the cōpasse in B and with the other I draw the arch D. E which I part into ij equall parts in F and thē draw a line frō B to F so I haue mine intēt THE IIII. CONCL. To deuide any measurable line into ij equall partes Open your compasse to the iust lēgth of the line And thē set one foote steddely at the one ende of the line with the other fote draw an arch of a circle against the midle of the line both ouer it and also vnder it then doo lyke waise at the other ende of the line And marke where those arche lines do meet crossewaies and betwene those ij pricks draw a line and it shall cut the first line in two equall portions Example The lyne is A. B. accordyng to which I open the compasse and make .iiij. arche lines whiche meete in C. and D then drawe I a lyne from C so haue I my purpose This conclusion serueth for makyng of quadrates and squires beside many other commodities howebeit it maye bee don more readylye by this conclusion that foloweth nexte THE FIFT CONCLVSION To make a plumme line or any pricke that you will in any right lyne appointed Example The lyne is A.B. the prick on whiche I shoulde make the plumme lyne is C. then open I the compasse as wyde as A C and sette one foote in C. and with the other doo I marke out C.A. and C. B then open I the compasse as wide as A. B and make ij arch lines which do crosse in D and so haue I doone Howe bee it it happeneth so sommetymes that the pricke on whiche you would make the perpendicular or plum line is so nere the eand of your line that you can not extende any notable length from it to th one end of the line and if so be it then that you maie not drawe your line lenger frō that end then doth this conclusion require a newe ayde for the last deuise will not serue In suche case therfore shall you dooe thus If your line be of any notable length deuide it into fiue partes And if it be not so long that it maie yelde fiue notable partes then make an other line at will and parte it into fiue equall portiōs so that thre of those partes maie be found in your line Then open your compas as wide as thre of these fiue measures be and sette the one foote of the compas in the pricke where you would haue the plumme line to lighte whiche I call the first pricke and with the other foote drawe an arche line righte ouer the pricke as you can ayme it then open youre compas as wide as all fiue measures be and set the one foote in the fourth pricke and with the other foote draw an other arch line crosse the first and where thei two do crosse thense draw a line to the poinct where you woulde haue the perpendicular line to light and you haue doone Example The line is A. B. and A. is the prick on whiche the perpendicular line must light Therfore I deuide A. B. into fiue partes equall then do I open the compas to the widenesse of three partes that is A. D. and let one foote staie in A. and with the other I make an arche line in C. Afterwarde I open the compas as wide as A.B. that is as wide as all fiue partes and set one foote in the .iiij. pricke which is E drawyng an arch line with the other foote in C. also Then do I draw thence a line vnto A and so haue I doone But and if the line be to shorte to be parted into fiue partes I shall deuide it into iij. partes only as you see the line F. G and then make D. an other line as is K. L. whiche I deuide into .v. suche diuisions as F. G. containeth .iij then open I the compaas as wide as .iiij. partes whiche is K. M. and so set I one foote of the compas in F and with the other I drawe an arch lyne toward H then open I the cōpas as wide as K. L. that is all .v. partes and set one foote in G that is the iij. pricke and with the other I draw an arch line toward H. also and where those .ij. arch lines do crosse whiche is by H. thence draw I a line vnto F and that maketh a very plumbe line to F. G as my desire was The maner of workyng of this conclusion is like to the second conclusion but the reason of it doth depēd of the .xlvi. proposiciō of the first boke of Euclide An other waie yet set one foote of the compas in the prick on whiche you would haue the plumbe line to light and stretche forth thother foote toward the longest end of the line as wide as you can for the length of the line and so draw a quarter of a compas or more then without stirryng of the compas set one foote of it in the same line where as the circularline did begin and extend thother in the circular line settyng a marke where it doth light then take half that quantitie more therevnto and by that prick that endeth the last part draw a line to the pricke assigned and it shall be a perpendicular Example A. B. is the line appointed to whiche I must make a perpendicular line to light in the pricke assigned which is A. Therfore doo I set one foote of the compas in A and extend the other vnto D. makyng a part of a circle more then a quarter that is D. E. Then do I set one foote of the compas vnaltered in D and stretch the other in the circular line and it doth light in F this space betwene D. and F. I deuide into halfe in the pricke G whiche halfe I take with the compas and set it beyond F. vnto H and therfore is H. the point by whiche the perpendicular line must be drawen so say I that the line H. A is a plumbe line to A. B as the conclusion would THE VI. CONCLVSION To drawe a streight line from any pricke that is not in a line and to make it perpendicular to an other line Open your compas so wide that it may extend somewhat farther thē from the prick to the line then sette the one foote of the compas in the pricke and with the other shall you draw a cōpassed line that shall crosse that other first line in .ij. places Now if you deuide that arch line into .ij. equall partes and from the middell pricke therof vnto the prick without the line you drawe a streight line it shal be a plumbe line to that firste lyne accordyng to the conclusion Example C. is the appointed pricke from whiche vnto the line A. B. I must draw a perpēdicular Therfore I open the cōpas so wide that it may haue one foote in C and thother to reach ouer the line and with that foote I draw an arch line as you see betwene A. and B which arch
in the angle appoincted equall to the triangle that is assigned Then with your compasse take the length of your line appointed and set out two lines of the same length in the second gemowe lines beginnyng at the one side of the likeiamme and by those two prickes shall you draw an other gemowe line whiche shall be parallele to two sides of the likeiamme Afterward shall you draw .ij. lines more for the accomplishement of your worke whiche better shall be perceaued by a shorte exaumple then by a greate numbre of wordes only without example therefore I wyl by example sette forth the whole worke Example Fyrst according to the last conclusion I make the likeiamme E. F. C. G equal to the triangle D in the appoynted angle whiche is E. Then take I the lengthe of the assigned line which is A. B and with my compas I sette forthe the same lēgth in the ij gemow lines N. F. and H. G setting one foot in E and the other in N and againe settyng one foote in C and the other in H. Afterward I draw a line from N. to H whiche is a gemow lyne to ij sydes of the likeiamme thenne drawe I a line also from N. vnto C and extend it vntyll it crosse the lines E. L. and F. G which both must be drawen forth longer then the sides of the likeiamme and where that lyne doeth crosse F. G there I sette M. Nowe to make an ende I make an other gemowe line whiche is parallel to N. F. and H. G and that gemowe line doth passe by the pricke M and then haue I done Now say I that H. C. K. L is a likeiamme equall to the triangle appointed whiche was D and is made of a line assigned that is A. B for H. C is equall vnto A. B and so is K. L The prose of the equalnes of this likeiam vnto the triāgle depēdeth of the thirty and two Theoreme as in the hoke of Theoremes doth appear where it is declared that in al likeiammes whē there are more then one made about one bias line the filsquares of euery of them muste needes be equall THE XVII CONCLVSION To make a likeiamme equal to any right lined figure and that on an angle appointed The readiest waye to worke this conclusion is to tourn that right lined figure into triangles and then for euery triangle to gether an equal likeiamme according vnto the eleuen cōclusion and then to ioine al those likeiammes into one if their sides