Now forasmuch as from the ende of the diameter AE namely from the point A is 〈◊〉 a right line AD making together with the right line AE a right angle therfore by the correllary of the 16. of the third the line AD toucheth the circle ABE And forasmuch as a certaine right line AD toucheth the circle ABE from the point A where the touch is is drawen into the circle a certaine right line AB therfore by the 32. of the third the angle DAB is equall to the angle AEB which is in the alternate segment of the circle But the angle DAB is equall to the angle C wherfore the angle C is equall to the angle AEB Wherfore vpon the right line geuen AB is described a segment of a circle which contayneth the angle AEB which is equall to the angle geuen namely to C. But now suppose that the angle C be a right angle It is againe required vpon the right line AB to describe a segment of a circle which shall contayne an angle equal to the right angle C. Describe againe vpon the right line AB and to the point in it A an angle BAD equal to the rectiline angle geuen C by the 23. of the first as it is set forth in the second description And by the 10. of the first deuide the line AB into two equall partes in the point F. And making the centre the point F and the space FA or FB describe by the 3. peticion the circle AEB Wherfore the right line AD toucheth the circle AEB for that the angle BAD is a right angle Wherfore the angle BAD is equall to the angle which is in the segment AEB for the angle which is in a semicircle is a right angle by the 31. of the third But the angle BAD is equal to the angle C. Wherfore there is againe described vpon the line AB a segment of a circle namely AEB which containeth an angle equall to the angle geuen namely to C. But now suppose that the angle C be an obtuse angle Vpon the right line AB and to the point in it A describe by the 23. of the first an angle BAD equall to the angle C as it is in the third description And from the point A rayse vp vnto the line AD a perpendiculer line AE by the 11. of the first And agayne by the 10. of the first deuide the line AB into two equall partes in the point F. And from the point F. raâ—se vp vnto the line AB a perpeÌ„dicular line FG by the 11. of the same drawe a line from G to B. And now forasmuch as the line AF is equal to the line FB and the line FG is common to them both therfore these two lines AF and FG are equall to these two lines BF and FG and the angle AFG is by the 4. peticion equall to the angle BFG wherfore by the 4. of the same the base AG is equall to the base GB Wherfore making the centre G and the space GA describe by the 3. peticion a circle and it shall passe by the point B let it be described as the circle AEB is And forasmuch as from the ende of the diameter AE is drawen a perpendiculer line AD therefore by the correllary of the 16. of the third the line AD toucheth the circle AEB from the point of the touche namely A is extended the line AB Wherfore by the 32. of the third the angle BAD is equall to the angle AHB which is in the alternate segment of the circle But the angle BAD is equall to the angle C. Wherefore the angle which is in the segment AHB is equall to the angle C. Wherfore vpon the right line geuen AB is described a segment of a circle AHB which contayneth an angle equall to the angle geuen namely C which was required to be done The 6. Probleme The 34. Proposition From a circle geuen to cut away a section which shal containe an angle equall to a rectiline angle geuen SVppose that the circle geuen be AC and let the rectiline angle geuen be D. It is required froÌ„ the circle ABC to cut away a segment which shall contayne an angle equall to the angle D. Draw by the 17. of the third a line touching the circle and let the same be EF and let it touche in the point B. And by the 23. of the first vpon the right line EF and to the point in it B describe the angle FBC equall to the angle D. Now forasmuch as a certayne right line EF toucheth the circle ABC in the point B and frora
proportion that the first of the three lines put is to the 〈◊〉 â—or tâ—e 〈◊〉 line to the third namely the line AE to the line EB is in double propoâ—tion that it is to the second by the 10. deâ—inition of the fiâ—t ¶ The second Corollary Hereby may we learne how from a rectiline â—igure geuen to take away a part appointed leaâ—ing the rest of the rectiline â—igure like vnto the whole For if froÌ„ the right line AB be cut of a part appoynted namely EB by the 9. of this booke as the line AE is to the line EB so is the rectiline â—igure described of the line AF to the rectiline figure described of the line FB the sayd â—igures being supposed to be like both the one to the other and also to the rectiline â—igure described of the line AB and being also in like sort situate Wherfore taking away â—rom the rectiline â—igure described of the line AB the rectiline figure described of the line FB the residue namely the rectiline figure described of the line AF shall be both like vnto the whole rectiline â—igure geuen described of the line AB and in like sort situate ¶ The third Corollary To compose two like rectiline â—igures into one rectiline figure like and equall to the same figures Let their sides of like proportioÌ„ be set so that they make a right angle as the lines AF and FB are And vpoÌ„ the line subtending the said angle namely the line AB describe a rectiline â—igure like vnto the rectiline figures geuen and in like sort situate by the 18. of this booke and the same shall be equall to the two rectiline figures geuen by the 31. of this booke ¶ The second Proposition If two right lines cut the one the other obtuseangled wise and from the endes of the lines which â—ut the one the other be drawen perpendicular lines to either line the lines which are betwene the endes and the perpendicular lines are cut reciprokally Suppose that there be two right lines AB and GD cutting the one the other in the point E and making an obtuse angle in the section E. And from the endes of the lines namely A and G let there be drawen to either line perpendicular lines namely from the point A to the line GD which let be AD and from the point G to the right line AB which let be GB Then I say that the right lines AB and GD do betwene the end A and the perpendicular B and the end G and the perpendicular D cut the one the other reciprokally in the point E so that as the line AE is to the line ED so is the line GE to the line EB For forasmuch as the angles ADE and GBE are right angles therfore they are equall But the angles AED and GEB are also equall by the 15. of the first Wherefore the angles remayning namely EAD EGB are equall by the Corollary of the 32. of the first Wherefore the triangles AED and GEH are equiangle Wherfore the sides about the equall angles shall be proportionall by the 4. of the sixt Wherfore as the line AE is to the line ED so is the line GE to the line EB If therefore two right lines cut the one the other obtuseangled wife c which was required to be proued ¶ The third Proposition If two right lines make an acute angle and from their endes be drawen to ech line perpendicular lines cutting them the two right lines geuen shall be reciprokally proportionall as the segmentes which are about the angle Suppose that there be two right lines AB and GB making an acute angle ABG And from the poyntes A and G let there be drawen vnto the lines AB and GB perpendicular lines AC and GE cutting the lines AB and GB in the poyntes E and â— Then I say that the lines namely AB to GB are reciprokally proportionall as the segmentes namely CB to EB which are about the acute angle B. For forasmuch as thâ— right angles ACB and GER are equall and the angleâ— ABG is common to the triangles ABC and GBE â— therefore the angles remayning BAC and EGB are equall by the Corollary of the 32. of the first Wherfore the triangles ABC and GBE are equiangle Wherefore the sideâ— about the equall angles are proportionall by the 4. of the sixe â— so that as the line AB is to the line FC so is the line GB to the line BE. Wherefore alternately as the line AB is to the line GB so is the line CB to the line BE. If therefore two right lines makâ— aâ—â—câ—te angleâ— câ— which was required to be proued â— The fourth Propositionâ— If in a circle be drawen two right lines cutting the one the other the sections of the one to the sections of the other shall be reciprokally proportionall In the circle AGB let these two right lines 〈…〉 one the other in the poynt E. Thâ—â— I say that reciprokally 〈◊〉 â—hâ— line AE is to the line ED so is the line GE to the line EB For forasmuch as by the 35. of the third the rectangle figure contayned vnder the lines AE and EB is equall to the rectangle figure contayned vnder the lines GE and ED but in equall rectangle parallelogrammes the sides about the equall angles are reciprokall by the 14. of the sixt Therefore the line AE is to the line ED reciprokally as the line GE is to the line EB by the second definition of the sixt If therefore in a circle be drawen two right lines c which was required to be proued ¶ The fift Proposition If from a poynt geuen be drawen in a plaine superâ—icies two right lines to the concaue circumference of a circle they shall be reciprokally proportionall with their partes takeÌ„ without the circle And moreouer a right line drawen from the sayd poynt touching the circle shall be a meane proportionall betwene the whole line and the vtter segment Suppose that there be a circle ABD and without it take a certayne poynt namely G. And from the point G drawe vnto the concaue circumference two right lines GB and GD cutting the circle in the poyntes C and E. And let the line GA touch the circle in the point A. TheÌ„ I say that the lines namely GB to GD are reciprokally as their parts taken without the circle namely as GC to GE. For forasmuch as by the Corollary of the 36. of the third the rectangle figure contayned vnder the lines GB and GE is equall to the rectangle figure contayned vnder the lines GD and GC therefore by the 14. of the sixt reciprokally as the line GB is to the line GD so is the line GC to the line GE for they are sides contayning equall angles I say moreouer that betwene the lines GB and GE or betwene the lines GD and GC the touch line GA is a meane proportionall For forasmuch as the rectangle
draw by the 11. of the firsâ— a perpeÌ„dicular line AB and let AB be a rationall line and make perfectâ— the parallelogramme BC. Wherefore BG is irrationall by that which was declared and proued in maner of an Assumpt in the end of the demonstration of the 38 and the line that containeth it iâ— power is also irrationall Let the line CD containe in power the superâ—icies BC. Wherefore CD is irrationall not of the selfe same kind with any of those that were before for the square of the line CD applied to a rationall line namely AB maketh the breadth a mediall line namely AC But the square of none of the foresaid lines applied to a rationall line maketh the breadth a mediall line Againe make perfecte the parallelogramme ED. Wherefore the parallelogramme ED is also irrationall by the sayd Assumpt in the end of the 98. his demonstration brieâ—ly proued and the line which containeth it in power is irrationallâ— let the line which containeth it in power be DF. Wherefore DF is irrationall and not of the selfe same kinde with any of the foresaid irrationall lines For the square of none of the foresayd irrationall lines applied vnto a rationall line maketh the breadth the line CD Wherefore of a mediall line are produced infinite irrationall lines of which none is of the selfe same kinde with any of those that were before which was required to be demonstrated ¶ The 92. Theoreme The 116. Proposition Now let vs proue that in square figures the diameter is incommensurable in length to the side SVppose that ABCD be a square and let the diameter therof be AC Then I say that the diameter AC is incommensurable in length to the side AB For if it be possible let it be coÌ„mensurable in leÌ„gth I say that theÌ„ this will follow that one and the selfe same nuÌ„ber shall be both an euen number an odde number It is manifest by the 47. of the first that the square of the line AC is double to the square of the line AB And for that the line AC is commensurable in length to the line AB by supposition therfore the lyne AC hath vnto the line AB that proportion that a number hath to a number by the 5. of the tenth Let the lyne AC haue vnto the line AB that proportion that the number EF hath to the number G. And let EF and G be the least numbers that haue one and the same proportion with them Wherfore EF is not vnitie For if EF be vnitie and it hath to the number G that proportion that the line AC hath to the lyne AB and the line AC is greater then the lyne AB Wherfore vnitie EF is greater then the number G which is impossible Wherfore FE is not vnitie wherfore it is a number And for that as the square of the line AC is to the square of the lyne AB so is the square number of the number EF to the square number of the number G for in eche is the proportion of their sides doubled by the corollary of the 20 of the sixt and 11. of the eight and the proportion of the line AC to the line AB doubled is equal to the proportioÌ„ of the nuÌ„ber EF to the number G doubled for as the line AC is to the line AB so is the nuÌ„ber EF to the number G. But the square of the line AC is double to the square of the line AB Wherfore the square number produced of the number EF is double to the square number produced of the number G. Wherefore the square number produced of EF is an euen number Wherfore EF is also an euen number For if EF were an odde number the square number also produced of it should by the 23. and 29. of the ninth be an odde number For if odde numbers how many soeuer be added together and if the multitude of theÌ„ be odde the whole also shal be odde Wherfore EF is an euen number Deuide the number EF into two equall partes in H. And forasmuch as the numbers EF and G are the lest numbers in that proportion therfore by the 24. of the seuenth they are prime numbers the one to the other And EF is an euen number Wherfore G is an odde number For if G were an euen number the number two should measure both the number EF and the number G for euery euen nuÌ„ber hath an halfe part by the definition but these numbers EF G are prime the one to the other Wherfore it is impossible that they should be measured by two or by any other number besides vnitie Wherfore G is an odde number And forasmuch as the number EF is double to the number EH therfore the square number produced of EF is quadruple to the square number produced of EH And the square number produced of EF is double to the square number produced of G. Wherfore the square number produced of G is double to the square number produced of EH Wherfore the square number produced of G is an euen number Wherfore also by those thinges which haue bene before spoken the number G is an euen number but it is proued that it is an odde number which is impossible Wherefore the line AC is not commensurable in length to the line AB wherfore it is incommensurable An other demonstration We may by an other demonstration proue that the diameter of a square is incommensurable to the side thereof Suppose that there be a square whose diameter let be A and let the side thereof be B. Then I say that the line A is incommensurable in length to the line B. For if it be possible let it be commensurable in length And agayne as the line A is to the line B so let the number EF be to the number G and let them be the least that haue one and the same proportion with them wherefore the numbers EF and G are prime the one to the other First I say that G is not vnitie For if it be possible let it be vnitie And for that the square of the line A is to the square of the line B as the square number produced of EF is to the square number produced of G as it was proued in the â—ormer demonstration but the square of the line A is double to the square of the line B. Wherfore the square nuÌ„ber produced of EF is double to the square number produced of G. And by your supposition G is vnitie Wherefore the square number produced of EF is the number two which is impossible Wherefore G is not vnitie Wherefore it is a number And for that as the square of the line A is to the square of the line B so is the square number produced of EF to the square number produced of G. Wherefore the square number produced of EF is double to the square number produced of G. Wherefore the square number produced of G.
