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
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
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
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
toucheth the circle For vnto the point where the line LM toucheth the circle whiche let be N drawe the lyne FN And it is manifest by the 18. of the third that either of the angleâ— at the poynt N is a right angle Wherâ—fore for asmuch as the angle L of the triangle FLN is equall to the angle M of the triangle FMN the angle N of the one is equall to the anglâ— N of the other and the lyne FN is coÌ„mon to theÌ„ both the line NL shall by the 26. of the first be equall to the line NM And so is the line ML deuided equally in the poynt N. And forasmuch as the three sides of the triangle FGP are equall to the three sides of the triangle FMP the angle P of the one shal be equall to the angle P of the other by the 8. of the first Wherfore either of theÌ„ is a right angle by the 13. of the same And forasmuch as the two angles FMP and FPM of the triangle FMP are equall to the two angles FMN FNM of the triangle FMN and the side FM is common to them both therefore the line FP is equall to the line FN But the line FN is drawen from the centre to the circumference Wherefore also the line FP is drawen from the centre to the circumference And forasmuch as the line MG is perpendicular to the line FP therefore by the Corollary of the 16. of the third it toucheth the circle Wherfore the Pentagon GHKLM circumscribed about the circle is equilater it is also equiangle aâ— it is easie to proue by the equalitie of the halfes which was required to be done The 13. Probleme The 13. Proposition An equilater and equiangle pentagon figure beyng geuen to describe in it a circle SVppose that the equilater equiangle PeÌ„tagoÌ„ figure geueÌ„ be ABCDE It is required in the said fiue angled figure ABCDE to describe a circle Deuide by the the 9. of the first either of these angles BCD and CDE into two equall partes by these right lines CF and FD and from the point F where thâ—se right lines CF and DF meete Draw these right lines FB FA Fâ— And forasmuch as BC is equal vnto CD â—nd CF is common to them both Wherefore these two lines BC and CF are equal to these two lines DC and CF and the angle BCF is equall to the angle DCF Wherfoâ—e by the 4. of the first the base BF is equall to the base DF and the triangle BCF is equall to the triangle DCF and the angles remayning are equall vnto the angles remaynyng the one to the other vnder which are subtended equall sides Wherefore the angle CBF is equall to the angle CDF And forasmuch as the angle CDE is double to the angle CDF But the angle CDE is equall to the angle ABC and the angle CDF is equall to the angle CBF Wherefore the angle CBA is double to the angle CBF Wherefore the angle ABF is equall to the angle FBC Wherefore the angle ABC is deuided into two equall partes by the right line BF In lyke sorte also may it be prouedâ— that either of these angles BAE and AED are deuided into two equall partes by either of these lines FA and EF. Drawe by the 12. of the first from the pointe E to the right lines AB BC CD and EA perpendiâ—ular lineâ— FG FH FK FL and FM And forasmuch as the angle HCF is eqâ—all 〈◊〉 the anglâ— KCF and the right angle FHC is equall to the right angle FKC now then there are two triangles FHC and FKCâ— hauyng two angles equall to two angles the one to the other and one side equall to one side for FC is commoâ— vnto them both and is subtended vnder one of the equal angles Wherfore by the 2â— of the first the sides remayâ—â—ng are eqâ—all vnto the sides remayning Wherfore the perpendiculer FH is equall vnto the perpendiculer FK And in like sort also may it be prouâ—d that euery one of these lines FL FM and FG is equall to euery one of thâ—â—â— lines FH and FK Whâ—â—fore these fiue right lines FG FH FK FL and FM are equall the one to the other Wherfore making the centre F and the space FG or FH or FK or FL or FM Describe a circle and it will passe by the poyntes G H K L M. And shall touche the right lines AB BC CD DE and EA by the correllary of the 16. of the third for the angles which are at the pointes G H K L M are right angles For if it do not touche them but cut then from the ende of the diameter of the circle shall be drawen a right line making two right angles and falling with in the circle whiche is by the 16. of the third proued to be impossible Wherefore makyng F the centre and the space one of these lynes FG FH FK FL FM describe a circle and it shall not cut the right lines AB BC CD DE and EA Wherefore by the corollary of the 16. of the third it shall touche them as it is manifest in the circle GHKLM Wherefore in the equilater and equiangle pentagon figure geuen is described a circle which was required to be doneâ— By this proposition and the former it is manifest that perpendicular lines drawn from the middle poyntes of the sides of an equilater and equiangle Pentagon figure and produced shall passe by the centre of the circle in which the sayd Pentagon figure is inscribed and shall also deuide the opposite angles equally as here AK is one right line and deuideth the side CD and the angle A equally And so of the rest which is thus proued The space about the centre F is equall to foure right angles which are deuided into ten equall angles by ten right lines metyng together in the point F. Wherfore the fiue angles AFM MFE EFL LFD and DFK are equall to two right angles Wherfore by the 14. of the first the lynes AF and FK make one right lyne The lyke proofe also will serue touching the reâ—t of the lynes And this is alwayes true in all equilater figures which consist of vneuen sides The 14. Probleme The 14. Proposition About a pentagon or figure of fiue angles geuen beyng equilater and equiangle to describe a circle SVppose that the Pentagon or figure of fiue angles geuen being of equall sides and of equall angles be ABCDE It is required about the sayd Pentagon ABCDE to describe a circle Deuide by the 9. of the first eyther of these angles BCD CDE into two equall partes by either of these right lines CF and DF. And from the poynt F where those right lines meete draw vnto the pointes B A E these right lines FB FA FE And in like sort by the Proposition going before may it be proued that euery one of these angles CBA BAE and AED is deuided into two equall partes by these right
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
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
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
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
last proposition in the second booke and also of the 31. in the sixth booke ¶ A Probleme 5. Two vnequall circles being geuen to finde a circle equall to the excesse of the greater to the lesse Suppose the two vnequal circles geueÌ„ to be ABC DEF let ABC be the greater whose diameter suppose to be AC the diameter of DEF suppose to be DF. I say a circle must be found equal to that excesse in magnitude by which ABC is greater thâ— DEF By the first of the fourth in the circle ABC Apply a right line equall to DF whose one end let be at C and the other let be at B. FroÌ„ B to A draw a right line By the 30. of the third it may appeare that ABC is a right angle and thereby ABC the triangle is rectangled wherfore by the first of the two corollaries here before the circle ABC is equall to the circle DEF For BC by construction is equall to DF and more ouer to the circle whose diameter is AB That circle therefore whose diameter is AB is the circle conteyning the magnitude by which ABC is greater then DEF Wherefore two vnequal circles being geuen we haue found a circle equall to the excesse of the greater to the lesse which ought to be doone A Probleme 6. A Circle being geuen to finde two Circles equall to the same which found Circles shall haue the one to the other any proportion geuen in two right lines Suppose ABC a circle geuen and the proportion geuen let it be that which is betwene the two right lines D and E. I say that two circles are to be found equall to ABC and with al one to the other in the proportioÌ„ of D to E. Let the diameter of ABC be AC As D is to E so let AC be deuided by the 10. of the sixth in the poynt F. At F to the line AC let a perpeÌ„dicular be drawne FB and let it mete the circuÌ„fereÌ„ce at the poynt B. From the poynt B to the points A and C let right lines be drawne BA and BC. I say that the circles whose diameteâ— are the lines BA and BC are equall to the circle ABC and that those circles hauing to their diameters the lines BA and BC are one to the other in the proportion of the line D to the line E. For first that they are equal it is euident by reason that ABC is a triangle rectangle wherfore by the 47. of the first the squares of BA and BC are equall to the square of AC And so by this second it is maniâ—est the two circles to be equall to the circle ABC Secondly as D is to â— so is AF to FC by construction And as the line AF is to the line FC so is the square of the line â—A to the square of the line BC. Which thing we will briefely pâ—oue thus The parallelogramme contayned vnder AC and AF is equall to the square of BA by the Lemma after the 32. of the tenth booke and by the same Lemma or Assumpt the parallelogramme contayned vnder AC and â—C is equall to the square of the line BC. Wherfore as the first parallelogramme hath it selfe to the secondâ— so hath the square of BA equall to the first parallelogramme it selfe to the square of BC equall to the second parallelogramme But both the parallelograÌ„mes haue one heigth namely the line AC and bases the lines AF and FC wherefore as AF is to FC so is the parallelogâ—amme contayned vnder AC AF to the parallelogramme contayned vnder AC FC by the fiâ—st of the sixth And therefore as AF is to FC so is the square of BA to the square of BC. And as the square of BA is to the square of BC so is the circle whose diameter is BA to the circle whose diameter is BC by this second of the twelfth Wherefore the circle whose diameter is BA is to the circle whose diameter is BC as D is to E. And before we proued them equall to the circle ABC Wherâ—fore a circle being geuen we haue found two circles equall to the same which haue the one to the other any proportion geuen in two right lines Which ought to be done Note Heâ—e may you perâ—eiue an other way how to execute my first probleme for if you make a right angle conteyned of the diameters geueÌ„ as in this figure suppose them BA and BC and then subtend the right angle with the line AC and from the right angle let fall a line perpendicular to the base AC that perpâ—ndicular at the point of his fall deuideth AC into AF and FC of the proportion required A Corollary It followeth of thinges manifestly proued in the demonstration of this probleme that in a triangle rectangle if from the right angle to the base a perpendicular be let fall the same perpendicular cutteth the base into two partes in that proportion one to the other that the squares of the righâ— lines conteyning the right angle are in one to the other those on the one side the perpendicular being compared to those on the other both square and segment A Probleme 7. Betwene two circles geuen to finde a circle middell proportionall Let the two circles geuen be ACD and BEF I say that a circle is to be fouÌ„d which betwene ACD and BEF is middell proportionall Let the diameter of ACD be AD and of BEF let Bâ— be the diameter betwene AD and BF finde a line middell proportionall by the 13. of the sixth which let be HK I say that a circle whose diameter is HK is middell proportionall betwene ACD and BEF To AD HK and BF three right lines in continuall proportion by construction let a fourth line be found to which BF shal haue that proportion that AD hath to HK by the 12. of the sixth let that line be â— It is manifest that the â—ower lines AD HK BF and L are in continuall proportion For by coÌ„struction as AD is to HK so is Bâ— to L. And by construction on as AD is to HK so is HK to BF wherefore HK is to BF as BF is to L by the 11. of the fifth wherfore the 4. lines are in continuall proportion Wherefore as the first is to the third that is AD to BF so is the square of the first to the square of the second that is the square of AD to the square of HK by the corollary of the 20. of the sixth And by the same corollary as HK is to L so is the square of HK to the square of BF But by alternate proportion the line AD is to BF as HK is to L wherefore the square of AD is to the square of HK as the square of HK is to the square of BF Wherefore the square of HK is middell proportionall betwene the square of AD and the square of BF But as the squares are
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
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
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
the 17. proposition of the 14. booke Construction â—rist part of the demonstration * For the 4. angles at the point K are equall to fower right angles by the Corollary of tâ—e 15. of the first and those 4. angle are equall the one to the other by the â— of the â—irst and theâ—efore ech is a right angle Second part of the demoÌ„stratioÌ„ * For the square of the line AB which is proued equall to the square of the line LM is double to the square of the line BD which is also equall to the square of the line LE. This Corollary is the 16. proposition of the 14. booke after Campane First part of the demonstration Second part of the ConstructioÌ„ Second part of the demonstraâ—ion * By the 2. Assumpt of the 13. of this booke Third part of the demonstration Second part of the demoÌ„stratioÌ„ For the line QW is equall to the line IZ the line ZW is commoÌ„ to them both This part is againe afterward demonstrated by Flussas The pentagon VBWCR proued to be in one and the selfe same playne superficies The pentagon VBWCZ it proued equiangle * Looke for a farther construction after Flussas at the ende of the demonstration That the side of the dodecahedron is a residuall line Draw in the former figure these lines ctA ctL câ—D The side of a pyramis The side of a cube The sides of a dodecahedron Comparison of the fiue sides of the foresayd bodies An other demoÌ„monstration to proue that the side of the IcosahedroÌ„ is greater then the side of the dodecahedron That 3. squares of the line FB are greater theÌ„ 6. squares of the line NB. That there can be no other solide besids these fiue contayned vnder equilater and equiangle bases That the angle of an equilater and equiangle Pentagon is one right angle and a 〈◊〉 part 〈◊〉 which thing was also before proued in the 〈◊〉 of the 32. of the â—irst The sides of the angle of the inclâ—â—â—tion of the 〈◊〉 of the 〈◊〉 are 〈◊〉 rationall The sides of the angle of the inclination of the 〈◊〉 â—f tâ—e 〈…〉 That the plaines of an octohedron are in liâ—e sort inclined That the plaines of an Icosahedron are in like sort inclined That the plaines of a Dâ—â—â—â—â—hedron are 〈◊〉 like sort inclined The sides of the angle of the inclination of the supeâ—ficieces of the TetrahedroÌ„ are proued rationall The sides of the angle of the inclination of the superficieces of the cube proued rationall The sides of thâ— angle c. of the octohedron proued rationall The sides of the angle c. of the Icosahedron proued irratioâ—all How to know whether the angle of the inclinatioÌ„ be a right angle an acute angle or an oblique angle The argument of the fourtenth booke First proposition after Flussas Construction Demonstration * This is manifest by the 12. propositioÌ„ of the thirtenh booke as Campane well gathereth in a Corollary of the same The 4. pâ—posâ—tioÌ„ after Flussas * This is afterward proued in the 4. proposition This Assumpt is the 3. proposition after Flussas Construction of the Assumpt Demonstration of the Assumpt Construction of the proposition Demonstration of the â—â—opâ—sition * Thâ— 5. proposition aâ—tâ—r 〈◊〉 Construction Demoâ—stration The 5. proposition aâ—ter Fâ—ussas Demonstration * This is the reason of the Corollary following A Corollary which also Flussas putteth as a Corollary after the 5. proposition in his order The 6. pâ—â—positioÌ„â—â—ter Flussas Coâ—struction Demonstration * This is not hard to proue by the 15. 16. and 19. of the â—â—â—eth †In the Corollary of the 17. of the tâ—irteÌ„th * 〈◊〉 againe is required the Assumpt which is afterward proued in this 4 proposition †But first the Assumpt following the construction whâ—reâ—f here beginneâ—h is to be proued The Assumpt which also Flussas putteth as an Assumpt aâ—ter the 6. propositioÌ„ Demonstration of the Assumpt Construction pertaining to the second dâ—monstration of the 4. propositioÌ„ Second demonstration oâ— the 4. proposition The 7â— proposition after Flussas Construction Demonstration * Here againe is required the Assumpt afterward proued in this 4. proposition †As may by the Assumpt afterward in this propositioÌ„ be plainely proued The 8. proâ—ition aâ—ter Flussas †By the Corollary added by Flussas after has Assumpt put after the 17. proposition of the 12. booke Corollary of the 8. after Flussas This Assumpt is the 3. propositioÌ„ aâ—ter â—lussas and is it which 〈◊〉 times hath bene taken aâ— gâ—aunted in this booke and oâ—ce also in the last proposition of the 13. booke as we haue beâ—ore noted Demonstration * In the 4. section â—f this proposition †In the 1. and 3 section of the same propositioÌ„ †In the 5. sectioÌ„ of the same proposition A Corollary The first proposition after Campane Construction Demonstration The 2. proposition after Campane Demonstration leading to an impossibilitie The 4. proposition after Campane Construâ—tâ—on Demonstration This Corollary Campane also â—utteth after the 4. proposition in his order The 5. proposition after Campane Construction Demonstration This is the 6. and 7. propositions after Campane Construction Demonsâ—ration This Corollary Campane also addeth after the 7. proposition iâ— his order The 5. proposition aâ—ter Campane Construction Demonstration This Assumpt Campane also hath after the 8. proposition in his order Construction Demonstration The 9. proposition after Campane Construction Demonstration This Campane putteth aâ— a Corollary in the 9. proposition after his order This Corollary is the 9. proposition after Campane The 12. proposition after Campane Construction Demonstration The 13. proposition after Campane The 14. proposition after Campane Demonstration of the first part Demonstration of the second part The 17. proposition after Campane Firâ—t part of the construction First part of the Demonstration Second parâ— of the câ—nstruction Second part of the Dâ—monstration The 18. proposition after Campane Demonstration of the first part Demonstration of the second part The Corollary of the 8. proposition after Campane Demâ—nstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration The argument of the 15. booke †In this proposition as also in all the other following by the name of a pyramis vnderstand a tetrahedron as I haue before admonished Construction Demonstration Construction Demonstration Construction Demonstrâ—tion Construction Demonstration Construction Demonstration That which here followeth concerning the inclination of the plaines of the fiue solides was before tought â—hough not altogether after the same maner out of Flussas in the latter â—nde of the 13 booke Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration First part of the construction First part of the demonsâ—ration Second part of the construction Second part of the Demonstration Third part of the construction Third part of the demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Produce in the figure the line TF to the point B. Construction Demonstration This proposition Campane hath is the last also in order of the 15. booke with him The argument of the 16. booke Construction Demonstration * By a Pyramis vnderstand a Tetrahedron throughout all this booke Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration †That is aâ— 18. to 1. Demonstration †That iâ— as 9. to 2. * That is as 18. to 2. or 9. to 1. Draw in the figure a line from B to H. * What the duple of an extreme and meane proportion is Construction Demonstration Demonstration Constrution Demonstration Construction Demonstration Demonstration * That is at 13. 1 â— is to â— Demonstration Demonstration Construction Demonstration Construction Demonstration Extend in the figure a line â—rom the point E to the point B. Extend in the figure a line from the point E to the point B. Demonstration Construction Demonstration Second part of the Demonstration Icosidodecahedron ExoctohedroÌ„ That the â—xoctohedron is contayned in a sphere That the exoctohedâ—on is contayned in the sphere geuen That the diaâ—â—â—ter of the sâ—here is doâ—ble to the side â—f the exoctohedron That the Icosidodecahedron is contayned in the sphere geuen * That is as 8. 103 â— Faultes escaped â—cl â—ag Line Faultesâ— Co 〈◊〉 〈◊〉    Errata Lib. 1.  1 2 41 point B. at Campane point C aâ— Campane 3 1 22 aâ—l lines drawne all righ◠〈…〉 3 1 28 lines drawen to the superficies right lines draweÌ„ to the circumference 9 1 42 liâ—es AB and AC lines AB and BC 15 1 35 are equall are proued equall 20 2 28 by the first by the fourth 21 1 39 tâ—e centre C. the centre E 2â— 2 â— Iâ—â—ower right If two right 25 2 3 fâ—â—â— petition fiueth petition 49 2 7 14. â—â— 32.64 c. 4.8.16.32.64 53 1 39 the triangle NG the triangle Kâ— 54 2 25 by the 44 by the 42 57 2 23 and Câ—G in the and CGB is thâ— Â Â Â In stede of â—lussates through out 〈◊〉 whole booke read â—lusâ—as    Errata Lib. 2.  60 2 29 Gnomon FGEH Gnomon AHKD   30 Gnomon EHFG Gnomon â—CKD 69 1 18 the whole line the whole â—igure 76 2 9 the diameter CD the diameter AHF    Errata Lib. 3.  82 2 36 angle equall to the angle 92 1 last the line AC is the line AF 〈◊〉    Errata Lib. 4.  110 2 10 CD toucheth the ED toucheth the   12 side of the other angle of the other 115 1 21 and HB and HE 117 2 44 the angle ACD the angle ACB 118 1 2 into ten equall into two equall 121 1 3â— CD and EA CD DE EA â—â—â— 1 29 the first the third    Errata Lib. 5.  126 1 43 it maketh 12. more then 17. by 5. it maketh 24. more then 17. by 7. 129 1  In stede of the figure of the 6. definition draw in the magâ—â— a figure like vnto thâ—s 134 2 4 As AB is to A so is CD to C As AB is to B so is CD to D 141 2 last But if K excede M But if H excede M LIEFE IS DEATHE AND DEATH IS LIEFE AETATIS SVAE XXXX AT LONDON Printed by Iohn Daye dwelling ouer Aldersgate beneath Saint Martins ¶ These Bookes are to be solde at his shop vnder the gate 1570.