happen to be equal which thing is euer certain when al the triangles happē iustly betwene one pair of gemow lines but and if they will not frame so then after that you haue for the firste triangle made his likeiamme you shall take the lēgth of one of his sides and set that as a line assigned on whiche you shal make al the other likeiams according to the twelft cōclusion and so shall you haue al your likeiammes with ij sides equal and ij like angles so that you maieasily ioyne thē into one figure Example If the right lined figure be like vnto A thē may it be turned into triangles that wil stād betwene ij parallels anye ways as you maise by C and D for ij sides of both the triāgls ar parallels Also if the right lined figure be like vnto E thē wil it be turned into triāgles liyng betwene two parallels also as the other did before as in the exāple of F. G. But and if the right lined figure be like vnto H and so turned into triāgles as you se in K. L. M wher it is parted into iij triāgles thē wil not all those triangles lye betwen one pair of parallels or gemow lines but must haue many for euery triangle must haue one paire of parallels seuerall yet it maye happen that when there bee three or sower triangles ij of theym maye happen to agre to one pair of parallels whiche thinge I remit to euery honest witte to serche for the manner of their draught wil declare how many paires of parallels they shall neede of which varietee bicause the examples ar infinite I haue set forth these few that by them you may coniecture duly of all other like Further explicacion you shal not greatly neede if you remembre what hath ben taught before and then diligētly behold how these sundry figures be turned into triāgles In the fyrst you se I haue made v. triangles and four paralleles in the seconde vij triangles and foure paralleles in the thirde thre triāgles and fiue parallels in the iiij you se fiue triāgles four parallels in the fift iiij triāgles and .iiij. parallels in the sixt ther ar fiue triāgles iiij paralels Howbeit a mā maye at liberty alter them into diuers formes of triāgles therefore I leue it to the discretion of the woorkmaister to do in al suche cases as he shal thinke best for by these examples if they bee well marked may all other like conclusions be wrought THE XVIII CONCLVSION To parte a line assigned after suche a sorte that the square that is made of the whole line and one of his parts shal be equal to the squar that cometh of the other parte alone First deuide your lyne into ij equal parts and of the length of one part make a perpendicular to light at one end of your line assigned then adde a bias line and make thereof a triangle this done if you take from this bias line the halfe lengthe ol your line appointed which is the iuste length of your perpendicular that part of the bias line whiche dothe remayne is the greater portion of the deuision that you seke for therefore if you cut your line according to the lengthe of it then will the square of that greater portior be equall to the square that is made of the whole line and his lesser portion And contrary wise the square of the whole line and his lesser parte wyll be equall to the square of the greater parte Example A. B is the lyne assigned E. is the middle pricke of A. B B. C. is the plumb line or perpendicular made of the halfe of A. B equall to A. E other B. E the byas line is C. A from whiche I cut a peece that is C. D equall to C. B and accordyng to the lengthe lothe peece that remaineth whiche is D. A I doo deuide the line A. B at whiche diuision I set F. Now say I that this line A B which was assigned vnto me is so diuided in this point F that that square of the hole line A. B of the one portiō that is F. B the lesser part is equall to the square of the other parte whiche is F. A and is the greater part of the first line The profe of this equalitie shall you learne by the .xl. Theoreme THE XIX CONCLVSION To make a square quadrate equall to any right lined figure appoincted First make a likeiamme equall to that right lined
figure with a right angle accordyng to the .xv. conclusion then consider the likeiamme whether it haue all his sides equall or not for yf they be all equall then haue you doone your conclusion but and if the sides be not all equall then shall you make one right line iuste as long as two of those vnequall sides that line shall you deuide in the middle and on that pricke drawe half a circle then cutte from that diameter of the halfe circle a certayne portion equall to the one side of the likeiamme and from that pointe of diuision shall you erecte a perpendicular which shall touche the edge of the circle And that perpendicular shall be the iuste side of the square quadrate equall both to the lykeiamme and also to the right lined figure appointed as the conclusion willed Example K is the right lined figure appointed and B. C. D. E is the likeiāme with right angles equall vnto K but because that this likeiamme is not a square quadrate I must turne it into such one after this sort I shall make one right line as long as .ij. vnequall sides of the likeiāme that line here is F. G whiche is equall to B. C and C. E. Then part I that line in the middle in the pricke M and on that pricke I make halfe a circle accordyng to the length of the diameter F. G. Afterward I cut awaie a peece from F. G equall to C. E markyng that point with H. And on that pricke I erecte a perpendicular H. K whiche is the iust side to the square quadrate that I seke for therfore accordyng to the doctrine of the .x. conclusion of that lyne I doe make a square quadrate and so haue I attained the practise of this conclusion THE XX. CONCLVSION when any .ij. square quadrates are set forth how you maie make one equall to them bothe First drawe a right line equall to the side of one of the quadrates and on the ende of it make a perpendicular equall in length to the side of the other quadrate then drawe a byas line betwene those .ij. other lines makyng thereof a right angeled triangle And that byas lyne wyll make a square quadrate equall to the other .ij. quadrates appointed Example A.B. and C. D are the two square quadrates appointed vnto which I must make one equall square quadrate First therfore I dooe make a righte line E. F equall to one of the sides of the square quadrate A.B. And on the one end of it I make a plumbe line E. G equall to the side of the other quadrate D. C. Then drawe I a byas line G. F whiche beyng made the side of a quadrate accordyng to the tenth conclusion will accomplishe the worke of this practise for the quadrate H. is as muche iust as the other two I meane A. B. and D. C. THE XXI CONCLVSION when any two quadrates be set forth howe to make a squire about the one quadrate whiche shall be equall to the other quadrate Determine with your selfe about whiche quadrate you wil make the squire and drawe one side of that quadrate forth in lengte accordyng to the measure of the side of the other quadrate whiche line you maie call the grounde line and then haue you a right angle made on this line by an other side of the same quadrate Therfore turne that into a right cornered triangle accordyng to the worke in the laste conclusion by makyng of a byas line and that byas lyne will performe the worke of your desire For if you take the length of that byas line with your compasse and then set one foote of the compas in the farthest angle of the first quadrate whiche is the one ende of the groundline and extend the other foote on the same line accordyng to the measure of the byas line and of that line make a quadrate enclosyng the first quadrate then will there appere the forme of a squire about the first quadrate which squire is equall to the second quadrate Example The first square quadrate is A. B. C. D and the seconde is E. Now would I maked squire about the quadrate A. B. C. D whiche shall bee equall vnto the quadrate E. Therfore first I draw the line A. D more in length accordyng to the measure of the side of E as you see from D. vnto F and so the hole line of bothe these seuerall sides is A. F thē make I a byas line from C to F whiche byas line is the measure of this woorke wherefore I open my compas accordyng to the length of that byas line C. F and set the one compas foote in A and extend thother foote of the compas toward F makyng this pricke G from whiche I erect a plumbe line G. H and so make out the