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
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
one of the circles ¶ An Assumpt added by Flussas If a Sphere be cut of a playne superficies the common section of the superficieces shall be the circumference of a circle Suppose that the sphere ABC be cut by the playne superficies AEB and let the centre of the sphere be the poynt D. And from the poynt D let there be drawne vnto the playne superficies AEB a perpendicular line by the 11. of the eleuenth which let be the line DE. And from the poynt E draw in the playne superficies AEB vnto the common section of the sayd superficies and the sphere lines how many so euer namely EA and EB And draw these lines DA and DB. Now forasmuch as the right angles DEA and DEB are equall for the line DE is erected perpendicularly to the playne superficies And the right lines DA DB which subtend those angles are by the 12. definiâ—ion of the eleuenth equall which right lines moreouer by the 47. of the first do contayne in power the squares of the lines DE EA and DE EB if therfore from the squares of the lines DE EA and DE EB ye take away the square of the line DE which is coÌ„mon vnto them the residue namely the squares of the lines EA EB shall be equal Wherfore also the lines EA and EB are equall And by the same reason may we proue that all the right lines drawne from the poynt E to the line which is the coÌ„mon section of the superficies of sphere and of the playne superficies are equall Wherefore that line shall be the circumference of a circle by the 15. definition of the first But if it happen the plaine superficies which cutteth the sphere to passe by the centre of the sphere the right lines drawne from the centre of the sphere to their common section shalâ— be equall by the 12. definition of the eleuenth For that common section is in the superficies of the sphere Wherefore of necesâ—itie the playne superficies comprehended vnder that line of the common section shall be a circle and his centre shall be one and the same with the centre of the sphere Iohn Dee Euclide hath among the definition of solides omitted certayne which were easy to conceaue by a kinde of Analogie As a segment of a sphere a sector of a sphere the vertex or toppe of the segment of a sphere with such like But that if nede be some fartheâ— light may be geueÌ„ in this figure next before vndersâ—and a segment of the sphere ABC to be that part of the sphere contayned betwen the circle AB whose center is E and the sphericall superficies AFB To which being a lesse segment adde the cone ADB whose base is the former circle and toppe the center of the sphere and you haue DAFB a sector of a sphere or solide sector as I call it DE extended to F sheweth the top or vertex of the segment to be the poynt F and EF is the altitude of the segment sphericall Of segmentes some are greater theÌ„ the halfe sphere some are lesse As before ABF is lesse the remanent ABC is a segment greater then the halfe sphere ¶ A Corollary added by the same Flussas By the foresayd assumpt it is manifest that if from the centre of a sphere the lines drawne perpendicularly vnto the circles which cutte the sphere be equall those circles are equall And the perpendicular lines so drawne fall vpon the centres of the same circles For the line which is drawne froÌ„ the centre of the sphere to the circumference containeth in power the power of the perpendicular line and the power of the line which ioyneth together the endes of those lines Wherfore froÌ„ that square or power of the line from the center of the sphere to the circumference or coÌ„mon sectioÌ„ drawne which is the semidiameter of the sphere taking away the power of the perpendicular which is coÌ„mon to them â—ound about it followeth that the residues how many so euer they be be equall powers and therefore the lines are equall the one to the other Wherefore they will describe equall circles by the first definition of the third And vpon their centers fall the perpendicular lines by the 9. of the third And those circles vpon which falleth the greater perpendicular lines are the lesse circles For the powers of the lines drawne from the centre of the sphere to the circumference being alwayes one and equall to the powers of the perpendicular lines and also to the powers of the lines drawne from the centres of the circles to their circumference the greater that the powers of the perpendicular lines taken away from the power contayning them both are the lesse are the powers and therefore the lines remayning which are the semidiameters of the circles and therefore the lesse are the circles which they describe Wherefore if the circles be equall the perpendicular lines falling from the centre of the sphere vpon theÌ„ shall also be equall For if they should be greater or lesse the circles should be vnequall as it is before manifest But we suppose the perpendiculars to be equall Also the perpendicular lines falling vpon those bases are the least of all that are drawne from the centre of the sphere for the other drawne from the centre of the sphere to the circumference of the circles are in power equall both to the powers of the perpendiculars and to the powers of the lines ioyning these perpendiculars and these subtendent lines together making triangles rectangleround about as most easily you may conceaue of the figure here annexed A the Center of the Sphere AB the lines from the Center of the Sphere to the Circumference of the Circles made by the Section BCB the Diameters of Circles made by the Sections AC the perpendiculars from the Center of the Sphere to the Circlesâ— whose diameters BCB are one both sides or in any situation els CB the Semidiameters of the Circles made by the Sections AO a perpendicular longe then AC and therefore the Semidiameter OB is lesse ACB AOB triangles rectangle ¶ The 2. Probleme The 17. Proposition Two spheres consisting both about one the selfe same ceÌ„tre being geueÌ„ to inscribe in the greater sphere a solide of many sides which is called a Polyhedron which shall not touch the superficies of the lesse sphere SVppose that there be two spheres about one the selfe same ceÌ„tre namely about A. It is required in the greater sphere to inscribe a Polyhedron or a solide of many sides which shal not with his superficies touch the superficies of the lesse sphere Let the spheres be cut by some one plaine superficies passing by the center A. Then shall their sectioÌ„s be circles For by the 12. definition of the eleuenth the Diameter remaining fixed and the semicircle being turned round about maketh a sphere Wherefore in what positioÌ„ so euer you imagine the
semicircle to be the playne superficies which passeth by it shal make in the superficies of the sphere a circle And it is manifest that is also a greater circle for the diameter of the sphere which is also the diameter of the semicircle and therefore also of the circle is by the 15. of the third greater then all the right lines drawne in the circle or sphere which circles shall haue both one center being both also in that one playne superficies by which the spheres were cut Suppose that that section or circle in the greater sphere be BCDE and in the lesse be the circle FGH Drawe the diameters of those two circles in such sorte that they make right angles and let those diameters be BD and CE. And let the line AG being part of the line AB be the semidiameter of the lesse sphere and circle as AB is the semidiameter of the greater sphere and greater circle both the spheres and circles hauing one and the same center Now two circles that is BCDE and FGH consisting both about one and the selfe same centre being geuen let there be described by the proposition nexte going before in the greater circle BCDE a poligonon figure consisting of equall and euen sides not touching the lesse circle FGH And let the sides of that figure in the fourth part of the circle namely in BE be BK KL LM and ME. And draw a right line from the point K to the point A and extende it to the point N. And by the 12. of the eleuenth from the A rayse vp to the superficies of the circle BCDE a perpeÌ„dicular line AX and let it light vppon the superficies of the greater sphere in the point X. And by the line AX and by either of these lines BD and KN extend playne superficieces Now by that which was before spokeÌ„ those plaine superficieces shal in the superficies of the sphere make two greater circles Let their semicircles consisting vpon the diameters BD and KN be BXD and KXN And forasmuch as the line XA is â—rected perpendicularly to the playne superficies of the circle BCDE Therfore al the plaine superficieces which are drawne by the line XA are erected perpendicularly to the superficies of the circle BCDE by the 18. of the eleuenth Wherefore the semicircles BXD and KXN are erected perpendicularly to the playne superficies of the circle BCDE And forasmuch as the semicircles BED BXD and KXN are equall for they consist vpon equall diameters â—D and KN therefore also the fourth parts or quarters of those circles namely BE BX and KX are equal the one to the other Wherefore how many sides of a poligonon figure there are in the fourth parte or quarter BE so many also are there in the other fourth partes or quarters BX and KX equall to the right lines BK KL LM and ME. Let those sides be described and let them be BO OP PR RX KS ST TV and VX and drawe these right lines SO Tâ— and Vâ— And from the pointes O and â— Drawe to the playne superficies of the circle BCDE perpendicular lines which perpendicular lines will fall vpon the common section of the plaine superficieces namely vpon the lines BD KN by the 38. of the eleuenth for that the playne superficieces of the semicircles BXD KXN are erected perpendicularly to the playne superficies of the circle BCDE Let those perpendicular lines be GZ and SW And drawe a right line from the point Z to the point W. And forasmuch as in the equall semicircles BXD and KXN the right lines BO and KS are equall from the ends wherof are drawne perpeÌ„dicular lines OZ and SW therfore by the corollary of the 35. of the eleueÌ„th the line OZ is equall to the line SW the line BZ is equal to the line KW Flussas proueth this an other way thus Forasmuch as in the triangles SWK and OZB the two angles SWK and OZB are equal for that by coÌ„struction they are right angles and by the 27. of third the angles WKS and ZBO are equall for they subtend equal circumferences SXN and OXD and the side SK is equall to the side OB as it hath before ben proued Wherefore by the 26. of the first the other sides angles are equall namely the line OZ to the lines SW and the line BZ to the line KW But the whole line BA is equall to the whole line KA by the definition of a circle Wherfore the residue ZA is equall to the residue WA Wherfore the line ZW is a parallel to the line BK by the 2. of the sixt And forasmuch as either of these lines OZ SW is erected perpendicularly to the playne superficies of the circle BCDE therefore the line OZ is a parallell to the line SW by the 6. of the eleueÌ„th and it is proued that it is also equal vnto it Wherfore the lines WZ and SO are also equall and parallels by the 7 of the eleuenth and the 33. of the first and by the 3. of the first And forasmuch as WZ is a parallell to SO But ZW is a parallell is to KB Wherefore SO is also a parallell to KB by the 9. of the eleuenth And the lines BO and KS do knit them together Wherefore the fower sided figure BOKS is in one and the selfe same playne superficies For by the 7. of the eleuenth if there be any two parallell right lines and if in either of them be taken a point at allauentures a right line drawn by these points is in one the selfe same playne superficies with the parallels And by the same reason also euery one of the fower sided figures SOPT and TPKV is in one and thâ— selfe same playne superficies And the triangle VRX is also in one and the selfe same plaine superficies by the â— of the eleuenth Now if we imagine right lines drawne froÌ„ the pointes O S P T R V to the point 〈◊〉 there shal be described a PolyhedroÌ„ or a solide figure of many sides betwene the circuÌ„ferences BX and KX composed of pyramids whose bases are the fower sided figures BKOS SOPT TPRV and the triangle VRX and toppe the point A. And if in euery one of the sides Kâ—L LM and ME we vse the selfe same construction that we did in BK and moreouer in the other three quadrants or quarters and also in the other halfe of the sphere there shall then be made a Polyhedron or solide figure consisting of many sides described in the sphere which Polyhedron is made of the pyramids whose bases are the foresayd fower sided figurâ—s and the triangle VRK and others which are in the selfe same order with them and common toppe to them all in the point A. Now I say that the forsayd polihedron solide of many sides toucheth not the superficies of the lesse sphere in which is the circle FGH Draw by the 11. of the eleuenth froÌ„ the poynt A to the
line KG into two equall parts in the point M. And draw a line from M to G. And forasmuch as by construction the line KH is equall to the line AC and the line HL to the line CB therefore the whole line AB is equal to the whole line KL Wherefore also the halfe of the line KL namely the line LM is equal to the semidiameter BN wherefore taking away from those equall lines equall parts BC and LH the residues NC and MH shal be equall Wherefore in the two triangles MHG and NCD the two sides about the equall right angles DCN and GHM namely the sides DC CN and GH HM are equall wherfore the bases MG and ND are equall by the 4. of the first And by the same reason may it be proued that right lines drawen from the poynt M to the points E and F are equal to the line ND But the right line ND is equall to the line AN which is drawen from the centre to the circumference wherefore the line MG is equall to the line MK also to the lines ME MF and ML Wherfore making the ceÌ„tre the poynt M and the space MK or MG describe a semicircle KGL and the diameter KL abiding fixed let the sayd semicircle KGL be moued rounde about vntill it returne to the same place from whence it began to be moued and there shal be described a sphere about the centre M by the 12. diffinition of the eleuenth touching euery one of the angles of the Pyramis which are at the points K E F G for those angles are equally distaÌ„t from the centre of the sphere namely by the semidiameter of the sayd sphere as hath before bene proued Wherefore in the sphere geuen whose diameter is the line KL or the line AB is inscribed a Tetrahedron EFGK Now I say that the diameter of the sphere is in power sesquialtera to the side of the Pyramis For forasmuch as the line AC is double to the line CB by coÌ„structioÌ„ therfore the line AB is treble to the line BC. Wherfore by conuersion by the corollary of the 19. of the fiueth the line AB is sesquialtera to the line AC But as the line BA is to the line AC so is the square of the line BA to the square of the line AD. For if we draw a right line froÌ„ the point B to the point D as the line BD is to the line AD so is the same AD to the line AC by reasoÌ„ of the likenes of the ttriangles DAB DAC by the 8. of the sixth by reason also that as the first is to th third so is the square of the first to the square of the second by the corollary of the 20. of the sixth Wherfore the square of the line BA is sesquialter to the square of the line AD. But the line BA is equal to the diameter of the sphere geueÌ„ namely to the line KL as hath bene proued the line AD is equal to the side of the pyramis inscribed in the sphere Wherfore the diameter of the sphere is in power sesquialter to the side of the pyramis Wherfore there is made a pyramis comprehended in a sphere geuen and the diameter of the sphere is sesquialtera to the side of the pyramis which was required to be done and proued ¶ An other demonstration to proue that as the line AB is to the line BC so is the square of the line AD to the square of the line DC Let the description of the semicircle ADB be as in the first description And vpon the line AC describe by the 46. of the first a square EC and make perfecte the parallelograÌ„me FB Now forasmuch as the triangle DAB is equiangle to the triangle DAC by the 32. of the sixt therfore as the line BA is to the line AD so is the line DA to the line AC by the 4. of the sixt Wherefore that which is contained vnder the lines BA and AC is equall to the square of the line AD by the 17. of the sixt And for that as the line AB is to the line BC so is the parallelogramme EB to the parallelogramme FB by the 1. of the sixt and the parallelogramme EB is that which is contained vnder the lines BA and AC for the line EA is equall to the line AC and the parallelogramme BF is that which is contained vnder the lines AC and BC. Wherefore as the line AB is to the line BC so is that which is contained vnder the lines BA and AC to that which is contained vnder the lines AC and CB. But that which is contained vnder the lines BA and AC is equall to the square of the line AD by the Corollary of the 8. of the sixt and that which is contained vnder the lines AC and CB is equall to the square of the line CD for the perpendicular line DC is the meane proportionall betwene the segmentes of the base namely AC and CB by the former Corollary of the 8. of the sixt for that the angle ADB is a right angle Wherefore as the line AB is to the line BC so is the square of the line AD to the square of the line DC by the 11. of the fift which was required to be proued ¶ Two Assumptes added by Campane First Assumpt Suppose that vpon the line AB be erected perpendicularly the line DC which line DC let be the meane proportionall betwene the partes of the line AB namely AC CB so that as the line AC is to the line CD so let the line CD be to the line CB. And vpon the line AB describe a semicircle TheÌ„ I say that the circumference of that semicircle shall passe by the point D which is the end of the perpeÌ„dicular line But if not then it shall either cut the line CD or it shall passe aboue it and include it not touching it First let it cut it in the point E. And drawe these right lines EB and EA Wherfore by the 31. of the third the whole angle AEB is a right angle Wherefore by the first part of the Corollary of the 8. of the sixt the line AC is to the line â—C as the line EC is to the line CB. But by the 8. of the fift the proportion of the line AC to the line EC is greater then the proportion of the same line AC to the line CD for the line CE is lesse then the line CD Now for that the line CE is to thâ— line CB as the line AC is to the line CE and the line CD is to the line CB as the line AC is to the line CD therefore by the 13. of the fift the proportion of the line EC to the line CB is greater then the proportion of the line CD to the line CB. Wherefore by the 10. of the fift the line EC is greater then the line DC
right linâ—s Now then multiply the 20. triangles into the sides of one of the triangles and so shall there be produced 6â— â—he halfe of which is 30. And so many sides hath an Icosahedron And in like sort in a dodecahedron forasmuch as 12. pentagons make a dodecahedron and euery pentagon contayneth â— right linesâ— multiply â—â— into 12. and there shall be produced 60. the halfe of which is 30. And so many are the sides of a dodecahedron And the reason why we take the halfe iâ— for that euery side whether it be of a triangle or of a pentagon or of a square as in a cube â—s taken twise And by the same reason may you finde out how many sides are in a cube and in a pyramis and in an octohedron But now agayne if ye will finde out the number of the angles of euery one of the solide figures when ye haue done the same multiplication that ye did before diâ—idâ— the same sides by the number of the plaine superficieces which comprehend one of the angles of the solides As for example forasmuch as 5. triangles contayne the solide angle of an Icosahedron diuide 60. by 5. and there will come forth 12. and so many solide angles hath an Icosahedâ—on In a dodecahedron forasmuch as three pentagons comprehend an angle diuide 60. by 3. and there will come forth 20 and so many are the angles of a dodecahedron And by the same reason may you finde out how many angles are in eche of the rest of the solide figures If it be required to be knowne how one of the plaines of any of the fiue solides being geuen there may be found out the inclination of the sayd plaines the one to the other which contayne eche of the solides This as sayth Isidorus our greate master is foâ—â—d out after this maner It is manifest that in a cube the plaines which contayne iâ— do◠〈◊〉 the one the other by a right angle But in a Tetrahedron one of the triangles being geuen let the endes of one of the sides of the sayd triangle be the centers and let the space be the perpendicular line drawne from the toppe of the triangle to the base and describe circumferâ—nces of a circle which shall cutte the one the other and from the intersection to the centers draw right lines which shall containe the inclination of the plaines coÌ„tayning the Tetrahedron In an Octoâ—edron take one of the sides of the triangle therâ—of and vpon it describe a square and draw the diagonall line and making the centres the endes of the diagonall line and the space likewise the perpendicular line drawne from the toppe of the triangle to the base describe circumferences and agayne from the common section to the centres draw right lines and they shall contayne the inclination sought for In an Icosahedron vpon the side of one of the triangles thereof describe a pentagon and draw the line which subtendeth one of the angles of the sayd pentagon and making the centres the endes of that line and the space the perpendicular line of the triangle describe circumferences and draw from the common intersectioâ— of the circumferences vnto the centres right lines and they shall contayne likewise the inclination of the plaines of the icosahedron In a dodecahedron take one of the pentagons and draw likewise the line which subtendeth one of the angles of the pentagon and making the centres the endes of that line and the space the perpendicular line drawne from the section into two equall partes of that line to the side of the pentagon which is parallel vnto it describe circumferences and from the point of the intersection of the circumferences draw vnto the centres right lines and they shall also containe the inclination of the plaines of the dodecahedron Thus did this most singular learned man reason thinking the deâ—onstration in euery one of them to be plaine and cleare But to make the demonstration of them manifest I think it good to declare and make open his wordesâ— and first in a Tâ—trahedronâ— The ende of the fiuetenth Booke of Euclides Elementes after 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ¶ The 6. Proposition The 6. Probleme In an Octohedron geuen to inscribe a trilater equilater Pyramis SVppose thaâ— the Octohedron whereâ—â— the Tetrahedron is required to be insâ—riâ—â—â— be ABGDEI Take 〈…〉 bases of the Octoâ—â—dron that is 〈…〉 close in the loweâ—â— triangle BGD namely AEâ— HED IGD and let the fourth be AIB which is opposite to the lowest triangle before put namely to EGD And take the centres of those fower bases which let be the pointes H C N â— And vpon the triangle HCN erecte a pyramis HCNL Now â—orasmuch as these two bases of the Octohedron namely AGE and ABI are set vpon the right lines EG and BI which are opposite the one to the otherâ— in the square GEBI of the Octohedron from the poinâ— A draâ—e by the centres of the bases namely by the centres H L perpendicular lines AHF ALK cutting the lines EG and BI 〈◊〉 two equall partes in the pointes F K by the Corollary of the 1â— of the thirtenth Wherfore a right line drawen froÌ„ the point F to the point K shall be a parallel and equall to the sides of the Octohedron namely to â—â— and GI by the 33. of the first And the right line HL which cutteth the 〈…〉 AF AK proportionally for AH and AL are drawen from the centres of equall circles to the circumferences is a parallel to the right line FK by the 2. of the sixth and also to the sides of the Octohedron namely to Eâ— and IG by the 9. of the eleuenth Wherefore as the line AF is to the line AH so is the line FK to the line HL by the 4. of the sixth For the triangles AFK and AHL are like by thâ— Corollary of the 2. of the sixth But the line AF is in sesquialter proportion to the line AH for the side EG maketh HF the halfe of the right line AH by the Corollary of the 12. of the thirtenth Wherfore FK or GI the side of the Octohedron is sesquialter to the righâ—line HL. And by the same reason may we proue that the sides of the Octohedron are sesquialter to the rest of the right lines which make the pyramis HNCI namely to the right lineâ—â— N NC CI LN and CH wherefore those right lines are equall and therefore the triangleâ— which are described of them namely the triangles HCN HNL NCL and CHL. which make the pyramis HNCL are equall and equilater And forasmuch as the angles of the same pyramis namely the angles H C N L do end in the centres of the bases of the Octohedron therefore it is inscribed â—o the same Octohedron by the first definition of this booke Wherefore in an Octohedron â—euen is inscribed a trilâ—ter equilaâ—â—â—â—â—â—amisâ— which was required to â—e donâ— A Corollary The bases of a Pyramis inscribed in an Octohedron are parallels