consider the diuerse trades of these two excellent Philosophers and am most sure both that Plato right well otherwise could teach and that Aristotle mought boldely with his hearers haue dealt in like sorte as Plato did I am in no little pang of perplexitie Bycause that which I mislike is most easy for me to performe and to haue Plato for my exaÌ„ple And that which I know to be most commendable and in this first bringyng into common handling the Artes Mathematicall to be most necessary is full of great difficultie and sundry daungers Yetâ— neither do I think it meâ—e for so straunge matter as now is ment to be published and to so straunge an audience to be bluntly at first put forth without a peculiar Preface Nor Imitatyng Aristotle well can I hope that accordyng to the amplenes and dignitie of the State Mathematicall I am able either playnly to prescribe the materiall boundes or precisely to expresse the chief purposes and most wonderfull applications therof And though I am sure that such as did shrinke from Plato his sâ—hole after they had percâ—iued his finall conclusion would in these thinges haue ben his most diligent hearers so infinitely mought their desires in fine and at length by our Artes Mathematicall be satisfied yet by this my Praeface forewarnyng Aswell all such may to their great behofe the soner hither be allured as also the Pythagoricall and Platonicall perfect scholer and the constant profound Philosopher with more ease and spede may like the Bee gather hereby both wax and hony Wherfore seyng I finde great occasion for the causes alleged and farder in respect of my Art Mathematike generall to vse a certaine forewarnyng and Praeface whose content shal be that mighty most plesaunt and frutefull Mathematicall Tree with his chief armes and second grifted braunches Both what euery one is and also what commodity in generall is to be looked for aswell of griff as stocke And forasmuch as this enterprise is so great that to this our tyme it neuer was to my knowledge by any achieued And also it is most hard in these our drery dayes to such rare and straunge Artes to wyn due and common credit Neuertheles if for my sincere endeuour to satisfie your honest expectation you will but lend me your thaÌ„kefull mynde a while and to such matter as for this time my penne with spede is hable to deliuer apply your eye or eare attentifely perchaunce at once and for the first salutyng this Preface you will finde a lesson long enough And either you will for a second by this be made much the apter or shortly become well hable your selues of the lyons claw to coniecture his royall symmetrie and farder propertie Now then gentle my frendes and countrey men Turne your eyes and bend your myndes to that doctrine which for our present purpose my simple talent is hable to yeld you All thinges which are haue beyng are found vnder a triple diuersitie generall For either they are demed Supernaturall Naturall or of a third being Thinges Supernaturall are immateriall simple indiuisible incorruptible vnchangeable Things Naturall are materiall compounded diuisible corruptible and chaungeable Thinges Supernaturall are of the minde onely comprehended Things Naturall of the sense exterior ar hable to be perceiued In thinges Naturall probabilitie and coniecture hath place But in things Supernaturall chief demoÌ„stration most sure Science is to be had By which properties comparasons of these two more easily may be described the state condition nature and property of those thinges which we before termed of a third being which by a peculier name also are called Thynges Mathematicall For these beyng in a maner middle betwene thinges supernaturall and naturall are not so absolute and excellent as thinges supernatural Nor yet so base and grosse as things naturall But are thinges immateriall and neuerthelesse by materiall things hable somewhat to be signified And though their particular Images by Art are aggâ—egable and diuisible yet the generall Formes notwithstandyng are constant vnchaungeable vntraÌ„sformable and incorruptible Neither of the sense can they at any tyme be perceiued or iudged Nor yet for all that in the royall mynde of man first conceiued But surmountyng the imperfectioÌ„ of coniecture weenyng and opinion and commyng short of high intellectuall coÌ„ceptioÌ„ are the Mercurial fruite of Dianoeticall discourse in perfect imagination subsistyng A meruaylous newtralitie haue these thinges Mathematicall and also a straunge participatioÌ„ betwene thinges supernaturall immortall intellectual simple and indiuisible and thynges naturall mortall sensible compounded and diuisible Probabilitie and sensible profe may well serue in thinges naturall and is commendable In Mathematicall reasoninges a probable Argument is nothyng regarded nor yet the testimony of sense any whit credited But onely a perfect demonstration of truthes certaine necessary and inuincible vniuersally and necessaryly concluded is allowed as sufficient for an Argument exactly and purely Mathematical Of Mathematicall thinges are two principall kindeâ—â— namely Number and Magnitude Number we define to be a certayne Mathematicall SuÌ„me of Vnits And an Vnit is that thing Mathematicall Indiuisible by participation of some likenes of whose property any thing which is in deâ—de or is counted One may resonably be called One. We account an Vnit a thing Mathematicall though it be no Number and also indiuisibleâ— because of it materially Number doth consistâ— which principally is a thing Mathematicall Magnitude is a thing Mathematicall by participation of some likenes of whose nature any thing is iudged long broade or thicke A thicke Magnitude we call a Solide or a Body What Magnitude so euer is Solide or Thicke is also broade long A broade magnitude we call a Superficies or a Plaine Euery playne magnitude hath also length A long magnitude we terme a Line A Line is neither thicke nor broade but onely long Euery certayne Line hath two endes The endes of a line are Pointes called A Point is a thing Mathematicall indiuisible which may haue a certayne determined situation If a Poynt moue from a determined situation the way wherein it moued is also a Line mathematically produced whereupon of the auncient Mathematiciens a Line is called the race or course of a Point A Poynt we define by the name of a thing Mathematicall though it be no Magnitude and indiuisible because it is the propre ende and bound of a Line which is a true Magnitude And Magnitude we may define to be that thing Mathematicall which is diuisible for euer in partes diuisible long broade or thicke Therefore though a Poynt be no Magnitude yet Terminatiuely we recken it a thing Mathematicall as I sayd by reason it is properly the end and bound of a line Neither Number nor Magnitude haue any Materialitie First we will consider of Number and of the Science Matheâ—aâ—icall to it appropriate called Arithmetike and afterward of Magnitude and his Science called Geometrie But that name contenteth me not whereof a
for their vse deuised The Rule of Alligation in how sundry cases doth it conclude for them such precise verities as neither by naturall witt nor other experience they were hable els to know And with the Marchant then to make an end how ample wonderfull is the Rule of False positions especially as it is now by two excellent Mathematiciens of my familer acquayntance in their life time enlarged I meane Gemma Frisius and Simon Iacob Who can either in brief conclude the generall and Capitall Rules or who can Imagine the Myriades of sundry Cases and particular examples in Act and earnest continually wrought tried and concluded by the forenamed Rules onely How sundry other Arithmeticall practises are commonly in Marchantes handes and knowledge They them selues can at large testifie The Mintmaster and Goldsmith in their Mixture of Metals either of diuerse kindes or diuerse values how are they or may they exactly be directed and meruailously pleasured if Arithmetike be their guide And the honorable PhisiciaÌ„s will gladly confesse them selues much beholding to the Science of Arithmetike and that sundry wayes But chiefly in their Art of Graduation and compounde Medicines And though Galenus Auerâ—ois Arnoldus Lullus and other haue published their positions aswell in the quantities of the Degrees aboue Temperament as in the Rules concluding the new Forme resulting yet a more precise commodious and easy Method is extant by a Countreyman of ours aboue 200 yeares ago inuented And forasmuch as I am vncertaine who hath the same or when that litle Latin treatise as the Author writ it shall come to be Printed Both to declare the desire I haue to pleasure my Countrey wherin I may and also for very good profe of Numbers vse in this most subtile and frutefull Philosophicall Conclusion I entend in the meane while most briefly and with my farder helpe to communicate the pith therof vnto you First describe a circle whose diameter let be an inch Diuide the Circumference into foure equall partes FroÌ„ the Center by those 4. sections extend 4. right lines eche of 4. inches and a halfe long or of as many as you liste aboue 4. without the circumference of the circle So that they shall be of 4. inches long at the leaâ—t without the Circleâ— Make good euident markes at euery inches end If you list you may subdiuide the inches againe into 10. or 12. smaller partes equall At the endes of the lines write the names of the 4. principall elementall Qualities Hote and Coldâ— one against the other And likewise Moyst and Dry one against the other And in the Circle write Temperate Which Temperature hath a good Latitude as appeareth by the Complexion of man And therefore we haue allowed vnto it the foresayd Circle and not a point Mathematicall or Physicall And though I here speake onely of two thyngs Miscible and most commonly mo then three foure fiue or six c. are to be Mixed and in one Compound to be reducedâ— the Forme resultyng of the same to serue the turne yet these Rules are sufficient duely repeated and iterated In procedyng first with any two and then with the Fonne Resulting and an other so forth For the last worke concludeth the Forme resultyng of them all I nede nothing to speake of the Mixture here supposed what it is Common Philosophie hath defined it saying Mixtio est miscibilium alteratorum per minima coniunctorum Vnio Euery word in the definition is of great importance I nede not also spend any time to shew how the other manner of distributing of degrees doth agree to these Rules Neither nede I of the farder vse belonging to the Crosse of Graduation before described in this place declare vnto such as are capable of that which I haue all ready sayd Neither yet with examples specifie the Manifold varieâ—ies by the foresayd two generall Rules to be ordered The witty and Studious here haue sufficient And they which are not hable to atteine to this without liuely teaching and more in particular would haue larger discoursing then is mete in this place to be dealt wâ—thall And other perchaunce with a proude snuffe will disdaine this litlâ—â— and would be vnthankefull for much more I therfore conclude and wish such as haue modest and earnest Philosophicall mindes to laude God highly for this and to Meruayle that the profoundest and subtilest point concerning Mixture of Formes and Qualities Naturall is so Matchâ— and maryed with the most simple easie and short way of the noble Rule of Algiebar Who can remaine therfore vnpersuaded to loue alow and honor the excellent Science of Arithmetike For here you may perceiue that the litle finger of Arithmetike is of more might and contriuing then a hunderd thousand mens wittes of the middle sorte are hable to perfourme or truely to conclude with out helpe thereof Now will we fâ—rder by the wise and valiant Capitaine be certified what helpe he hath by the Rules of Arithmetike in one of the Artes to him appertaining And of the Grekes named 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 That is the Skill of Oâ—dring Souldiers in Battell ray after the best maner to all purposes This Art so much dependeth vppon Numbers vse and the Mathematicals that AElianus the best writer therof in his worke tö the Emperour Hadrianus by his perfection in the Mathematicals beyng greater then other before him had thinketh his booke to passe all other the excellent workes written of that Art vnto his dayes For of it had written AEneas Gyneas of Thessaly Pyrrhus Epirota and Alexander his sonne Clearchus Pausanias Euangelus Polybiâ—s familier frende to Scipio Eupolemus Iphicrates Possidonius and very many other worthy Capitaines Philosophers and Princes of Immortall fame and memory Whose fayrest floure of their garland in this feat was Arithmetike and a litle perceiuerance in Geometricall Figures But in many other cases doth Arithmetike stand the Capitaine in great stede As in proportionyng of vittayles for the Army either remaining at a stay or suddenly to be encreased with a certaine number of Souldiers and for a certain tyme Or by good Art to diminish his company to make the victuals longer to serue the remanent for a certaine determined tyme if nede so require And so in sundry his other accountes Reckeninges Measurynges and proportionynges the wise expert and Circumspect â—apitaine will affirme the Science of Arithmetike to be one of his chief Counsaylors directers and aiders Which thing by good meanes was euident to the Noble the Couragious the loyall and Curteous Iohn late Earle of Warwicke Who was a yong Gentleman throughly knowne to very few Albeit his lusty valiantnes force and Skill in Chiualrous feates and exerciâ—es his humblenes and frendelynes to all men were thinges openly of the world perceiued But what rotes otherwise vertue had fastened in his brest what Rules of godly and honorable life he had framed to him selfe what vices in some then liuing notable he tooke great care to eschew
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
superficies ABCD. whose length is taken by the lyne AB or CD and breadth by the lyne AC or BD and by reason of those two dimensions a superficies may be deuided two wayes namely by his length and by hys breadth but not by thicknesse for it hath none For that is attributed onely to a body which is the third kynde of quantitie and hath all three dimensions length breadth and thicknes and may be deuided according to any of them Others define a superficies thus A superficies is the terms or ende of a body As a line is the ende and terme of a superficies 6 Extremes of a superficies are lynes As the endes limites or borders of a lyne are pointes inclosing the line so are lines the limites borders and endes inclosing a superficies As in the figure aforesayde you maye see the superficies inclosed with foure lynes The extremes or limites of a bodye are superficiesles And therfore a superficies is of some thus defined A superficies is that which endeth or incloseâ—h a body as is to be sene in the sides of a die or of any other body 7 A plaine superficies is that which lieth equally betwene his lines A plaine superficies is the shortest extension or drâ—ught from one lyne to an other like as a right lyne is the shortest extension or draught from one point to an other Euclide also leaueth out here to speake of a crooked and hollow superficies because it may easely be vnderstand by the diffinition of a plaine superficies being hys contrary And euen as from one point to an other may be drawen infinite crooked lines but one right line which is the shortest so from one lyne to an other may be drawen infinite croked superficiesses but one plain superficies which is the shortest Here must you consider when there is in Geometry mention made of pointes lines circles triangles or of any other figures ye may not conceyue of them as they be in matter as in woode in mettall in paper or in any such lyke for so is there no lyne but hath some breadth and may be deuidedâ— nor points but that shal haue some partes and may also be deuided and so of others But you must conceiue them in mynde plucking them by imagination from all matter so shall ye vnderstande them truely and perfectly in their owne nature as they are defined As a lyne to be long and not broade and a poynte to be so little that it shall haue no part at all Others otherwyse define a playne superficies A plaine superficies is that which is firmly set betwene his extremes as before was sayd of a right lyne Agayne A plaine superficies is that vnto all whose partes a right line may well be applied Again A plaine superficies is that which is the shortest of al superficies which haue one the self extremes As a rigâ—t line was the shortest line that can be drawen betwene two pointes Againe A playne superficies is that whose middle darkeneth the extremes as was also sayd of a right lyne 8 A plaine angle is an inclination or bowing of two lines the one to the other and the one touching the other and not beyng directly ioyned together As the two lines AB BC incline the one to the other and touch the one the other in the point B in which point by reason of the inclination of the sayd lines is made the angle ABC But if the two lines which touch the one the other be without all inclination of the one to the other and be drawne directly the one to the other then make they not any angle at all as the lines CD and DE touch the one the other in the point D and yet as ye see they make no angle 9 And if the lines which containe the angle be right lynes then is it called a rightlyned angle As the angle ABC in the former figures is a rightlined angle because it is contained of right lines where note that an angle is for the most part described by thre letters of which the second or middle letter representeth the very angle and therfore is set at the angle By the contrary a crooked lyned angle is that which is contained of crooked lines which may be diuersly figured Also a mixt angle is that which is caused of them both namely of a right line and a crooked which may also be diuersly figured as in the figures before set ye may see There are of angles thre kindes a right angle an acute angle and an obtuse angle the definitions of which now follow 10 VVhen a right line standing vpon a right line maketh the angles on either side equallâ— then either of those angles is a right angle And the right lyne which standeth erected is called a perpendiculer line to that vpon which it standeth As vpon the right line CD suppose there do stand an other line AB in such sort that it maketh the angles on either side therof equall namely the angle ABC on the one side equall to the angle ABD on the other side then is eche of the two angles ABC and ABD a right angle and the line AB which standeth erected vpon the line CD without inclination to either part is a perpendicular line commonly called among artificers a plumbe lyne 11 An obtuse angle is that which is greater then a right angle As the angle CBE in the example is an obtuse angle for it is greater then the angle ABC which is a right angle because it contayneth it and containeth moreouer the angle ABE 12 An acute angle is that which is lesse then a right angle As the angle EBD in the figure before put is an acute angle for that it is lesse then the angle ABD which is a right angle for the right angle containeth it and moreouer the angle ABE 13 A limite or terme is the ende of euery thing For as much as of thinges infinite as Plato saith there is no science therefore must magnitude or quantitie wherof Geometry entreateth be finite and haue borders and limites to inclose it which are here defined to be the endes therof As a point is the limite or terme of a line because it is th end therof A line likewise is the limite terme of a superficies and likewise a superficies is the limite and terme of a body as is before declared 14 A figure is that which is contayned vnder one limite or terme or many As the figure A is contained vnder one limit which is the round line also the figure B is contayned vnder three right lines And the figure C vnder foure and so of others which are their limites or termes 15 A circle is a plaine figure conteyned vnder one line which is called a circumference vnto which all lynes drawen from one poynt within the figure and falling vpon