square quadrate A. G. H. K whose sides are equall eche of them to A. G. And this square doth contain the first quadrate A. B. C. D and also a squire G. H.K whiche is equall to the second quadrate E for as the last conclusion declareth the quadrate A. G.H. K is equall to bothe the other quadrates proposed that is A. B. C.D and E. Then muste the squire G. H.K needes be equall to E consideryng that all the rest of that great quadrate is nothyng els but the quadrate self A. B. C. D and so haue I thintent of this conclusion THE XXI CONCLVSION To find out the cētre of any circle assigned Draw a corde or stryng line crosse the circle then deuide into .ij. equall partes both that corde and also the bowe line or arche line that serueth to that corde and from the prickes of those diuisions if you drawe an other line crosse the circle it must nedes passe by the centre Therfore deuide that line in the middle and that middle pricke is the centre of the circle proposed Example Let the circle be A. B.C.D whose centre I shall seke First therfore I draw a corde crosse the circle that is A. C. Then do I deuide that corde in the middle in E and likewaies also do I deuide his arche line A. B.C in the middle in the pointe B. Afterward I drawe a line from B. to E and so crosse the circle whiche line is B. D in which line is the centre that I seeke for Therefore if I parte that line B. D in the middle in to two equall portions that middle pricke whiche here is F is the verye centre of the sayde circle that I seke This conclusion may other waies be wrought as the moste part of conclusions haue sondry formes of practise and that is by makinge thre prickes in the circūference of the circle at liberty where you wyll and then findinge the centre to those thre prickes Which worke bicause it serueth for sondry vses I thinke meet to make it a seuerall conclusion by it selfe THE XXIII CONCLVSION To find the commen centre belongyng to anye three prickes appointed if they be not in an exacte right line
for it is of lyke distance as is the line M.N. Nowe saie I that A. D beyng the diameter is the longest of all those lynes and also of any other that maie be drawen within that circle And the other line M. N is longer then F. G because it is nerer to the centre of the circle then F. G. Also the line F. G is shorter then the line B. C. for because it is farther from the centre then is the lyne B. C. And thus maie you iudge of al lines drawen in any circle how to know the proportion of their length by the proportion of their distance and contrary waies howe to discerne the proportion of their distance by their lengthes if you knowe the proportion of their length And to speake of it by the waie it is a maruaylouse thyng to consider that a man maie knowe an exacte proportion betwene two thynges and yet can not name nor attayne the precise quantitee of those two thynges As for exaunple If two squares be sette foorthe whereof the one containeth in it fiue square seete and the other contayneth fiue and fortie foote of like square feete I am not able to tell no nor yet anye manne liuyng what is the precyse measure of the sides of any of those .ij. squares and yet I can proue by vnfallible reason that their sides be in a triple proportion that is to saie that the side of the greater square whiche containeth .xlv. foote is three tymes so long iuste as the side of the lesser square that includeth but fiue foote But this seemeth to be spoken out of ceason in this place therfore I will omitte it now reseruyng the exacter declaration therof to a more conuenient place and time and will procede with the residew of the Theoremes appointed for this boke The .lxi. Theoreme If a right line be drawen at any end of a diameter in perpendicular forme and do make a right angle with the diameter that right line shall light without the circle and yet so iointly knitte to it that it is not possible to draw any other right line betwene that saide line and the circumfereÌ„ce of the circle And the angle that is made in the semicircle is greater then any sharpe angle that may be made of right lines but the other angle without is lesser then any that can be made of right lines Example In this circle A. B.C the diameter is A. C the perpendicular line which maketh a right angle with the diameter is E. A whiche line falleth without the circle and yet ioyneth so exactly vnto it that it is not possible to draw an other right line betwene the circumference of the circle and it whiche thyng is so plainly seene of the eye that it needeth no farther declaracion For euery man wil easily consent that betwene the croked line A. F whiche is a parte of the circumfereÌ„ce of the circle and A. E which is the said perpeÌ„dicular line there can none other line bee drawen in that place where they make the angle Nowe for the residue of the theoreme The angle D. A. B which is made in the semicircle is greater then anye sharpe angle that maye bee made of ryghte lines and yet is it a sharpe angle also in as much as it is lesser then a right angle which is the angle E. A.D and the residue of that right angle which lieth without the circle that is to saye E. A.B is lesser then any sharpe angle that can be made of right lines also For as it was before rehersed there canne no right line be drawen to the angle betwene the circumference and the right line E.A. Then must it needes folow that there can be made no lesser angle of righte lines And againe if ther canne be no lesser then the one then doth it sone appear that there canne be no greatter then the other for they twoo doo make the whole right angle so that if anye corner coulde bee made greater then the one parte then shoulde the residue bee lesser then the other parte so that other bothe partes muste be false or els bothe graunted to be true The lxij Theoreme If a right line doo touche a circle and an other right line drawen frome the centre of tge circle to the point where they touch that line whiche is drawenne frome the centre shall be a perpendicular line to the touch line Example The circle is A. B. C and his centre is F. The touche line is D. E and the point wher they touch is C. Now by reason that a right line is drawen frome the centre F. vnto C which is the point of the touche therefore saith the theoreme that the sayde line F. C muste needes bee a perpendicular line vnto the touche line D.E. The lxiij Theoreme If a righte line doo touche a circle and an other right line be drawen from the pointe of their touchinge so that it doo make righte corners with the touche line then shal the centre of the circle bee in that same line so drawen Example The circle is A. B. C and the centre of it is G. The touche line is D. C.E and the pointe where it toucheth is C. Nowe it appeareth manifest that if a righte line be drawen from the pointe where the touch line doth ioine with the circle and that the said lyne doo make righte corners with the touche line then muste it needes go by the centre of the circle and then consequently it must haue the sayde ceÌ„tre in him For if the saide line shoulde go beside the centre as F. C. doth then dothe it not make righte angles with the touche line which in the â—heoreme is supposed The lxiiij Theoreme If an angle be made on the centre of a circle and an other angle made on the circumference of the same circle and their grounde line be one common portion of the circumference then is the angle on the centre twise so great as the other angle on the circuÌ„fereÌ„ce Example The cirle is A. B. C. D and his centre is E the angle on the centre is C. E.D and the angle on the circumference is C. A. D t their commen ground line is C. F.D Now say I that the angle C. E. D whiche is one the centre is twise so greate as the angle C. A.D which is on the circumference The lxv Theoreme Those angles whiche be made in one cantle of a circle must needes be equal togither Example Before I declare this theoreme by an example it shall bee needefull to declare what is to be vnderstande by the wordes in this theoreme For the sentence canne not be knowen onles the uery meaning of the wordes be firste vnderstand Therefore when it speaketh of angâ—es made in one cantle of a circle it is this to be vnderstand that the angle muste touch the circumference and the lines that doo inclose that angle muste be drawen to the extremities of that line which maketh the cantle of the