coÌ„pareth them all with Triangles also together the one with the other In it also is taught how a figure of any forme may be chaunged into a Figure of an other forme And for that it entreateth of these most common and generall thynges thys booke is more vniuersall then is the seconde third or any other and therefore iustly occupieth the first place in order as that without which the other bookes of Eâ—clide which follow and also the workes of others which haue written in Geometry cannot be perceaued nor vnderstanded And forasmuch â—s all the demonstrations and proofes of all the propositions in this whole booke depende of these groundes and principles following which by reason of their playnnes neede no greate declaration yet to remoue all be it neuer so litle obscuritie there are here set certayne shorte and manifesâ— expositions of them Definitions 1. A signe or point is that which hath no part The better to vnderstand what manâ—r of thing a signe or point is ye must note that the nature and propertie of quantitie wherof Geometry entreateth is to be deuided so that whatsoeuer may be deuided into sundâ—y partes is called quantitie But a point although it pertayne to quantitie and hath his beyng in quantitie yet is it no quantitie for that it cannot be deuided Because as the definition saith it hath no partes into which it should be deuided So that a pointe is the least thing that by minde and vnderstanding can be imagined and conceyued then which there can be nothing lesse as the point A in the margent A signe or point is of Pithagoras Scholers after this manner defined A poynt is an vnitie which hath position NuÌ„bers are conceaued in mynde without any forme figure and therfore without matter wheron to 〈◊〉 figure consequently without place and position Wherfore vnitie beyng a part of number hath no position or determinate place Wherby it is manifest that â—umbâ—â— iâ— more simple and pure then is magnitude and also immateriall and so vnity which iâ— the bâ—ginning of number is lesse materiall then a â—igne or poyâ—â— which is the beginnyng of magnitude For a poynt is maâ—eriall and requireth position and place and â—â—â—rby differeth from vnitie â— A line is length â—ithout breadth There pertaine to quantiâ—â—e three dimensions length bredth thicknes or depth and by these thre are all quaÌ„titieâ— measured made known There are also according to these three dimensions three kyndes of continuall quantities a lyne a superficies or plaine and a body The first kynde namely a line is here defined in these wordes A lyne is length without breadth A point for that it is no quantitie nor hath any partes into which it may be deuided but remaineth indiuisible hath not nor can haue any of these three dimensions It neither hath length breadth nor thickenes But to a line which is the first kynde of quantitie is attributed the first dimension namely length and onely that for it hath neither breadth nor thicknes but is conceaued to be drawne in length onely and by it it may be deuided into partes as many as ye list equall or vnequall But as touching breadth it remaineth indiuisible As the lyne AB which is onely drawen in length may be deuided in the pointe C equally or in the point D vnequally and so into as many partes as ye list There are also of diuers other geuen other definitions of a lyne as these which follow A lyne is the mouyng of a poynte as the motion or draught of a pinne or a penne to your sence maketh a lyne Agayne A lyne is a magnitude hauing one onely space or dimension namely length wantyng breadth and thicâ—â—es 3 The endes or limites of a lyne are pointes For a line hath his beginning from a point and likewise endeth in a point so that by this also it is manifest that pointes for their simplicitie and lacke of composition are neither quantitie nor partes of quantitie but only the termes and endes of quantitie As the pointes A B are onely the endes of the line AB and no partes thereof And herein differeth a poynte in quantitie from vnitie in numberâ— for that although vnitie be the beginning of nombers and no number as a point is the beginning of quantitie and no quantitie yet is vnitie a part of number For number is nothyng els but a collection of vnities and therfore may be deuided into them as into his partes But a point is no part of quantitie or of a lyneâ— neither is a lyne composed of pointes as number is of vnities For things indiuisible being neuer so many added together can neuer make a thing diuisible as an instant in time is neither tyme nor part of tyme but only the beginning and end of time and coupleth ioyneth partes of tyme together 4 A right lyne is that which lieth equally betwene his pointes As the whole line AB lyeth straight and equally betwene the poyntes AB without any going vp or comming downe on eyther side A right line is the shortest extension or draught that is or may be from one poynt to an other Archimedes defineth it thus Plato defineth a right line after this maner A right line is that whose middle part shadoweth the exâ—remeâ— As if you put any thyng in the middle of a right lyne you shall not see from the one ende to the other which thyng happeneth not in a crooked lyne The Ecclipse of the Sunne say Astronomers then happeneth when the Sunne the Moone our eye are in one right line For the Moone then being in the midst betwene vs and the Sunne causeth it to be darkened Diuers other define a right line diuersly as followeth A right lyne is that which standeth firme betwene his extremes Agayne A right line is that which with an other line of lyke forme cannot make a figure Agayne A right lyne is that which hath not one part in a plaine superficies and an other erected on high Agayne A right lyne is that all whose partes agree together with all his other partes Agayne A right lyne is that whose extremes abiding cannot be altered Euclide doth not here define a crooked lyne for it neded not It may easely be vnderstand by the definition of a right lyne for euery contrary is well manifested set forth by hys contrary One crooked lyne may be more crooked then an other and from one poynt to an other may be drawen infinite crooked lynes but one right lyne cannot be righter then an other and therfore from one point to an other there may be drawen but one tight lyne As by figure aboue set you may see 5 A superficies is that which hath onely length and breadth A superficies is the second kinde of quantitie and to it are attributed two â—imensions namely length and breadth As in the
be in one and the selfe same plaine superficies wherfore from the one to the other there is some shortest draught whiche is a right line Likewise any two right lines howsoeuer they be set are imagined to be in one superficies and therefore from any one line to any one line may be drawen a superficies 2 To produce a right line finite straight forth continually As to draw in length continually the right line AB who will not graunt For there is no magnitude so great but that there may be a greter nor any so litle but that there may be a lesse And a line is a draught from one point to an other therfore from the point B which is the ende of the line AB may be drawn a line to some other point as to the point C and from that to an otherâ— and so infinitely 3 Vpon any centre and at any distance â—o describe a circle A playne superficies may in compasse be extended infinitely as from any pointe to any pointe may be drawen a right line by reason wherof it commeth to passe that a circle may be described vpon any centre and at any space or distance As vpon the centre A and vpon the space AB ye may describe the circle BC vpon the same centre vpon the distance AD ye may describe the circle DE or vppon the same centre A according to the distaunce AF ye may describe the circle FG and so infinitely extendyng your space 4 All right angles are equall the one to the other This peticion is most plaine and offreth it selfe euen to the sence For as much as a right angle is 〈…〉 riâ—ht lyne falling perpendicularly vppon an other and no one line can fall more perpendicularly vpoÌ„ a line then an otherâ— therfore no one right angle can be greater 〈◊〉 an otherâ— neither dâ— the length or shortenes of the lines alter the greatnes of the angle For in the example the right angle ABC though it be made of much longer lines then the right angle DEF whose lines are much shorter yet is that angle no greater then the other For if ye set the point E â—ust vpon the point B â— then shal the line ED euenly and iustly fall vpon the line AB â— and the line EF shall also fall equally vpon the line BC and so shal the angle DEF be equall to the angle ABC for that the lines which cause them are of like inclination It may euidently also be sene at the centre of a circle For if ye draw in a circle two diameters the one cutting the other in the centre by right angles ye shall deuide the circle into fowre equall partes of which eche contayneth one right angle so are all the foure right angles about the centre of the circle equall 5 VVhen a right line falling vpon â—wo right lines doth make â—n one the selfe same syde the two inwarde angles lesse then two right angles then shal these two right lines beyng produced 〈◊〉 length concurre on that part in which are the two angles lesse then two right angles As if the right line AB fall vpon two right lines namely CD and EF so that it make the two inward angles on the one side as the angles DHI FIH lesse then two right angles as in the example they do the said two lines CD and EF being drawen forth in leÌ„gth on that part wheron the two angles being less◠〈◊〉 â—wo right angleâ— consist shal 〈◊〉 leÌ„gth concurre and meete together as in the point D as it is easie to see For the partes of the lines towardes DF are more enclined the one to the other then the partes of the lines towardes CE are Wherfore the more these parts are produced the more they shall approch neare and neare till at length they shal mete in one point Contrariwise the same lines drawn in leÌ„gth on the other side for that the angles on that side namely the angle CHB and the angle EIA are greater then two right angles so much as the other two angles are lesse thâ—n two right angles shall neuer mete but the further they are drawen the further they shal be distant the one from the other 6 That two right lines include not a superficies If the lines AB and AC being right lines should inclose a superficies they muste of necessitie bee ioyned together at both the endes and the superficies must be betwene theÌ„ Ioyne them on the one side together in the pointe A and imagine the point B to be drawen to the point C so shall the line AB fall on the line AC and couer it and so be all one with it and neuer inclose a space or superficies Common sentences 1 Thinges equall to one and the selfe same thyng are equall also the one to the other After definitions and petitions now are set common sentences which are the third and last kynd of principles Which are certaine general propositioÌ„s commonly known of all men of themselues most manifest cleare therfore are called also dignities not able to be denied of any Peticions also are very manifest but not so fully as are the coÌ„mon sentences and therfore are required or desired to be graunted Peticions also are more peculiar to the arte whereof they are as those before put are proper to Geometry but common sentences are generall to all things wherunto they can be applied Agayne peticions consist in actions or doing of somewhat most easy to be done but common sentences consist in consideration of mynde but yet of such thinges which are most easy to be vnderstanded as is that before set As if the line A be equall to the line B And if the line C be also equall to the line B then of necessitie the lines A and C shal be equal the one to the other So is it in all superâ—iciesses angles numbers in all other things of one kynde that may be compared together 2 And if ye adde equall thinges to equall thinges the whole shal be equall As if the line AB be equal to the line CD to the line AB be added the line BE to the line CD be added also an other line DF being equal to the line BE so that two equal lines namely BE and DF be added to two equall lynes AB CD then shal the whole lyne AE be equall to the whole lyne CF and so of all quantities generally 3 And if from equall thinges ye take away equall thinges the thinges remayning shall be equall As if from the two lines AB and CD being equal ye take away two equall lines namely EB and FD then maye you conclude by this common sentence that the partes remayning namely AE and CF are equall the one to the other and so of all other quantities 4 And if from vnequall thinges ye take away equall thinges the thynges which remayne shall be vnequall As if the lines AB and CD be vnequall
namely AB AC and BC and therfore it is an equilater triangle by the definition of an equilater triangle and this is the first reason That the three lines be equall is thus proued The lines AC and CB are equall to the line AB wherfore they are equall the one to the other and this is the second reason That the lines ABâ— and BC are equal is thus proued The lines AB and AC are drawen from the centre of the circle ACE to the circumference of the same wherfore they are equall by the definitioÌ„ of a circle and this is the third reason Likewise that the lines AC and AB are equall is proued by the same reason For the lines AC and AB are drawn from the centre of the circle BCD wherfore they are equall by the same definition of a circle this is the fourth reason or sillogisme And thus is ended the whole resolution for that you are come to a principle which is indemoÌ„strable can not be resolued Of a â—emonstration leading to an impossibilitie or to an absurditie you may haue an example in the â—ourth proposition of this booke An addition of Campanus BVt nowe if vpon the same line geuen namely AB ye wil describe the other two kinds of triangles namely an Isosceles or a triaÌ„gle of two equal sides a Scalenon or a triangle of three vnequall sides First for the describing of an Isosceles triangle produce the line AB on ether side vntill it concur with the circumferences of both the circles in the pointes D and F and making the centre the point A describe a circle HFG according to the quaÌ„tity of the line AF. Likewise making the centre the poynte B describe a circle HDG according to the quantitie of the line BD. Now theÌ„ these circles shall cut the one the other in two poyntes which let be H and G And lâ—t the â—ndâ—s of the line geuen be ioyned with one of the sayd sections by two right lines which let be AG and BG And forasmuche as these two lines AB and AD are drawen froÌ„ the centre of the circle CDE vnto the circumfereÌ„ce therof therfore ar they equal Likewise the lines BA and BF for that they are drawen from the centre of the circle EACF to the circumference therof aâ—e equal And forasmuch as ether of the lines AD and BF is equall to the line AB therfore they are equal the one to the other Wherfore putting the line AB coÌ„moÌ„ to theÌ„ both the whole line BD shal be equall to the whole line AF. But BD is equal to BG for they are both drawen froÌ„ the ceÌ„tre of the circle HDG to the circumfereÌ„ce therof And likewise by the same reason the line AF is equal to the line AG. Wherfore by the coÌ„moÌ„ senteÌ„ce the lines AG and BG are equal the one to the other and either of them is greater then the line AB for that either of the two lines BD and AF is greater then the line AB Wherfore vpon the line geuen is described an Isosceles or triangle of two equall sides Ye may also describe vpon the selfe same line a Scaleâ—on or triangle of three vnequall sides if by two right lines ye ioyne both the endes of the line gâ—uen to some one point that is in the circumference of one of the two greater circles so that that poynt be not in one of the two sections and that the line DF do not concur with it when it is on either side produced continuallye and directlye For let the poynte K be taken in the circumference of the circle HDG and let it not be in any of the sections neyther let the line DF concur with it when it is produced continually and directly vnto the circumference therof And draw these lines AK and BK and the line AK shal cut the circumference of the circle HFG Let it cut it in the poynte L now then by the common sentence the line BK shal be equal to the line AL for by the definition of a circle the line BK is equall to the line BG and the line AL is equall to the line AG which is equal to the line BG Wherfore the line AK is greater then the line BK and by the same reason maye it be proued that the line BK is greater then the line AB Wherfore the triangle ABK consisteth of three vnequal sides And so haue ye vpon the line geuen described all the kindes of triangles This is to be noted that if a man will mechanically and redely not regarding demonstration vpon a line geuen describe a triangle of three equall sides he needeth not to describe the whole foresayd circle but onely a little part of eche namely where they cut the one the other and so from the point of the section to draw the lines to the endes of the line geuenâ— As in this figure here put And likewise if vpon the said line he will describe a triangle of two equall sydes let him extende the compasse according to the quantitie that he will haue the syde to be whether longer then the line geuen or shorter and so draw onely a litle part of eche circle where they cut the one the other froÌ„ the point of the section draw the lines to the ende of the line geuen As in the figures here put Note that in this the two sydes must be such that beyng ioyned together they be longer then the line geuen And so also if vpon the sayd right line he will describe a triangle of three vnequal sydes let him extend the compasse First according to the quantitie that he will haue one of the vnequall sydes to be and so draw a little part of the circle then extend it according to the quantitie that he wil haue the other vnequal syde to be and draw likewyse a little part of the circle and that done from the point of the section draâ— the lines to the endes of the line geuen as in the figure here put Note that in this the two sides must be such that the circles described according to their quaÌ„titie may cut the one the other The second Probleme The second Proposition FroÌ„ a point geuen to draw a right line equal to a rightline geuen SVppose that the point geueÌ„ be A let the right line geuen be BC. It is required froÌ„ the point A to drawe a right lyne equall to the line BC. Draw by the first peticioÌ„ from the point A to the poynte B a right line AB and vpon the line AB describe by the first propositioÌ„ an equilater triangle and let the same be DAB and exteÌ„d by the second peticioÌ„ the right lines DA DB to the poyntes E and F by the third peticioÌ„ making the centre B and the space BC describe a circle CGH againe by the same making the centre D and the space DG describe a circle GKL
And forasmuch as the pointe B is the centre of the circle CGH therfore by the 15. definitioÌ„ the line BC is equal to the line BG and forasmuch as the poynt D is the centre of the circle GKL therefore by the same the line DL is equall to the line DG of which the line DA is equall to a line DB by the propositioÌ„ going before wherfore the residue namely the line AL is equal to the residue namely to the line BG by the third common sentence And it is proued that the line BC is equall to the line BG VVherfore eyther of these lines AL BC is equal to the line BG But things which are equall to one and the same thing are also equall the one to the other by the first commoÌ„ sentence VVherfore the line AL is equal to the line BC. VVherfore from the poynt geueÌ„ namely A is drawn a right line AL equall to the right line geuen BC which was required to be done Of Problemes and Theoremesâ— as we haue before noted some haue no cases at all which are those which haue onely one position and constructionâ— and other some haue many and diuers cases which are such propositions which haue diuers descriptions constructions and chaunge their positions Of which sorte is this second proposition which is also a Probleme This proposition hath two thinges geuen Namely a pointe and a line the thing reâ—uired is that from the pointe geuen wheresoeuer it be put be drawen a line equall to the line geuen Now this poynt geuen may haue diuers positioÌ„s For it may be placed eyther without the right line geuen or in some point in it If it be without it either it is on the side of it so that the right line drawen from it to the ende of the right line geuen maketh an angleâ— or els it is put directly vnto itâ— so that the right line geuen being produced shall fall vpon the point geuen which is without But if it be in the line geuen then either it is in one of the endes or extreames thereof or in some place betwene the extremes So are there foure diuers positions of the poynt in respect of the line Wherupon follow diuers delineations and constructions and consequently varietie of cases To the second case the figure here on the side set belongeth And as touching the order both oâ— construction and of demonstration it is â—ll one with the first The fourth case as touching construction herein differeth from the two firste for thâ—t whereas in theÌ„ you are willed to draw a right line from the poynt geuen namely A to the poynt B which is one of the endes of the line geueÌ„ here you shal not nede to draw that line for that it is already drawen As touching the rest both in construction and demonstration you may proceede as in the two firsteâ— As it is manifeste to see in thys figure here on the side put The 3. Probleme The 3. Proposition Two vnequal right lines being geuen to cut of from the greater a right lyne equall to the lesse SVppose that the two vnequal right lines geuen be AB C of which lâ—t the lyne AB be the greater It is required from the line AB being the greater to cut of a right line equal to the right line C which is the lesse line-drawâ— on by the second proposition froÌ„ the point A a right line equall to the line C and let the same be AD and making the centre A and the space AD describe by the third peticion a circle DEF And forasmuche as the point A is the centre of the circle DEF therfore AE is equal to AD but the line C is equal to the line AD. VVherfore either of these lines AE and C is equall to AD wherfore the line AE is equall to the line C wherfore two vnequall right lines being geuen namely AB and C there is cut of from AB being the greater a right line AE equall to the lesse lyne namely to C which was required to be done This proposition which is a Probleme hath two thinges geuen namely two vnequall right lines the thing required is from the greater to cut of a line equal to the lesse It hath also diuers cases For the lines geuen either are distinct th' one from the other or are ioyned together at one of their endes or they cut the one the other or the one cutteth the other in one of the extreames VVhich may be two wayes For ether the greater cutteth the lesse or the lesse the greater If they cut the one the other either â—ch cutteth th' other into equall partes or into vnequall partes or the one into equall partes and the other into vnequall partes VVhich may happâ—n in two sorts for the greater may be cut into equall partes and the lesse into vnequall partes or contrariwise When the vnequall lines geuen are distinct the one from the other the figure before put serueth But if the one cut the other in one of the extremes As for exaÌ„ple Suppose that the vnequall right lines geuen be AB and CD of which let the line CD be the greater And let the line CD cut the line AB in his extreame C. Then making the centre A and the space AB describe a circle BE. And vpon the line AC describe an equilater triangle by the â—irst which let bâ— AEC produce the lines EA and EC And againe making the ceÌ„tre E and the space EF describe a ciâ—â—â—e GF Likewise making the centre C and the space CG describe a circle GL Now forasmuch as the line EF is equall to the line EG â—or E is the centre of which the line EA is equall to the line Eâ— thâ—rfore the residue AF is equalâ— to the residue CG â—ut the line AF is equall to the line AB for A is the centre whereâ—oâ—e also the line CG is equall to the line AB But the line CG is also equâ—ll â—o the line CL for the point C is the centre Wherefore the line AB is equall to the line CL. Wherefore from the line CD is cut of the line CL which is equall to the line AB Or it is lesse then the halfe and then making the centre C the space CD describe a circle which shall cut of from the line AB a line equal to the line CD Or it is greater then the half And theÌ„ vnto the point A put the line AF equall to the line CD by the second And making the centre A the space AF describe a circle which shall cut of from the line AB a line equall to the line AF that is vnto the line CD But if it be greater then the halfe then againe vnto the point A put the line AF equal to the line CD by the second propositioÌ„ making the centre A and the space AF describe a circle which shall cut of from the line