AB a line equall to the line AF that is to the line CD But if they cut the one the other as the lines CD AB do TheÌ„ making the ceÌ„tre B the space BA describe a circle AF draw a line from the point B to the point C produce it to the point F. And forasmuch as the two right lines BF and CD are vnequall and the line CD cutteth the line BF by one of his extreames therefore it is possible to cut of from CD a line equall to the line BF For how to do it we haue before declared wherefore it is possible from the line CD to cut of a line equall to the line AB or AB and BF are equall the one to the other This is to be noted that in all these cases a man may both as touchyng construction and demonstration procede as in the first case For it is possible in any position to put to the ende of the greater lyne a line equall to the lesse lyne and so makyng the centre the sayd ende and the space the lesse line to describe a circle which shall cut of froÌ„ the greater lyne a lyne equall to the line put namely to the lesse line geuen as it is manifest to see in the figures partly here vnder set and partly in the beginning of the other side put The first Theoreme The 4. Proposition If there be two triangles of which two sides of th' one be equal to two sides of the other eche side to his correspondent side and hauing also on angle of the one equal to one angle of the other namely that angle which is contayned vnder the equall right lines the base also of the one shall be equall to the base of the other and the one triangle shall be equal to the other triangle and the other angles remayning shal be equall to the other angles remayning the one to the other vnder which are subtended equall sides SVppose that there be two triangles ABC DEF hauing two sides of the one namely AB and AC equall to two sides of the other namely to DE and DF the one to the other that is AB to DE and AC to DF hauyng also the angle BAC equall to the angle EDF Then I say that the base also BC is equall to the base EF that the triangle ABC is equall to the triangle DEF and that the other angles remainyng are equall to the other angles remayning the one to the other vnder which are subtended equall sydes that is that the angle ABC is equall to the angle DEF and that the angle ACB is equall to to the angle DFE For the triangle ABC exactly agreyng with the triangle DEF and the point A being put vpoÌ„ the point D the right line AB vpon the right line DE the pointe B also shall exactly agree with the pointe E for that by supposition the line AB is equal to the line DE. And the line AB exactly agreeyng with the line DE the right line also AC exactly agreeth with the right line DF for that by supposition the angle BAC is equall to the angle EDF And forasmuch as the right line AC is also by supposition equall to the right line DF therfore the pointe C exactly agreeth with the pointe F. Againe forasmuch as the pointe C exactly agreeth with the poynte F and the point B exactly agreeth with the point E therefore the base BC shall exactly agree with the base EF. For if the point B do exactly agree with the point E and the point C with the point F and the base BC do not exactly agre wyth the base EF then two right lines do include a superficies which by the 10. coÌ„mon sentence is impossible VVherfore the base BC exactly agreeth with the base EF and therfore is equall vnto it VVherfore the whole triangle ABC exactly agreeth with the whole triangle DEF therfore by the 8. common sentence is equall vnto it And by the same the other angles remayning exactly agree with the other angles remayning and are equall the one to the other that is the angle ABC to the angle DEF and the angle ACB to the angle DFE If therfore there be two triangles of which two sides of the one be equall to two sydes of the other eche to his correspondent side and hauing also one angle of the one equall to one angle of the other namely that angle which is contayned vnder the equall right lines the base also of the one shall be equall to the base of the other and the one triangle shall be equall to the other triangle and the other angles remainyng shall be equall to the other angles remayning the one to the other vnder which are subtended eqaull sydes whiche thing was required to be demonstrated This Proposition which is a Theoreme hath two things geuen namely the equality of two sides of the one triangle to two sides of the other triangle and the equalitie of two angles contayned vnder the equall sydes In it also are thre thinges required The equality of base to base the equality of field to field and the equality of the other angles of the one triangle to the other angles of the other triangle vnder which are subtended equall sides One side of a playne figure is equall to an other and so generally one right lyne is equall to an other when the one being applied to the other theyr extreames agree together For otherwise euery righte line applied to any right lyne agreeth therwith but equall right lines only agree in the extremes One rectilined angle is equall to an other rectilined angle when one of the sides which comprehendeth the one angle being set vpon one of the sides which comprehendeth the other angle the other side of the one agreeth with the other syde of the other And that angle is the greater whose syde falleth without and that the lesse whose syde falleth within VVhere as in this proposition is put this particle eche to his correspondent side instede wherof often times afterward is vsed this phrase the one to the other it is of necessity so put For otherwise two sydes of one triangle added together may be equall to two sydes of an other triangle added together and the angles also contayned vnder the equall sydes may be equall and yet the two triangles may notwithstanding be vnequall VVhere note that a triangle is sayd to be equall to an other triangle when the field or area of the one is equall to the area of the other And the area of a triangle is that space which is contayned within the sydes of a triangle And the circuite or compasse of a triangle is a line composed of all the sides of a triangle And so may you think of all oâ—her rectilined figures And now to proue that there may be two triangles two sydes of one of which being added together
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
diameter is double to that square whose diameter it is The 34. Theoreme The 48. Proposition If the square which is made of one of the sides of a triangle be equall to the squares which are made of the two other sides of the same triangle the angle comprehended vnder those two other sides is a right angle SVppose that ABC be a triangle and let the square which is made of one of the sides there namely of the side BC be equall to the squares which are made of the sides BA and AC Then I say that the angle BAC is a right angle Rayse vp by the 11. propositioÌ„ from the point A vnto the right line AC a perpendicular line AD. And by the thirde proposition vnto the line AB put an equall line AD. And by the first peticion draw a right line from the point D to the poinâ— C. And forasmuch as the line DA is equall to the line AB the square which is made of the line DA is equall to the square whiche is made of the line AB Put the square of the line AC common to them both VVherefore the squares of the lines DA and AC are equal to the squares of the lines BA and AC But by the proposition going before the square of the line DC is equal to the squares of the lines AD and AC For the angle DAC is a right angle and the square of BC is by supposition equall to the squares of AB and AC VVherefore the square of DC is equall to the square of BC wherefore the side DC is equall to the side BC. And forasmuch as AB is equall to AD â—nd AC is common to them both therefore these two sides DA and AC are equall to these two sides BA and AC the one to the other and the base DC is equall to the base BCâ— wherfore by the 8. proposition the angle DAC is equall to the angle BAC But the angle DAC is a right angle wherefore also the angle BAC is a right angle If therefore the square which is made of one of the sides of a triangle be equall to the squares which are made of the two other sides of the same triangle the angle comprehended vnder those two other sides is a right angle which was required to be proued This proposition is the conuerse of the former and is of Pelitarius demonstrated by an argument leading to an impossibilitie after this maner The ende of the first booke of Euclides Elementes ¶ The second booke of Euclides Elementes IN this second booke Euclide sheweth what is a GnomoÌ„ and a right angled parallelogramme Also in this booke are set forth the powers of lines deuided euenly and vneuenly and of lines added one to an other The power of a line is the square of the same line that is a square euery side of which is equall to the line So that here are set forth the qualities and proprieties of the squares and right lined figures which are made of lines of their parts The Arithmetician also our of this booke gathereth many compendious rules of reckoning and many rules also of Algebra with the equatioÌ„s therein vsed The groundes also of those rules are for the most part by this second booke demonstrated This booke moreouer contayneth two wonderfull propositions one of an obtuse angled triangle and the other of an acute which with the ayde of the 47. proposition of the first booke of Euclide which is of a rectangle triangle of how great force and profite they are in matters of astronomy they knowe which haue trauayled in that arte VVherefore if this booke had none other profite be side onely for these 2. propositions sake it were diligently to be embraced and studied The definitions 1. Euery rectangled parallelogramme is sayde to be contayned vnder two right lines comprehending a right angle A parallelogramme is a figure of fower sides whose two opposite or contrary sides are equall the one to the other There are of parallelogrammes fower kyndes a square a figure of one side longer a Rombus or diamond and a Romboides or diamond like figure as before was sayde in the 33. definition of the first booke Of these fower sortes the square and the figure of one side longer are onely right angled Parallelogrammes for that all their angles are right angles And either of them is contayned according to this definition vnder two right lynes whiâ—h concurre together and cause the right angle and containe the same Of which two lines the one is the length of the figure the other the breadth The parallelogramme is imagined to be made by the draught or motion of one of the lines into the length of the other As if two numbers shoulde be multiplied the one into the other As the figure ABCD is a parallelograme and is sayde to be contayned vnder the two right lines AB and AC which contayne the right angle BAC or vnder the two right lines AC and CD for they likewise contayne the right angle ACD of which 2. lines the one namely AB is the length and the other namely AC is the breadth And if we imagine the line AC to be drawen or moued directly according to the leÌ„gth of the line AB or contrary wise the line AB to be moued directly according to the length of the line AC you shall produce the whole rectangle parallelogramme ABCD which is sayde to be contayned of them euen as one number multiplied by an other produceth a plaine and righte angled superficiall number as ye see in the figure here set where the number of sixe or sixe vnities is multiplied by the number of fiue or by fiue vnities of which multiplication are produced 30. which number being set downe and described by his vnities representeth a playne and a right angled number VVherefore euen as equall numbers multipled by equal numbers produce numbers equall the one to the other so rectangle parallelogrames which are comprehended vnder equal lines are equal the one to the other 2. In euery parallelogramme one of those parallelogrammes which soeuer it be which are about the diameter together with the two supplementes is called a Gnomon Those perticuler parallelogrames are sayde to be about the diameter of the parallelograme which haue the same diameter which the whole parallelograme hath And supplementes are such which are without the diameter of the whole parallelograme As of the parallelograme ABCD the partial or perticuler parallelogrames GKCF and EBKH are parallelogrames about the diameter for that ech of them hath for his diameter a part of the diameter of the whole parallelogramme As CK and KB the perticuler diameters are partes of the line CB which is the diameter of the whole parallelogramme And the two parallelogrammes AEGK and KHFD are supplementes because they are wythout the diameter of the whole parallelogramme namely CB. Now any one of those partiall parallelogrammes
of other â—ectiline figures consider By these two definitions it is manifest that the inscription and circumscription of rectiline figures here spoken of pertayne to such rectiline figures onely which haue equall sides and equall angles which are commonly called regular It is also to be noted that rectiline figures only of one kinde or forme can be inscribed or circumscribed the one within or about the other As a triangle within or about a triangle A square within or about a square and so a Pentagon within or about a PentagoÌ„ likewise of others of one forme But a triangle can not be inscribed or circumscribed within or aboute a square nor a square within or about a Pentagon And so of others of diuers kyndes For euery playne rectiline figure hath so many angles as it hath sides Wherfore the figure inscrided must haue so many angles as the figure in which it is inscribed hath sides and the angles of the one as is sayd must touche the sides of the other And contrariwise in circumscription of figures the sides of the figure circumscribed must touch the angles of the figure about which it is circumscribed A rectiline figure is sayd to be inscribed in a circle when euery one of the angles of the inscribed figure toucheth the circumference of the circle A circle by reason of his vniforme and regular distance which it hath from the centre to the circumference may easily touche all the angles of any regular rectiline figure within it and also all the sides of any figure without it And therfore any regular rectiline figure may be inscribed within it and also be circumscribed about it And agayne a circle may be both inscribed within any regular rectiline figure and also be circumscribed about it As the triangle ABC is inscribed in the circle ABC â— for that euery angle toucheth some one pointe of the circumference of the circle As the angle CAB of the triangle ABC toucheth the point A of the circumference of the circle And the angle ABC of the triangle toucheth the pointe B of the circumference of the circle And also the angle ACB of the triâ—ngle 〈…〉 the pointe â— of the circumference of the circle In like manner the square ADEF is inscribed in the same circle ABC for that euery angle of the square inscribed toucheth some one poynte of the circle in which it is inscribed And so imagine you of rectilined figures A circle is sayd to be circumscribed about a rectiline figure wheÌ„ the circumference of the circle toucheth euery one of the angles of the figure about which it is circumscribed As in the former example of the third definition The circle ADEF is circumscribed about the triangle ABC because the circumference of the circle which is circumscribed toucheth euery angle of the triangle about which it is circumscribedâ— namely the angles CAB ABC and BCA Likewise the same circle ADEF is circumscribed about the square ADEF by the same definition as you may see A circle is sayd to be inscribed in a rectiline figure when the circumference of the circle toucheth euery one of the sides of the figure within which it is inscribed As the circle ABCD is inscribed within the triangle EFG because the circumference of the circle toucheth euery side of the triangle in which it is inscribedâ— namely the side EF in the point B and the side GF in the pointe C and the side GE in the point D. Likewise by the same reason the same circle is inscribed within the square HIKL And so may you iudge of other rectiline figures A rectilined figure is said to be circumscribed about a circle when euery one of the sides of the figure circumscribed toucheth the circumference of the circle As in the former figure of the fift definition the triangle EFG is circumscribed about the circle ABCD for that euery side of the same triangle beyng circumscribed toucheth the circumference of the circle about which it is circumscribed As the side EG of the triangle EFG toucheth the circumference of the circle in the point D and the side EF toucheth it in the point B and the side GF in the point C. Likewise also the square HIKL is circumscribed about the circle ABCD for euery one of his sides toucheth the circumference of the circle namely in the pointes A B C D. And thus consider of all other regular right lined figures for of them onely are vnderstanded these definitions to be circumscribed about a circle or to be inscribed within a circle or of a circle to be circumscribed or inscribed about or within any of them A right lyne is sayd to be coapted or applied in a circle when the extremes or endes therof fall vppon the circumference of the circle As the line BC is sayd to be coapted or to be applied to the circle ABC for that both his extremes fall vpon the circumference of the circle in the pointes B and C. Likewise the line DE. This definition is very necessary and is properly to be taken of any lyne geuen to be coapted and applied into a circle so â—hat it exceede not the diameter of the circle geuen The 1. Probleme The 1. Proposition In a circle geuen to apply a right line equall vnto a right line geuen which excedeth not the diameter of a circle SVppose that the circle geuen be ABC and let the right line geuen exceding not the diameter of the same circle be D. Now it is required in the circle geuen ABC to apply a right line equall vnto the right line D. Draw the diameter of the circle ABC and let the same be BC. Now if the line BC be equall vnto the line D then is that done which was required For in the circle geuen ABC is applyed a right line BC equall vnto the right line D. But if not then is the line BC greater then the line D. And by the third of the first put vnto the line D an equall line CE. And making the centre C and the space CE describe by the third petition a circle EGF cutting the circle ABC in the point F draw a line from C to F. And for asmuch as the point C is the centre of the circle EGF therefore by the 13. definition of the first the line CF is equall vnto the line CE. But the line CE is equall vnto the line D. Wherefore by the first common sentence the line CF also is equall vnto the line D. Wherefore in the circle geuen ABC is applyed a right line CA equall vnto the right line geuen D which was required to be done The 2. Probleme The 2. Proposition In a circle geuen to describe a triangle equiangle vnto a triangle geuen SVppose that the circle geuen be ABC and let the triangle geuen be DEF Now it is required in the circle geuen ABC to describe a triangle equiangle vnto the triangle geuen DEF
the diameter of the circle shall be drawen a right line making two right angles falling within the circle which is impossible as it was manifest by the 16. of the third Wherefore the circle described D being the centre therof and the space therof being either DE or DF or DG cutteth not these right lines AB BC CA. Wherfore by the Corollary of the same it toucheth them and the circle is described in the triangle ABC Wherfore in the triangle geuen ABC is described a circle EFG which was required to be done The 5. Probleme The 5. Proposition About a triangle geuen to describe a circle SVppose that the triangle geuen be ABC It is required about the triangle ABC to describe a circle Deuide by the 10. of the first the right lâ—nes AB and AC into two equall partes in the pointes D and E. And from the pointes D and E by the 11. of the first rayse vp vnto the lines AB AC two perpendicular lines DF and EF. Now these perpendicular lines meete together either within the triangle ABC or in the right line BC or els withoâ— the right line BC. But now suppose that the right lines DF and EF do meete together without the triangle ABC in the point F. Againe as it is in the third description draw right lines from F to A from F to B and from F to C. And forasmuch as the line AD is equall vnto the line DB and the line DF is common vnto them both and maketh a right angle on eche side of him wherfore by the 4. of the first the base AF is equall vnto the base BF And in like sort may we proue that thâ— line CF is equal vnto the line AF. Wherfore agayne making F the centre and the space FA or FB or FC describe a circle and it shal passe by the pointes A B C and so is there a circle described about the triangle ABC as ye see it iâ— in the third description Wherfore about a triangle geuen is described a circle which was required to be done Correlary Hereby it is manifest that when the centre of the circle falleth within the triangle the angle BAC being in a greater segment of a circle is lesse theÌ„ a right angle But when it falleth vpon the right line BC the angle BAC being in a semicircle is a right angle But when the centre falleth without the right line BC the angle BAC being in a lesse segment of a circle is greater then a right angle Wherfore also when the angle geuen is lesse then a right angle the right lines DF and EF will meete together within the sayd triangle But when it is a right angle they will meete together vpon the line BC. But wheÌ„ it is greater then a right angle they will meete together without the right line BC. The 6. Probleme The 6. Proposition In a circle geuen to describe a square SVppose that the circle geuen be ABCD. It is required in the circle ABCD to describe a square Draw in the circle ABCD two diameters making right angles and let the same be AC and BD and drawe right lines from A to B from B to C from C to D and from D to A. And forasmuch as the line BE is equall vnto the line ED by the 15. definition of the first for the point E is the centre And the line EA is common to them both making on eche side a right angle therfore by the 4. of the first the base AB is equall vnto the base AD. And by the same reason also either of these lines BC and CD is equall to either of these lines AB and AD wherefore ABCD is a figure of foure equal sides I say also that it is a rectangle figure For forasmuch as the right line BD is the diameter of the circle ABCD therfore the angle BAD beyng in the semicircle is a right angle by the 31. of the third And by the same reason euery one of these angles ABC BCD and CDA is a right angle Wherfore the foure sided figure ABCD is a rectangle figure and it is proued that it consisteth of equall sides Wherfore by the 30. definition of the first it is a square and it is described in the circle ABCD which was required to be done The 7. Probleme The 7. Proposition About a circle geuen to describe a square SVppose that the circle geuen be ABCD. It is required about the circle ABCD to describe a square Draw in the circle ABCD two diameters making right angles where they cut the one the other and let the same be AC and BD. And by the pointes A B C D draw by the 17. of the third right lines touching the circle ABCD and let the same be FG GH HK and KF Now for asmuch as the right line FG toucheth the circle ABCD in the point A and from the centre E to the point A where the touch is is drawen a right line EA therfore by the 18. of the third the angles at the point A are right angles and by the same reason the angles which are at the pointes B C D are also right angles And forasmuch as the angle AEB is a right angle the angle EBG is also a right angle therfore by the 28. of the first the line GH is a parallel vnto the line AC and by the same reason the lyne AC is a parallel vnto the lyne FK In like sorte also may we proue that either of these lines GF and HK is a parallel vnto the lyne BED Wherfore these figures GK GC AK FB and BK are parallelogramesâ— Wherfore by the 34. of the first the line GF is equall vnto the line HK and the line GH is equall vnto the line FK And forasmuch as the line AC is equall vnto the lyne BD but the line AC is equall vnto either of these lines GH and FK anâ— the line BD is equall to either of these lines GF and HK Wherfore either of these lines GH and FK is equall to either of these lines GF and HK Wherfore the figure FGHK consisteth of foure equall sides I say also that it is a rectangle figure For forasmuch as GBEA is a parallelogramme and the angle AEâ— is a right angle therfore by the 34. of the first the angle AGB is a right angle In like sorte may we proue that the angles at the poyntes H K and F are right angles Wherfore FGHK is a rectangle foure sided figure and it is proued that it consisteth of equall sides wherfore it is a square and it is described about the circle ABCD. Wherfore about a circle geuen is described a square which was required to be done The 8. Probleme The 8. Proposition In a square geuen to describe a circle SVppose that the square geuen be ABCD. It is required in the square ABCD to describe a circle Deuide by the 10.