circle as no circle can be made by cōpasse without it then is it called a centre And thereof doe masons and other worke menne call that patron A centre a centre whereby thei drawe the lines for iust hewyng of stones for arches vaultes and chimneies because the chefe vse of that patron is wrought by findyng that pricke or centre from whiche all the lynes are drawen as in the thirde booke it doeth appere Paralleles or gomowe lynes be suche lines as be drawen foorth still in one distaunce Parallelys Gemowe lynes and are no nerer in one place then in an other for and if they be nerer at one ende then at the other then are they no paralleles but maie bee called bought lynes and loe here exaumples of them bothe tortuouse paralleles parallelis parallelis circular Concentrikes bought lines A twine line And to returne to my matter an other fashioned line is there which is named a twine or twist line and it goeth as a wreyth about some other bodie A spirall line A worme line And an other sorte of lines is there that is called a spirall line or a worm line whiche representeth anapparant forme of many circles where there is not one in dede of these .ii. kindes of lines these be examples A twiste lyne A spirall lyne A touche lyne And when that a line doth crosse the edg of the circle A corde thē is it called a cord as you shall see anon in the speakynge of circles Matche corner Marche corner Where A. and B. are matche corners so are C. and D. but not A. and C. nother D. and A. Nowe will I beginne to speak of figures that be properly so called of whiche all be made of diuerse lines except onely a circle an egge forme and a tunne forme which .iij. haue no angle and haue but one line for their bounde and an eye fourme whiche is made of one lyne and hath an angle onely A circle is a figure made and enclosed with one line A circle and hath in the middell of it a pricke or centre from whiche all the lines that be drawen to the circumfernece are equall all in length as here you see A diameter And all the lines that bee drawen crosse the circle and goe by the centre are named diameters whose halfe I meane from the center to the circumference any waie Semidiameter is called the semidiameter or halfe diameter An arche An egge forme A tunne forme A tunne or barre form For if it be lyke the figure of a circle pressed in length and bothe endes lyke bygge then is it called a tunne forme or barrell forme the right makyng of whiche figures I wyll declare hereafter in the thirde booke An other forme there is whiche you maie call a nutte forme and is made of one lyne muche lyke an egge forme saue that it hath a sharpe angle And it chaunceth sometyme that there is a right line drawen crosse these figures and that is called an axelyne An axtre or axe lyne or axtre Howebeit properly that line that is called an axtre whiche goeth thoroughe the myddell of a Globe for as a diameter is in a circle so is an axe lyne or axtre in a Globe that lyne that goeth from side to syde and passeth by the middell of it And the two poyntes that suche a lyne maketh in the vtter bounde or platte of the globe are named polis which you may call aptly in englysh tourne pointes of whiche I do more largely intreate in the booke that I haue written of the vse of the globe An other hath two compassed lines and one right lyne and is as the portion of halfe a globe example of B. And thus I make an eand to speake of platte formes and will briefelye saye somwhat touching the figures of bode is which partly haue one platte forme fortheir bound and that iust roūd as a globe hath or ended long as in an egge and a tunne The globe as is before fourme whose pictures are these Howe be it you must marke that I meane not the very figure of a tunne when I saye tunne form but a figure like a tunne for a tune fourme hath but one plat forme and therfore must needs be round at the endes where as a tunne hath thre platte formes and is flatte at eche end as partly these pictures do shewe But as these formes be harde to be iudged by their pycturs so I doe entende to passe them ouer with a great number of other formes of bodyes which afterwarde shall be set forth in the boke of Perspectiue bicause that without perspectiue knowledge it is not easy to iudge truly the formes of them in flatte protacture And thus I make an ende for this tyme of the definitions Geometricall appertayning to this parte of practise and the rest wil I prosecute as cause shall serue THE PRACTIKE WORKINGE OF sondry conclusions Geometrical THE FYRST CONCLVSION To make a threlike triangle or any lyne measurable TAKE THE IVSTE lēgth of the lyne with your cōpasse and stay the one foot of the compas in one of the endes of that line turning the other vp or doun at your will drawyng the arche of a circle against the midle of the line and doo likewise with the same cōpasse vnaltered at the other end of the line and wher these ij croked lynes doth crosse frome thence drawe a lyne to ech end of your first line and there shall appear a threlike triangle drawen on that line Example A.B. is the first line on which I wold make the threlike triangle therfore I open the compasse as wyde as that line is long and draw two arch lines that mete in C then from C. I draw ij other lines one to A another to B and than I haue my purpose THE II. CONCLVSION If you wil make a twileke or a nouelike triangle on ani certaine line Consider fyrst the length that yow will haue the other sides to containe and to that length open your compasse and then worke as you did in the threleke triangle remembryng this that in a nouelike triangle you must take ij lengthes besyde the fyrste lyne and draw an arche lyne with one of thē at the one ende and with the other at the other end the exāple is as in the other before THE III. CONCL. To diuide an angle of right lines into ij equal partes First open your compasseas largely as you can so that it do not excede the length of the shortest line that incloseth the angle Then set one foote of the compasse in the verye point of the angle and with the other fote draw a compassed arch frō the one lyne of the angle to the other that arch shall you deuide in halfe and thē draw a line frō the āgle to the middle of that arch and so the angle is diuided into ij equall partes Example Let the triāgle be A. B.C thē set I one foot of
line I deuide in the middell in the point D. Then drawe I a line from C. to D and it is perpendicular to the line A. B accordyng as my desire was THE VII CONCLVSION To make a plumbe lyne or any porcion of a circle and that on the vtter or inner bughte Mark first the prick where the plūbe line shal lyght and prick out on ech side of it .ij. other poinctes equally distant from that first pricke Then set the one foote of the cōpas in one of those side prickes and the other foote in the other side pricke and first moue one of the feete and drawe an arche line ouer the middell pricke then set the compas steddie with the one foote in the other side pricke and with the other foote drawe an other arche line that shall cut that first arche and from the very poincte of their meetyng drawe a right line vnto the firste pricke where you do minde that the plumbe line shall lyghte And so haue you performed thintent of this conclusion Example The arche of the circle on whiche I would erect a plumbe line is A. B. C. and B. is the pricke where I would haue the plumbe line to light Therfore I meate out two equall distaunces on eche side of that pricke B. and they are A. C. Then open I the compas as wide as A. C. and settyng one of the feete in A. with the other I drawe an arch line which goeth by G. Likewaies I set one foote of the compas steddily in C. and with the other I drawe an arche line goyng by G. also ▪ Now consideryng that G. is the pricke of their meetyng it shall be also the poinct from whiche I must drawe the plūbe line Then draw I aright line from G. to B. and so haue mine intent Now as A. B. C. hath a plumbe line erected on his vtter bought so may I erect a plumbe line on the inner bught of D. E. F doynge with it as I did with the other that is to saye fyrste settyng for the the pricke where the plumbe line shall light which is E and then markyng one other on eche syde as are D. and F. And then proceding as I dyd in the example before THE VIII CONCLVSYON How to deuide the arche of a circle into two equall partes without measuring the arche Deuide the corde of