supposed that the angles at the base be equall VVhich in the former proposition was the conclusion And the conclusion is that the two sydes subtending the two angles are equall which in the former proposition was the supposition This is the chiefest kind of conuersion vniforme and certayne There is an other kind of conuersion but not so full a conuersion nor so perfect as the first is VVhich happeneth in composed propositions that is in such which haue mo suppositions then one and passe from these suppositions to one conclusion In the coÌ„nuerses of such propositioÌ„s you passe from the conclusion of the first proposition with one or mo of the suppositions of the same conclude some other supposition of the selfe first proposition of this kinde there are many in Euclide Therof you may haue an example in the 8. proposition being the conuerse of the fourâ—h This conuersion is not so vniforme as the other but more diuers and vncertaine according to the multitude of the things geuen or suppositions in the proposition But because in the fifth proposition there are two conclusions the first that the two angles at the baâ—e be equall the second that the angles vnder the baâ—e are equall this is to be noted that this sixt proposition is the conuerse of the â—ame fifth as touching the first conclusion onely You may in like maner make a conuerse of the same proposition touching the second conclusion therof And that after this maner THe two sides of a triangle beyng produced if the angles vnder the base be equall the said triangle shall be an Isosceles triangle In which propositioÌ„ the supposition is that the angles vnder the base are equall which in the fifth proposition was the conclusionâ— the conclusion in this proposition is that the two sides of the triangle are equal which in the fift proposition was the supposition But now for proofe of the said proposition For suppose that AC be equall to AD and produce the line CA to the poynt E and put the line AE equall to the line DB by the third proposition wherefore the whole line CE is equall to the whole line AB by the second common senteÌ„ce Draw a line from the poynt E to the point B. And forasmuch as the line AB is equall to the line EC and the line BC is common to them both and the angle ACB is supposed to be equall to the angle ABC Wherfore by the fourth proposition the triangle EBC is equall to the triangle ABC namelye the whole to the part which is impossible The 4. Theoreme The 7. Proposition If from the endes of one line be drawn two right lynes to any pointe there can not froÌ„ the self same endes on the same side be drawn two other lines equal to the two first lines the one to the other vnto any other point FOr if it be possible then from the endes of one the self same right line namely AB from the pointes I say A and B let there be drawn two right lines AC and CB to the point C and from the same endes of the line AB let there be drawen two other right right lines AD and DB equall to the lines AC and CB the one to the other is eche to his correspondent line and on one and the same side and to an other pointe namely to D so that let CA be equall to DA beyng both drawen from one end that is A let CB be equall to DB beyng both also drawn from one ende that is B. And by the first peticion draw a right line from the point C to the point D. Now forasmuch as AC is equal to AD the angle ACD also is by the 5. proposition equall to the angle ADC wherfore the angle ACD is lesse theÌ„ the angle BDC VVherfore the angle BCD is much lesse then the angle BDC Againe forasmuch as BC is equall to BD and therfore also the angle BCD is equall to the angle BDC And it is proued that it is much lesse then it which is impossible If therfore from the endes of one line be drawen â—wo right lines to any pointe there can not from the selfe same endes on the same side be drawen two other lines equall to the two first lines the one to the other vnto any other point VVhich was required to be demonstrated In this proposition the conclusion is a negation which very rarely happeneth in the mathematicall artes For they euer for the most part vse to conclude affirmatiuely not negatiuely For a propositioÌ„ vniuersall affirmatiue is most agreable to sciences as saith Aristotle and is of it selfe strong and nedeth no negatiue to his proofe But an vniuersall proposition negatiue must of necessitie haue to his proofe an affirmatiue For of onely negatiue propositions there can be no demonstrations And therfore sciences vsing demonstration conclude affirmatiuely and very seldome vse negatiue conclusions An other demonstration after Campanus Suppose that there be a line AB from whose ends A and B let there be drawen two lines AC and BC on one side which let concur in the poynt C. Then I say that on the same side there cannot be drawen two other lines from the endes of the line AB which shall concur at any other poynt so that that which is draweÌ„ from the point A shall be equall to the line AC and that which is drawen from the point B shal be equall to the line BC. For if it be possible let there be drawn two other lines on the selfe same side which let concurre in the point D and let the line AD be equall to the line AC the line BD equall to the line BC. Wherfore the poynt D shall fall either within the triangle ABC or without For it cannot fall in one of the sides for then a parte should be equall to his whole If therfore it fall withoutâ— then either one of the lines AD and DB shall cut one of the lines AC and CB or els neither shall cut neyther Firste let one cut the other and draw a right line from C to D. Now forasmuch as in the triangle ACD the two sides AC and AD are equall therfore the angle ACD is equall to the angle ADC by the fifth propositioÌ„ likewise forasmuch as in the triaÌ„gle BCD the two sides BC and BD are equall therfore by the same the angles BCD BDC are also equall And forasmuch as the angle BDC is greter theÌ„ the angle ADC it followeth that the angle BCD is greater then the angle ACD namely the part greater then the whole which is impossible But if the point D fal without the triangle ABC so that the lines cut not the one the other draw a line from D to C. And produce the lines BD BC beyond the base CD vnto the points E F. And forasmuch as the lines AC and AD are equall the angles
ACD and ADC shall also be equall by the fifth propositionâ— likewise for asmuch as the lines BC and BD are equal the angles vnder the base namely the angles FDC and ECD are equall by the seconde part of the same proposition And for as much as the angle ECD is lesse then the angle ACD It followeth that the angle FDC is lesse theÌ„ the angle ADC which is impossible for that the angle AD C is a part of the angle FD C. And the same inconuenience will follow if the poynt D fall within the triangle ABC The fift Theoreme The 8. Proposition If two triangles haue two sides of th' one equall to two sides of the other eche to his correspondent side haue also the base of the one equall to the base of the other they shall haue also the angle contained vnder the equall right lines of the one equall to the angle contayned vnder the equall right lynes of the other SVppose that there be two triangles ABC and DEF let these two sides of the one AB and AC be equall to these two sides of the other DE and DF ech to his correspondent side that is AB to D E and AC to DF let the base of the one namely BC be equal to the base of the other namely to EF. Then I say that the angle BAC is equall to the angle EDF For the triangle ABC exactly agreing with the triangle DE F and the point B being put vpon the point E and the right line BC vpon the right line EF the point C shall exactly agree with the point F for the line BC is equall to the line EF And BC exactly agreeing with EF the lines also BA and AC shall exactly agree with the lines ED DF. For if the base BC do exactly agree with the base FE but the sides BA AC doo not exactly agree with the sides ED DF but differ as FG GF do theÌ„ from the endes of one lyne shal be drawn two right lines to a poynt from the self same endes on the same side shal be drawn two other lines equal to the two first lines the one to the other and vnto an other poynt but that is impossible by the seuenth propositioÌ„ VVherfore the base BC exactly agreeing with the base EF the sides also BA and AC do exactly agre with the sides ED and DF. VVherfore also the angle BAC shall exactly agre with the angle EDF and therfore shall also be equal to it If theâ—fore two triangles haue two sides of the one equall to two sides of the other ech to his correspondent side and haue also the base of the one equall to the base of the other they shall haue also the angle contayned vnder the equall right lines of the one equall to the angle contayned vnder the equall right lines of the other which was required to be proued This Theoreme is the conuerse of the fourth but it is not the chiefest and principall kind oâ— conuersion For it turneth not the whole supposition into the conclusion and the whole conclusion into the supposition For the fourth propositioÌ„ whose conuerse this is is a coÌ„pound â—heoreme hauing two things geueÌ„ or sâ—pposed which are these the one that two sides of the one triaÌ„gle be equal to two sides of the other triaÌ„gle th' other that the angle coÌ„tained of the two sides of th' one is equal to the angle contained of the two sides of th' one but hath amongest other one thing required whiche is that the base of the one is equal to the base of the other Now in this 8. propositioÌ„ being the conuerse therofâ— that the base of the one is equal to the base of th' other is the supposition or the thing geueÌ„ which in the former propositioÌ„ was the conclusioÌ„ And this that two sides of the one are equall to two sides of the other is in this proposition also a supposition like as it was in the former proposition so that it is a thing geuen in either proposition The conclusion of this proposition is that the angle enclosed of the two equall sides of the one triangle is equall to the angle enclosed of the two equall sides of the other triangle which in the former proposition was one of the things geuen Philo and his scholas demonstrate this proposition without the helpe of the former proposition in this maner First let it fall directlye And forasmuche as the line DE is equall to the line E G and DFG is one righte line therfore DEG is an Isosceles triaÌ„gle and so by the fifth proposition the angle at the point D is equal to the angle at the poynt G which was required to be proued The 4. Probleme The 9. Proposition To deuide a rectiline angle geuen into two equall partes SVppose that the rectiline angle geuen be BAC It is required to deuide the angle BAC into two equal partes In the line AB take a point at all aduentures let the same be D. And by the third proposition from the lyne AC cutte of the line AE equall to AD. And by the first peticion draw a right line from the point D to the point E. And by the first proposition vpon the line DE describe an equilater triangle and let the same be DFE and by the first peticion drawe a right line from the poynte A to the point F. Then I say that the angle BAC is by the line AF deuided into two equal partes For forasmuch as AD is equall to AE and AF is coÌ„mon to them both therfore these two DA and AF are equall to these two EA and AF the one to the other But by the first proposition the base DF is equall to the base EF wherfore by the 8. proposition the angle DAF is equal to the angle FAE VVherfore the rectiline angle geuen namely B AC is deuided into two equal partes by the right line AFâ— VVhich was required to be done In this proposition is not taught to deuide a right lined angle into mo partes then two albeit to deuide an angle so it be a right angle into three partes it is not hard And it is taught of Viâ—ellio in his first boke of Perspectiue the 28. Proposition â—or to deuide an acute angle into three equal partes is as saith Proclus impossible vnles it be by the helpe of other lines which are of a mixt nature Which thing Nicomedes did by such lines which are called Concoideâ— linea who first serched out the inuention nature properties of such lines And others did it by other meanes as by the helpe of quadrant lines inuented by Hippias Nicomedes Others by Helices or Spiral lines inuented of Archimedes But these are things of much difficulty and hardnes and not here to be intreated of Here against this proposition may of the aduersary be brought an instance For he may cauill that the
hed of the equilater triangle shall not fall betwene the two right lines but in one of them or without them both As for example There may also in this proposition be diuers casesâ— â—f it so happen that there be no space vnder the base DE to describe an equilater triangle but that of necessitie you must describe it on the same side that the lines AB and AC are For then the sides of the equilater triangle either exactly agree with the lines AD and AE if the said lines AD and AE be equall with the base DE. Or they fall without them if the lines AD and AE be lesse then the base DE. Or they fall within them if the said lines be greater theâ— the base DE. The 5. Probleme The 10. Proposition To deuide a right line geuen being finite into two equall partes SVppose that the right line geuen be AB It is required to deuide the line AB into two equal partes Describe by the first proposition vpon the line AB an equilater triangle and let the same be ABC And by the former proposition deuide the angle ACB into two equall partes by the right line CD Then I say that the right line AB is deuided into two equall partes in the poynt D. For forasmuch as by the first proposition AC is equall to CB and CD is common to theÌ„ both therfore these two lines AC CD are equal to these two lines BC CD the one to the other and the angle ACD is equall to the angle BCD VVherfore by the 4. proposition the base AD is equall to the base BD. VVherefore the righte line geuen AB is deuided into two equall partes in the poynt D which was required to be done Apollonius â—eacheth to deuide a right line being finite into two equall partes after this manner The 6. Probleme The 11. Proposition Vpon a right line geuen to rayse vp from a poynt geuen in the same line a perpendicular line SVppose that the right line geuen be AB let the point in it geuen be C. It is required from the poynte C to rayse vp vnto the right line AB a perpendicular line Take in the line AC a poynt at all aduentures let the same be D and by the 3. proposition put vnto DC an equall line CE. And by the first proposition vpon the line DE describe an equilater triangle FDE draw a line froÌ„ F to C. Then I say that vnto the right line geuen AB and from the poynte in it geuen namely C is raysed vp a perpendicular line FC For forasmuch as DC is equal to CE the line CF is coÌ„mon to them both therfore these two DC and CF are equal to these two EC CF the one to the other and by the first proposition the base DF is equal to the base EF wherefore by the 8. proposition the angle DCF is equall to the angle ECF and they be side angles But wheÌ„ a right line standing vpon a right line doth make the two side angles equall the one to the other ether of those equall angles is by the. 10. definition a right angle the line standing vpon the right line is called a perpeÌ„dicular line VVherfore the angle DCF thangle FCE are right angles VVherfore vnto the right line geueÌ„ AB froÌ„ the poynt in it C is raysed vp a perpendicular line CF which was required to be done Although the poynte geuen should be set in one of the endes of the righte line geuen it is easy so do it as it was before For producing the line in length from the poynt by the second peticion you may worke as you did before But if one require to erect a right line perpendicularly from the poynt at the end of the lyne without producing the rightlyne that also may well bee done after thys maner Appollonius teacheth to rayse vp vnto a line geuen from a point in it geuen a perpendiculer line after this maner The 7. Probleme The 12. Proposition Vnto a right line geuen being infinite and from a point geuen not being in the same line to draw a perpendicular line LEt the right line geuen being infinite be AB let the point geuen not being in the said line AB be C. It is required from the point geueÌ„ namely C to draw vnto the right line geuen AB a perpendiculer line Take on the other syde of the line AB namely on that syde wherein is not the pointe C a pointe at all aduentures and let the same be D. And making the centre C and the space CD describe by the third peticion a circle and let the same be EFG which let cutte the line AB in the pointes E and G. And by the x. proposition deuide the lyne EG into two equal partes in the point H. And by the first peticion draw these right lines CG CH and CE. Then I say that vnto the right line geuen AB from the point geuen not being in it namely C is drawen a perpendiculer lyne CH. For forasmuch as GH is equall to HE and HC is common to them both therfore these two sydes GH and HC are equall to these two sydes EH HC the one to the otherâ— and by the 15 definitioÌ„ the base CG is equal to the base CE wherfore by the 8. proposition the angle CHG is equall to the angle CHE and they are syde angles but when a right line standing vpon a right line maketh the two syde angles equall the one to the other either of those equall angles is by the 10. definition a right angle and the line standing vpon the sayde right line is called a perpendiculer line VVherfore vnto the right line geueÌ„ AB and from the point geuen C which is not in the line AB is drawn a perpendiculer line CH which was required to be done This Probleme did Oenâ—pides first finde out considering the necessary vse therof to the study of Astronomy There arâ— two kindes of perpendiculer lines wherof one is a plaine perpendiculer lyne the other is a solide A plaine perpendiculer line is when the point from whence the perpendiâ—uler line iâ— draâ—en is in the same plaine superficies with the line wherunto it is a perpendicular A solide perpendiculer line is wheÌ„ the point from whence the perpendiculer is drawne is on high and wiâ—hout the plaine superficies So that a plaine perpendiculer line is drawen to a right line a solide perpendiculer line is drawn to a superficies A plainâ— perpendiculer line causeth right angles with one onely line namely with that vpon whome it falleth But a solide perpendiculer line causeth right anâ—leâ— not only with one line but with as many lynes as may be drawn in that superficies by the touch therof This proposition teacheth to draw a plaine perpendiculer line for it is drawn to one line and supposed to
be in the selfe same plaine superficies The 6. Theoreme The 13. Proposition When a right line standing vpon a right line maketh any angles those angles shall be either two right angles or equall to two right angles SVppose that the right line AB standing vppon the right line CD do make these angles CBA and ABD Then I say that the angles CBA and ABD are eyther two right angles or its â—quall to two right angles If the angle CBA be equall to the angle ABD then are they two right angles by the tenth difinition But if not raise vp by the 11. proposition vnto the right line CD and from the pointe geuen in it namelyâ— B a perpendiculer line BEâ— VVherfore by the x. definition the angle CBE and EBD arâ— right angles Now forasmuch as the angle CBE is equall to these two angles CBA and ABE put the angle EBD common to them bothâ— wherfore the angles CBE and EBD are equal to thesâ— three angles CBA ABE and EBD Agayne forasmuch as the angle DBA is equall vnto these two angles DBE and EBA put the angle ABC common to them both wherfore the angles DBA and ABC are equal to these three angles DBE EBA and ABC And it is proued that the angles CBE and EBD are equal to the selfe same three angles but thinges equall to one the self same thingâ— are also by the first commoÌ„ sentence equall the oâ—e to the othe VVherfore the angles CBE and EBD are equall to the angles DBA ABC But the angles CBE and EBD are two right angles wherfore also the angles DBA and ABC are equall to two right angles VVherfore when a right line standing vpon a right line maketh any angles those angles shal be either two right angles or equall to two right angles which was required to be demonstrated An otheâ— demonstration after Pelitarius Suppose that the right line AB do stand vpon the right line CD Then I say that the two angles ABC and ABD are either two right angles or equal to two right angles For if AB be perpeÌ„diculer vnto CD theÌ„ is it manifest that they are right angles by the conuersion of the definition But if it incline towardes the end C then by the 11. proposition from the point B erect vnto the line CD a perpendiculâ—r line BE. By whichâ— construction the propositioÌ„ is very manifest For forasmuch as the angle ABD is greâ—ter then the right angle DBE by the angle ABE â— and the other angle ABC is lesse then the right angle CBE by the selfe same angle ABE if from the greatâ—r bâ—e taken away the excesse and the same bee added to the lesse they shall be made two right angles That is if from the obtuse angle ABD be taken away the anglâ— ABE there shal remayne the right angle DBE And then if the same angle ABE be added to the â—cute angle CBA there shall bee made the right angle CBE Wherefore it is manifest that the two angles namely the obtuse angle ABD the acute angle ABC are equall to the two right angles CBE and DBE which was requâ—red to be proued The 7. Theoreme The 14. Proposition If vnto a right line and to a point in the same line be drawn two right lines noâ— both on one and the same side making the side angles equall to two right angles those two right lynes shall make directly one right line VNto the right line AB to y point in it Bâ— let therâ— be drawn two right lines BC and BD vnto contrary sides making the syde angles namely ABC ABD equall to two right angles Then I say that the right lines BD and BC make both one right line For if CB and BD do not make both one right line let the right line BE be so drawn to BC that they both make one right line Now forasmuch as the right line AB standeth vpon the right line CBE therfore the angles ABC and ABE are equall to two right angles by the 13. proposition But by supposition the angles ABC and ABD are equall to two right angles wherfore the angles CBA and ABE are equall to the angles CBA and ABD take away the angle ABC which is common to them both VVherfore the angle remayning ABE is equall to the angle remaining ABD namely the lesse to the greater which is impossible VVherefore the line BE is not so directly drawen to BC that they both make one right line In like sorte may we proue that no other line besides BD can so be drawneâ— VVherfore the lines CB and BD make both one right line If therfore vnto a right line to a point in the same line be drawn two right lines not both on one and the same side making the side angles equall to two right angles those two lines shal make directly one right line which was required to be proued An other demonstration after Pelitarius Suppose that there be a right line AB vnto whose poiâ—te â— let there be drawen two right lines CB and BD vnto contrary sides and let the two angles CBA and DBA be either two right angles or equall to two right angles Then I say that the two lines CB and BD do make directly one right line namely CD For if they do not theÌ„ lâ—t â—E bâ—â—o dâ—awn vnto CB that they both make directly one right line Câ—â— which shall passe â—ither aboue the line BD or vnder it First lâ—t it passe aboue it And for aâ— much as the two angles CBA and ABE are by the former proposition equall to two right angles and are a part of the two angles CBA and ABD but the angles CBA and Aâ—D aâ—e by supposition equall also to two right anglesâ— therefore the parâ—â— is equall to the whole which is imâ—ossible And the like absurditie will follow if CB E passe vnder the line BD namely that the whole shal be equall to the partâ— which is also impossible Wherefore CD is one right line which was required to be proued The 8. Theoreme The 15. Proposition If two right lines cut the one the other the hed angles shal be equal the one to the other SVppose that these two right lines AB and CD do cut the one the other in the point E. Then I say that the angle AEC is equall to the angle DEB For forasmuch as the right line AE standeth vpon the right line DC making these angles CEA and AED therefore by the 13. propositioÌ„ the angles CEA and AED are equall to two right angles Agayne forasmuch as the right line DE standeth vpon the right line AB making these angles AED and DEB therfore by the same proposition the angles AED and DEB are equall to two right angles and it is proued that the angles CEA and AED are also equall to two right angles VVherfore the angles CEA and AED are equall to the angles AED and DEB Take away the angle AED which