13. of the sixth take the meane proportionall betwene the lines BC and CF which let be GH And by the 18. of the sixth vpon the line GH let there be described a rectiline figure KHG like vnto the rectiline figure ABC and in like sorte situate And for that as the line BC is to the line GH so is the line GH to the line CF but if there be thre right lines proportional as the first is to the third so is the figure which is described of the first vnto the figure which is described of the second the said figures being like and in like sort situate by the second correllary of the 20. of the sixth wherfore as the line BC is to the line CF so is the triangle ABC to the triangle KGH But as the line BC is to the lyne CF so is the parallelogramme BE to the parallelogramme EF by the 1. of the sixth Wherfore as the triangle A BC is to the triangle KGH so is the parallelogramme BE to the parallelogramme EF. Wherfore alternately also by the 16. of the fifth as the triangle ABC is to the parallelograÌ„me BE so is the triangle KGH to the parallelogramme EF but the triangle ABC is equal vnto the parallelograÌ„me BE wherfore also the triangle KGH is equall vnto the parallelogramme EF but the parallelogramme FE is equall vnto the rectiline figure D. Wherfore also the rectiline figure KGH is equall vnto the rectiline figure D and the rectiline figure KGH is by supposition like vnto the rectiline figure ABC Wherefore there is described a rectiline figure KGH lyke vnto the rectiline figure geuen ABC and equall vnto the other rectiline figure geuen D which was required to be done The 19. Theoreme The 26. Proposition If from a parallelogramme be taken away a parallelograme like vnto the whole and in like sorte set hauing also an angle common with it then is the parallelogramme about one and the selfe same dimecient with the whole SVppose that there be a parallelogramme ABCD and from the parallelogramme ABCD take away a parallelogramme AF like vnto the parallelogramme ABCD and in like sort situate hauing also the angle DAB common with it Then I say that the parallelogrammes ABCD and AF are both about one and the self same dimecient AFC that is that the dimecient AFC of the whole parallelogramme ABCD passeth by the angle F of the parallelogramme AF and is common to either of the parallelogrammes For if AC do not passe by the point F then if it be possible let it passe by some other point as AHC doth Now then the dimetient AHC shall cut eyther the side GF or the side EF of the parallelogramme AF. Let it cut the side GF in the point H. And by the 31. of the first by the point H let there be drawen to either of these lines AD and BC a parallel line HK wherfore GK is a parallelogramme and is about one and the selfe same dimetient with the parallelogramme ABCD. And forasmuch as the parallelogrammes ABCD and GK are about one and the self same dimecient therfore by the 24. of the sixth the parallelogramme ABCD is like vnto the parallelogramme GK Wherfore as the line DA is to the line AB so is the line GA to the line AK by the conuersion of the first definition of the sixth And for that the parallelogrammes ABCD and EG are by supposition like therfore as the line DA is to the lyne AB so is the line GA to the line AE Wherfore the line GA hath one and the selfe proportion to either of these lines AK and AE Wherfore by the 9. of the fifth the line AK is equall vnto the line AE namely the lesse to the greater which is impossible The selfe same inconuenience also will follow if you put the dimetient AC to cut the side FE Wherfore AC the dimetieâ—t of the whole parallelogramme ABCD passeth by the angle and poynt F. And therfore the parallelogramme AEFG is about one and the selfe same dimetient with the whole parallelogramme ABCD. Wherfore if from a parallelogramme be taken away a parallelograme lyke vnto the whole and in lyke sorte situate hauing also an angle common with it then is that parallelogramme about one and the selfe same dimetient with the whole which was required to be proued ¶ An other demonstration after Flussates which proueth this proposition affirmatiuely From the parallelogramme ABGD let there be taken away the parallelogramme AEZK like and in like sorte situate with the whole parallelogramme ABGD and hauing also the angle A common with the whole parallelogramme Then I say that both their diameters namely AZ and AZG do make one and the selfe same right line Deuide the sides AB and Bâ— into two equall partes in the pointes C and F by the 10. of the first And drawe a line from C to F. Wherefore the line CF is a parallel to the right line AG by the corollary added by Campane after the 29. of the first Wherfore the angles BAG and BCF are equall by the 29. of the first but the angle EAZ is equall vnto the angle BAG by reasoÌ„ the parallelogrammes are supposed to be like wherefore the same angle EAZ is equall to the angle BCF namely the outward angle to the inward and opposite angle Wherfore by the 28. of the first the lines AZ and CF are parallel lynes Now then the lines AZ and AG being parallels to one and the selfe same lyne namely to CF do concurre in the point A. Wherefore they are set directly the one to the other so that they both make one right line by that which was added in the ende of the 30. proposition of the first wherfore the parallelogrammes ABGD and AEZK are about one and the selfe fame dimetient which was required to be proued The 20. Theoreme The 27. Proposition Of all parallelogrammes applied to a right line wanting in figure by parallelogrammes like and in like sort situate to that parallelograme which is described of the halfe line the greatest parallelogramme is that which is described of the halfe line being like vnto the want LEt there be a right line AB and by the 10. of the first deuide it in two equall partes in the point C. And vnto the right line AB apply a parallelogramme AD wanting in figure by the parallelogramme DB which let be like and in like sort described vnto the parallelogramme described of halfe the line AB which is BC. Then I say that of all the parallelogrammes which may be applied vnto the line AB and which waÌ„t in figure by parallelograÌ„mes like and in like sort situate vnto the parallelogramme DB the greatest is the parallelogramme AD. For vnto the right line AB let there be applied a parallelogramme AF wanting in figure by the parallelogramme FB which let be like and in like sort situate vnto the parallelogramme DB. Then I say that the parallelogramme AD is
all those numbers A B C D vnitie added together and let E multiplying D which is the last number produce the number FG. Then I say that FG is a perfect number How many in multitude A B C D are so many in continuall double proportion take beginning at E which let be the numbers E HK L and M. VVherefore of equalitie by the 13. of the seuenth as A is to D so is E to M. VVherefore that which is produced of E into D is equall to that which is produced of A into M. But that which is produced of E into D is the number FG. VVherefore that which is produced of A into M is equall vnto FG. VVherefore A multiplying M produceth FG. VVherefore M measureth FG by those vnities which are in A. But A is the number two VVherefore FG is double to M. And the numbers M L HK and E are also in continuall double proportion VVherefore all the numbers E HK L M and FG are continually proportionall in double proportion Take from the second number KH and from the last FG a number equall vnto the first namely to E and let those numbers taken be HN FX VVherefore by the Proposition going before as the excesse of the second number is to the first number so is the excesse of the last to all the numbers going before it VVherefore as NK is to E so is XG to these numbers M L KH and E. But NK is equall vnto E for it is the halfe of HK which is supposed to be double to E VVherefore XG is equall vnto these numbers M L HK and E. But XF is equall vnto E and E is equall vnto these numbers A B C D and vnto vnitie Wherfore the whole number FG is equall vnto these numbers E HK L M and also vnto these numbers A B C D and vnto vnitie Moreouer I say that vnitie and all the numbers A B C D E HK L and M do measure the number FG. That vnitie measureth it it needeth no proufe And forasmuch as FG is produced of D into E therefore D and E do measure it And forasmuch as the double from vnitie namely the nūbers A B C do measure the number D by the 13. of this booke therefore they shall also measure the number FG whom D measureth by the ◠common sentence By the same reason forasmuch as the nūbers E HK L and M are vnto FG as vnitie and the numbers A ◠C are vnto D namely in subduple proportion and vnitie and the numbers A B C do 〈◊〉 D therefore also the numbers E HK L and M shall measure the number FG No◠I say also that no other number measureth FG besides these numbers A B C D E HK L M and vnitie For if it be possible let O measure FG. And let O not be any of these numbers A B C D E HK L and M. And how often O measureth FG 〈◊〉 vnities let there be in P. Wherefore O multiplying P produceth FG. But E also multiplying D produced FG. Wherefore by the 19. of the seuenth as E is to O so is P to D. Wherefore alternately by the 9. of the seuenth as E is to P so is O to D. And forasmuch as from vnitie are these numbers in continuall proportion A B C D and the number A which is next after vnitie is a prime number therfore by the 13. of the ninth no other number measureth D besides the numbers A B C. And it is supposed that O is not one and the same with any of these nūbers A B C. Wherefore O measureth not D. But as O is to D so is E to P. Wherefore neither doth E measure P. And E is a prime number But by the 31. of the seuenth euery prime number is to euery number that it measureth not a prime number Wherefore E and P are prime the one to the other yea they are prime and the least But by the 21. of the seuenth the least measure the numbers that haue one and the same proportion with them equally the antecedent the antecedent and the consequent the consequent And as E is to P so is O to D. Wherefore how many times E measureth O so many times P measureth D. But no other number measureth D besides the numbers A B C by the 13. of this booke Wherefore P is one and the same with one of these numbers A B C. Suppose that P be one and the same with B how many B C D are in multitude so many take from E vpward namely E HK and L. But E HK and L are in the same proportion that B C D are VVherefore of equalitie as B is to D so is E to L. VVherefore that which is produced of B into L is equall to that which is produced of D into E. But that which is produced of D into E is equall to that which is produced of P into O. VVherefore that which is produced of P into O is equall to that which is produced of B into L. VVherefore as P is to B so is L to O and P is one the same with B wherefore L also is one and the same with O which is impossible For O is supposed not to be one and the same with any of the numbers geuen VVherefore no number measureth FG besides these numbers A B C D E HK L M and vnitie And it is proued that FG is equall vnto these numbers A B C D E HK L M and vnitie which are the partes therof by the 39. of the seuenth But a perfect nūber by the definition is that which is equall vnto all his partes VVherfore FG is a perfect number◠which was required to be proued The end of the ninth booke of Euclides Elementes ¶ The tenth booke of Euclides Elementes IN THIS TENTH BOOKE doth Euclide entreat of lines and other magnitudes rationall irrationall but chiefly of irrationall magnitudes commensurable and incommensurable of which hitherto in al his former 9. bokes he hath made no mention at all And herein differeth number from magnitude or Arithmetike from Geometry for that although in Arithmetike certayne numbers be called prime numbers in consideration of themselues or in respect of an other and so are called incommensurable for that no one number measureth them but onely vnitie Yet in dede and to speake absolutely and truely there are no two numbers incommensurable but haue one common measure which measureth thē both if none other yet haue they vnitie which is a common part and measure to all numbers and all numbers are made of vnities as of their partes As hath before bene shewed in the declaration of the definitiōs of the seuenth booke But in magnitude it is farre otherwise for although many lines plaine figures and bodies are commensurable and may haue one measure to measure them yet all haue not so nor can haue For that a line is not made of pointes as
rationall part rather then of the mediall part ¶ The 28. Theoreme The 40. Proposition If two right lines incoÌ„mensurable in power be added together hauing that which is made of the squares of them added together mediall and the parallelogramme contayned vnder them rationall the whole right line is irrationall and is called a line contayning in power a rationall and a mediall superficies In this Proposition is taught the generation of the sixt irrationall line which is called a line whose power is rationall and mediall The definition of which is gathered of thys Proposition after this maner A line whose power is rationall and mediall is an irrationall line which is made of two right lines incommensurable in power added together whose squares added together make a mediall superficies but that supersicies which they containe is rationall The reason of the name is before set forth in the Proposition ¶ The 29. Theoreme The 41. Proposition If two right lines incommensurable in power be added together hauyng that which is composed of the squares of them added together mediall and the parallelogramme contayned vnder them mediall and also incommensurable to that which is composed of the squares of them added togetherâ— the whole right line is irrationall and is called a line contayning in power two medials In this proposition is taught the nature of the 7. kinde of irrationall lines which is called a line whose power is two medials The definition whereof is taken of this proposition after this maner A line whose power is two medials is an irrationall line which is composed of two right lines incommensurable in power the squares of which added together make a mediall superficies and that which is contained vnder them is also mediall and moreouer it is incommensurable to that which is composed of the two squares added together The reason why this line is called a line whose power is two medials was before in the ende of the demonstration declared And that the said irrationall lines are deuided one way onely that is in one point onely into the right lines of which they are composed and which make euery one of the kindes of those irrationall lines shall straight way be demonstrated but first will we demonstrate two assumptes here following ¶ An Assumpt Take a right line and let the same be AB and deuide it into two vnequall partes in the point C and againe deuide the same line AB into two other vnequal partes in an other point namely in D and let the line AC by supposition be greater then the line DB. Then I say that the squares of the lines AC and BC added together are greater then the squares of the lines AD and DB added together Deuide the line AB by the 10. of the first into two equall partes in the point E. And forasmuch as the line AC is greater then the line DB take away the line DC which is common to them both wherfore the residue AD is greater then the residue CB but the line AE is equall to the line EB Wherfore the line DE is lesse then the line EC Wherfore the pointes C and D are not equally distant from the point E which is the point of the section into two equall partes And forasmuch as by the 5. of the second that which is contayned vnder the lines AC and CB together with the square of the line EC is equall to the square of the line EB And by the same reason that which is contayned vnder the lynes AD and DB together with the square of the line DE is also equall to the self same square of the line EB wherfore that which is contained vnder the lines AC and CB together with the square of the line EC is equall to that which is contained vnder the lines AD and DB together with the square of the line DE of which the square of the line DE is lesse then the square of the line EC for it was proued that the line DE is lesse then the line EC Wherfore the parallelogramme remayning contayned vnder the lines AC and CB is lesse theÌ„ the parallelogramme remayning contayned vnder the lines AD and DB. Wherfore also that which is contayned vnder the lines AC and CB twise is lesse then that which is contayned vnder the lines AD and DB twise But by the fourth of the second the square of the whole line AB is equall to that which is composed of the squares of the lines AC and CB together with that which is contained vnder the lynes AC and CB twise and by the same reason the square of the whole line AB is equall to that which is composed of the squares of the lines AD and DB together with that which is contayned vnder the lynes AD and DB twise wherfore that which is composed of the squares of the lynes AC and CB together with that which is contayned vnder the lynes AC and CB twise is equall to that which is composed of the squares of the lynes AD and DB together with that which is contayned vnder the lynes AD and DB twise But it is already proued that that which is contayned vnder the lynes AC and CB twise is lesse then that which is contayned vnder the lines AD DB twise Wherfore the residue namely that which is composed of the squares of the lines AC and CB is greater then the residue namely then that which is composed of the squares of the lines AD and DB which was required to be demonstrated ¶ An Assumpt A rationall superficies exceedeth a rationall superficies by a rationall superficies Let AD be a rationall superficies and let it exceede AF being also a rationall superficies by the superficies ED. Then I say that the superficies ED is also rationall For the parallelogramme AD is commensurable to the parallelogramme AF for that either of them is rationall Wherefore by the second part of the 15. of the tenth the parallelogramme AF is commensurable to the parallelogramme ED. But the the parallelogramme AF is rationall Wherfore also the parallelogramme ED is rationall ¶ The 30. Theoreme The 42. Proposition A binomiall line is in one point onely deuided into his names SVppose that AB be a binomiall line and in the point G let it be deuided into his names that is into the lines wherof the whole line AB is composed Wherefore these lines AC and CB are rationall commensurable in power onely Now I say that the line AB cannot in any other point besides C be deuided into two rationall lines commensurable in power onely For if it be possible let it be deuided in the point D so that let the lines AD and DE bâ— rationall commensurable in power onely First this manifest that neithâ—â— of these poinâ—es C and D deuideth the right line AB into two equall partes Otherwise the lines AC and CB should be rationall commensurable in
length and so likewise should the lines AD and DB be For euery line measureth it selfe and any other line equall to it selfe Moreouer the line DB is either one and the same with the line ACâ— that is is equall to the line AC oâ— els it is greater-then the line AC either els it is lesse then it If DB be equall to the line AC then putting the line DB vpon the line AC eche endes of the one shall agree with eche endes of the other Wherfore putting the point B vpon the point A the point D also shall fall vpon the point C and the line AD which is the rest of the line AC shall also be equall to the line CB which is the rest of the line DB. Wherfore the line AB is deuided into his names in the point C. And so also shal the line AB being deuided in the point D be deuided in the self â—ame point that the self same line AB was before deuided in the point C which is coÌ„trary to the suppositioÌ„ For by suppositioÌ„ it was deuided in sundry pointes namely in C D. But if the line DB be greaterâ— the the line AC let the line AB be deâ—ided into two equal partes in the point E. Wherfore the points C D shal not equally be distant froÌ„ the point E Now by the first assupt going before this propositioÌ„ that which is coÌ„posed of the squares of the lines AD DB is greater theÌ„ that which is composed of the squares of the lines AC CBâ— But that which is composed of the squares of the lines AD DB together with that which is coÌ„tained vnder the lines AD DB twise is equall to that which is composed of the squares of the lines AC CB together with that which is contained vnder the lines AC and CB twise for either of them is equall to the square of the whole line AB by the 4. of the second wherefore how much that which is coÌ„posed of the squares of the lines AD and DB added together is greater then that which is composed of the squares of the lines AC and CB added together so much is that which is contained vnder the lines AC and CB twise greater then that which is contained vnder the lines AD and DB twise But that which is composed of the squares of the lines AD and DB excedeth that which is composed of the squares of the lines AC and CB by a rationall superficies by the 2. assumpt going before this propositionâ— For that which is composed of the squares of the lines AD and DB is rationall and so also is that which is composed of the squares of the lines AC and CB for the lines AD and DB are put to be rationall commensurable in power onely and so likewise are the lines AC and CB. Wherfore also that which is contained vnder the lines AC and CB twise exceedeth that which is contained vnder the lines AD DB twise by a rational superficies wheÌ„ yet notwithstaÌ„ding they are both medial superficieces by the 21. of the tenth which by the 26. of the same is impossible And if the line DB be lesse then the line AC we may by the like demonstration proue the selfe same impossibilitie Wherfore a binomiall line is in one point onely deuided into his names Which was required to be demonstrated 〈…〉 ollary added by Flussates Two ration 〈…〉 surable in power onely being added together cannot be equall to two other rationall line 〈…〉 in power onely added together For either of them should make a binomia 〈…〉 so should a binomiall line be deuided into his names in moe poyntes then onâ—â—â—ch by this proposition is proued to be impossible The like shall follow in the fiue 〈◊〉 irrationall lines as touching their two names ¶ The 31. Probleme The 43. Proposition A first bimediall line is in one poynt onely deuided into his names SVppose that AB be a first bimediall line and let it be deuided into his partes in the point C so that let the lines AC and CB be mediall coÌ„mensurable in power onely and containing a rationall superficies Then I say that the line AB can not be deuided into his names in any other poynt then in C. For if it be possible let it be deuided into his names in the poynt D so that let AD DB be mediall lines commensurable in power onely comprehending a rationall superficies Now forasmuch as how much that which is contayned vnder the lines AD and DB twise diâ—ferreth from that which is contayned vnder the lines AC and CB twise so much differreth that which is composed of the squares of the lines AD and DB from that which is composed of the squares of the lines AC and CB but that which is contayned vnder the lines AD and DB twise differreth from that which is contayned vnder the lines AC and CB twise by a rationall superficies by the second assumpt going before the 41. of the tenth For either of those superficieces is rationall Wherefore that which is composed of the squares of the lines AC and CB differeth from that which is composed of the squares of the lines AD and DB by a rationall superficies when yet they are both mediall superficieces which is impossible Wherefore a first bimediall line is in one poynt onely deuided into his names which was required to be proued ¶ The 32. Theoreme The 44. Proposition A second bimediall line is in one poynt onely deuided into his names SVppose that the line AB being a second bimediall line be deuided into hys names in the poynt C so that let the lines AC and CB be mediall lines commensurable in power onely comprehending a mediall superficies It is manifest that the poynt C deuideth not the whole line AB into two equall partes For the lines AC and CB are not commensurable in length the one to the other Now I say that the line AB cannot be deuided into his names in any other poynt but onely in C. For if it be possible let it be deuided into his names in the poynt D so that let not the line AC be one and the same that is let it not be equall with the line DB. But let it be greater then it Now it is manifest by the first assumpt going before the 42. proposition of this booke that the squares of the lines AC and CB are greater then the squares of the lines AD and DB. And also that the lines AD and DB are mediall lines commensurable in power onely comprehending a mediall supersicies Take a rationall line EF. And by the 44. of the first vpon the line EF apply a rectangle parallelograme EK equall to the square of the line AB From which parallelograme take away the parallelograme EG equall to the squares of the lines AC and CB Wherefore the residue namely the parallelograme HK