that line into ij equall portions and then from the middle prycke erecte a plumbe line and it shal parte that arche in the middle Example The arch to be diuided ys A. D.C the corde is A B. C this corde is diuided in the middle with B from which prick if I erecte a plum line as B. D thē will it diuide the arch in the middle that is to say in D. THE IX CONCLVSION To do the same thynge other wise And for shortenes of worke if you wyl make a plumbe line without much labour you may do it with your squyre so that it be iustly made for yf you applye the edge of the squyre to the line in which the prick is and foresee the very corner of the squyre doo touche the pricke And than frome that corner if you drawe a lyne by the other edge of the squyre yt will be a perpendicular to the former line Example A.B. is the line on which I wold make the plumme line or perpendicular And therefore I marke the prick from which the plumbe lyne muste rise which here is C. Then do I sette one edg of my squyre that is B.C. to the line A. B so that the corner of the squyre do touche C. iustly And from C. I drawe a line by the other edge of the squire which is C. D. And so haue I made the plumme line D. C which I sought for THE X. CONCLVSION How to do the same thinge an other way yet If so be it that you haue an arche of suche greatnes that your squyre wyll not suffice therto as the arche of a brydge or of a house or window then may you do this Mete vnderneth the arch where the midle of his cord wyl be and ther set a mark Then take a long line with a plummet and holde the line in suche a place of the arch that the plummet do hang iustely ouer the middle of the corde that you didde diuide before and then the line doth shewe you the middle of the arche Example The arch is A. D.B of which I trye the midle thus I draw a corde from one syde to the other as here is A. B which I diuide in the middle in C. Thē take I a line with a plummet that is D. E and so hold I the line that the plummet E dooth hange ouer C And then I say that D. is the middle of the arche And to thenien◠that my plummet shall point the more iustely I doo make it sharpe at the nether ende and so may I trust this woorke for certaine THE XI CONCLVSION when any line is appointed and without it a pricke whereby a parallel must be drawen howe you shall doo it Take the iuste measure beetwene the line and the pricke accordinge to which you shal open your compasse Thē pitch one foote of your compasse at the one ende of the line and with the other foote draw a bowe line right ouer the pytche of the compasse lykewise doo at the other ende of the lyne then draw a line that shall touche the vttermoste edge of bothe those bowe lines and it will bee a true parallele to the fyrste lyne appointed Example A. B is the line vnto which I must draw an other gemow line which muste passe by the prick C first I meate with my compasse the smallest distance that is from C. to the line and that is C. F wherfore staying the compasse at that distaunce I sette the one foote in A and with the other foot I make a bowe lyne which is D thē like wise set I the one foote of the compasse in B and with the other I make the second bow line which is E. And then draw I a line so that it toucheth the vttermost edge of bothe these bowe lines and that lyne passeth by the pricke C end is a gemowe line to A. B as my sekyng was THE XII CONCLVSION To make a triangle of any .iij. lines so that the lines be suche that any .ij. of them be longer then the thirde For this rule is generall that any two sides of euerie triangle taken together are longer then the other side that remaineth If you do remember the first and seconde conclusions then is there no difficultie in this for it is in maner the same woorke First cōsiuer the .iij. lines that you must take and set one of thē for the ground line then worke with the other .ij. lines as you did in the first and second conclusions Example I haue .iij. lynes A. B. and C. D. and E. F. of whiche I put C.D. for my ground line then
with my compas I take the length of A. B. and set the one foote of my compas in C and draw an arch line with the other foote Likewaies I take the leÌ„gth of E. F and set one foote in D and with the other foote I make an arch line crosse the other arche and the pricke of their metyng whiche is G. shall be the thirde corner of the triangle for in all suche kyndes of woorkynge to make a tryangle if you haue one line drawen there remayneth nothyng els but to fynde where the pitche of the thirde corner shall bee for two of them must needes be at the two candes of the lyne that is drawen THE XIII CONCLVSION If you haue a line appointed and a pointe in it limited howe you maye make on it a righte lined angle equall to an other right lined angle all ready assigned Fyrste draw a line against the corner assigned and so is it a triangle then take heede to the line and the pointe in it assigned and consider if that line from the pricke to this end bee as long as any of the sides that make the triangle assigned and if it bee longe inoughe then prick out there the length of one of the lines and then woorke with the other two lines accordinge to the laste conclusion makynge a triangle of thre like lynes to that assigned triangle If it bee not longe inoughe thenne lengthen it fyrste and afterwarde doo as I haue sayde beefore Example Lette the angle appoynted bee A. B. C and the corner assigned B. Farthermore let the lymited line bee D. G and the pricke assigned D. Fyrste therefore by drawinge the line A. C I make the triangle A.B.C. Then consideringe that D. G is longer thanne A. B you shall cut out a line froÌ„ D. toward G equâ—l to A. B as for exaÌ„ple D F. TheÌ„ measure oute the other ij lines and worke with theÌ„ according as the conclusion with the fyrste also and the second teacheth yow and then haue you done THE XIIII CONCLVSION To make a square quadrate of any righte lyne appoincted First make a plumbe line vnto your line appointed whiche shall light at one of the endes of it accordyng to the fifth conclusion and let it be of like length as your first line is then opeÌ„ your compasse to the iuste length of one of them and sette one foote of the compasse in the ende of the one line and with the other foote draw an arche line there as you thinke that the fowerth corner shall be after that set the one foote of the same compasse vnsturred in the cande of the other line and drawe an other arche line crosse the first arche line and the poincte that they do crosse in is the pricke of the fourth corner of the square quadrate which you seke for therfore draw a line from that pricke to the eande of eche line and you shall therby haue made a square quadrate Example A. B. is the line proposed of whiche I shall make a square quadrate therefore firste I make a pluÌ„be line vnto it whiche shall lighte in A and that pluÌ„b line is A. B then open I my compasse as wide as the length of A. B or B. C for they must be bothe equall and I set the one foote of th end in C and with the other I make an arche line nigh vnto D afterward I set the compas again with one foote in B A and with the other foote I make an arche line crosse the first arche line in D and from the prick of their crossyng I draw .ij. lines one to B A and an other to C and so haue I made the square quadrate that I entended THE XV. CONCLVSION To make a likeiaÌ„me equall to a triangle appointed and that in a right lined aÌ„gle limited First from one of the angles of the triangle you shall drawe a gemowe line whiche shall be a parallele to that syde of the triangle on whiche you will make that likeiamme Then on one end of the side of the triangle whiche lieth against the gemowe lyne you shall draw forth a line vnto the gemow line so that one angle that commeth of those .ij. lines be like to the angle whiche is limited vnto you Then shall you deuide into ij equall partes that side of the triangle whiche beareth that line and from the pricke of that deuision you shall raise an other line parallele to that former line and continewe it vnto the first gemowe line and theÌ„ of those .ij. last gemowe lynes and the first gemowe line with the halfe side of the triangle is made a lykeiamme equall to the triangle appointed and hath an angle lyke to an angle limited accordyng to the conclusion Example B. C. G is the triangle appoincted vnto whiche I muste make an equall likeiamme