was required to be demonstrated This Proposition may also be demonstrated without producing any of the sides aâ—ter this maner The same may yet also be demonstrated an other way This proposition may yet moreouer be demonstrated by an argument leading to an absurditie and that after this manner A man may also more briefely demonstrate this proposition by Campanus definition of a right line which as we haue before declared is thus A right line is the shortest extension or drawght that is or may be from one point to another Wherfore any one side of a triangle for that it is a right line drawen from some one point to some other one point is of necessitie shorter then the other two sides drawen from and to the same pointes Epicurus and such as followed him derided this proposition not counting it worthy to be added in the number of propositions of Geometry for the easines thereof for that it is manifest euen to the sense But not all thinges manifest to sense are straight wayes manifest to reason and vnderstanding It pertayneth to one that is a teacher of sciences by profe and demonstration to render a certayne and vndoubted reason why it so appeareth to the senseâ— and in that onely consisteth science The 14. Theoreme The 21. Proposition If from the endes of one of the sides of a triangle be drawen to any point within the sayde triangle two right lines those right lines so drawen shal be lesse then the two other sides of the triangle but shall containe the greater angle SVppose that ABC be a triangle and froÌ„ the endes of the side BC namely froÌ„ the pointes B and C let there be drawen within the triangle two right lines BD and CD to the point D. Then I say that the lines BD and CD are lesse then the other sides of the triangle namely then the sides BA and AC and that the angle which they contayne namely BDC is greater then the angle BAC Extend by the second peticion the line BD to the point E. And forasmuch as by the 20. proposition in euery triangle the two sides are greater then the side remayning therefore the two sides of the triangle ABE namely the sides AB and AE are greater then the side EB Put the line EC common to them both VVherefore the lines BA and AC are greater then the lines BE and ECâ— Againe forasmuch as by the same in the triangle CED the two sides CE and ED are greater then the side DC put the line DB common to them bothâ— wherfore the lines CE and ED are greater then the lines CD and DB. But it is proued that the lines BA and AC are greater then the lines BE and EC VVherefore the lines BA and AC are much greater then the lines BD and DC Agayne forasmuch as by the 16. proposition in euery triangle the outward angle is greater then the inward and opposite angle therefore the outward angle of the triangle CDE namely BDC is greater then the angle CED VVherefore also by the same the outward angle of the triangle ABE namely the angle CEB is greater then the angle BAC But it is proued that the angle BDC is greater then the angle CEB VVherfore the angle BDC is much greater then the angle BAC VVherefore if from the endes of one of the sides of a triangle be drawen to any point within the sayde triangle two right lines those right lines so drawn shal be lesse then the two other sides of the triangle but shall contayne the greater angle which was required to be demonstrated In this proposition is expressed that the two right lines drawen within the triangle haue their beginning at the extremes of the side of the triangle For froÌ„ the one extreme of the side of the triangle and from some one point of the same side may be drawen two right lines within the triangle which shall be longer theÌ„ the two outward lines which is wonderfull and seemeth straunge that two right lines drawen vpon a parte of a line should be greater then two right lines drawen vpon the whole line And agayne it is possible from the one extreme of the side of a triangle and from some one point of the same side to drawe two right lynes within the triangle which shall containe an angle lesse then the angle contayned vnder the two outward lines As touching the first part The 8. Probleme The 22. Proposition Of thre right lines which are equall to thre right lines geueÌ„ to make a triangle But it behoueth two of those lines which two soeuer be taken to be greater then the third For that in euery triangle two sides which two sides soeuer be taken are greater then the side remayning SVppose that the three right lines geueÌ„ be A B C of which let two of them which two soeuer be taken be greater then the third that is let the lines A B be greater then the line C and the lines A C then the line B and the lines B C then the line A. It is required of three right lines equall to the right lines A B C to make a triangle Take a right line hauing an appointed ende on the side D and being infinite on the side E. And by the 3. proposition put vnto the line A an equall line DF and put vnto the line B an equall line FG and vnto the line C an equall line GH And making the centre F and the space DF describe by the 3. peticioÌ„ a circle DKL Agayne making the centre G and the space G H describe by the same a circle HKL and let the point of the intersection of the sayd circles be K and by the first peticion draw a right line from the point K to the point F an other from the point K to the point G. Then I say that of thre right lines equall to the lines A B C is made a triangle KFG For forasmuch as the point F is the centre of the circle DKL therefore by the 15. definition the line FD is equall to the line FK But the line A is equall to the line FD VVherfore by the first common sentence the line FK is equall to the line A. Agayne forasmuch as the point G is the centre of the circle LKH therefore by the same definition the line GK is equall to the line GHâ— But the line C is equall to the line GH wherefore by the first common sentence the line KG is equall to the line C. But the line FG is by supposition equal to the line B wherefore these three right lines GF FK and KG are equall to these three right lines A B C. VVherefore of three right lines that is KF FG and GK which are equall to the thre right lines geuen that is to A B C is made a triangle KFG which was required to be done An other construction and demonstration after Flussates Suppose that the
squares which are made of the lines AB and AC Describe by the 46. proposition vpon the line BC a square BDCE and by the same vpon the lines BA and AC describe the squares ABFG and ACKH And by the point A draw by the 31. proposition to either of these lynes BD and CE a parallel line AL. And by the first peticion draw a right lyne from the point A to the point D and an other from the point C to the point E. And forasmuch as the angles BAC and BAG are right angles therfore vnto a right line BA and to a point in it geuen A are drawen two right lines AC and AG not both on one and the same side makyng the two side angles equall to two right angles wherfore by the 14. proposition the lines AC and AG make directly one right line And by the same reason the lines BA and AH make also directly one right line And forasmuch as the angle DBC is equall to the angle FBA for either of theÌ„ is a right angle put the angle ABC common to them both wherfore the whole angle DBA is equall to the whole angle FBC And forasmuch as these two lines AB and BD are equal to these two lines BF and BC the one to the other and the angle DBA is equal to the angle FBC therfore by the 4. proposition the base AD is equall to the base FC and the triangle ABD is equall to the triangle FBC But by the 31. propositionâ— the parallelogramme BL is double to the triangle ABD for they haue both one aud the same base namely BD and are in the selfe same parallel lynes that is BD and AL and by the same the square GB is double to the triangle FBC for they haue both one and the selfe same base that is BF and are in the selfe same parallel lynes that is FB and GC But the doubles of thinges equall are by the sixte common sentence equall the one to the other VVherfore the parallelograme BL is equall to the square GB And in like sorte if by the first peticion there be drawen a right line from the point A to the point E and an other from the point B to the point K we may proue that the parallelograme CL is equal to the square HC VVherfore the whole square BDEC is equall to the two squares GB and HC But the square BDEC is described vpon the line BC and the squares GB and HC are described vppon the lines BA AC wherfore the square of the side BC is equal to the squares of the sides BA and AC VVherefore in rectangle triangles the square whiche is made of the side that subtendeth the right angle is equal to the squares which are made of the sides contayning the right angle which was required to be demonstrated This most excellent and notable Theoreme was first inuented of the greate philosopher Pithagoras who for the exceeding ioy conceiued of the inuention therof offered in sacrifice an Oxe as recorde Hiâ—rone Proclus Lycius Vitruutus And it hath bene commoÌ„ly called of barbarous writers of the latter time Dulcarnon An addition of Pelitarius To reduce two vnequall squares to two equall squares Suppose that the squares of the lines AB and AC be vnequall It is required to reduce them to two equall squares Ioyne the two lines AB and AC at their endes in such sort that they make a right angle BAC And draw a line from B to C. Then vppon the two endes B and C make two angles eche of which may be equal to halfe a right angle This is done by erecting vpon the line BC perpeÌ„diculer lines from the pointes B and C and so by the 9. proposition deuiding â—che of the right angles into two equall partes and let the angles BCD and CBD be either of theÌ„ halfe of a right angle And let the lines BD and CD concurre in the point D. Then I say that the two squares of the sides BD and CD are equall to the two squares of the sides AB and AC For by the 6 proposition the two sides DB and DC are equall and the angle at the pointe D is by the 32. proposition a right angle Wherefore the square of the side BC is equal to the squares of the two sides DB and DC by the 47. proposition but it is also equall to the squares of the two sides AB and AC by the self same proposition wherfore by the common sentence the squares of the two sides BD and DC are equall to the squares of the two sides AB and AC which was required to be done An other addition of Pelitarius If two right angled triangles haue equall bases the squares of the two sides of the one are equall to the squares of the two sides of the other This is manifest by the former construction and demonstration An other addition of Pelitarius Two vnequall lines being geuen to know how much the square of the one is greater then the square of the other Suppose that there be two vnequal lines AB and BC of which let AB be the greater It is required to search out how much the square of AB excedeth the square of BC. That is I wil finde out the square which with the square of the line BC shal be equal to the square of the line AB Put the lines A B and BC directly that they make both one right line and making the centre the point B and the space BA describe a circle ADE And produce the line AC to the circumference and let it concurre with it in the point E. And vpon the lyne AE and froÌ„ the point C erect by the 11. proposition a perpendiculer line CD which produce till it concurre with the circumference in the point D draw a line from B to D. Then I say that the square of the line CD is the excesse of the square of the line AB aboue the square of the line BC. For forasmuch as in the triangle BCD the angle at the point C is a right angle the square of the base BD is equall to the squares of the two sides BC and CD by this 47. proposition Wherefore also the square of the line AB is equall to the selfe same squares of the lines BC and CD Wherefore the square of the line BC is so much lesse then the square of the line AB as is the square of the line CD which was required to search out An other addition of Pelitarius The diameter of a square being geuen to geue the square thereof This is easie to be done For if vpon the two endes of the line be drawen two halfe right angles and so be made perfect the triangle then shal be described half of the square the other halfe whereof also is after the same manner easie to be described Hereby it is manifest that the square of the
the other in the poynt D and that onely poynt is common to them both neither doth the one enter into the other If any part of the one enter into any part of the other then the one cutteth and deuideth the other and toucheth the one the other not in one poynt onely as in the other before but in two pointâ—s and haue also a superficies common to them both As the circles GHK and HLK cut the one the other in two poyntes H and K and the one entreth into the other Also the superficies HK is common to them both For it is a part of the circle GHK and also it is a part of the circle HLK. Right lines in a circle are sayd to be equally distant from the centre when perpendicular lines drawen from the centre vnto those lines are equall And that line is sayd to be more distant vpon whom falleth the greater perpendicular line As in the circle ABCD whose centre is E the two lynes AB and CD haue equall distance from the centre E bycause that the lyne EF drawen from the centre E perpendicularly vpon the lyne AB and the lyne EG drawen likewise perpendilarly from the centre E vpon the lyne CD are equall the one to the other But in the circle HKLM whose centre is N the lyne HK hath greater distance from the centre N then hath the lyne LM for that the lyne ON drawen from the centre N perpendicularly vppon the lyne HK is greater then the lyne NP which is drawen froÌ„ the centre N perpendicularly vpon the lyne LM So likewise in the other figure the lynes AB and DC in the circle ABCD are equidistâ—nt from the centre G â— bycause the lynes OG and GP perpendicularly drawen from the centre G vppon the sayd lynes AB and DC are equall And the lyne AB hath greater distance from the centre G then hath the the lyne EF bycause the lyne OG perpendiculâ—rly drâ—wen from the centre G to the lyne AB is greâ—ter then the lyne HG whiche is perpendicularly drawen from the câ—â—tre G to the lyne EF. A section or segment of a circle is a figure coÌ„prehended vnder a right line and a portion of the circumference of a circle As the figure ABC is a section of a circle bycause it is comprehended vnder the right lyne AC and the circumference of a circle ABC Likewise the figure DEF is a section of a circle for that it is comprehended vnder the right lyne DF and the circuÌ„ference DEF And the figure ABC for that it coÌ„taineth within it the centre of the circle is called the greater section of a circle and the figure DEF is the lesse section of a circle bycause it is wholy without the centre of the circle as it was noted in the 16. Definition of the first booke An angle of a section or segment is that angle which is contayned vnder a right line and the circuÌ„ference of the circle As the angle ABC in the section ABC is an angle of a section bycause it is contained of the circumference BAC and the right lyne BC. Likewise the angle CBD is an angle of the section BDC bycause it is contayned vnder the circumference BDC and the right lyne BC. And these angles are commonly called mixte angles bycause they are contayned vnder a right lyne and a crooked And these portions of circumferences are commonly called arkes and the right lynes are called chordes or right lynes subtended And the greater section hath euer the greater angle and the lesse section the lesse angle An angle is sayd to be in a section wheÌ„ in the circumference is taken any poynt and from that poynt are drawen right lines to the endes of the right line which is the base of the segment the angle which is contayned vnder the right lines drawen from the poynt is I say sayd to be an angle in a section As the angle ABC is an angle is the section ABC bycause from the poynt B beyng a poynt in the circumference ABC are drawen two right lynes BC and BA to the endes of the lyne AC which is the base of the section ABC Likewise the angle ADC is an angle in the section ADC bycause from the poynt D beyng in the circuÌ„ference ADC are drawen two right lynes namely DC DA to the endes of the right line AC which is also the base to the sayd section ADC So you see it is not all one to say an angle of a section and an angle in a section An angle of a section coÌ„sisteth of the touch of a right lyne and a crooked And an angle in a section is placed on the circumference and is contayned of two right lynes Also the greater section hath in it the lesse angle and the lesse section hath in it the greater angle But when the right lines which comprehend the angle do receaue any circumference of a circle then that angle is sayd to be correspondent and to pertaine to that circumference As the right lynes BA and BC which containe the angle AB C and receaue the circumference ADC therfore the angle ABC is sayd to subtend and to pertaine to the circuÌ„ference ADC And if the right lynes whiche cause the angle concurre in the centre of a circle then the angle is sayd to be in the centre of a circle As the angle EFD is sayd to be in the centre of a circle for that it is comprehended of two right lynes FE and FD whiche concurre and touch in the centre F. And this angle likewise subtendeth the circumference EGD whiche circumference also is the measure of the greatnes of the angle EFD A Sector of a circle is an angle being set at the centre of a circle a figure contayned vnder the right lines which make that angle and the part of the circumference receaued of them As the figure ABC is a sector of a circle for that it hath an angle at the centre namely the angle BAC is coÌ„tained of the two right lynes AB and AC whiche contayne that angle and the circumference receaued by them Like segmentes or sections of a circle are those which haue equall angles or in whom are equall angles Here are set two definitions of like sections of a circle The one pertaineth to the angles whiche are set in the centre of the circle and receaue the circumfereÌ„ce of the sayd sections the other pertaineth to the angle in the section whiche as before was sayd is euer in the circumference As if the angle BAC beyng in the centre A and receaued of the circumference BLC be equall to the angle FEG beyng also in the centre E and receaued of the circumference FKG then are the two sections BCL and FGK lyke by the first definition By the same definition also are the other two sections like namely BCD and FGH for that the angle BAC is equall to the
a circle be taken any poynt which is not the centre of the circle and from that poynt be drawen vnto the circumference certaine right lines the greatest of those lines shall be that line wherein is the centre and the lest shall be the residue of the same line And of all the other lines that which is nigher to the line which passeth by the centre is greater then that which is more distant And from that point can fall within the circle on ech side of the least line onely two equall right lines SVppose that there be a circle ABCD and let the diameter thereof be AD. And take in it any poynt besides the centre of the circle and let the same be F. And let the centre of the circle by the 1. of the third be the poynt E. And from the poynt F let there be drawen vnto the circumference ABCD these right lines FD FC and FG. Then I say that the line FA is the greatest and the line FD is the lest And of the other lines the line FB is greater then the line FC and the line FC is greater then the line FG. Drawe by the first petition these right lines BE CE and GE. And for asmuch as by the 20. of the first in euery triangle two sides are greater then the third therefore the lines EB and EF are greater then the residue namely then the line FB But the line AE is equall vnto the line BE by the 15. definition of the first VVherefore the lines BE and EF are equall vnto the line AF. VVherefore the line AF is greater then then the line BF Agayne for asmuch as the line BE is equall vnto CE by the 15. definition of the first and the line FE is common vnto them both therefore these two lines BE and EF are equall vnto these two CE and EF. But the angle BEF is greater then the angle CEF VVherefore by the 24. of the first the base BF is greater then the base CF and by the same reason the line CF is greater then the line FG. Agayne for asmuch as the lines GF and FE are greater then the line EG by the 20. of the first But by the 15. definition of the first the line EG is equall vnto the line ED VVherefore the lines GF and FE are greater then the line ED take away EF which is cōmon to thē both wherfore the residue GF is greater then the residue FD. VVherefore the line FA is the greatest and the line FD is the lest and the line FB is greater then the line FC and the line FC is greater then the line FG. Now also I say that from the poynt F there can be drawen onely two equall right lines into the circle ABCD on eche side of the least line namely FD. For by the 23. of the first vpon the right line geuen EF and to the poynt in it namely E make vnto the angle GEF an equall angle FEH and by the first petition draw a line from F to H. Now for asmuch as by the 15. definition of the first the line EG is equall vnto the line EH and the line EF is common vnto them both therefore these two lines GE and EF are equall vnto these two lines HE and EF and by construction the angle GEF is equall vnto the angle HEF VVherefore by the 4. of the first the base FG is equall vnto the base FH I say moreouer that from the poynt F can be drawen into the circle no other right line equall vnto the line FG. For if it possible let the line◠FK be equall vnto the line FG. And for asmuch as FK is equall vnto FG. But the line FH is equall vnto the line FG therefore the line FK is equall vnto the line FH VVherfore the line which is nigher to the line which passeth by the centre is equall to that which is farther of which we haue before proued to be impossible Or els it may thus be demonstrated Draw by the first petition a line from E to K and for asmuch as by the 15. definitiō of the first the line GE is equall vnto the line EK and the line FE is common to them both and the base GF is equall vnto the base FK therefore by the 8. of the first the angle GEF is equall to the angle KEF But the angle GEF is equall to the angle HEF VVherefore by the first common sentence the angle HEF is equall to the angle KEF the lesse vnto the greater which is impossible VVherefore from the poynt F there can be drawen into the circle no other right line equall vnto the line GF VVherefore but one onely If therefore in the diameter of a circle be taken any poynt which is not the centre of the circle and from that poynt be drawen vnto the circumference certaine right lines the greatest of those right lines shall be that wherein is the centre and the least shall be the residue And of all the other lines that which is nigher to the line which passeth by the centre is greater then that which is more distant And from that poynt can fall within the circle on ech side of the least line onely two equall right lines which was required to be proued ¶ A Corollary Hereby it is manifest that two right lines being drawen frō any one poynt of the diameter the one of one side and the other of the other side if with the diameter they make equall angles the sayd two right lines are equall As in thys place are the two lines FG and FH The 7. Theoreme The 8. Proposition If without a circle be taken any poynt and from that poynt be drawen into the circle vnto the circumference certayne right lines of which let one be drawen by the centre and let the rest be drawen at all aduentures the greatest of those lines which fall in the concauitie or hollownes of the circumference of the circle is that which passeth by the centre and of all the other lines that line which is nigher to the line which passeth by the centre is greater then that which is more distant But of those right lines which end in the conuexe part of the circumference that is the least which is drawen from the poynt to the diameter and of the other lines that which is nigher to the least is alwaies lesse then that which is more distant And from that poynt can be drawen vnto the circumference on ech side of the least onely two equall right lines Now also I say that from the poynt D can be drawen vnto the circumference on eche side of DG the least onely two equall right lines Vpon the right line MD and vnto the poynt in it M make by the 23. of the first vnto the angle KMD an equall angle DMB. And by the first petition drawe a line from D to B. And for asmuch as by the 15.