which is contained vnder the lines AD and DB twise by a rationall superâ—icies for either of them is rationall Wherfore also the squares of the lines AC and CB added together exceede the squares of the lines AD and DB added together by a rationall superficies when yet ech of them is a mediall superficies which is impossible Wherefore a line containing in power a rationall and a mediall is in one point onely deuided into his names which was required to be demonstrated ¶ The 35. Theoreme The 47. Proposition A line contayning in power two medials is in one point onely deuided into his names SVppose that AB being a line containing in power two medialls be deuided into his names in the point C so that let the lines AC and CB be incommensurable in power hauing that which is composed of the squares of the lines AC CB mediall and that also which is contained vnder the lines AC and CB mediall and moreouer incommensurable â—o that which is composed of the squares of the lines AC and CB. Then I say that the line AB can in no other point be deuided into his names but onely in the point C. For if it be possible let it be deuided into his names in the point D so that let not the line AC be one and the same that is equall with the line DB but by supposition let the line AC be the greater And take a rationall line EF. And by the 43. of the first vpon the line EF apply a rectangle parallelograÌ„me EG equall to that which is coÌ„posed of the squares of the lines AC and CB and likewise vpon the line HC which is equall to the line EF apply the parallelogramme HK equall to that which is contained vnder the lines AC and CB twise Wherefore the whole parallelogramme EK is equall to the square of the line AB Againe vpon the same line EF describe the parallelogramme EL equall to the squares of the lines AD and DB. Wherefore the residue namely that which is contayned vnder the lines AD and DB twise is equall to the parallelogramme remaining namely to MK And forasmuch as that which is coÌ„posed of the squares of the lines AC and CB is by supposition mediall therefore the parallelograÌ„me EG which is equall vnto it is also mediall and it is applied vpon the rationall line EF. Wherefore by the 22. of the tenth the line HE is rationall and incommensurable in length vnto the line EF. And by the same reason also the line HN is rationall and incommensurable in length to the same line EF. And forasmuch as that which is composed of the squares of the lines AC and CB is incommensurable to that which is contained vnder the lines AC and CB twise for it is supposed to be incommensurable to that which is coÌ„tained vnder the lines AC and CB once therefore the parallelogramme EG is incommensurable to the parallelogramme H â— Wherefore the line EH also is incommensurable in length to the line HN and they are rationall lines wherfore the lines EH and HN are rationall commensurable in power onely Wherefore the whole line EN is a binomiall line and is deuided into his names in the point H. And in like sort may we proue that the same binomiall line EN is deuided into his names in the point M and that the line EH is not one and the same that is equall with the line MN as it was proued in the end of the demonstration of the 44. of this booke Wherefore a binomiall line is deuided into his names in two sundry pointes which is impossible by the 42. of the tenth Wherefore a line containing in power two medials is not in sundry pointes deuided into his names Wherefore it is deuided in one point onely which was required to be demonstrated ¶ Second Definitions IT was shewed before that of binomiall lines there were sixe kindes the definitions of all which are here now set and are called second definitioÌ„s All binomiall lines as all other kindes of irrationall lines are coÌ„ceaued coÌ„sidered and perfectly vnderstanded onely in respecte of a rationall line whose partes as before is taught are certayne and knowen and may be distinctly expressed by number vnto which line they are compared Thys rationalâ— line must ye euer haue before your eyes in all these definitions so shall they all be â—asie inough A binomiall line ye know is made of two partes or names wherof the one is greater then the other Wherfore the power or square also of the one is greater then the power or square oâ— the other The three first kindes of binomiall lines namely the first the seconâ— the third are produced when the square of the greater name or part of a binomâ—all eâ—cedeth the square of the lesse name or part by the square of a line which is commâ—nsurable in length to it namely to the greater The three last kindes namely the fourth the â—iâ—t and the sixt are produced when the square of the greater name or part â—â—â—â—edeth the square of the lesse name or part by the square of a line incommensurable in length vnto it that is to the greater part A first binomiall line is whose square of the greater part exceedeth the square of tâ—e lesse part â—y the square of a line commensurable in length to the greater part and the greater part is also commensurable in length to tâ—e rationall line first set As lâ—t the raâ—ionâ—ll line first set be AB whose partes are distinctly knowen suppose also that the line CE be a binomiall line whose names or partes let be CD and DE. And let the square of the line CD the greater part excede the square of the line DE the lesse part by the square of the line FG which line FG let bâ—e commensuâ—able in length to the line CD which is the greater part of the binomiall line And moreouer let the line CD the greater paâ—t be commensurble in length to the rationall line first set namely to AB So by this dâ—â—inition the binomiall line CE is a first binomiall line A second binomiall line is when the square of the greater part exceedeth the square of the lesse part by the square of a line commensurable in length vnto it and the lesse part is commensurable in length to the rationall line first set As supposing euer the rationall line let CE be a binomiall line deuided in the poynt D. The square of whose greater part CD let exceede the square of the lesse part DE by the square of the line FG which line â—G let be coÌ„mensurable in length vnto the line CD tâ—e grâ—ater pâ—â—â— oâ— the binomiall line And let also the line DE the lesse part of the binomiall line be commensuâ—able in lâ—ngth to the rationall line first set AB So by this definition the binomiall line CE is a second binomiall line A third binomiall
FG is commensurable to the square of the line GH But the square of the line FG is rationall wherfore the square also of the line GH is rationall wherfore the line GH is also rationall And for that the number Bâ— hath not to the number CD that proportion that a square number hath to a square number therfore the line FG is incommensurable in length to the line GH and they are both rationall Wherefore the lines FG and GH are rationall commensurable in power onely Wherfore the lyne FH is a residuall line I say moreouer that it is a sixt residuall line For for that as the number E is to the number BC so is the square of the line A to the square of the line FG and as the number BC is to the nuÌ„ber CD so is the square of the line FG to the square of the line GH therefore by equalitie of proportion as the number E is to the number CD so is the square of the line A to the square of the line GH But the number E hath not to the number CD that proportion that a square number hath to a square number Wherefore the line A is inâ—â—mmensurable in length to the line GH and neither of these lines FG nor Gâ— is commensurable in length to the rationall line A. And forasmuch as the square of the line FG is greater then the square of the line GH vnto the square of the line FG let the the squares of the lines GH and K be equall Now therfore for that as the number Bâ— is to the number CD so is the square of the line FG to the square of the line GH therefore by conuersion of proportion as the number BC is to the number BD so is the square of the line FG to the square of the line K. But the number BC hath not to the number BD that proportion that a square number hath to a square number therfore the line FG is incommensurable in length to the line K. Wherfore the line FG is in power more then the lyne GH by the square of a line incommensurable in length to the line FG and neither of the lines FG nor GH is commensurable in length to the rationall line A. Wherfore the line FH is a sixt residual line Wherfore there is found out a sixt residuall line which was required to be done There is also a certayne other redier way to finde out euery one of the forsayd sixe residuall lines which is after this maner Suppose that it were required to finde out a first residuall line Take a first binomiall line AC let the greater name thereof be AB And vnto the line BC let the line BD be equall Wherefore the lines AB and BC that is the lines AB and BD are rationall commensurable in power onely and the line AB is in power more then the line BC that is then the line BD by the square of a line commensurable in length to the line AB And the line AB is commensurable in length to the rationall line geuen For the line AC is put to be a first binomiall line Wherefore the line AD is a first residual line And in like maner may ye finde out a second a third a fourth a fift and a sixt residuall line if ye take for eche a binomiall line of the same order ¶ The 67. Theoreme The 91. Proposition If a superficies be contayned vnder a rationall line a first residuall line the line which contayneth in power that superficies is a residuall line SVppose that there be a rectangle superficies AB contayned vnder a rationall line AC and a first residuall line AD. Then I say that the line which contayneth in power the superficies AB is a residuall line For forasmuch as AD is a first residuall line let the line ioyned vnto it be DG by the line ioyned vnto it vnderstand such a line as was spoken of in the end of the 79. proposition Wherefore the lines AG and GD are rationall coÌ„meÌ„surable in power only the whole line AG is coÌ„mensurable in length to the rationall line AC and the line AG is in power more then the line GD by the square of a line commensurable in length vnto AG by the definition of a first residuall line Deuide the line GD into two equall partes in the poynt E. And vpon the line AG apply a parallelogramme equall to the square of the line EG and wanting in figure by a square and let the sayd parallelogramme be that which is coÌ„tayned vnder the lines AF and FG. Wherefore the line AF is commensurable in length to the line FG by the 17. of the tenth â— And by the poyntes E F and G draw vnto the line AC these parallel lines EH FI and GK And make perfect the parallelograme AK And forâ—as much as the line AF is commensurable in length to the line FG therefore also the whole line AG is commensurable iâ— length to either of the lines AF and FG by the 15. of the tenth But the line AG is commensurable in length to the line AC Wherefore either of the lines AF and FG is commensurable in length to the line AC But the line AC is rationall wherefore either of the lines AF and FG is also rationall Wherefore by the 19. of the tenth either of the parallelogrammes AI and FK is also rationall And forasmuch as the line DE is commeÌ„surable in length to the line EG therfore also by the 15. of the tenth the line DG is commensurable in length to either of the lines DE and EG But the line DG is rationall wherefore either of the lines DE and EG is rationall and the selfe same line DG is incommensurable in length to the line AC by the definition of a first residuall line or by the 13. of the tenthâ— For the line DG is incommensurable in length to the line AG which line AG is coÌ„mensurable in length to the line AC wherfore either of the lines DE and EG is rationall and incommensurable in length to the line AC â— Wherefore by the 21. of the tenth either of these parallelogrammes DH and EK is mediall Vnto the parallelogramme AI let the square LM be equall and vnto the parallelogramme FK let the square NX be equall being taken away from the square LM â— and haâ—ing the angle LOM common to them both And to doo this there must be founde out the meane proportionall betwene the lines FI and FG. For the square of the meane proportionall is equall to the parallelogramme contayned vnder the lines FI and FG. And from the line LO cut of a line equall to the meane proportionall so founde out and descriâ—e the square thereof Wherefore both the squares LM and NX are about one and the selfe same diameter by the 20. of the sixth let their diameter be OR and describe the figure as
parallelogramme FK wherfore also the square of the lyne LO is incommensurable to the square of the line ON Wherfore the lines LO and ON are incommensurable in power hauing that which is made of their squares added together mediall and that which is contayned vnder them twise rationall Wherfore the line LN is that irrationall line which is called a lyne making with a rationall superâ—icies the whole superficies mediall and it contayneth in power the superficies AB Wherfore the line contayning in power the superâ—icies AB is a line making with a rationall superficies the whole superficies mediall If therfore a superficies be contayned vnder a rationall lyne a fift residuall line the line that contayneth in power the same superficies is a line makyng with a rationall superficies the whole superâ—icies mediall which was required to be proued ¶ The 72. Theoreme The 96. Proposition If a superficies be contayned vnder a rationall line and a sixth residuall line the line which contayneth in power the same superficies is a line making with a mediall superficies the whole superficies mediall SVppose that AB be a superficies contayned vnder a rationall line AC a sixt residuall line AD. Then I say shal the line which contayneth in power the superficies AB is a line making with a medial superficies the whole superficies mediall For vnto the line AD let the line DG be ioyned And let the rest be as in the propositions going before And forasmuch as the line AF is incommensurable in length to the line FG therefore the parallelogramme AI is incoÌ„mensurable to the parallelogramme FKâ— And forasmuch as the lines AG and AC are rationall commensurable in power onely therefore the parallelogramme AK is mediall and in like maner the parallelogramme DK is mediall Now forasmuch as the lines AG and GD are commensurable in power onely therefore they are incommensurable in length the one to the other But as the line AG is to the line GD so is the parallelogramme AK to the parallelogramme DK therefore the parallelogramme AK is incommensurable to the parallelogramme DK Describe the like figure that was described in the former propositions and we may in like sort proue that the line LN contayneth in power the superficies AB I say moreouer that it is a line making wâ—th a mediall superficies the whole superficies mediall For the parallelograme AK is mediall wherefore that which is equal vnâ—â— it namely that which is made of the sqâ—ares of the lines LO and ON added together is also mediall And forasmuch as the parallelogramme â—K is mediall therefore that which is equall vnto â—t namely that which is contayned vnder the lines LO and ON twise is also mediall And forasmuch as the parallogramme AK is incommensurable to the parallelogramme DK therefore the squares of the lines LO and ON are incommensurable to that which is contained vnder the lines LO and ON twise And forasmuch as the parallelogramme AI is incommensurable to the parallelogramme FK therefore also the square of the line LO is incommensurable to the square of the line ON Wherefore the lines LO and ON are incommensurable in power hauing that which is made of the squares of the lines LO and ON mediall and that which is contayned vnder them twise mediall and moreouer that which is made of the squares of them is incommensurable to thatâ— which is contayned vnder them twise Wherfore the line LN is that irrationall line which is called a line making with a mediall superficies the whole superficies mediall and it contayneth in power the superficies AB Wherefore the line which contayneth in power the superficies AB is a line making with a mediall superficies the whole superficies mediall If therefore a superficies be contayned vnder a rationall line and a sixth residuall line the line which contayneth in power the same superficies is a line making which a mediall superficies the whole superficies mediallâ— which was required to be demonstrated ¶ The 73. Theoreme The 97. Proposition The square of a residuall line applyed vnto a rationall line maketh the breadth or other side a first reâ—iduall line SVppose that AB be a residuall line and let CD be a rationall line And vnto the line CD apply the parallelogramme CE equall to the square of the line AB and making the breadth the line CF. Then I say that the line CF is a first residuall lineâ— For vnto the line AB let the line coÌ„ueniently ioyned be supposed to be Bâ— which selfe line is also called a line ioyned as we declared in the end of the 79. proposition Wherefore the â—ines AG and GB are rationall â—â—â—mensurable in power onely And vnto the line CD apply the parallelogramme CH equall to the square of the line AG and vnto the line KH which is equall to the line CD apply the parallelogramme KL equall to the square of the line BG Wherfore the whole parallelogramme CL is equall to the squares of the linâ—s Aâ— and GBâ— And the parallelogrâ—mmâ— CE is equall to the square of the line AB wherefore the parallelogramme remayning namely the parallelogramme FL is equall to that which is contayned vnder the lines AG and GB twise For by the 7. of the second the squares of the lines AG and GB are equall to that which is contayned vnder the lines AG GB twise and to the square of the line AB Deuide the line FM into two equall partes in the point N. And by the poynt N draw vnto the line CD a parallell line NX Wherfore either of the parallelogrammes FX and NL is equall to that which is contayned vnder the lines AG and GB once And forasmuch as the squares of the lines AG and GB are rationall vnto which squares the parallelogramme CL is equall therefore the parallelogramme CL also is rationall wherefore the line GM is rationall and commensurable in length to the line CD Agayne forasmuch as that which is contayned vnder the lines AG and GB twise is mediall therefore the parallelogramme equall vnto it namely the paralâ—elogramme FL is also mediall Wherefore the line FM is rationall and incommensurable in length to the line CD And forasmuch as the squares of the lines AG and GB are rationall and that which is contayned vnder the lines AG and GB twise is mediall therefore the squares of the lines AG and GB are incoÌ„mensurable to that which is contayned vnder the lines AG and GB twise But vnto the squares of the lines AG and GB is equall the parallelogramme CL and to that which is contayned vnder the lines AG and GB twise is equall the parallelogramme FL wherefore the parallelogramme CL is incommensurable to the parallelogramme FL. Wherefore also the line CM is incommensurable in length to the line FM and they are both rationall Wherefore the lines CM and FM are rationall commensurable in power onely and therefore the line CF is