And D is the angle that the likeiamme must haue Therfore first entendyng to erecte the likeiaÌ„me on the one side that the ground line of the triangle whiche is B. G. I do draw a gemow line by C and make it parallele to the ground line B. G and that new gemow line is A. H. Then do I raise a line from B. vnto the gemowe line whiche line is A. B and make an angle equall to D that is the appointed angle accordyng as the .viij. coÌ„clusion teacheth and that angle is B. A. E. Then to procede I doo parte in the middle the said grouÌ„d line B. G in the prick F froÌ„ which prick I draw to the first gemowe line A. H. an other line that is parallale to A. B and that line is E. F. Now saie I that the likeiaÌ„me B. A. E. F is equall to the triangle B. C. G. And also that it hath one angle that is B. A. E. like to D. the angle that was limitted And so haue I mine intent The prose of the equalnes of those two figures doeth depend of the .xli. proposition of Euclides first boke and is the .xxxi. proposition of this second boke of Theoremis whiche saieth that whan a tryangle and a likeiamme be made betwene .ij. selfe same gemow lines and haue their ground line of one length then is the likeiamme double to the triangle wherof it foloweth that if .ij. suche figures so drawen differ in their ground line onely so that the ground line of the likeiamme be but halfe the ground line of the triangle then be those .ij. figures equall as you shall more at large perceiue by the boke of Theoremis in the .xxxi. theoreme THE XVI CONCLVSION To make a likeiamme equall to a triangle appoincted accordyng to an angle limitted and on a line also assigned In the last conclusion the sides of your likeiamme wer left to your libertie though you had an angle appoincted Nowe in this conclusion you are somwhat more restrained of libertie sith the line is limitted which must be the side of the likeiaÌ„me Therfore thus shall you procede Firste accordyng to the laste conclusion make a likeiamme
in that one point F and those iij. angles be equal to the iij. angles of the triangle assigned whiche thinge doth plainely appeare in so muche as they bee equall to ij right angles as you may gesse by the sixt theoreme And the thre angles of euerye triangle are equall also to ij righte angles as the two and twenty theoreme dothe show so that bicause they be equall to one thirde thinge they must needes be equal togither as the cōmon sentence saith Thē do I draw a line frome G. to H and that line maketh a triangle F.G.H. whose angles be equall to the angles of the triangle appointed And this triangle is drawen in a circle as the conclusion didde wyll The proofe of this conclusion doth appeare in the seuenty and iiij Theoreme THE XXX CONCLVSION To make a triangle about a circle assigned whiche shall haue corners equall to the corners of any triangle appointed First draw forth in length the one side of the triangle assigned so that therby you may haue ij vtter angles vnto which two vtter angles you shall make ij other equall on the centre of the circle proposed drawing thre halfe diameters frome the circumference whiche shal enclose those ij angles thē draw iij. touche lines which shall make ij right angles eche of them with one of those semidiameters Those iij. lines will make a triangle equally cornered to the triangle assigned and that triangle is drawē about a circle apointed as the cōclusiō did wil. Example A. B.C is the triangle assigned and G. H.K is the circle appointed about which I muste make a triangle hauing equall angles to the angles of that triangle A.B.C. Fyrst therefore I draw A.C. which is one of the sides of the triangle in length that there may appeare two vtter angles in that triangle as you se B. A. D and B. C E. Then drawe I in the circle appointed a semidiameter whiche is here H. F for F. is the cētre of the circle G. H.K. Then make I on that centre an angle equall to the vtter angle B. A. D and that angle is H.F. K. Likewaies on the same cētre by drawyng an other semidiameter I make an other angle H. F. G equall to the second vtter angle of the triangle whiche is B. C. E. And thus haue I made .iij. semidiameters in the circle appointed Then at the ende of eche semidiameter I draw a touche line whiche shall make righte angles with the semidiameter And those .iij. touch lines mete as you see and make the triangle L. M. N whiche is the triangle that I should make for it is drawen about a circle assigned and hath corners equall to the corners of the triangle appointed for the corner M. is equall to C. Likewaies L. to A and N. to B whiche thyng you shall better perceiue by the vi Theoreme as I will declare in the booke of proofes THE XXXI CONCLVSION To make a portion of a circle on any right line assigned whiche shall conteine an angle equall to a right lined angle appointed The angle appointed maie be a sharpe angle a right angle other a blunte angle so that the worke must be diuersely handeled according to the diuersities of the angles but consideringe the hardenes of those seuerall woorkes I wyll omitte them for a more meter time and at this tyme wyll she we you one light waye which serueth for all kindes of angles and that is this When the line is proposed and the angle assigned you shall ioyne that line proposed so to the other twoo lines contayninge the angle assigned that you shall make a triangle of theym for the easy dooinge whereof you may enlarge or shorten as you see cause nye of the two lynes contayninge the angle appointed And when you haue made a triangle of those iij. lines then accordinge to the doctrine of the seuē and twety coclusiō make acircle about that triangle And so haue you wroughte the request of this conclusion whyche yet you maye woorke by the twenty and eight conclusion also so that of your line appointed you make one side of the triāgle be equal to the āgleassigned as youre selfe mai easily gesse Example First for example of a sharpe āgle let A. stād B.C. shal be that lyne assigned Thē do I make a triangle by adding B. C as a thirde side to those other ij which doo include the āgle assigned and that triāgle is D E. F so that E. F. is the line appointed and D. is the angle assigned Then doe I drawe a portion of a circle about that triangle from the one ende of that line assigned vnto the other that is to saie from E. a long by D. vnto F whiche portion is euermore greatter then the halfe of the circle by reason that the angle is a sharpe angle But if the angle be right as in the second exaumple you see it then shall the portion of the circle that containeth that angle euer more be the iuste halfe of a circle And when the angle is a blunte angle as the thirde exaumpse dooeth propounde then shall the portion of the circle euermore be lesse then the halfe circle So in the seconde example G. is the right angle assigned and H. K. is the lyne appointed and L.M.N. the portion of the circle aunsweryng thereto In the third exaumple O. is the blunte corner assigned P. Q. is the line and R. S. T. is the portion of the circle that containeth that blūt corner and is drawen on R. T. the lyne appointed THE XXXII CONCLVSION To cutte of from any circle appoineed a portion containyng an angle equall to a right lyned angle assigned When the angle and the circle are assigned first draw a touch line vnto that circle and then drawe an other line from the pricke of the touchyng to one side of the circle so that thereby those two lynes do make an angle equall to the angle assigned Then saie I that the portion of the circle of the contrarie side to the angle drawen is the parte that you scke for Example A. is the angle appointed and D. E. F. is the circle assigned frō which I must cut away a portiō that doth contain an angle equall to this angle A. Therfore first I do draw a touche line to the circle assigned and that touch line is B. C the very pricke of the touche is D from whiche D. J. drawe a lyne D. E so that the angle made of those two lines be equall to the angle appointed Then say I that the arch of the circle D. F. E is the arche that I seke after For if I doo deuide that arche in the middle as here it is done in F. and so draw thence two lines one to A and the other to E then will the angle F be equall to the angle assigned THE XXXIII CONCLVSION To make a square quadrate in a circle assigned Draw .ij. diameters in the circle so that they runne a crosse and that they make .iiij.