definition of the first the line MB is equall vnto the line MK put the line MD common to the both wherfore these two lines MK and MD are equall to these two lines BM and MD the one to the other and the angle KMD is by the 23. of the first equall to the angle BMD VVhereâ—ore by the 4. of the first the base DK is equall to the base DB. Or it may thus be demonstrated Draw by the first petition a line from M to N. And for asmuch as by the 15. definition of the first the line KM is equall vnto the line MN and the line MD is common to them both And the base KD is equall to the base DN by supposition therefore by the 8. of the first the angle KMD is equall to the angle DMN But the angle KMD is equall to the angle BMD Wherfore the angle BMD is equall to the angle NMD the lesse vnto the greater which is impossible Wherefore from the poynt D can not be drawen vnto the circumference ABC on eche side of DG the lest more then two equall right lines If therefore without a circle be taken any poynt and from that poynt be drawen into the circle vnto the circumference certaine right lines of which let one be drawen by the centre and let the rest be drawen at all adventures the greatest of those right lines which fall in the concauitie or hollownes of the circumference of the circle is that which passeth by the centre And of all the other lines that line which is nigher to the line which passeth by the centre is greater then that which is more distant But of those right lines which end in the conuexe part of the circumference that line is the lest which is drawen from the poynt to the dimetient and of the other lines that which is nigher to the least is alwayes lesse then that which is more distant And from that poynt can be drawen vnto the circumference on ech side of the lest only two equall right lines which was required to be proued Thys Proposition is called commonly in old bookes amongest the barbarous Caâ—dâ— Panonis that is the Peacockes taile ¶ A Corollary Hereby it is manifest that the right lines which being drawen from the poynt geuen without the circle and fall within the circle are equally distant from the least or from the greatest which is drawen by the centre are equall the one to the other but contrarywyse if they be vnequally distant whether they light vpon the concaue or conuexe circumference of the circle they are vnequall The 8. Theoreme The 9. Proposition If within a circle be taken a poynt and from that poynt be drawen vnto the circumference moe then two equall right lines the poynt taken is the centre of the circle SVppose that the circle be ABC and within it let there be taken the poynt D. And from D let there be drawen vnto the circumference ABC moe then two equall right lines that is DA DB and DC Then I say that the poynt D is the centre of the circle ABC Draw by the first petition these right lines AB and BC and by the 10. of the first deuide theÌ„ into two equall partes in the poyntes E and F namely the line AB in the poynt E and the line BC in the poynt F. And draw the lines ED and FD and by the second petition extend the lines ED and FD on eche side to the poyntes K G and H L. And for asmuch as the line AE is equall vnto the line EB and the line ED is common to them both therefore these two sides AE and ED are equall vnto these two sides BE and ED and by supposition the base DA is equall to the base DB. Wherfore by the 8. of the first the angle AED is equall to the angle BED Wherfore eyther of these angles AED and BED is a right angle Wherefore the line GK deuideth the line AB into two equall partes and maketh right angles And for asmuch as if in a circle a right line deuide an other right line into two equall partes in such sort that it maketh also right angles in the line that deuideth is the centre of the circle by the Correllary of the first of the third Therfore by the same Correllary in the line GK is the centre of the circle ABC And by the same reason may we proue that in the line HL is the centre of the circle ABC and the right lines GK and HL haue no other poynt common to them both besides the poynt D. Wherefore the poynt D is the centre of the circle ABC If therefore within a circle be taken a poynt and from that point be drawen vnto the circumference more then two equall right lines the poynt taken is the centre of the circle which was required to be proued ¶ An other demonstration Let there be taken within the circle ABC the poynt D. And from the poynt D let there be drawen vnto the circumference more then two equall right lines namely DA DB and DC Then I say that the poynt D is the centre of the circle For if not then if it be possible let the point E be the centre and draw a line from D to E and extend DE to the poyntes F and G. Wherefore the line FG is the diameter of the circle ABC And for asmuch as in FG the diameter of the circle ABC is taken a poynt namely D which is not the centre of that circle therefore by the 7. of the third the line DG is the greatest and the line DC is greater then the line DB and the line DB is greateâ— then the line DA. But the lines DC DB DA are also equall by supposition which is impossible Wherefore the poynt E is not the centre of the circle ABC And in like sort may we proue that no other poynt besides D. Wherefore the poynt D is the centre of the circle ABC which was required to be proued The 9. Theoreme The 10. Proposition A circle cutteth not a circle in moe pointes then two FOr if it be possible let the circle ABC cut the circle DEF in mo pointes then two that is in B G H F. And drawe lines froÌ„ B to G and from B to H. And by the 10. of the first deuide either of the lines BG BH into two equall partes in the pointes K and L. And by the 11. of the first from the poynt K raise vp vnto the line BH a perpendicular line KC and likewise from the poynt L raise vp vnto the line BG a perpendicular line LM and extend the line CK to the poynt A and LNM to the poyntes X and E. And for asmuch as in the circle ABC the right line AC deuideth the right line BH into two equall partes and maketh right angles therfore by the 3. of the third in the line AC is the centre of
when perpendicular lines drawen frō the centre to those lines are equall by the 4. definition of the third Wherfore the lines AB and CD are equally distant from the centre But now suppose that the right lines AB and CD be equally distant from the centre that is let the perpendicular line EF be equall to the perpendicular line EG Then I say that the line AB is equall to the line CD For the same order of construction remayning we may in like sort proue that the line AB is double to the line AF and that the line CD is double to the line CG And for asmuch as the line AE is equall to the line CE for they are drawen from the centre to the circumference therfore the square of the line AE is equall to the square of the line CE. But by the 47. of the first to the square of the line AE are equall the squares of the lines EF and FA. And by the selfe same to the square of the line CE are equall the squares of the lines EG and GC Wherfore the squares of the lines EF and FA are equall to the squares of the lines EG and GC Of which the square of the line EG is equall to the square of the line EF for the line EF is equall to the line EG Wherefore by the third common sentence the square remayning namely the square of the line AF is equall to the square of the line CG Wherefore the line AC is equall vnto the line CG But the line AB is double to the line AF and the line CD is double to the line CG Wherefore the line AB is equall to the line CD Wherefore in a circle equall right lines are equally distant from the centre And lines equally distant from the centre are equall the one to the other which was required to be proued ¶ An other demonstration for the first part after Campane Suppose that there be a circle ABDC whose centre let be the poynt E. And draw in it two equall lines AB and CD Then I say that they are equally distant from the centre Draw from the centre vnto the lines AB and CD these perpendicular lines EF and EG And by the 2. part of the 3. of this booke the line AB shall be equally deuided in the poynt F. and the line CD shall be equally deuided in the poynt G. And draw these right lines EA EB EC and ED. And for asmuch as in the triangle AEB the two sides AB and AE are equall to the two sides CD and CE of the triangle CED the base EB is equall to the base ED. therefore by the 8. of the first the angle at the point A shall be equall to the angle at the point C. And for asmuch as in the triangle AEF the two sides AE and AF are equall to the two sides CE and CG of the triangle CEG and the angle EAF is equall to the angle CEG therefore by the 4. of the first the base EF i◠equall to the base EG which for asmuch as they are perpendicular lines therefore the lines AB CD are equally distant frō the centre by the 4. definition of this booke The 14. Theoreme The 15. Proposition In a circle the greatest line is the diameter and of all other lines that line which is nigher to the centre is alwayes greater then that line which is more distant SVppose that there be a circle ABCD and let the diameter thereof be the line AD and let the centre thereof be the poynt E. And vnto the diameter AD let the line BC be nigher then the line FG. Then I say that the line AD is the greatest and the line BC is greater then the line FG. Draw by the 12. of the first from the centre E to the lines BC and FG perpendicular lines EH and EK And for asmuch as the line BC is nigher vnto the centre then the line FG therfore by the 4. definition of the third the line EK is greater then the line EH And by the third of the first put vnto the line EH an equall line EL. And by the 11. of the first from the point L raise vp vnto the line EK a perpendicular line LM and extend the line LM to the poynt N. And by the first petition draw these right lines EM EN EF and EG And for asmuch as the line EH is equall to the line EL therefore by the 14. of the third and by the 4. definition of the same the line BC is equall to the line MN Againe for asmuch as the line AE is equall to the line EM and the line ED to the line EN therefore the line AD is equall to the lines ME and EN But the lines ME and EN are by the 20. of the first greater then the line MN Wherefore the line AD is greater then the line MN And for asmuch as these two lines ME and EN are equall to these two lines FE and EG by the 15. definition of the first for they are drawen from the centre to the circumference and the angle MEN is greater then the angle FEG therefore by the 24. of the first the base MN is greater then the base FG. But it is proued that the line MN is equall to the line BC Wherefore the line BC also is greater then the line FG. Wherefore the diameter AD is the greatest and the line BC is greater then the line FG. Wherefore in a circle the greatest line is the diameter and of all the other lines that line which is nigher to the centre is alwaies greater then that line which is more distant which was required to be proued ¶ An other demonstration after Campane In the circle ABCD whose centre let be the poynt E draw these lines AB AC AD FG and HK of which let the line AD be the diameter of the circle Then I say that the line AD is the greatest of all the lines And the other lines eche of the one is so much greater then ech of the other how much nigher it is vnto the centre Ioyne together the endes of all these lines with the centre by drawing these right lines EB EC EG EK EH and EF. And by the 20. of the first the two sides EF and EG of the triangle EFG shall be greater then the third side FG. And for asmuch as the sayd sides EF EG are equall to the line AD by the definition of a circle therefore the line AD is greater then the line FG. And by the same reason it is greater then euery one of the rest of the lines if they be put to be bases of triangles for that euery two sides drawen frō the centre are equall to the line AD. Which is the first part of the Proposition Agayne for asmuch as the two sides EF and EG of the triangle EFG are equall to the
touching the circle ABC and let the same be DE. And by the first of the same let the point F be the centre of the circle ABC and draw these right lines FE FB and FD. Wherfore the angle FED is a right angle And for asmuch as the right line DE toucheth the circle ABC and the right line DCA cutteth the same therfore by the Proposition going before that which is contayned vnder the lines AD and DC is equall to the square of the line DE. But that which is contayned vnder the lines AD and DC is supposed to be equall to the square of the line DB. Wherefore the square of the line DE is equall to the square of the line DB. Wherefore also the line DE is equall to the line DB. And the line FE is equall to the line FB for they are drawen from the centre to the circumference Now therefore these two lines DE and EF are equall to these two lines DB and BF and FD is a common base to them both Wherefore by the 8. of the first the angle DEF is equall to the angle DBF But the angle DEF is a right angle Wherefore also the angle DBF is a right angle And the line FB being produced shall be the diameter of the circle But if from the end of the diameter of a circle be drawen a right line making right angles the right line so drawen toucheth the circle by the Correllary of the 16. of the third Wherfore the right line DB toucheth the circle ABC And the like demonstration will serue if the centre be in the line AC If therefore without a circle be taken a certaine point and from that poynt be drawen to the circle two right lines of which the one doth cut the circle and the other falleth vpon the circle and that in such sort that the rectangle parallelogramme which is contayned vnder the whole right line which cutteth the circle and that portion of the same line that lieth betwene the poynt and the vtter circumference of the circle is equall to the square made of the line that falleth vpon the circle then the line that so falleth vpon the circle shall touch the circle which was required to be proued ¶ An other demonstration after Pelitarius Suppose that there be a circle BCD whose centre let be E and take a point without it namely A And frō the poynt A drawe two right lines ABD and AC of which let ABD cut the circle in the poynt B let the other fall vpon it And let that which is contained vnder the lines AD and AB be equall to the square of the line AC Then I say that the line AC toucheth the circle For first if the line ABD do passe by the centre draw the right line CE. And by the 6. of the second that which is contayned vnder the lines AD and AB together with the square of the line EB that is with the square of the line EC for the lines EB and EC are equall is equall to the square of the line AE But that which is contained vnder the lines AD and AB is supposed to be equall to the square of the line AC Wherefore the square of the line AC together with the square of the line CE is equall to the square of the line AE Wherefore by the last of the first the angle at the point C is a right angle Wherfore by the 18. of this boke the line AC toucheth the circle But if the line ABD doo not passe by the centre drawe from the point A the line AD in which let be the centre E. And forasmuch as that which is contained vnder this whole line and his outward part is equall to that which is contained vnder the lines AD and AB by the first Corollary before put therefore the same is equall to the square of the line AC wherefore the angle ECA is a right angle as hath before bene proued in the first part of this Proposition And therfore the line AC toucheth the circle Which was required to be proued The ende of the third booke of Euclides Elementes ¶ The fourth booke of Euclides Elementes THIS FOVRTH BOOKE intreateth of the inscription circumscription of rectiline figures how one right lined figure may be inscribed within an other right lined figure and how a right lined figure may be circumscribed about an other right lined figure in such as may be inscribed and circumscribed within or about the other For all right lined figures cannot so be inscribed or circumscribed within or about the other Also it teacheth how a triangle a square and certayne other rectiline figures being regular may be inscribed within a circle Also how they may be circumscribed about a circle Likewise how a circle may be inscribed within them And how it may be circumscribed about them And because the maner of entreatie in this booke is diuers from the entreaty of the former bookes he vseth in this other wordes and termes then he vsed in them The definitions of which in order here after follow Definitions A rectiline figure is sayd to be inscribed in a rectiline figure when euery one of the angles of the inscribed figure toucheth euery one of the sides of the figure wherin it is inscribed As the triangle ABC is inscribed in the triangle DEF because that euery angle of the triangle inscribed namely the triangle ABC toucheth euery side of the triangle within which it is described namely of the triangle DEF As the angle CAB toucheth the side ED the angle ABC toucheth the side DF and the angle ACB toucheth the side EF. So likewise the square ABCD is said to be inscribed within the square EFGH for euery angle of it toucheth some one side of the other So also the Pentagon or fiue angled figure ABCDE is inscribed within the Pentagon or fiue angled figure FGHIK ◠As you see in the figure◠Likewise a rectiline figure is said to be circumscribed about a rectiline figure when euery one of the sides of the figure circumscribed toucheth euery one of the angles of the figure about which it is circumscribed As in the former descriptions the triangle DEF is said to be circumscribed about the triangle ABC for that euery side of the figure circumscribed namely of the triangle DEF toucheth euery angle of the figure wherabout it is circumscribed As the side DF of the triangle DEF circumscribed toucheth the angle ABC of the triangle ABC about which it is circumscribed and the side EF toucheth the angle BCA and the side CD toucheth the angle CAB Likewise vnderstand you of the square EFGH which is circumscribed about the square ABCD for euery side of the one toucheth some one side of the other Euē so by the same reason the Pentagon FGHIK is circumscribed about the Pentagon ABCDE as you see in the figure on the other side And thus may you
of the first either of these lines AB and AD into two equall partes in the pointes E and F. And by the point E by the 31. of the first draw a line EH parallel vnto either of these lines AB and DC and by the same by the point F draw a line FK parallel vnto either of these lines AD and BC. Wherfore euery one of these figures AK KB AH HD AG GC BG and GD is a parallelograme and the sides which are opposite the one to the other are by the 34. of the first equall the one to the other And forasmuch as the line AD is equall vnto the line AB and the halfe of the line AD is the line AE and the halfe of the line AB is the line AF therefore the line AE is equall vnto the line AF wherefore by the same the sides which are opposite are equall Wherefore the line FG is equal vnto the line EG In like sort may we proue that either of these lines GH and GK is equall to either of these lines FG and GE. Wherfore by the first common sentence these foure lines GE GF GH and GK are equall the one to the other Wherfore making the centre G and the space either GE or GF GH or GK describe a circle and it will passe by the pointes E F H K and will touche the right lines AB BC CD and DA. For the angles at the pointes E F H K are right angles For if the circle do cut the right lines AB BC CD and DA then the line which is drawen by the ende of the diameter of the circle making right angles should fall within the circle which is impossible by the 16. of the third Wherfore the centre being the poynt G and the space beyng GE or GF or GH or GK if a circle be described it shall not cut the rigâ—t lines AB BC CD and DA. Wherefore it shall touch them And it is described in the square ABCD wherefore in a square geuen is described a circle which was required to be done The 9. Probleme The 9. Proposition About a square geuen to describe a circle SVppose that the square geuen be AB CD It is required about the square ABCD to describe a circle Drawe right lines from A to C and from D to B let them cut the one the other in the poynt E. And forasmuch as the lyne DA is equall vnto the lyne AB and the line AC is common vnto them both therfore these two lines DA and AC are equall vnto these two lynes BA and AC the one to the other And the base DC is equall vnto the base BC. Wherefore by the 8. of the first the angle DAC is equall vnto the angle BAC Wherefore the angle DAB is deuided into two equall partes by the line AC And in like sorâ— may we proue that euery one of these angles ABC BCD and CDA is deuided into two equall partes by the right lines AC and DB. And forasmuch as the angle DAB is equall vnto the angle ABC and of the angle DAB the angle EAB is the halfe and of the angle ABC the angle EBA is the halfe Therfore the angle EAB is equall vnto the angle EBA wherfore by the 6. of the first the side EA is equall vnto the side EB In like sorte may we proue that either of these right lines EA and EB is equall vnto either of these lines EC and ED. Wherfore these foure lines EA EB EC and ED are equall the one to the other Wherfore making the centre E and the space any of these lines EA EB EC or ED. Describe a circle and it will passe by the pointes A B C D and shall be described about the square ABCD as it is euident in the figure ABCD. Wherfore about a square geueÌ„ is described a circle which was required to be done ¶ A Proposition added by Pelitarius A square circumscribed about a circle is double to the square inscribed in the same circle Thys may also be demonstrated by the equalite of the triangles and squares contayned in the great squares The 10. Probleme The 10. Proposition To make a triangle of two equall sides called Isosceles which shall haue eyther of the angles at the base double to the other angle TAke a right line at all aduentures which let be AB by the 11. of the second let it be so deuided in the pointe C that the rectangle figure comprehended vnder the lines AB and BC be equall vnto the square which is made of the line AC And making the centre the point A the space AB describe by the 3. peticion a circle BDE and by the 1. of the fourth into the circle BDE apply a right line BD equall to the right lyne AC which is not greater then the diameter of the circle BDE And draw lines from A to D and from D to C. And by the 5. of the fourth about the triangle ACD describe a circle ACDF And forasmuch as the rectangle figure contained vnder the lines AB and BC is equall to the square which is made of the line AC For that is by supposition But the line AC is equall vnto the line BD. Wherfore that which is contayned vnder the lines AB and BC is equall to the square which is made of the line BD. And forasmuch as without the circle ACDF is taken a poynt B and from B vnto the circle ACDF are drawen two right lines BCA and BD in such sort that the one of them cutteth the circle and the other endeth at the circumference and that which is contained vnder the lines AB and BC is equall to the square which is made of the line BD therfore by the 17. of the third the line BD toucheth the circle ACDF And forasmuch as the line BD toucheth in the point D and from D where the touche is is drawen a right line DC therefore by the 32. of the same the angle BDC is equall vnto the angle DAC which is in the alternate segment of the circle And forasmuch as the angle BDC is equal vnto the angle DAC put the angle CDA common vnto theÌ„ both Wherfore the whole angle BDA is equal to these two angles CDA DAC But vnto the angles CDA DAC is equall the outward angle BCD by the 32. of the 1. Wherfore the angle BDA is equal vnto the angle BCD But the angle BDA is by the 5. of the first equall vnto the angle CBD for by the 15. definition of y first the side AD is equall vnto the side AB wherfore by the 1. common sentence the angle DBA is equall vnto the angle BCD Wherefore these three angles BDA DBA and BCD are equall the one to the other And forasmuch as the angle DBC is equall vnto the angle BCD the side therfore BD is equall vnto the side DC But the line BD is by supposition