of the tenth the line DE is a first binomiall line Deuide it into his names in the point G. And let DG be the greater name Wherfore the lines DG and â—E are rationall commensurable in power onely And the line DG is in power more then the line GE by the square of a line commensurable in length to the lyne DG and the line DG is commensurable in length to the rationall line put DC Wherefore the line DF is commensurable in length to the line DG Wherfore by the 13. of the tenth the whole line DF is commensurable in leÌ„gth to the line remaining namely â—o the line GF And forasmuch as the line DF is coÌ„meÌ„surable to the line FG but the line FD is rationall Wherfore the line FG is also rationall And forasmuch as the line FD is commensurable in length to the line FG but the line DF is incommensurable in length to the line FE Wherfore the line FG is incommensurable in length to the line FE by the 13. of the tenth and they are both rationall lines Wherfore the lines GF and FE are rationall commensurable in power onely Wherfore by the 73. of the tenth the line EG is a residuall line but it is also rationall as before hath bene proued which is impossible namely that one the same line should be both rationall and irrationall Wherfore a residuall line is not one and the same with a binomiall line that is is not a binomiall line which was required to be demonstrated ¶ A Corollary A residuall lyne and the other fiue irrationall lynes following it are neither mediall lines nor one and the same betwene themseluesâ— that is one is vtterly of a diuers kinde froÌ„ an other For the square of a mediall line applied to a rationall line maketh the breadth rationall and incommensurable in length to the rationall lyne whereunto it is applied by the 22. of the tenth The square of a residuall line applied to a rationall line maketh the breadth a first residuall line by the 97. of the tenth The square of a first mediall residuall line applied to a rationall line maketh the breadth a second residuall lyne by the 98. of the tenth The square of a second mediall residuall line applied vnto a rationall line maketh the breadth a third residuall line by the 99. of the tenth The square of a lesse line applied to a rationall line maketh the breadth a fourth residuall line by the 100. of the tenth The square of a line making with a rationall superficies the whole superficies mediall applied to a rationall line maketh the breadth a â—ift residuall line by the 101. of the tenth And the square of a line making with a mediall superâ—icies the whole superficies mediall applied to a rationall line maketh the breadth a sixt residuall line by the 102. of the teÌ„th Now forasmuch as these foresaid sides which are the breadthes differ both from the first breadth sor that it is rational and differ also the one froÌ„ the other for that they are residuals of diuers orders and kindes it is manifest that those irrationall lines differ also the one from the other And forasmuch as it hath bene proued in the 111. proposition that 〈◊〉 residual 〈◊〉 is not one and the same with a binomiall line and it hath also bene proued that the 〈…〉 of a residuall line and of the fiue irrationall lines that follow it being applied to a rational line do make their breadthes one of the residuals of that order of which they were whose squareâ— were applied to the rationall line likewise also the squares of a binomiall line and of the fiue irrationall lines which follow it being applied to a rationall line do make the breadthes one of the binomials of that order of which they were whose squares were applied to the rationall line Wherfore the irrationall lines which follow the binomiall line and the irrationall lines which follow the residuall line differ the one from the other so that all the irrationall lynes are 13. in number namely these 1 A mediall line 2 A binomiall line 3 A first bimediall line 4 A second bimediall line 5 A greater line 6 A line containing in power a rationall superficies and a mediall superficies 7 A line contayning in power two mediall superficieces 8 A residuall line 9 A first mediall residuall line 10 A second mediall residuall line 11 A lesse line 12 A line making with a rationall superficies the whole superficies mediall 13 A line making with a mediall superficies the whole superficies mediall ¶ The 88. Theoreme The 112. Proposition The square of a rationall line applyed vnto a binomiall line maketh the breadth or other side a residuall line whose names are commensurable to the names of the binomiall line in the selfe same proportioÌ„ moreouer that residuall line is in the selfe same order of residuall lines that the binomiall line is of binomiall lines SVppose that A be a rationall line and BC a binomiall line whose greater name let be CD And vnto the square of the line A let the parallelogramme contayned vnder the lines BC and EF so that EF be the breadth be equall Then I say that EF is a residuall line whose names are commensurable to the names of the binomiall line BC which names let be CD and DB and are in the same proportion with them and moreouer the line EF is in the selfe same order of residual lines that the line BC is of binomiall lines Vnto the square of the line A let the parallelogramme contayned vnder the lines BD and G be equall Now forasmuch as that which is coÌ„tayned vnder the lines BC EF is equal to that which is contayned vnder the lines BD and G therfore reciprocally by the 14. of the sixth as the line CB is to the BD so is the line G to the line EF. But the line BC is greater then the line BD wherefore the line G is greater then the line EF. Vnto the line G let the line EH be equall Wherefore by the 11. of the fift as the line CB is to the line BD so is the line HE to the line FE Wherefore by deuision by the 17. of the fifth as the line CD is to the line BD so is the line HF to the line FE As the line HF is to the FE so let the line FK be to the line KE how this is to be done we will declare at the end of this demonstration Wherefore by the 12. of the fift the whole line HK is to the whole line KF as the line FK is to the line KE For as one of the antecedentes is to one of the consequentes so are all the antecedentes to all the consequentes But as the line FK is the line KE so is the line CD to the line DB for FK is to EK as HF is to FE and HF is to FE as CD is DB Wherfore by the
of the one be commensurable in length to the rationall line put neither name of the other also shall be commensurable in length to the same rationall line put by the 13. of the same Wherefore the residuall line GZ shall be in the selfe same order of residuall lines that the binomiall line GB is of binomiall lines by the definitions of residuall and binomiall lines The square therefore of a rationall line applied to a binomiall lineâ— â— which was required 〈◊〉 be proued ¶ The 89. Theoreme The 113. Proposition The square of a rational line applied vnto a residuall maketh the breadth or other side a binomial line whose names are commensurable to the names of the residuall line and in the selfe same proportion and moreouer that binomiall line is in the selfe same order of binomiall lynes that the residual line is of residuall lynes SVppose that A be a rationall line and BD a residuall line And vnto the square of the line A let that which is contained vnder the lines BD and KH be equal Wherfore the square of the rationall line A applied vnto the residuall line BD maketh the breadth or other side KH Then I say that the line KH is a binomiall line whose names are commensurable to the names of the residuall line BD and in the selfe same proportion and that the line KH is in the selfe same order of binomiall lines that the line BD is of residuall lines Vnto the line BD let the line conueniently ioyned be DC Wherfore the lines BC and DC are rationall commensurable in power onely And vnto the square of the line A let the parallelogramme contained vnder the lines BC and G be equall But the square of the line A is rationall Wherfore the parallelogramme contained vnder the lines BC and G is also rationall Wherfore also the line G is rationall and commensurable in length to the line BC by the 20. of the tenth Now forasmuch as the parallelogramme contained vnder the lines BC and G is equall to that which is contained vnder the lines BD and KH therfore by the 16. of the sixt as the line BC is to the line BD so iâ— the line KH to the line G. But the line BC is greater then the line BD. Wherfore also the line KH is greater then the line G. Vnto the line G lâ—t the line KE be equall Wherfore the line KE is rationall and coÌ„mensurable in length to the line BC as also the line G was by the 12. of the tenth And for that as BC is to BD so is KH to KE Wherfore by â—duersion of proportioÌ„ by the corollary of the 19. of the fift as BC is to DC so is KH to EH â—KH into EH so let the line FH be to the line EF how this is to be done we will decare at the ende of this demonstration Wherfore the residue KF is to the residue FH as the whole KH is to the whole HE by the 19. of the fift that is as the line BC is to the line CD But the lines BC and CD are commensurable in power onely Wherefore also the lines KF and FH are commensurable in power only And for that as KH is to HE so is KF to FH but as KH is to HE so is also HF to FE therfore as KF is to FH so is FH to FE Wherfore by the corollary of the 19. of the sixt as the first is to the third so is the square of the first to the square of the second Wherefore as KF is to FE so is the square of the line KF to the square of the line FH but these squares are commensurable for the lines KF and FH are commensurable in power Wherfore the lines KF and FE are commensurable in length Wherfore by the second part of the 15. of the tenth the lines KE and EF are commensurable in length Wherfore by the same the lines KF and FE are commensurable in length But the line KE is rationall and commensurable in length to the line BC wherefore the line KF is also rationall and commensurable in leÌ„gth to the line BC. And for that as the line BC is to the line CD so it KF to EH therfore alternately by the 16. of the fift as BC is to KF so is CD to FH But the line BC is commensurable in length to the line KF Wherfore the line CD is commensurable in length to the line FH But the line CD is rationall Wherfore also the line FH is rationall And the lines BC and CD are rationall commensurable in power onely Wherfore the lines KF and FH are rationall commensurable in power onely Wherfore the line KH is a binomiall line whose names are commensurable to the names of the residuall line and in the same proportion I say moreouer that it is a binomiall of the selfe same order of binomial lines that the line BD is of residual lines For if the line BC be in power more then the line CD by the square of a line commensurable in length to the line BC the line KF is also in power more then the line FH by the square of a line commensurable in length to the line KF by the 14. of the tenth And if the line BC be commensurable in length to the rationall line put the line KF is also by the 12. of the tenth commensurable in length to the rationall line and so the lyne BD is a first residuall lyne and the line KH is in like sort a first binomiall line If the line CD be commensurable in length to the rational line the line FH is also commensurable in length to the same line and so the line BD is a second residuall line and the line KH a second binomiall line And if neither of the lines BC nor CD be commensurable in length to the rationall line neither also of the lines KF nor FH is commensurable in length to the same and so the line BD is a third residuall line and the line KH a third binomiall line But if the line BC be in power more then the line CD by the square of a line incommensurable in length to the line BC the line KF is in power more theÌ„ the line FH by the square of a line incommensurable in length to the line KF by the 14. of the tenth And if the line BC be commensurable in length to the rationall line put the line KF is also commensurable in length to the same line and so the line BD is a fourth residuall line and the line KH a fourth binomiall line And if the line CD be coÌ„meÌ„surable in leÌ„gth to the rational line the line FH is also coÌ„meÌ„surable in leÌ„gth to the same so the line BD is a fift residuall line the line KH a fift binomiall line And if neither of the lines BC nor CD be commensurable in length to the rationall line neither also of the lines KF nor FH is commensurable in length
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.
the wordes of Psellus in his Epitome of Geometrie where he entreateth of the production and constitution of these bodyes His wordes are these All râ—ctiliâ—e figures being erected vpon their playnes or bases by right angles make Prismes Who perceaueth not but that a Pentagon erected vpoÌ„ his base of â—iue sides maketh by his motion a sided Columne of fiue sides Likewise an Hexagon erected at right angles produceth a Columne hauing sixe sides and so of all other rectillne figures All which solides or bodyes so produced whether they be sided Columnes or Parallelipipedons be here in most plaine words of this excelleÌ„t and auncient Greke author Psellus called Prismes Wherfore if the definitioÌ„ of a Prisme geueÌ„ of Euclide should extend it selfe so largely as Flussas imagineth and should enclude such figures or bodyes as he noted he ought not yet for all that so much to be offended and so narowly to haue sought faultes For Euclide in so defining mought haue that meaning sense of a Prisme which Psellus had So ye see that Euclide may be defended either of these two wayes either by that that the definition extendeth not to these figures and so not to be ouer generall nor stretch farther then it ought or ells by that that if it should stretch so far it is not so haynous For that as ye se many haue takâ— it in that sense In deede coÌ„monly a Prisme is taken in that significatioÌ„ and meaning in which Campaâ—â—â— Flussas and others take it In which sense it semeth also that in diuers propositions in these bookes following it ought of necessitie to be taken 12 A Sphere is a figure which is made when the diameter of a semicircle abiding fixed the semicircle is turned round about vntill it returne vnto the selfe same place from whence it began to be moued To the end we may fully and perfectly vnderstand this definition how a Sphere is produced of the motion of a semicircle it shall be expedient to coÌ„sider how quantities Mathematically are by imagination conceaued to be produced by flowing and motion as was somewhat touched in the beginning of the first booke Euer the lesse quantitie by his motion bringeth forâ—h the quaÌ„titie next aboue it As a point mouing flowing or gliding bringeth forth a line which is the first quantitie and next to a point A line mouing produceth a superficies which is the second quantitie and next vnto a line And last of all a superficies mouing bringeth forth a solide or body which is the third last quantitie These thinges well marked it shall not be very hard to attaine to the right vnderstanding of this definition Vpon the line AB being the diameter describe a semicircle ACB whose centre let be D the diameter AB being sixed on his endes orâ— pointes imagine the whole superficies of the semicircle to moue round from some one point assigned till it returne to the same point againe So shall it produce a perfect Sphere or Globe the forme whereof you see in a ball or bowle And it is fully round and solide for that it is described of a semicircle which is perfectly round as our countrey man Iohannes de Sacro Busco in his booke of the Sphere of this definition which he taketh out of Euclide doth well collecte But it is to be noted and taken heede of that none be deceaued by the definition of a Sphere geuen by Iohannes de Sacro Busco A Sphere sayth he is the passage or mouing of the circumference of a semicircle till it returne vnto the place where it beganne which agreeth not with Euclide Euclide plainly sayth that a Sphere is the passage or motion of a semicircle and not the passage or motion of the circumference of a semicircle neither can it be true that the circumference of a semicircle which is a line should describe a body It was before noted that euery quantitie moued describeth and produceth the quantitie next vnto it Wherefore a line moued can not bring forth a body but a superficies onely As if ye imagine a right line fastened at one of his endes to moue about from some one point till it returne to the same againe it shall describe a plaine superficies namely a circle So also if ye likewise conceaue of a crooked line such as is the circumference of a semicircle that his diameter fastened on both the endes it should moue from a point assigned till it returne to the same againe it should describe produce a â—ound superficies onely which is the superficies and limite of the Sphere and should not produce the body and soliditie of the Sphere But the whole semicircle which is a superficies by his motion as is before said produceth a body that is a perfect Sphere So see you the errour of this definition of the author of the Sphere which whether it happened by the author him selfe which I thinke not or that that particle was thrust in by some one after him which is more likely it it not certaine But it is certaine that it is vnaptly put in and maketh an vntrue definition which thing is not here spoken any thing to derogate the author of the booke which assuredly was a man of excellent knowledgeâ— neither to the hindrance or diminishing of the worthines of the booke which vndoubtedly is a very necessary booke then which I know none more meere to be taught and red in scholes touching the groundes and principles of Astronomie and Geographie but onely to admonishe the young and vnskilâ—ull reader of not falling into errour Theodosius in his booke De Sphericis a booke very necessary for all those which will see the groundes and principles of Geometrie and Astronomie which also I haue translated into our vulgare tounge ready to the presse defineth a Sphere after thys maner A Sphere is a solide or body contained vnder one superficies in the midle wherof there is a point froÌ„ which all lines drawen to the circumference are equall This definition of Theodosius is more essentiall and naturall then is the other geuen by Euclide The other did not so much declare the inward nature and substance of a Sphere as it shewed the industry and knowledge of the producing of a Sphere and therfore is a causall definition geuen by the cause efficient or rather a description then a definition But this definition is very esâ—entiall declaring the natuâ—e and substance of a Sphere As if a circle should be thus defined as it well may A circle is the passage or mouing of a line from a point till it returne to the same point againeâ— it is a causall definition shewing the efficient cause wherof a circle is produced namely of the motion of a line And it is a very good description fully shewing what a circle is Such like description is the deâ—inition of a Sphere geuen oâ— Euclide â— by the motion of a semicircle But when a circle is defined to be a plaine superficies
onely one axe 16 A cone is a solide or bodely figure which is made when one of the sides of a rectangle triangle namely one of the sides which contayne the right angle abiding fixed the triangle is moued about vntill it returne vnto the selfe same place from whence it began first to be moued Now if the right line which abideth fixed be equall to the other side which is moued about and containeth the right angle then the cone is a rectangle cone But if it be lesse then is it an obtuse angle cone And if it be greater theÌ„ is it an a cuteangle cone If the right line which abideth fixed be equall to the other side which moueth roâ—â—d about and containeth the right angle then the Cone is a rectangle Cone As suppose in the former example that the line AB which is fixed and about which the triangle was moued and after the motion yet remayneth be equall to the line BC which is the other line contayning the right angle which also is moued about together with the whole triangleâ— then is the Cone described as the Cone ADC in this example a right angled Cone so called for that the angle at the toppe of the Cone is a right angle For forasmuch as the lines AB and BC of the triangle ABC are equall the angle BAC is equall to the angle BCA by the 5. of the first And eche of them is the halfe of the right angle ABC by the 32. of the first In like sort may it be shewed in the triangle ABD that the angle â—DA is equall to the angle â—AD and that eche of them is the halfe of a right angle Wherefore the whole angle CAD which is composed of the two halfe right angles namely DAâ— and CAâ— is a right angle And so haue ye what is a right angled Cone This figure of a Cone is of Campane of Vitellio and of others which haue written in these latter times called a round Pyramis which is not so aptly For a Pyramis and a Cone are farre distant of sundry natures A Cone is a regular body produced of one circumuolution of a rectangle triangle and limited and bordered with one onely round superficies But a Pyramis is terminated and bordered with diuers superficieces Therefore can not a Cone by any iust reason beare the name of a Pyramis This solide of many is called Turbo which to our purpose may be Englished a Top or Ghyg and moreouer peculiarly Campane calleth a Cone the Pyramis of a round Columne namely of that Columne which is produced of the motion of a parallelogramme contained of the lines AB and BC moued about the line AB being fixed Of which Columnes shall be shewed hereafter 17 The axe of a Cone is that line which abideth fixed about which the triangle is moued And the base of the Cone is the circle which is described by the right line which is moued about As in the example the line AB is supposed to be the line about which the right angled triangle ABC to the production of the Cone was moued and that line is here of Euclide called the axe of the Cone described The base of the Cone is the circle which is described by the right line which is moued about As the line AB was fixed and slayed so was the line BC together with the whole triangle ABC moued and turned about A line moued as hath bene sayd before produceth a superficies and because the line BC is moued about a point namely the point B being the end of the axe of the Cone AB it produceth by his motion and reuolution a circle which circle is the base of the Cone as in this example the circle CDE The line which produceth the base of the Cone is the line of the triangle which together with the axe of the Cone contayneth the right angle The other side also of the triangle namely the line AC is moued about also with the motion of the triangle which with his reuolution describeth also a superficies which is a round superficies is erected vpon the base of the Cone endeth in a point namely in the higher part or toppe of the Cone And it is commonly called a Conicall superficies 18 A cylinder is a solide or bodely figure which is made when one of the sides of a rectangle parallelogramme abiding fixed the parallelogramme is moued about vntill it returne to the selfe same place from whence it began to be moued This definition also is of the same sort and condition that the two definitionâ— before geuâ— were namely the definition of a Sphere and the definition of a Cone For all are geuen by mouing of a superficies about a right line fixed the one of a semicircle about his diameter the other of a rectangle triangle about one of his sides And this solide or body here deâ—ined is caused of the motion of a rectangle parallelograÌ„me hauing one of his sides contayning the right angle fixed from some one poynt till it returne to the same agayne where it began As suppose ABCD to be a rectangle parallelogramme hauing his side AB fastned about which imagine the whole parallelogramme to be turned till it returne to the poynt where it began then is that solide or body by this motion described a Cylinder which because of his roundnes can not at full be described in a playne superficies yet haue you for an example thereof a sufficient designation therof in the margent 〈◊〉 as in a plaine may be If you wil perfectly behold the forme of a cilinder Consider a round piller that is perfectly round 19 The axe of a cilinder is that right line which abydeth fixed about which the parallelogramme is moued And the bases of the cilinder are the circles described of the two opposite sides which are moued about Euen as in the description of a Sphere the line fastened was the axe of the Sphere proâ—uced and in the description of a cone the line fastened was the axe of the cone brought forth so in this description of a cilinder the line abiding which was fixed about which the rectangle parallelogramme was moued is the axe of that cilinder As in this example is the line AB The bases of the cilinder â—â—c In the reuolution of a parallelogramme onely one side is fixed therefore the three other sides are moued about of which the two sides which with the axe make right angles and which also are opposite sides in their motion describe eche of them a circle which two circles are called the bases of the cilinder As ye see in the figure before put two circles described of the motioÌ„ of the two opposit lines AD and BC which are the bases of the Cilinder The other line of the rectangle parallelogramme moued by his motion describeth the round superficies about the Cilinder As the third line or side of a rectangle triangle