right angles Then drawe .iiij. lines that may ioyne the .iiij. endes of those diameters one to an other and then haue you made a square quadrate in the circle appointed Example A. B. C. D. is the circle assigned and A. C. and B. D. are the two diameters whiche crosse in the centre E and make .iiij. right corners Then do I make fowre other lines that is A. B B. C C. D and D. A which do ioyne together the fowre endes of the ij diameters And so is the square quadrate made in the circle assigned as the conclusion willeth THE XXXIIII CONCLVSION To make a square quadrate aboute annye circle assigned Drawe two diameters in crosse waies so that they make foure righte angles in the centre Then with your compasse take the length of the halfe diameter and set one foote of the compas in eche end of those diameters drawing twoo arche lines at euery pitchinge of the compas so shall you haue viij arche lines Then yf you marke the prickes wherin those arch lines do crosse and draw betwene those iiij prickes iiij right lines then haue you made the square quadrate accordinge to the request of the conclusion Example A.B.C. is the circle assigned in which first I draw two diameters in crosse waies making iiij righte angles and those ij diameters are A.C. and B.D. Then sette I my compasse whiche is opened according to the semidiameter of the said circle fixing one foote in the end of euery semidiameter and drawe with the other foote twoo arche lines one on euery side As firste when I sette the one foote in A then with the other foote I doo make twoo arche lines one in E and an other in F. Then sette I the one foote of the compasse in B and drawe twoo arche lines F. and G. Like wise settinge the compasse foote in C I drawe twoo other arche lines G. and H and on D. I make twoo other H. and E. Then frome the crossinges of those eighte arche lines I drawe iiij straighte lynes that is to saye E. F and F. G. also G. H and H. E whiche iiij straight lines do make the square quadrate that I should draw about the circle assigned THE XXXV CONCLVSION To drawe a circle in any square quadrate appointed Fyrste deuide euery side of the quadrate into twoo equall partes and so drawe two lynes betwene eche two contrary poinctes and where those twoo lines doo crosse there is the centre of the circle Then sette the one foote of the compasse in that point and stretch forth the other foot according to the length to halfe one of those lines and so make a compas in the square quadrate assigned Example A B.C.D. is the quadrate appointed in whiche I muste make a circle Therfore first I do deuide euery side in ij equal partes and draw ij lines acrosse betwene eche ij cōtrary prickes as you se E. G and F. H whiche mete in K and therfore shal K be the centre of the circle Then do I set one foote of the compas in K. and opē the other as wide as K. E and so draw a circle whiche is made ancordinge to the conclusion THE XXXVI CONCLVSION To draw a circle about a square quadrate Draw ij lines betwene the iiij corners of the quadrate and where they mete in crosse ther is the centre of the circle that you seeke for Thē set one foot of the compas in that centre and extend the other foote vnto one corner of the quadrate and so may you draw a circle which shall iustely inclose the quadrate proposed Example A. B. C. D. is the square quadrate proposed about which I must make a circle Therefore do I draw ij lines crosse the square quadrate from angle to angle as you se A. C. B. D. And where they ij do crosse that is to say in E. there set I the one foote of the compas as in the centre and the other foote I do extend vnto one angle of the quadrate as for exāpse to A and so make a compas whiche doth iustly inclose the quadrate according to the minde of the conclusion THE XXXVII CONCLVSION To make a twileke triangle whiche shall haue euery of the ij angles that lye about the ground line double to the other corner Fyrste make a circle and deuide the circumference of it into fyue equall partes And thenne drawe frome one pricke which you will two lines to ij other prickes that is to say to the iij. and iiij pricke counting that for the first wherhence you drewe both those lines Then drawe the thyrde lyne to make a triangle with those other twoo and you haue doone according to the conclusion and haue made a twelike triāgle whose ij corners about the grounde line are eche of theym double to the other corner Example A B. C. is the circle whiche I haue deuided into fiue equal portions And from one of the prickes which is A I haue drawē ij lines A. B. and A. C whiche are drawen to the third and iiij prickes Then draw I the third line C. B. which is the grounde line and maketh the triangle that I would haue for the āgle C. is double to the angle A and so is the angle B. also THE XXXVII CONCLVSION To make a cinkangle of equall sides and equall corners in any circle appointed Deuide the circle appointed into fiue equall partes as you didde in the laste conclusion and drawe ij lines from euery pricke to the other ij that are nexte vnto it And so shall you make a cinkangle after the meanynge of the conclusion Example Yow se here this circle A. B. C. D. E. deuided into fiue equall portions And from eche pricke ij lines drawen to the other ij nexte prickes so from A. are drawen ij lines one to B and the other to E and so from C. one to B. and an other to D and likewise of the reste So that you haue not only learned hereby how to make a sinkangle in anye circle but also how you shal make a like figure spedely whanne and where you will onlye drawinge the circle for the intente readylye to make the other figure I meane the cinkangle thereby THE XXXIX CONCLVSION How to make a cinkangle of equall sides and equall angles about any circle appointed Deuide firste the circle as you did in the laste conclusion into fiue equall portions and draw fiue semidiameters in the circle Then make fiue touche lines in suche sorte that euery touche line make two right angles with one of the semidiameters And those fiue touche lines will make a cinkangle of equall sides and equall angles Example A. B. C. D. E. is the circle appointed which is deuided into fiue equal partes And vnto euery prycke is drawē a semidiameter as you see Then doo I make a touche line in the pricke B whiche is F. G makinge ij right angles with the semidiameter B and lyke waies on C. is made G. H on D. standeth H. K