in the middest wherof is a point from which all lines drawen to the circumference therof are equall this definition is essentiall and formall and declareth the very nature of a circle And vnto this definition of a circle is correspondent the deâ—inition of a Sphere geueÌ„ by Theodosius saying that it is a solide oâ— body in the middest whereof there is a point from which all the lines drawen to the circumference are equall So see you the affinitie betwene a circle and a Sphere For what a circle is in a plaine that is a Sphere in a Solide The fulnes and content of a circle is described by the motion of a line moued about but the circumference therof which is the limite and border thereof is described of the end and point of the same line moued about So the fulnes content and body of a Sphere or Globe is described of a semicircle moued about But the Sphericall superficies which is the limite and border of a Sphere is described of the circumference of the same semicircle moued about And this is the superficies ment in the definition when it is sayd that it is contained vnder one superficies which superficies is called of Iohannes de â—acro Busco others the circumference of the Sphere Galene in his booke de diffinitionibus mediciâ— â— geueth yet an other definitioÌ„ of a Sphere by his propertie or coÌ„mon accideÌ„ce of mouing which is thus A Sphere is a figure most apt to all motion as hauing no base whereon th stay This is a very plaine and witty deâ—inition declaring the dignitie thereof aboue all figures generally All other bodyes or solides as Cubes Pyramids and others haue sides bases and angles all which are stayes to rest vpon or impedimentes and lets to motion But the Sphere hauing no side or base to stay one nor angle to let the course thereof but onely in a poynt touching the playne wherein 〈◊〉 standeth moueth freely and fully with out let And for the dignity and worthines thereof this circular and Sphericall motion is attributed to the heauens which are the most worthy bodyes Wherefore there is ascribed vnto them this chiefe kinde of motion This solide or bodely figure is also commonly called a Globe 13 The axe of a Sphere is that right line which abideth fixed about which the semicircle was moued As in the example before geuen in the definition of a Sphere the line AB about which his endes being fixed the semicircle was moued which line also yet remayneth after the motion ended is the axe of the Sphere described of that semicircle Theodosius defineth the axe of a Sphere after this maner The axe of a Sphere is a certayne right line drawen by the centre ending on either side in the superficies of the Sphere about which being fixed the Sphere is turned As the line AB in the former example There nedeth to this definition no other declaration but onely to consider that the whole Sphere turneth vpon that line AB which passeth by the centre D and is extended one either side to the superficies of the Sphere wherefore by this definition of Theodosius it is the axe of the Sphere 14 The centre of a Sphere is that poynt which is also the centre of the semicircle This definition of the centre of a Sphere is geuen as was the other definition of the axe namely hauing a relation to the definition of a Sphere here geuen of Euclide where it was sayd that a Sphere is made by the reuolution of a semicircle whose diameter abideth fixed The diameter of a circle and of a semicrcle is all one And in the diameter either of a circle or of a semicircle is contayned the center of either of them for that they diameter of eche euer passeth by the centre Now sayth Euclide the poynt which is the center of the semicircle by whose motion the Sphere was described is also the centre of the Sphere As in the example there geuen the poynt D is the centre both of the semicircle also of the Sphere Theodosius geueth as other definition of the centre of a Sphere which is thus The centre of a Sphere is a poynt with in the Sphere from which all lines drawen to the superficies of the Sphere are equall As in a circle being a playne figure there is a poynt in the middest from which all lines drawen to the circumfrence are equall which is the centre of the circle so in like maner with in a Sphere which is a solide and bodely figure there must be conceaued a poynt in the middest thereof from which all lines drawen to the superficies thereof are equall And this poynt is the centre of the Sphere by this definition of Theodosius Flussas in defining the centre of a Sphere comprehendeth both those definitions in one after this sort The centre of a Sphere is a poynt assigned in a Sphere from which all the lines drawen to the superficies are equall and it is the same which was also the centre of the semicircle which described the Sphere This definition is superfluous and contayneth more theÌ„ nedeth For either part thereof is a full and sufficient diffinition as before hath bene shewed Or ells had Euclide bene insufficient for leauing out the one part or Theodosius for leauing out the other Paraduenture Flussas did it for the more explication of either that the one part might open the other 15 The diameter of a Sphere is a certayne right line drawen by the ceÌ„tre and one eche side ending at the superficies of the same Sphere This definitioÌ„ also is not hard but may easely be couceaued by the definitioÌ„ of the diameter of a circle For as the diameter of a circle is a right line drawne froÌ„ one side of the circuÌ„frence of a circle to the other passing by the centre of the circle so imagine you a right line to be drawen from one side of the superficies of a Sphere to the other passing by the center of the Sphere and that line is the diameter of the Sphere So it is not all one to say the axe of a Sphere and the diameter of a Sphere Any line in a Sphere drawen from side to side by the centre is a diameter But not euery line so drawen by the centre is the axe of the Sphere but onely one right line about which the Sphere is imagined to be mouedâ— So that the name of a diameter of a Sphere is more general then is the name of an axe For euery axe in a Sphere is a diameter of the same but not euery diameter of a Sphere is an axe of the same And therefore Flussas setteth a diameter in the definition of an axe as a more generall word â—n this maner The axe of a Sphere is that fixed diameter aboue which the Sphere is moued A Sphere as also a circle may haue infinite diameters but it can haue but
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
same superficies Wherefore these right lines AB BD and DC are in one and the selfe same superficies and either of these angles ABD and BDC is a right angle by supposition Wherefore by the 28. of the first the line AB is a parallel to the line CD If therefore two right lines be erected perpendicularly to one and the selfe same playne superficies those right lines are parallels the one to the other which was required to be proued Here for the better vnderstanding of this 6. proposition I haue described an other figure as touching which if ye erect the superficies ABD perpendicularly to the superficies BDE and imagine only a line to be drawne from the poynt A to the poynt E if ye will ye may extend a thred from the saide poynt A to the poynt E and so compare it with the demonstration it will make both the proposition and also the demonstration most cleare vnto you ¶ An other demonstration of the sixth proposition by M. Dee Suppose that the two right lines AB CD be perpendicularly erected to one the same playne superficies namely the playne superficies OP Then I say that â—â— and CD are parallels Let the end points of the right lines AB and CD which touch the plaine supâ—â—â—â—cies Oâ— be the poyntes â— and D froÌ„â— to D let a straight line be drawne by the first petition and by the second petition let the straight line â—D be exteÌ„ded as to the poynts M N. Now forasmuch as the right line AB from the poynt â— produced doth cutte the line MN by construction Therefore by the second proposition of this eleuenth booke the right lines AB MN are in one plainâ— superficies Which let be QR cutting the superficies OP in the right line MN By the same meanes may we conclude the right line CD to be in one playne superficies with the right line MN But the right line MN by supposition is in the plaine superficies QR wherefore CD is in the plaine superficies QR And Aâ— the right line was proued to be in the same plaine superficies QR Therfore AB and CD are in one playne superficieâ— namely QR And forasmuch as the lines Aâ— and CD by supposition are perpendicular vpon the playne superficies OP therefore by the second definition of this booke with all the right lines drawne in the superficies OP and touching AB and CD the same perpeÌ„diculars Aâ— and CD do make right angles But by construction MN being drawne in the plaine superficies OP toucheth the perpendiculars AB and CD at the poyntes â— and D. Therefore the perpendiculars Aâ— and CD make with the right line MN two right angles namely ABN and CDM and MN the right line is proued to be in the one and the same playne superficies with the right lines AB CD namely in the playne superficies QR Whâ—refore by the second part of the 28. proposition of the first booke the right lineâ— AB and CD are parallelâ— If therefore two right lines be erected perpendicularly to one and the selfe same playne superficies those right lines are parallels the one to the other which was required to be demonstrated A Corollary added by M. Dee Hereby it is euident that any two right lines perpendicularly erected to one and the selfe same playne superficies are also them selues in one and the same playne superficies which is likewisâ— perpendicularly erected to the same playne superficies vnto which the two right lines are perpendicular The first part hereof is proued by the former construction and demonstration that the right lines AB and CD are in one and the same playne superficies Qâ— The second part is also manifest that is that the playne superficies QR is perpendicularly erected vpon the playne superficies OP for that Aâ— and CD being in the playne superficies QR are by supposition perpendicular to the playne superficies OP wherefore by the third definition of this booke QR is perpendicularly erected to or vpon OP which was required to be proued Io. d ee his aduise vpon the Assumpt of the 6. As concerning the making of the line DE equall to the right line AB verely the second of the first without some farther consideration is not properly enough alledged And no wonder it is for that in the former bookeâ— whatsoeâ—â—â—â—aâ—h of lines bene spoken the same hath alwayâ—s bene imagined to be in one onely playne superficies considered or executed But here the perpendicular line AB is not in the same playnâ— superficies that the right line DB is Therfore some other helpe must be put into the handes of young beginners how to bring this probleme to execution which is this most playne and briefe Vnderstand that BD the right line is the common section of the playne superficies wherein the perpendiculars AB and CD are of the other playne superficies to which they are perpendiculars The first of these in my former demonstration of the 6 â— I noted by the playne superficies QR and the other I noted by the plaine superficies OP Wherfore BD being a right line common to both the playne supâ—rficieces QR OP therby the ponits B and D are coÌ„mon to the playnes QR and OP Now from BD sufficiently extended cutte a right line equall to AB which suppose to be BF by the third of the first and orderly to BF make DE equall by the 3. oâ— the first if DE be greater then BF Which alwayes you may cause so to be by producing of DE sufficiently Now forasmuch as BF by construction is cutte equall to AB and DE also by construction put equâ—ll to BF therefore by the 1. common sentence DE is put equall to AB which was required to be done In like sort if DE were a line geuen to whome AB were to be cutte and made equall first out of the line DB suâ—â—iciently produced cutting of DG equall to DE by the third of the first and by the same 3. cutting from BA sufficiently produced BA equall to DG then is it euideÌ„t that to the right line DE the perpeÌ„dicular line AB is put equall And though this be easy to conceaue yet I haue designed the figure accordingly wherby you may instruct your imagination Many such helpes are in this booke requisite as well to enforme the young studentes therewith as also to master the froward gaynesayer of our conclusion or interrupter of our demonstrations course ¶ The 7. Theoreme The 7. Proposition If there be two parallel right lines and in either of them be taken a point at all aduentures a right line drawen by the said pointes is in the self same superficies with the parallel right lines SVppose that these two right lines AB and CD be parallels and in either of theÌ„ take a point at all aduentures namely E and F. Then I say that a right line drawen from the point E to the point F is in the selfe same plaine superficies that the
at all aduentures namely D V G S and a right line is drawen from the point D to the point G and an other from the point V to the point S. Wherefore by the 7. of the eleuenth the lines DG and VS are in one and the selfe same plaine superficies And forasmuch as the line DE is a parallel to the line BG therefore by the 24. of the first the angle EDT is equall to the angle BGT for they are alternate angles and likewise the angle DTV is equall to the angle GTS Now then there are two triangles that is DTV and GTS hauing two angles of the one equall to two angles of the other and one side of the one equall to one side of the other namely the side which subtendeth the equall angles that is the side DV to the side GS for they are the halfes of the lines DE and BG Wherefore the sides remayning are equall to the sides remayning Wherfore the line DT is equall to the line TG and the line VT to the line T S If therefore the opposite sides of a Parallelipipedon be deuided into two equall partes and by their sections be extended plaine superficieces the common section of those plaine superficieces and the diameter of the Parallelipipedon do deuide the one the other into two equall partes which was required to be demonstrated A Corollary added by Flussas Euery playne superficies extended by the center of a parallelipipedon diuideth that solide into two equall partes and so doth not any other playne superficies not extended by the center For euery playne extended by the center cutteth the diameter of the parallelipipedon in the center into two equall partes For it is proued that playne superficieces which cutte the solide into two equall partes do cut the dimetient into two equall partes in the center Wherefore all the lines drawen by the center in that playne superficies shall make angles with the dimetient And forasmuch as the diameter falleth vpon the parallel right lines of the solide which describe the opposite sides of the sayde solide or vpon the parallel playne superficieces of the solide which make angels at the endes of the diameter the triangles contayned vnder the diameter and the right line extended in that playne by the center and the right line which being drawen in the opposite superficieces of the solide ioyneth together the endes of the foresayde right lines namely the ende of the diameter and the ende of the line drawen by the center in the superficies extended by the center shall alwayes be equall and equiangle by the 26. of the first For the opposite right lines drawen by the opposite playne superficieces of the solide do make equall angles with the diameter forasmuch as they are parallel lines by the 16. of this booke But the angles at the ceÌ„ter are equall by the 15. of the first for they are head angles one side is equall to one side namely halfe the dimetient Wherefore the triangles contayned vnder euery right line drawen by the center of the parallelipipedon in the superficies which is extended also by the sayd center and the diameter thereof whose endes are the angles of the solide are equall equilater equiangle by the 26. of the first Wherfore it followeth that the playne superficies which cutteth the parallelipipedon doth make the partes of the bases on the opposite side equall and equiangle and therefore like and equall both in multitude and in magnitude wherefore the two solide sections of that solide shal be equall and like by the 8. diffinition of this booke And now that no other playne superficies besides that which is extended by the center deuideth the parallelipipedon into two equall partes it is manifest if vnto the playne superficies which is not extended by the center we extend by the center a parallel playne superficies by the Corollary of the 15. of this booke For forasmuch as that superficies which is extended by the center doth deuide the parallelipipedoÌ„ into two equall parâ—â— it is manifest that the other playne superficies which is parallel to the superficies which deuideth the solide into two equall partes is in one of the equall partes of the solide wherefore seing that the whole is euer greater then his partes it must nedes be that one of these sections is lesse then the halfe of the solide and therefore the other is greater For the better vnderstanding of this former proposition also of this Corollary added by Flussas it shal be very nedefull for you to describe of pasted paper or such like matter a parallelipipedoÌ„ or a Cube and to deuide all the parallelograÌ„mes therof into two equall parts by drawing by the câ—Ì„ters of the sayd parallelogrammes which centers are the poynts made by the cutting of diagonall lines drawen froÌ„ thâ— opposite angles of the sayd parallelograÌ„mes lines parallels to the sides of the parallelograÌ„mes as in the former figure described in a plaine ye may see are the sixe parallelograÌ„mes DE EH HA AD DH and CG whom these parallel lines drawen by the ceÌ„ters of the sayd parallelograÌ„mes namely XO OR PR and PX do deuide into two equall parts by which fower lines ye must imagine a playne superficies to be extended also these parallel lynes KL LN NM and Mâ— by which fower lines likewise yâ— must imagine a playne superficies to be extended ye may if ye will put within your body made thus of pasted paper two superficieces made also of the sayd paper hauing to their limites lines equall to the foresayde parallel lines which superficieces must also be deuided into two equall partes by parallel lines drawen by their centers and must cut the one the other by these parallel lines And for the diameter of this body exteÌ„d a thred from one angle in the base of the solide to his opposite angle which shall passe by the center of the parallelipipedon as doth the line DG in the figure before described in the playne And draw in the base and the opposite superficies vnto it Diagonall lines from the angles from which is extended the diameter of the solide as in the former description are the lines BG and DE. And when you haue thus described this body compare it with the former demonstration and it will make it very playne vnto you so your letters agree with the letters of the figure described in the booke And this description will playnely set forth vnto you the corollary following that proposition For where as to the vnderstanding of the demonstration of the proposition the superficieces put within the body were extended by parallel lynes drawen by the ceÌ„ters of the bases of the parallelipipedon to the vnderstanding of the sayd Corollary ye may extende a superficies by any other lines drawen in the sayd bases so that yet it passe through the middest of the thred which is supposed to be the center of the parallelipipedon ¶ The 35. Theoreme The 40. Proposition If there be