â—ignifieth Last of all a Dodecahedron for that it is made of Pâ—ntagoâ— whose angles are more ample and large then the angles of the other bodies and by that â—eaâ—â—â— draw more â—â— rounâ—nes 〈◊〉 to the forme and nature of a sphere they assigned to a sphere namely 〈…〉 Who so will 〈…〉 in his Tineus shall â—ead of these figures and of their mutuall proportionâ—â—â—raunge maâ—terâ— which hâ—re are not to be entreated of this which is sayd shall be sufficient for the 〈◊〉 of them and for thâ— declaration of their diffinitions After all these diffinitions here set of Euclide Flussas hath added an other diffinition which 〈◊〉 of a Parallelipipedon which bicause it hath not hitherto of Euclide in any place bene defined and because it is very good and necessary to be had I thought good not to omitte it thus it is A parallelipipedon is a solide figure comprehended vnder foure playne quadrangle figures of which those which are opposite are parallels Because these fiue regular bodies here defined are not by these figures here set so fully and liuely expressed that the studious beholder can throughly according to their definitions conceyue them I haue here geuen of them other descriptions drawn in a playne by which ye may easily attayne to the knowledge of them For if ye draw the like formes in matter that wil bow and geue place as most aptly ye may do in fine pasted paper such as pastwiues make womeÌ„s pastes of theÌ„ with a knife cut euery line finely not through but halfe way only if theÌ„ ye bow and bende them accordingly ye shall most plainly and manifestly see the formes and shapes of these bodies euen as their definitions shew And it shall be very necessary for you to hadâ—â—tore of that pasted paper by you for so shal yoâ— vpon it 〈…〉 the formes of other bodies as Prismes and Parallelipopedons 〈…〉 set forth in these fiue bookes following and see the very 〈◊〉 of thâ—se bodies there meÌ„cioned which will make these bokes concerning bodies as easy vnto you as were the other bookes whose figures you might plainly see vpon a playne superficies Describe thiâ— figurâ— which consistâ—th of twâ—luâ—â—quilâ—â—â—â— and â—quianglâ— Pâ—ntâ—â—â—â—â— vpoâ— the foresaid mattâ—r and finely cut as before was â—â—ught tâ—â—â—lâ—uâ—n lines containâ—d within thâ— figurâ— and bow and folde the Penâ—â—gonâ— accordingly And they will so close toâ—ethâ—â— thaâ— thâ—y will â—â—kâ— thâ— very forme of a Dodecahedron If ye describe this figure which consisteth of twentie equilater and equiangle triangles vpon the foresaid matter and finely cut as before was shewed the ninâ—tâ—ne lines which are contayned within the figure and then bowe and folde them accordingly they will in such sort close together that therâ— will be made a perfecte forme of an Icosahedron Because in these fiue bookes there are sometimes required other bodies besides the foresaid fiue regular bodies as Pyramises of diuers formes Prismes and others I haue here set forth three figures of three sundry Pyramises one hauing to his base a triangle an other a quadrangle figure the other â— Pentagonâ— which if ye describe vpon the foresaid matter finely cut as it was before taught the lines contained within ech figure namely in the first three lines in the second fower lines and in the third fiue lines and so bend and folde them accordingly they will so close together at the toppes that they will â—ake Pyramids of that forme that their bases are of And if ye conceaue well the describing of these ye may most easily describe the body of a Pyramis of what forme so euer ye will. Because these fiue bookes following are somewhat hard for young beginners by reason they must in the figures described in a plaine imagine lines and superficieces to be eleuated and erected the one to the other and also conceaue solides or bodies which for that they haue not hitherto bene acquainted with will at the first sight be somwhat sâ—raunge vnto theÌ„ I haue for their more â—ase in this eleuenth booke at the end of the demonstration of euery Proposition either set new figures if they concerne the eleuating or erecting of lines or superficieces or els if they concerne bodies I haue shewed how they shall describe bodies to be compared with the constructions and demonstrations of the Propositions to them belonging And if they diligently weigh the maner obserued in this eleuenth booke touching the description of new figures agreing with the figures described in the plaine it shall not be hard for them of them selues to do the like in the other bookes following when they come to a Proposition which concerneth either the eleuating or erecting of lines and superficieces or any kindes of bodies to be imagined ¶ The 1. Theoreme The 1. Proposition That part of a right line should be in a ground playne superficies part eleuated vpward is impossible FOr if it be possible let part of the right line ABC namely the part AB be in a ground playne superficies and the other part therof namely BC be eleuated vpwarde And produce directly vpoÌ„ the ground playne superficies the right line AB beyond the point B vnto the point D. Wherfore vnto two right lines geuen ABC and ABD the line AB is a common section or part which is impossible For a right line can not touche a right line in 〈◊〉 pointes then one vâ—lesse those right be exactly agreing and laid the one vpon the other Wherfore that part of a right line should be in a ground plaine superficies and part eleuated vpward is impossible which was required to be proued This figure more plainly setteth forth the foresaid demonstratioÌ„ if ye eleuate the superficies wheriâ— the line BC. An other demonstration after Flâ—sâ—â—s If it be possible let there be a right line ABG whose part AB let be in the ground playne superficies AED and let the rest therof BG be eleuated on high that is without the playne AED Then I say that ABG is not one right line For forasmuch as AED is a plaine superficies produce directly equally vpon the sayd playne AED the right lyne AB towardes D which by the 4. definition of the first shall be a right line And from some one point of the right line ABD namely from C draâ— vnto the point G a right lyne CG Wherefore in the triangle 〈…〉 the outward angâ—â— ABâ— is eqâ—â—ll to the two inward and opposite angles by the 32. of the first and therfore it is lesse then two right angles by the 17. of the same Wherfore the lyne ABG forasmuch as it maketh an angle is not â— right line Whâ—refore that part of a right line should be in a ground playne superficies and part eleuated vpward is impossible If ye marke well the figure before added for the playâ—er declaration of Euclides demonstration iâ— will not be hard for you to coâ—â—â—â—e this figure which â—lussâ—s putteth for his demonstâ—â—tion â— wherein
superficies geuen to be â— and the rectangle parallelipipedon geuen to be AM. Vppon â— as a base must AM be applyed that is a rectangle parâ—llipipedon must be erected vppon â— as a base whiche shall be equall to AM. By the laste of the second to the right lyned figure â— let an equall square be made which suppose to be FRX produce one side of the base of the parallelipipedoÌ„ AM which let be AC extended to the point P. Let the other side of the sayde base concurring with AC be CG As CG is to FR the side of the square FRX so let the same FR be to a line cut of from CP sufficiently extended by the 11. of the sixth and let that third proportionall line be CP Let the rectangle parallelogramme be made perfect as CD It is euident that CD is equall to the square FRX by the 17. of the sixth and by construction FRX is equall to B. Wherfore CD is equall to â— By the 12. of the sixth as CP is to AC so let AN the heith of AM be to the right line O. I say that a solide perpendicularly erected vppon the base â— hauinge the heith of the line O is equall to the parallelipipedon AM. For CD is to AG as CP is to Aâ— by the firste of the sixth and â— is proued equall to CD Wherfore by the 7. of the fifth B is to AG as CP is to AC But as CP is to AC so is AN to O by construction Wherefore B is to AG as AN is to O. So than the bases â— and AG are reciprocally in proportion with the heithes AN and O. By this 34 therefore a solide erected perpendicularly vppon â— as a base hauing the height O is equall to AM. Wherefore vppon a right lyned playn superficies geuen we haue applied a rectangle parallelipipedon geuen Which was requisite to be donne A Probleme 2. A rectangle parallelipipedon being geuen to make an other equall to it of any heith assigned Suppose the rectangle parallelipipedon geuen to be A and the heith assigned to be the right line â— Now must we make a rectangle parallelipipedon equal to A Whose heith must be equall to â— According to the manner before vsed we must frame our coÌ„struction to a reciprokall proportioÌ„ betwene the bases and heithes Which will be done if as the heith assigned beareth it selfe in proportion to the heith of the parallelipipedon giuen so one of the sides of the base of the parallelipipedon giuen be to a fourth line by the 12. of the sixth found For that line founde and the other side of the base of the geuen parallelipipedon contayne a parallelogramme which doth serue for the base which onely we wanted to vse with our giuen heith and so is the Probleme to be executed Note Euclide in the 27. of this eleuenth hath taught how of a right line geueÌ„ to describe a parallepipedoÌ„ like likewise situated to a parallelipipedoÌ„ geueÌ„ I haue also added How to a parallepipedon geuen an other may be made equall vppon any right lined base geuen or of any heith assigned But if either Euclide or any other before our time answerably to the 25. of the sixth in playns had among solids inuented this proposition Two vnequall and vnlike parallelipipedons being geuen to describe a parallelipipedon equall to the one and like to the other we would haue geuen them their deserued praise and I would also haue ben right glad to haue ben eased of my great trauayles and discourses about the inuenting thereof Here ende I. Dee his additions vppon this 34. Proposition The 30. Theoreme The 35. Proposition If there be two superficiall angles equall and from the pointes of those angles be eleuated on high right lines comprehending together with those right lines which containe the superficiall angles equall angles eche to his corespoÌ„dent angle and if in eche of the eleuated lines be takeÌ„ a point at all auentures and from those pointes be drawen perpendicular lines to the ground playne superficieces in which are the angles geuen at the beginning and from the pointes which are by those perpendicular lines made in the two playne superficieces be ioyned to those angles which were put at the beginning right lines those right lines together with the lines eleuated on high shall contayne equall angles SVppose that these two rectiline superficiall angles BAC and EDF be equall the one to the other and from the pointes A and D let there be eleuated vpward these right lines AG and DM comprehendinge together with the lines put at the beginninge equall angles ech to his correspondent angle that is the angle MDE to the angle GAB and the angle MDF to the angle GAC and take in the lines AG and DM pointes at all aueÌ„tures and let the same be G and M. And by the 11. of the eleueÌ„th from the pointes G and M draw vnto the ground playne superficieces wherein are the angles BAC and E DF perpendicular lines GL and MN and let them fall in the sayd playne superâ—icieces in the pointes N and L and drawe a right line from the point L to the point A and an other from the pointe N to the pointe D. Then I say that the angle GAL is equall to the angle MDN. FroÌ„ the greater of the two lines AG and DM which let be AG cut of by the 3. of the first the line AH equall vnto the line DM And by the 31. of the first by the point H drawe vnto the line GL a parallel line and let the same be HK Now the line GL is erected perpendicularly to the grounde playne superficies BAL Wherfore also by the 8. of the eleuenth the line HK is erected perpeÌ„dicularly to the same grounde plaine superficies BAC Drawe by the 12. of the first froÌ„ the pointes K and N vnto the right lines AB AC DF DE perpeÌ„dicular right lines and let the same be KC NF KB NE. And drawe these right lines HC CB MF FE Now forasmuch aâ— by the 47. of the first the square of the line HA is equall to the squares of the lines HK and KA but vnto the square of the line KA are equall the squares of the lines KC and CA Wherefore the square of the line HA is equall to the squares of the lines HK KC and CA. But by the same vnto the squares of the lines HK and KC is equall the square of the line HC Wherefore the square of the line HA is equall to the squares of the lines HC and CA wherfore the angle HCA is by the 48. of the first a right angle And by the same reason also the angle MFD is a right angle Wherefore the angle HCA is equall to the angle MFD But the angle HAC is by suppositioÌ„ equal to the angle MDF Wherfore there are two triangles MDF and HAC hauing two angles of the one equall to twoo angles of the other eche to his correspondent angle
and one side of the one equall to one side of the other namely that side which subtendeth one of the equall angles that is the side HA is equall to the side DM by construction Wherefore the sides remayning are by the 26. of the first equall to the sides remayning Wherefore the side AC is equall to the side DF. In like sort may we proue that the side AB is equall to the side DE if ye drawe a right line from the point H to the point B and an other from the point M to the point E. For forasmuch as the square of the line AH is by the 47. of the firste equall to the squares of the lines AK and KH and by the same vnto the square of the line AK are equall the squares of the lines AB and BK Wherefore the squares of the lines AB BK and KH are equall to the square of the line AH But vnto the squares of the lines BK and KH is equall the square of the line BH by the 47. of the first for the angle HKB is a right angle for that the line HK is erected perpeÌ„dicularly to the ground playne superficies Wherefore the square of the line AH is equall to the squares of the lines AB and BH Wherefore by the 48. of the first the angle ABH is a right angle And by the same reason the angle DEM is a right angle Now the angle BAH is equall to the angle EDM for it is so supposed and the line AH is equall to the line DM Wherefore by the 26. of the firste the line AB is equall to the line DE. Now forasmuch as the line AC is equall to the line DF and the line AB to the line DE therefore these two lines AC and AB are equall to these two lines FD and DE. But the angle also CAB is by supposition equall to the angle FDE Wherefore by the 4. of the firste the base BC is equall to the base EF and the triangle to the triangle and the rest of the angles to the reste of the angles Wherefore the angle ACB is equall to the angle DFE And the right angle ACK is equal to the right angle DFN. Wherfore the angle remayning namely BCK is equall to the angle remayning namely to EFN And by the same reasoÌ„ also the angle CBK is equal to the angle FEN Wherfore there are two triangles BCK EFN hauing two angles of the one equal to two angles of the other eche to his correspondent angle and one side of the one equall to one side of the other namely that side that lieth betwene the equall angles that is the side BC is equall to the side EF Wherefore by the 26. of the first the sides remaininge are equall to the sides remayning Wherfore the side CK is equall to the side FN but the side AC is equall to the side DF. Wherefore these two sides AC and CK are equall to these two sides DF and FN and they contayne equall angles Wherefore by the 4. of the first the base AK is equall to the base DN And forasmuch as the line AH is equall to the line DM therefore the square of the line AH is equall to the square of the line DM But vnto the square of the line AH are equall the squares of the lines AK and KH by the 47. of the first for the angle AKH is a right angle And to the square of the line DM are equall the squares of the lines DN and NM for the angle DNM is a right angle Wherefore the squares of the lines AK and KH are equall to the squares of the lines DN and NM of which two the square of the line AK is equall to the square of the line DN for the line AK is proued equall to the line AN Wherefore the residue namely the square of the line KH is equal to the residue namely to the square of the line NM Wherefore the line HK is equall to the line MN And forasmuch as these two lines HA and AK are equall to these two lines MD and DN the one to the other and the base HK is equall to the base MN therfore by the 8. of the first the angle HAK is equall to the angle MDN. If therefore there be two superficiall angles equall and froÌ„ the pointes of those angles be eleuated on high right lines comprehending together with those right lines which were put at the beginning equall angles ech to his corespondent angle and if in ech of the erected lines be taken a point at all aduentures and from those pointes be drawen perpendicular lines to the plaine superficieces in which are the angles geuen at the beginning and frâ—â— the pointes which are by the perpendicular lines made in the two plaine superficieces be ioyned right lines to those angles which were put at the beginning those right lines shall together with the lines eleuated on high make equall angles which was required to be proued Because the figures of the former demonstration are somewhat hard to conceaue as they are there drawen in a plaine by reason of the lines that are imagined to be eleuated on high I haue here set other figures wherein you must erecte perpendicularly to the ground superficieces the two triangles BHK and EMN and then eleuate the triangles DFM ACH in such sort that the angles M and H of these triangles may concurre with the angles M and H of the other erected triangles And then imagining only a line to be drawen from the point G of the line AG to the point L in the ground superficies compare it with the former construction demonstration and it will make it very easye to conceaue ¶ Corollary By this it is manifest that if there be two rectiline superficiall angles equall and vpon those angles be eleuated on high equall right lines contayning together with the right lines put at the bâ—ginning equall angles perpendicular lines drawen from those eleuated lines to the ground plaine superficieces wherein are the angles put at the beginning are equall the one to the other For it is manifest that the perpendicular lines HK MN which are drawen from the endes of the equall eleuated lines AH and DM to the ground superficieces are equall ¶ The 31. Theoreme The 36. Proposition If there be three right lines proportionall a Parallelipipedon described of those three right lines is equall to the Parallelipipedon described of the middle line so that it consiste of equall sides and also be equiangle to the foresayd Parallelipipedon SVppose that these three lines A B C be proportionall as A is to B so let B be to C. Then I say that the Parallelipipedon made of the lines A B C is equall to the Parallelipipedon made of the line B so that the solide made of the line B consist of equall sides and be also equiangle to the solide made of the lines A B C. Describe by the 23. of the
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
and meane proportion but in one onely point which was requisite to be demonstrated A Theoreme 2. What right line so euer being deuided into two partes hath those his two partes proportionall to the two segmentes of a line deuided by extreame and meane proportion is also it selfe deuided by an extreame and meane proportion and those his two partes are his two segments of the sayd proportion Suppose AB to be a line deuided by an extreame and meane proportion in the point C and AC to be the greater segment Suppose also the right line DE to be deuided into two partes in the point F and that the part DF is to FE as the segment AC is to CB or DF to be to AC as FE is to CB. For so these paâ—tes are proportionall to the sayd segmentes I say now that DE is also deuided by an extreame and meane proportion in the point F. And that DF FE are his segmentes of the sayd proportion For seing as AC is to CB so is DF to FE by supposition Therfore as AC and CB which is AB are to CB so is DF and FE which is DE to FE by the 18. of the fifth Wherefore alternately as AB is to DE so is CB to FE And therefore the residue AC is to the residue DF as AB is to DE by the fifth of the fift And then alternately AC is to AB as DE is to DF. Now therefore backward AB is to AC as DE is to DF. But as AB is to AC so is AC to CB by the third definition of the sixth booke Wherefore DE is to DF as AC is to CB by the 11. of the fifth And by supposition as AC is to CB so is DF to FE wherefore by the 11. of the fifth as DE is to DF so is DF to FE Wherefore by the 3. definition of the sixth DE is deuided by an extreame and meane proportion in the point F. Wherefore DF and FE are the segmentes of the sayd proportion Therefore what right line so euer being deuided into two partes hath those his two partes proportionall to the two segmentes of a line deuided by extreame and meane proportionâ— is also it selfe deuided by an extreme and meane proportion and those his two partes are his two segmentes of the sayd proportionâ— which was requisite to be demonstrated Note Many wayes these two Theoremes may be demonstrated which I leaue to the exercise of young studentes But vtterly to want these two Theoremes and their demonstrations in so principall a line or rather the chiefe piller of Euclides Geometricall pallace was hetherto and so would remayne a great disgrace Also I thinke it good to note vnto you what we meant by one onely poynt We mâ—â—â—â— that the quantities of the two segmentes can not be altered the whole line being once geuen And though from either end of the whole line the greater segment may begin And so as it were the point of section may seeme to be altered yet with vs that is no alteration forasmuch as the quantities of the segmentes remayne all one I meane the quantitie of the greater segment is all one at which end so euer it be taken And therefore likewise the quantitie of the lesse segment is all one c. The like confideration may be had in Euclides tenth booke in the Binomiall lines c. Ioâ—n Dee 1569. Decemb. 18. The 3. Theoreme The 3. Proposition If a right line be deuided by an extreme and meane proportion and to the lesse segment be added the halfe of the gerater segment the square made of those two lines added together is quintuple to the square made of the half line of the greater segment SVppose that the right line AB be deuided by an extreme and meane proportion in the point C. And let the greater segment thereof be AC And deuide AC into two equall partes in the point D. Then I say that the square of the line BD it quintuple to the square of the line DC Describe by the 46. of the first vpon the line AB a square AE And describe and make perfect the figure that is diuide the lyne AT like vnto the diuision of the line AB by the 10. of the sixth in the pointes R H by which pointes drawe by the 31. of the first vnto the line AB parallel lines RM and HN. So likewise draw by the pointes D C vnto the line BE these parallel lines DL and CS draw the diameter BT And forasmuch as the line AC is double to the line DC therefore the square of AC is quadruple to the square of DC by the 20. of the sixth that is the square RS to the square FG And forasmuch as that which is contayned vnder the lines AB and BC is equall to the square of the line AC and that which is contayned vnder the lines AB and BC is equall to the parallelogramme CE the square of the line AC is the square RS wherefore the parallelogramme CE is equall to the square RS. But the square RS is quadruple to the square FG wherefore the parallelogramme CE also is quadruple to the square FG. Agayne forasmuch as the line AD is equall to the line DC therfore the line HK is equall the line KF wherefore also the square GF is equall to the square HL wherefore the line GK is equall to the line KL that is the line MN to the line NE wherefore the parallelogramme MF is equall to the parallelogramme FE But the parallelogramme MF is equall to the parallelogramme CG wherfore the parallelogramme CG is also equal to the parallelogramme FE Put the parallelogramme CN common to theÌ„ both Wherefore the gnomon XOP is equall to the parallelogramme CK But the parallelogramme CE is proued to be quadruple to Gâ— the square wherefore the gnomon XOP is quadruple to the square GF Wherefore the square DN is quintuple to the square FG And DN is the square of the line DB and GF the square of the line DC Wherefore the square of the line DB is quintuple to the square of the line DC If therefore a right line be deuided by an extreme and meane proportion and to the lesse segment be added the halfe of the greater segment the square made of those two lines added together is quintuple to the square made of the halfe line of the greater segmeÌ„t Which was required to be demonstrated Ye shall finde this proposition an other way demonstrated after the fiueth proposition of this booke Here foloweth M. Dee his additions ¶ A Theoreme 1. If a right line geuen be quintuple in power to the powre of a segment of him self the double of that segment and the other part remayning of the first geuen line make a line diuided by extreme and meane proportion and that double of the segment is the greater part thereof Forasmuch as this is the conuerse of Euclides