vpright cylinder is cuâ— by a plaine parallel to his bases by construction therefore as the cylinder DC is to the cylinder BE so is the axe of DC to the axe of BE by the 13. of this twelfth Wherefore as the axe is to the axe so is cylinder to cylinder But axe is to axe as side to side namely CE to QE because the axe is parallel to any side of an vpright cylinder by the definition of a cylinder And the circle of the section is parallel to the bases by construction Wherefore in the parallelogramme made of the axe and of two semidiameters on one side parallels one to the other being coupled together by a line drawen betwene their endes in their circumferences which line is the side QC it is euident that the axe of BC is cut in like proportion that the side QC is cut Wherfore the cylinder DC is to the cylinder BE as EC is to QE Wherefore by composition the cylinders DC and BE that is whole BC are to the cylinder BE as CE and QE the whole right line QC are to QE But by coÌ„struction QC is of 3. such partes as QE containeth 2. Wherefore the cylinder BC is of 3. such partes as BE contayneth 2. Wherefore BC the cylinder is to BE as 3. to 2 which is sesquialtera proportion But by the former Theoreme BC is sesquialtera to the Sphere A Wherefore by the 7. of the fift BE is equall to A. Therefore to a Sphere geuen we haue made a cylinder equall Or thus more briefely omitting all cutting of the Cylinder Forasmuch as BC is an vpright Cylinder his sides are equall to his axe or heith therefore the two cylinders whereof one hath the heith QC and the other the heith QE hauing both their bases the greatest circle in the Sphere A are one to the other as QC is to QE by the 14. of this twelfth but QC is to QE as 3. to 2 by construction and 3. to 2. is in Sesquialtera proportion therefore the cylinder BC hauing his heith QC his base the greatest circle in A coÌ„teyned is Sesquialtera to the cylinder which hath his base the greatest circle in A conteyned and heith the line QE But by the former Theoreme BC is also Sesquialtera to A wherfore the cylinder hauing the base BQ which by supposition is equal to the greatest circle in A conteyned and heith QE is equall to the sphere A by the 7. of the fift And now it can not be hard to geue a cylinder to A in that proportion which is betwene X and Y. For let the side QE be to QP as Y is to X by the 12. of the sixt Therefore backeward QP is to QE as X to Y. Wherefore the cylinder hauing the base the greatest circle in A and heith the line QP is to the cylinder hauing the same base and heith the line QE as X is to Y by the 14. of this twelfth but the cylinder hauing the heith QE his base the greatest circle in A conteyned is proued equall to the Sphere A. Wherefore by the 7. of the fift the cylinder whose heith is QP and base the greatest circle in A conteyned is to the sphere A as X to Y. Therefore to a sphere geuen we haue made a cylinder in any proportion geuen bâ—twene two right lines and also before we haue to a sphere geuen made a cylinder equall Therefore to a sphere geuen we haue made a cylinder equall or in any proportion geuen betwene two right lines A Probleme 5. A Sphere being geuen and a circle vpon that circle as a base to rere a cylinder equall to the sphere geuen or in any proportion geuen betwene two right lines A Probleme 6. A Sphere being geuen and a right line to make a cylinder equall to the sphere geuen or in any other proportion betwene two right lines geuen In this 5. and 6. probleme first make a cylinder equall to the sphere geuen by the 4. probleme and then by the order of the 2. and 3. problemes in cones execuâ—e these accordingly in cylinders A Probleme 7. Two vnlike Cones or Cylinders being geuen to finde two right lines which haue the same proportion one to the other that the two geuen cones or cylinders haue one to the other Vpon one of their bases rere a cone if cones be compared or a cylinder if cylinders be compared equall to the other by the order of the second and third problemeâ— and the heith of the cone or cylinder on whose base you rered an equall cone or cylinder with the new heith found haue that proportion which the cones or cylinders haue one to the other by the 14. of this twelfth booke A Probleme 8. An vpright Cone and Cylinder being geuen to finde two right lines hauing that proportion the one to the other which the Cone and Cylinder haue one to the other Suppose QEK an vpright cone and AB an vpright cylinder geuen I say two right lines are to be geueÌ„ which shall haue that proportion one to the other which QEK and AB haue one to the other Vpon the base BH erecte a Cone equall to QEK by the order of the secoÌ„d probleme which let be OBH and his heith let be OC and let the heith of AB the cylinder be CS produce CS to P so that CP be tripla to CS make peâ—fect the cone PBH I say that FC and OC haue that proportion which AB hath to QEK For by constrâ—ction OBH is equall to QEK and PBH is equal to AB as we will proue Assumpt wise And PBH and OBH are vpon one base namely BH wherfore by the 14. of this twelfth â—BH and OBH are one to the other as their heithes PC and OC are one to the other wherefore the cylinder and cone equall to PBH and OBH are as PC is to OC by the 7. of the fifth But AB the cylinder QEK the cone are equall to PBH and OBH by construction wherefore AB the cylinder is to QEK the cone as PC is to OC Wherefore we haue found two right lines hauing that proportion that a cone and a cylinder geuen haue one to the other Which thing we may execute vpon the base of the cone geuen as we did vpon the base of the cylinder geuâ—n on this maner Vpon the base of the cone QEK which base let be EK erect a cylinder equall to AB by the order of my second probleme Which cylinder let be ED and GT his heith and let the heith of the cone QEK be QG Take the line GR the â—hird part oâ— QG by the 9. of the sixth and with a playne passing by R parallel to EK cut of the cylinder Eâ— which â—hall be equall to the cone QEK by the assumpt followingâ— I say now that AB the cylinder is to QEK the cone as GT is to GR. For the cylinder ED is to the cylinder EF as GT is
superficies or soliditie in the hole or in partâ— such certaine knowledge demonstratiue may arise and such mechanical exercise thereby be deuised that sure I am to the sincere true student great light ayde and comfortable courage farther to wade will enter into his hart and to the Mechanicall witty and industrous deuiser new maner of inuentions executions in his workes will with small trauayle for fete application come to his perceiueraunce and vnderstanding Therefore euen a manifolde speculations practises may be had with the circle his quantitie being not knowne in any kinde of smallest certayne measure So likewise of the sphere many Problemes may be executed and his precise quantitie in certaine measure not determined or knowne yet because both one of the first humane occasioÌ„s of inuenting and stablishing this Arte was measuring of the earth and therfore called Geometria that is Earthmeasuring and also the chiefe and generall end in deede is measure and measure requireth a determination of quantitie in a certayne measure by nuÌ„ber expressed It was nedefull for Mechanicall earthmeasures not to be ignorant of the measure and contents of the circle neither of the sphere his measure and quantitie as neere as sense can imagine or wish And in very deede the quantitie and measure of the circle being knowne maketh not onely the cone and cylinder but also the sphere his quantitie to be as precisely knowne and certayne Therefore seing in respect of the circles quantitie by Archimedes specified this Theoreme is noted vnto you I wil by order vpon that as a supposition inferre the conclusion of this our Theoremes Note 1. Wherfore if you deuide the one side as TQ of the cube TX into 21. equall partes and where 11. partes do end reckening from T suppose the point P and by that point P imagine a plaine passing parallel to the opposite bases to cut the cube TX and therby the cube TX to be deuided into two rectangle parallelipipedons namely TN and PX It is manifest TN to be equall to the Sphere A by construction and the 7. of the fift Note 2. Secondly the whole quantitie of the Sphere A being coÌ„tayned in the rectangle parallelipipedon TN you may easilie transforme the same quantitie into other parallelipipedons rectangles of what height and of what parallelogramme base you list by my first and second Problemes vpon the 34. of this booke And the like may you do to any assigned part of the Sphere A by the like meanes deuiding the parallelipipedon TN as the part assigned doth require As if a third fourth fifth or sixth part of the Sphere A were to be had in a parallelipipedon of any parallelograâ—â—e base assigned or of any heith assigned then deuiding TP into so many partes as into 4. if a fourth part be to be transformed or into fiue if a fifth part be to be transformed c. and then proceede â—s you did with cutting of TN from TX And that I say of parallelipipedons may in like sort by my â—â—yd two problemes added to the 34. of this booke be done in any sided columnes pyramids and prismeâ— so thâ—â— in pyramids and some prismes you vse the cautions necessary in respect of their quan 〈…〉 odyes hauing parallel equall and opposite bases whose partes 〈…〉 re in their propositions is by Euclide demonstrated And finally 〈…〉 additions you haue the wayes and orders how to geue to a Sphere or any segmeâ—â— oâ— the same Cones or Cylinders equall or in any proportion betwene two right lines geuen with many other most necessary speculations and practises about the Sphere I trust that I haue sufficiently â—raughted your imagination for your honest and profitable studie herein and also geuen you reaâ—â— â—â—tter wheâ—â— with to sâ—â—p the mouthes of the malycious ignorant and arrogant despisers of the most excellent discourses trauayles and inuentions mathematicall Sting aswel the heauenly spheres sterres their sphericall soliditie with their conueâ—e spherical superficies to the earth at all times respecting and their distances from the earth as also the whole earthly Sphere and globe it selfe and infinite other cases concerning Spheres or globes may hereby with as much ease and certainety be determined of as of the quantitie of any bowle ball or bullet which we may gripe in our handes reason and experience being our witnesses and without these aydes such thinges of importance neuer hable of vs certainely to be knowne or attayned vnto Here ende M. Iohn d ee his additions vpon the last proposition of the twelfth booke A proposition added by Flussas If a Sphere touche a playne superficiesâ— a right line drawne from the center to the touche shall be erected perpendicularly to the playne superficies Suppose that there be a Sphere BCDL whose centre let be the poynt A. And let the playne superficies GCI touch the Spere in the poynt C and extend a right line from the centre A to the poynt C. Then I say that the line AC is erected perpendicularly to tâ—e playne GIC. Let the sphere be cutte by playne superficieces passing by the right line LAC which playnes let be ABCDL and ACEL which let cut the playne GCI by the right lines GCH and KCI Now it is manifest by the assumpt put before the 17. of this booke that the two sections of the sphere shall be circles hauing to their diameter the line LAC which is also the diameter of the sphere Wherefore the right lines GCH and KCI which are drawne in the playne GCI do at the poynt C fall without the circles BCDL and ECL. Wherefore they touch the circles in the poynt C by the second definition of the third Wherefore the right line LAC maketh right angles with the lines GCH and KCI by the 16. of the third Wherefore by the 4. of the eleuenth the right line AC is erected perpendicularly to to the playne superficies GCI wherein are drawne the lines GCH and KCI If therefore a Sphere touch a playne superficies a right line drawne from the centre to the touche shall be erected perpendicularly to the playne superficies which was required to be proued The ende of the twelfth booke of Euclides Elementes ¶ The thirtenth booke of Euclides Elementes IN THIS THIRTENTH BOOKE are set forth certayne most wonderfull and excellent passions of a lyne deuided by an extreme and meane proportion a matter vndoubtedly of great and infinite vse in Geometry as ye shall both in thys booke and in the other bookes following most euidently perceaue It teacheth moreouer the composition of the fiue regular solides and how to inscribe them in a Sphere geuen and also setteth forth certayne comparisons of the sayd bodyes both the one to the other and also to the Sphere wherein they are described The 1. Theoreme The 1. Proposition If a right line be deuided by an extreme and meane proportion and to the greater segment be added the halfe of the whole line the square made of those two
line thereby adioyned let be BQ I say that CQ is a line also deuided by an extreame and meane proportion in the point B and that BQ the line adioyned is the lesse segment For by the thirde it is proued that halfe AC which let be CD with CB as one line composed hath his powre or square quintuple to the powre of the segment CD Wherfore by the second of this booke the double of C D is deuided by extreme and middell proportionâ— and the greater segment thereof shal be CB. But by construction CQ is the double of CD for it is equall to AC Wherefore CQ is deuided by extreme and middle proportion in the point B and the greater segment thereof shal be CB. Wherefore BQ is the lesse segment which is the line adioyned Therefore a line being deuided by extreme and middell proportion if the lesse segment be produced equally to the length of the greater segment the line thereby adioyned together with the sayd lesse segment make a new line deuided by extreme meane proportion whoâ—e lesse segment is the line adioyned Which was to be demonstrated ¶ A Corollary 2. Ifâ— from the greater segment of a line diuided by extreme and middle proportion a line equall to the lesse segment be cut of the greater segment thereby is also deuided by extreme and meane proportion whose greater segmentâ— shall be 〈◊〉 that part of it which is cut of For taking from AC a line equall to CB let AR remayne I say that AC is deuided by an extreme and meane proportion in the point R and that CR the line cut of is the greater segment For it is proued in the former Corollary that CQ is deuided by extreme and meane proportion in the point B. But AC is equall to CQ by construction and CR is equall to CB by construction Wherefore the reâ—idue AR is equall to BQ the residue Seing therfore the whole AC is equall to the whole CQ and the greater part of AC which is CR is equal to CB the greater part of CQ and the lesse segmeÌ„t also equall to the lesse and withall seing CQ is proued to be diuided by extreme meane proportion in the point B it foloweth of necessity that AC is diuided by extreme and meane proportion in the point R. And seing CB is the greater segment of CQ CR shall be the greater segment of AC Which was to be demonstrated A Corollary 3. It is euident thereby a line being diuided by extreme and meane proportion that the line wheâ—â—by the greater segment excedeth the lesse together with the lesse segment do make a line diuided by extreme and meane proportion whose lesse segment is the sayd line of exceesse or difference betwene the segments Iohn Dee ¶ Two new wayes to deuide any right line geuen by an extreme and meane proportion demonstrated and added by M. Dee A Probleme To deuide by an extreme and meane proportion any right line geuen in length and position Suppose a line geuen in length and positionâ— to be AB I say that AB is to be deuided by an extreme and meane proportion Deuide AB into two equall parts as in the point C. Produce AB directly from the point B to the point D making BD equal to BC. To the line AD and at the point D let a line be drawen perpendicular by the 11. of the first which let be DF of what length you will From DF and at the point D cut of the sixth parte of DF by the 9. of the sixth And let that sixth part be the line DG Vppon DF as a diameter describe a semicircle which let be DHF From the point G rere a line perpendicular to DF which suppose to be GH and let it come to the circumference of DHF in the point H. Draw right lines HD and HF. Produce DH from the point H so long till a line adioyned with DH be equall to HF which let be DI equall to HF. From the point H to the point B the one ende of our line geuen let a right line be drawen as HB From the point I let a line be drawen to the line AB so that it be also parallel to the line HB Which parallel line suppose to be IK cutting the line AB at the point K. I say that AB is deuided by an extreme meane proportion in the point K. For the triangle DKI hauing HB parallel to IK hath his sides DK and DI cut proportionally by the 2. of the sixth Wherefore as IH is to HD so is KB to BD. And therfore compoundingly by the 18. of the fiueth as DI is to DH so is DK to DB. But by construction DI is equall to HF wherefore by the 7. of the fifth DI is to DH as HF is to DH Wherefore by the 11. of the fifth DK is to DB as HF is to DH Wherefore the square of DK is to the square of DB as the square of HF is to the square of DH by the 22. of the sixth But the square of HF is to the square of DH as the line GF is to the line GDâ— by my corrollary vpon the 5 probleme of my additions to the second proposition of the twelfth Wherefore by the 11. of the fifth the square of DK is to the square of DB as the line GF is to the line GD But by construction GF is quintuple to GD Wherefore the square of DK is quintuple to the square of DB and therefore the double of DB is deuided by an extreme and meane proportioÌ„ and BK is the greater segment therof by the 2. of this thirtenth Wherefore seing AB is the double of DB by construction the line AB is deuided by an extreme and meane proportion and his greater segment is the line BK Wherefore AB is deuided by an extreme and meane proportion in the point K. We haue therefore deuided by extreme and meane proportion any line geuen in length and position Which was requisite to be done The second way to execute this probleme Suppose the line geuen to be AB Deuide Aâ— into two equall parts as suppose it to be done in the point C. Produce AB from the point B adioyning a line equall to BC which let be BD. To the right line AD and at the point D erect a perpendicular line equall to BD let that be DE. Produce ED froÌ„ the point D to the point F making DF to contayne fiue such equall partes as DE is one Now vpon EF as a diameter describe a semicircle which let 〈◊〉 EKF and let the point where the circumference of EKF doth cut the line AB be the point K. I say that AB is deuided in the point K by an extreme and meane proportion For by the 13. of the sixth ED DK DF are three lines in continuall proportion DK being the middle proportionall â— Wherefore by the corollary of the 20. of the sixth as ED is to DF so is
the circumference on which the Icosahedron is described And that the diameter of the sphere is composed of the side of an hexagon and of two sides of a decagon described in one and the selfe same circle Flussas proueth the Icosahedron described to be coÌ„tayned in a sphere by drawing right lines from the poynt a to the poyntes P and G after this maner Forasmuch as the lines ZW WP are put equal to the line drawen from the centre to the circumferenceâ— and the line drawen from the centre to the circumference is double to the line aW by construction therefore the line WP is also double to the same line aW Wherefore the square of the line WP is quadruple to the square of the line aW by the corollary of the 20. of the sixth And those lines PW and Wa contayning a right angle PWa as hath before bene proued are subtended of the right line aP Wherefore by the 47. of the first the line aP contayneth in power the lines PW and Wa. Wherefore the right line aP is in power quintuple to the line Wa. Wherefore the right lines aP and aQ being quintuple to one and the same line Wa are by the 9. of the flueth equall In like sorte also may we proue that vnto those lines aP and aQ are equall the rest of the lines drawen from the poynt a to the rest of the angles R S T V. For they subtend right angles contayned of the line Wa and of the lines drawen from the centre to the circumference And forasmuch as vnto the line Wa is equall the line Va which is likewise erected perpendicularly vnto the other plaine superficies OLMNX therefore lines drawen from the point a to the angles O L M N X and subtending right angles at the point Z contayned vnder lines drawen froÌ„ the centre to the circumference and vnder the line aZ are equal not onely the one to the other but also to the lines drawen froÌ„ the sayde poynt a to the former angles at the poynts P R S T Vâ— For the lines drawen froÌ„ the centre to the circumference of ech circle are equall the line aW is equall to the line aZ. But the line aP is proued equal to the line aQ which is the halfe of the whole QY Therefore the residue aY is equall to the foresayd lines aP aQ c. Wherefore making the centre the poynt a and the space one of those lines aQ aP c. extende the superficies of a sphere it shal touch the 12. angles of the Icosahedron which are at the pointes O L M. N X P K S T V Q Y which sphere is described if vpoÌ„ the diameter QY be drawen a semicircle and the sayd semicircle be moued about till it returne vnto the same place from whense it began first to be moued ¶ A Corollary added by Flussas The opposite sides of an Icosahedron are parallels For the diameters of the sphere do fall vpon the opposite angles of the Icosahedron as it was manifest by the right line QY If therefore there be imagined to be drawen the two diameters PN and OM they shall concurre in the point F wherefore the right lines which ioyne them together PV and LN are in one and the selfe same playne superficies by the 2. of the eleuenth And forasmuch as the alternate angles at the endes of the diameters are equall by the 8. of the first for the triangles contayned vnder equall semidiameteâ—s and the side of the Icosahedron are equiangle therefore by the 28. of the first the lines PV and LN are paralles ¶ The 5. Probleme The 17. Proposition To make a Dodecahedron and to comprehend it in the sphere geuen wherin were comprehended the foresayd solides and to proue that the side of the dodecahedron is an irrationall line of that kind which is called a residuall line Now I say that it is in one and the self same playne superâ—icies Forasmuch as the line ZV is a parallell to the line SR as was before proued but vnto the same line SR is the line CB a parallell by the 28. of the first Wherfore by the 9. of the eleuenth the line VZ is a parallell to the line CB. Wherefore by the seueÌ„th of the eleuenth the right lines which ioyne theÌ„ together are in the selfe same playne wherein are the parallell lines Wherefore the Trapesium BVZC is in one playne And the triangle BWC is in one playne by the 2. of the eleuenth Now to proue that the Trapesium BVZC the triangle BWC are in one and the self same plaine we must proue that the right lines YH and HW are made directly one right line which thing is thus proued Forasmuch as the line HP is diuided by an extreme and meane proportion in the point T and his greater segment is the line PT therefore as the line HP is to the line PT so is the line PT to the line TH. But the line HP is equall to the line HO and the line PT to either of these lines TW and OY Wherefore as the line HO is to the line OY so is the line WT to the line TH. But the lines HO and TW being sides of like proportion are parallels by the 6. of the eleuenth For either of them is erected perpendicularly to the plaine superficies BD â— and the lines TH and OY are parallels which are also sides of like proportion by the same 6. of the eleuenth For either of them is also erected perpendicularly to the playne superficies BF But when there are two triangles hauing two sides proportionall to two sides so set vpon one angle that their sides of like proportioÌ„ are also parallels as the triangles YOH and HTW are â— whose two sides OH HT being in the two bases of the cube making an angle at the point H the sides remayning of those triangles shal by the 32. of the sixth be in one right line Wherfore the lines YH HW make both one right line But euery right line is by the 3. of the eleuenth in one the self same plaine superficies Wherefore if ye draw a right line from B to Y there shal be made a triangle BWY which shal be in one and the selfe same plaine by the 2. of the eleuenth And therefore the whole pentagon figure VBWCZ is in one and the selfe same playne superficies Now also I say that it is equiangle For forasmuch as the right line NO is diuided by an extreame and meane proportion in the point R and his greater segment is OR therefore as both the lines NO and OR added together is to the line ON so by the 5. of this booke is the line ON to the line OR But the line OR is equall to the line OS Wherefore as the line SN is to the line NO so is the line NO to the line OS Wherfore the line SN is diuided by an extreme and meane proportion in the point O