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
a right line coupling their centres being diuided by an extreame and meane proportion maketh the greater segment the right line which coupleth the centres of the next bases If by the centres of fiue bases set vppon one base be drawne a playne superficies and by the centres of the bases which are set vpon the opposite base be drawne also a playne superficies and then be drawne a right line coupling the centres of the opposite bases that right line is so cut that eche of his partes set without the playne superficies is the greater segment of that part which is contayned betwene the playnes The side of the dodecahedron is the greater segment of the line which subtendeth the angle of the pentagon A perpendicular line drawne from the centre of the dodecahedron to one of the bases is in power quintuple to half the line which is betwene the playnes And therfore the whole line which coupleth the centres of the opposite bases is in power quintuple to the whole line which is betwene the sayd playnes The line which subtâ—deth the angle of the base of the dodecahedroÌ„ together with the side of the base are in power quintuple to the line which is drawne from the ceÌ„tre of the circle which contayneth the base to the circumference A section of a sphere contayning three bases of the dodecahedron taketh a third part of the diameter of the sayd sphere The side of the dodecahedron and the line which subtendeth the angle of the pentagon are equall to the right line which coupleth the middle sections of the opposite sides of the dodecahedron ¶ The ende of the Elementes of Geometrie of the most auncient Philosopher 〈◊〉 of Megara The intent of this Preface Number Note the worde Vnit to expresse the Greke Monaâ— not Vnitie as we hauâ— all commonly till now vsed Magnitude A point A Line Magnitude Ano. 1488. ☞ Arithmetike Note * Anno. 1550. R. B. Note 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ☞ This noble Earle dyed Anno. 1554. skarse of 24. yeares of ageâ— hauing no issue by his wife Daughter to the Duke of Somerset Iustice. ☞ * Plato 7. de Rep. ☞ * Note The difference betwene Strataruhmetrie and Tacticie I.D. Fâ—endâ—â— you will finde it hard to performâ— my descriptiâ—n of â—his Fâ—ate But by Châ—râ—graphieâ— you may helpe your selfe some â—hat wherâ— thâ— Figures knowne in Sidâ—â—â—nd Angles are not Regular And whereâ— Resolution into Triangles can sâ—â—uâ— c. And yet you will finde it strange to deale thus generâ—lly with Arithmeticall figures and that for Battayle â—ay Their coâ—tentâ—â—â— differ so much from like Geometrâ—call Figurâ—s A marueilous Glasse ☞ S.W.P. ☞ Note 1. 2. 3. 5. 6. 7. 8. I.D. Read in Aristotle his 8. booke of Politikes the 5 6 and 7. chapters Where you shall haue some occasion farder to thinke of Musike than commonly is thought ☜ * Anno. 1548 and 1549. in Louayn Note I.D. The Cutting of a Sphare according to any proportion assigned may by this proposition be done Mechanically by tempering Lâ—quor to a certayne waight in respect of the waight of the Sphare 〈◊〉 Swyâ—â—â—ng A common errorâ— noted A paradox N. T. The wonderfull vse of these Propositions The practise Staticall to know the proportion betwene the Cube and the Sphare I. D. * For so haue you 256. partes of a Graine * The proportion of the Square to the Circle insâ—ribed * The Squaâ—ing of the Cirâ—le Mâ—â—haniâ—ally * To any Squirâ— gâ—uenâ— to 〈…〉 Note Squaring of the Circle without knowledge of the proportion betwene Circumference and Diameter To Dubble the Cube redily by Art Mechanicall depending vppon Demonstration Mathematicall I. D. The 4. sides of this Pyramiâ— must be 4. Isosceles Triangles â— likâ— and â—quall I. D. * In all workâ—ngeâ— with this Pyramis or Cone Let their Situationâ— be in all Pointeâ— and Conditions a like oâ— all one while you are about â—ne worke Els you will 〈◊〉 I. D. * Consider well whan you must put your wateâ—â— togyther and whan you must empty youâ— first waterâ— out of your Pyramiâ— or Cone Elâ— you will 〈◊〉 * Vitruuius Lib. 9. Cap. 3. ☞ God bâ— thanked â—or this Inuentionâ— the fruiâ—e â—nsuing * Note Note as concerning the Sphaericall Superâ—icies of the water ☞ * Note Note this Abridgeâ—ent of Dubbling 〈◊〉 Cube â—â— * Note * ☜ To giue Cubes one to the other in any proportion Rationall or Irrationall * Emptying the first The demonstrations of this Dubbling of the Cube and oâ— the rest I.D. * Here 〈…〉 of the water * By the 33. of the eleuenth books of Euclide I.D. * And your diligence in praâ—â—ise can â—o in waight of wateâ—â— peâ—forme it Therefore now you arâ— able to â—eue good reason of your whole doing * Note this Corollary * The great Commodities following of these new Inuentions * ☞ Such is the Fruite of the Mathematicall Sciences and Artes. MAN is the Lesse World. * ☜ Microcosmus * Lib. 3. Cap. 1. ☞ Saw Milles. * Atheneus Lib. 5. cap. 8. Proclus Pag. 18. To go to the bottom of the Sea without daunger Plutâ—â—â—bus in Marco Mâ—rcello Syâ—asius in Epistolis Polybius Plinius Quintâ—lianus T. Liuius * Athenaâ—s * Galeâ—us Anthemius Burning Glasses Gunnes 4. Reg. 20. A perpetuall Motion An obiection The Answer ☜ A Mathematicien Vitrunius VVho is an Architect * The Immaterialitie of perfect Architecture What Lineament is Note Anno. 1559. * Anno. 1551 De his quae Mundo mirabiliter eueniunt cap. 8. Tusc. â— * ☞ A Digression Apologeticall * A prouerb Fayre fisht and caught a Frog ☞ Psal. 140. Act. 7. C. Lib. 30. Cap. 1. ☜ R. B. ☞ Vniuersities ☜ ☞ The Ground platt of this Praeface in a Table The argumâ—â—â— of the first Booke Definition of a poynt Definition of a poynt after Pithagoras Definition of a liâ—â— An other definition of a line An other The endes of a line Difference of a point frâ—â—nity Vnitie is a part of number A poynt is no part of quantitie Definition of a right line DefinitioÌ„ therof after Plato An other definition An other An other An other An other VVhy Euclide here defineth not a crooked lyne Definition of a superficies A superficies may be deuided two wayes An other definition of a superficies The extremes of a superficies Another definition of a superficies Definition of a plaine superficies Another definition of a playne superficies NOTE Another definition of a playne superficies An other definition An other definition An other definition Definition of a playne angle Definition of a â—ecâ—ilined angle 〈◊〉 of angles VVhat a right angle VVhat also a perpendicular lyne iâ— VVhat an obtuse angle â—â— VVhat an acute angle is The limite of any thing No science of thinges infinite Definition of a figure Definition of a circle A circle the most perfect of all figures The centre of a circle Definition of a diameter Definition of a semicircle Definition of a section of a circle Definition of râ—â—â—â—lined figures
casâ— Demonstâ—atâ—on Construction Demonstration Demonstration leading to an impossibilitie Note this maner of imagination Mathematicall Demonstration leading to an impossibilitie Construction Demonstration Demonstration leading to an absurditieâ— In tâ—is â—ronoâ—â—â—oâ—â—â— must vndâ—rsâ—and the propâ—rtioâ—â—ll â—artes or sâ—â—â—ions to be thâ—se which are câ—ntaiâ—ed 〈◊〉 the parallel superâ—â—cies Construction Demâ—nstration Construction Demonstration Demonstration leading to an impossibilitie Demonstration Construction Demonstraâ—ion Two cases in this proposition The first case Second case Constructiâ—â—â— Demonstration An other demonstration Construction Demonstraâ—â—â—n Construction Three cases in this proposition The first case A necessary thing to be proued before he pâ—oceede any â—arther in the construction of the Problemâ— * Which how to finde out is taught at the end of this demonstration and also was taught in the asâ—umpt put before the 14. proposition of the teÌ„th boke Demonstration of the first case Second case Third case An other demoÌ„stratioÌ„ to proue that thâ— line AB is not lesse theÌ„ the line LX. This was before taâ—ght in the tenth booke in the assumpt put before the 14. proposition * M. Dee to auoide cauillation addeth to Euclides proposition this worde sixe whome I haue followed accordingly and not Zamberts in this This kinde of body mencioned in the proposition is called a ParallelipipedoÌ„ according to the diâ—finition before geuen thereof Demonstration that the opposite sides are parallelogrammes DemonstratioÌ„ that the opposite superficies are equall * AB is equall to DC because the superficies AC is proued a parallelograÌ„me and by the same reason is BH equall to CF because the superficies FB is proued a parallelogramme therefore the 34. of the first is our proofe Fâ—â—â—â— Corollary Second Corolry These solides which he speaketh of in this Corollary are of some called sidâ—d columnâ—s Third Corollary Constrâ—ction Demonstrâ—tion * Looke at the end of the demonstratioâ— what is vnderstanded by staÌ„ding lines Iohn d ee his figure By this figure it appâ—arâ—th why â—uch Prismes were called â—â—edges of 〈◊〉 vâ—ry shape of a wedge as is the solide DEFGAC c. StaÌ„ding lines Construction Demonstration I. Dees figure Two cases in this proposition Thâ— first case Construction We are beholding to M. d ee for inuenting this figure with other which till his reforming were as much mishappen as this was and so both in the Greeke and Latine copies remaine Demonstration * Note now how the base respectiuely is takenâ— for so may alteratioÌ„ of respects alter the name of the bowndes eyther of solides or playnes Second case * There you perceaue how the base is diuersly considered chosen as before we aduertised you Construction Demonstration Construction Demonstration * * Note this famous Lâ—mma The doubling oâ—â—he Cube * Note * 〈…〉 Lemma Note what iâ— yet lacking requisite to the doubling of the Cube The conuerse of both the partes of the first case The conuerse of the second case The generall conclusion Construction DemonstratioÌ„ of the first part DemonstratioÌ„ of the second part To finde two middâ—e proportionals betwene two numbers geueÌ„ Note the practise of appâ—oching to precieâ—es in Cubik rootes * This is the way to apply any square geuen to a line also geueÌ„ sufficienty extended A probleme worth the searching for Construction Demonstration Construction Demonstration * It is euideÌ„t that those perpendiculars are all one with the staâ—ding lines of the solides if their solide angles be made of superâ—icâ—â—il right angles onely Dâ—uâ—ling of the 〈◊〉 c. * Dâ—mâ—â—â—â—ratioâ—â—f pâ—sâ—ibilitie in the â—â—oblem An other argumeâ—t to comâ—oâ—t the studious Demonstration of the first part Demonstration of the second part which is the conuerse of the first part Demonstration leading to an impossibilitie Construction Demonstration Construction Demonstration * Which of some are called sided Columnes Sided Columnes Construction Demonstration The squaring of the circle Demonstratiâ—n â—eading to an impossibiâ—itie Two cases in this proposition The first case That a square within any circle described is bigger than halfe the circle That the Isosceles triangles without the square are greater then halfe the segments wherin they are Second case * This As 〈…〉 afteâ—ward at the end of the demâ—straâ—ion proued Construction Demonstration Construction Demonstration Diâ—â—reence betwene the first probleme and the second Construction Demonstration Consâ—ruction Demonstration Consâ—ruction Demonstration Consâ—â—uction Note this well for it iâ— of great vse An other way of demonstâ—atâ—oÌ„ of the fâ—â—st â—robleme of thâ—â— addition Note this properâ—ie of a triangle rectangle Construction Demonstration * Though I say without the square yet you must thinke that it may be also within the square that diuersly Wherfore this Probleme may haue diuerse cases so but briefly to aâ—â—yde all may thus be said cut any side of that square into 3. partsâ— in the proportion of X to Y. Note the maner of the drift in this demonstration and construction mixtly and with no determination to the constructioÌ„â— as commoÌ„ly iâ— in problemeâ—â— which is here of me so vsedâ— for an example to young studeÌ„tes of variety in art Construction Demonstration * Note and remember one â—eâ—th in these solids The conclusion of the first part DemonstratioÌ„ of the second part namely that it is deuided moreouer into two equal Prismes Conclusion of the second part DemonstratioÌ„ of the last part that the two Prismes are greater then the halfe of the whole Pyramis Conclusion of the last part Conclusion of the whole proposition * An AssuÌ„pt An Assumpt Conclusion of the whole Demonstration leading to an impossibilitie * In the Assuâ—pâ—â—â—llowinâ— the second â—ropâ—â—ition of this bâ—â—ke Construction Demonstration Demonstration Note Sided Columnes sometime called prismes are triple to pyramids hauing one base and equall heâ—th with them Note â—arallelipipedons treble to pyramids of one base and heith with them Construction Demonstration An addition by Campane and Flussas Demonstration of the first part Demonstration â—f the second part which iâ— the conueâ—se of the first Constr 〈◊〉 〈◊〉 Parallelipipedons called Prismes * By this it is manifest that Euclide comprehended sided Columnes also vnder the name of a Prisme * A prisme hauing for his base a poligonon figure as we haue often before noted vnto you Note M. d ee his chiefe purpose in his additions Demonstration touching cylinders Second case Second parâ— which concerneth Cillinders Construction Demonstraâ—ion Construction Demonstration touching Cylinders Demonstration touching Cones First part of the propositioÌ„ demonstrated touching Cones Two cases in this proposition The first case Second case Construction Demonstration touching cylinders Second part demonstraâ—ed Construction * Note this LM because of KZ in the next proposition and here the point M for the point Z in the next demonstration I. Dee * For that the sections were made by the number two that is by taking halues and of the residue the halâ—eâ— and so to LD being an halfe and a residue which shall be a coÌ„mon measure backe againe to make sides of the Poligonon figure Construction
Draw by the 17. of the third a right line touching the circle ABC and let the same be GAH and let it touch in the point A. And by the 23. of the first vnto the right line AH and vnto the point in it A describe an angle HAC equall vnto the angle DEF And by the selfe same vnto the right line AG and vnto the point in it A make an angle GAB equall vnto the angle DFE And draw a right line from B to C. And for asmuch as a certaine right line GAH toucheth the circle ABC and from the point where it toucheth namely A is drawen into the circle a right line AC therefore by the 31. of the third the angle HAC is equall vnto the angle ABC which is in the alternate segment of the circle But the angle HAC is equall to the angle DEF Wherfore the angle ABC is equall to the angle DEF And by the same reason the angle ACB is equall to the angle DFE Wherefore the angle remayning BAC is equall vnto the angle remayning EDF Wherefore the triangle ABC is equiangle vnto the triangle DEF And it is described in the circle geuen ABC Wherefore in a circle geuen is described a triangle equiangle vnto a triangle geuen which was required to be done The 3. Probleme The 3. Proposition About a circle geuen to describe a triangle equiangle vnto a triangle geuen SVppose that the circle geuen be ABC and let the triangle geuen be DEF It is required about the circle ABC to describe a triangle equiangle vnto the triangle DEF Extend the line EF on ech side to the poyntes G and H. And by the first of the third take the centre of the circle ABC and let the same be the point K. And then draw a right line KB And by the 23. of the first vnto the right line KB and vnto the point in it K make an angle BKA equall vnto the angle DEG and likewise make the angle BKC equall vnto the angle DFH. And by the 17. of the third draw right lines touching the circle A B C in the pointes A B C. And let the same be LAM MBN and NCL And for asmuch as the right lines LM MN NL do touch the circle ABC in the pointes A B C and from the centre K vnto the pointes A B C are drawen right lines KA KB and KC therefore the angles which are at the pointes A B C are right angles by the 18. of the third And for asmuch as the fower angles of the fower sided figure AMBK are equall vnto fower right angles whose angles KAM KBM are two right angles therfore the angles remayning AKB and AMB are equall to two right angles And the angles DEG DEF are by the 13. of the first equall to two right angles Wherefore the angles AKB and AMB are equall vnto the angles DEG and DEF of which two angles the angle AKB is equall vnto the angle DEG Wherefore the angle remayning AMB is equall vnto the angle remayning DEF In like sort may it be proued that the angle LNM is equall to the angle DFE Wherfore the angle remayning MLN is equall vnto the angle remayning EDF Wherefore the triangle LMN is equiangle vnto the triangle DEF and it is described about the circle ABC Wherefore about a circle geuen is described a triangle equiangle vnto a triangle geuen which was required to be done ¶ An other way after Pelitarius In the circle ABC inscribe a triangle GHK equiangle to the triangle EDF by the former Proposition so that let the angle at the poynt G be equall to the angle D and let the angle at the point H be equall to the angle E and let also the angle at the poynt K be equall to the angle F. Then drawe the line LM parallel to the line GH which let touch the circle in the poynt A which may be done by the Proposition added of the sayd Pelitarius after the 17. Proposition Draw likewyse the line MN parallel vnto the line HK and touching the circle in the poynt B And also draw the line LN parallel vnto the line GK and touching the circle in the poynt C. And these three lines shall vndoubtedly concurre as in the poyntes L M and N which may easily be proued if you produce on either side the lines GH GK and HK vntill they cut the lines LM LN and MN in the poyntes O P Q R S T. Now I say that the triangle LMN circuÌ„scribed about the circle ABC is equiangle to the triangle DEF For it is manifest that it is equiangle vnto the triaÌ„gle GHK by the proprietie of parallel lines For the angle MTQ is equall to the angle at the poynt G of the triangle GHK by the 29. of the first and therefore also the angle at the poynt L is equall to the selfe same angle at the poynt G for the angle at the point L is by the same 29. Proposition equall to the angle MTQ And by the same reason the angle at the poynt M is equall to the angle at the poynt H of the selfe same triangle and the angle at the poynt N to the angle at the poynt K. Wherfore the whole triangle LMN is equiangle to the whole triangle GHK Wherfore also it is equiangle to the triangle DEF which was required to be done The 4. Probleme The 4. Proposition In a triangle geuen to describe a circle SVppose that the triangle geuen be ABC It is required to describe a circle in the triangle ABC Deuide by the 9. of the first the angles ABC and ACB into two equall parâ—es by two right lines BD and CD And let these right lines meete together in the point D. And by the 12. of the first from the point D draw vnto the right lines AB BC and CA perpendicular lines namely DE DF and DG And forasmuch as the angle ABD is equall to tho angle CBD and the right angle BED is equall vnto the right angle BFD Now then there are two triangles EBD and FBD hauing two angles equall to two angles and one side equall to one side namely BD which is common to them both and subtendeth one of the equall angles Wherfore by the 26. of the first the rest of the sides are equall vnto the rest of the sides Wherfore the line DE is equall vnto the line DF and by the same reason also the â—ne DG is equall vnto the line DF. Wherfore these three right lines DE DF DG are equall the one to the other by the first common sentence Wherefore making the centre the point D and the space DE or DF or DG describe a circle and it will passe through the pointes E F G and will touch the right lines AB BC and CA. For the angles made at the pointes E F G are right angles For if the circle cut those right lines then froÌ„ the end of