and on E is set K. L so that of those .v. touche lynes are made the .v. sides of a cinkeangle accordyng to the conclusion An other waie An other waie also maie you drawe a cinkeangle aboute a circle drawyng first a cinkeangle in the circle whiche is an easie thyng to doe by the doctrine of the .xxxvij. conclusion and then drawyng .v. touche lines whiche shall be iuste paralleles to the .v. sides of the cinkeangle in the circle forseeyng that one of them do not crosse ouerthwarte an other and then haue you done The exaumple of this because it is easie I leaue to your owne exercise THE XL. CONCLVSION To make a circle in any appointed cinkeangle of equall sides and equall corners Drawe a plumbe line from any one corner of the cinkeangle vnto the middle of the side that lieth iuste against that angle And do likewaies in drawyng an other line from some other corner to the middle of the side that lieth against that corner also And those two lines wyll meete in crosse in the pricke of their crossyng shall you iudge the centre of the circle to be Therfore set one foote of the compas in that pricke and extend the other to the ende of the line that toucheth the middle of one side whiche you liste and so drawe a circle And it shall be iustly made in the cinkeangle accordyng to the conclusion Example The cinkeangle assigned is A. B. C. D. E in whiche I muste make a circle wherfore I draw a right line from the one angle as from B to the middle of the contrary side whiche is E. D and that middle pricke is F. Then lykewaies from an other corner as from E I drawe a right line to the middle of the side that lieth against it whiche is B. C. and that pricke is G. Nowe because that these two lines do crosse in H I saie that H. is the centre of the circle whiche I would make Therfore I set one foote of the compasse in H and extend the other foote vnto G or F. whiche are the endes of the lynes that lighte in the middle of the side of that cinkeangle and so make I a circle in the cinkangle right as the coÌ„clusion meaneth THE XLI CONCLVSION To make a circle about any assigned cinkeangle of equall sides and equall corners Drawe .ij. lines within the cinkeangle from .ij. corners to the middle on the .ij. contrary sides as the last conclusion teacheth and the pointe of their crossyng shall be the centre of the circle that I seke for Then sette I one foote of the compas in that centre and the other foote I extend to one of the angles of the cinkangle and so draw I a circle about the cinkangle assigned Example An other waye also An other waye maye I do it thus presupposing any three corners of the cinkangle to be three prickes appointed vnto whiche I shoulde finde the centre and then drawinge a circle touchynge them all thre accordinge to the doctrine of the seuentene one and twenty and two and twenty conclusions And when I haue founde the centre then doo I drawe the circle as the samc conclusions do teache and this forty conclusion also THE XLII CONCLVSION To make a siseangle of equall sides and equall angles in any circle assigned Yf the centre of the circle be not knowen then seeke oute the centre according to the doctrine of the sixetenth conclusion And with your compas take the quantitee of the semidiameter iustly And then sette one foote in one pricke of the circuÌ„ference of the circle and with the other make a marke in the circumference also towarde both sides Then sette one foote of the compas stedily in eche of those newe prickes and point out two other prickes And if you haue done well you shal peceaue that there will be but euen sixe such diuisions in the circumference Whereby it dothe well appeare that the side of anye siseangle made in a circle is equalle to the semidiameter of the same circle Example The circle is B. C. D. E. F. G whose centre I finde to bee A. Therefore I sette one foote of the compas in A and do exteÌ„d the other foote to B thereby takinge the semidiameter Then sette I one foote of the compas vnremoued in B and marke with the other foote on eche side C. and G. Then from C. I marke D and froÌ„ D E from E. marke I F. And then haue I but one space iuste vnto G. and so haue I made a iuste siseangle of equall sides and equall angles in a circle appointed THE XLIII CONCLVSION To make a siseangle of equall sides and equall angles about any circle assigned THE XLIIII CONCLVSION To make a circle in any siseangle appointed of equall sides and equal angles THE XLV CONCLVSION To make a circle about any siseangle limited of equall sides and equall angles Bicause you maye easily coniecture the makinge of these figures by that that is saide before of cinkangles only consideringe that there is a difference in the numbre of the sides I thought beste to leue these vnto your owne deuice that you should study in some thinges to exercise your witte withall and that you mighte haue the better occasion to perceaue what difference there is betwene eche twoo of those conclusions For thoughe it seeme one thing to make a siseangle in a circ e and to make a circle about a siseangle yet shall you perceaue that it is not one thinge nother are those twoo conclusions wrought one way Like waise shall you thinke of those other two conclusions To make a siseangle about a circle and to make a circle in a siseangle thoughe the figures be one in fashion when they are made yet are they not one in working as you may well perceaue by the xxx vij xxx viij xxxix and xl conclusions in whiche the same workes are taught touching a circle and a cinkangle yet this muche wyll I saye for your helpe in workyng that when you shall seeke the centre in a sise angle whether it be to make a cirâ—le in it other about it you shall drawe the two crosse lines from one angle to the other angle that lieth againste it and not to the middle of any side as you did in the cinkangle THE XLVI CONCLVSION To make a figure of fifteene equall sides and angles in any circle appointed This rule is generall that how many sides the figure shall haue that shall be drawen in any circle into so many partes iustely muste the circles bee deuided And therefore it is the more easier woorke commonly to drawe a figure in a circle then to make a circle in an other figure Now therfore to end this conclusion deuide the circle firste into fiue partes and then eche of them into thre partes againe Or els first deuide it into thre partes and then ech of theÌ„ into fiue other partes as you list and canne most readilye Then draw lines betwene euery two
sensible principles And this is the cause why all learned menne dooth approue the certenty of geometry and cōsequently of the other artes mathematical which haue the grounds as Arithmetike musike and astronomy aboue all other artes and sciences that be vsed amōgest men Thus muche haue I sayd of the first principles and now will I go on with the theoremes whiche I do only by examples declae minding to reserue the proofes to a peculiar boke which I will then set forth when I perceaue this to be thankfully taken of the readers of it The theoremes of Geometry brieflye declared by shorte examples The firste Theoreme When .ij. triangles be so drawen that the one of thē hath ij sides equal to ij sides of the other triangle and that the angles enclosed with those sides bee equal also in bothe triangles then is the thirde side likewise equall in them And the whole triangles be of one greatnes and euery angle in the one equall to his matche angle in the other I meane those angles that be inclosed with like sides Example This triangle A.B.C. hath ij sides that is to say C.A. and C. B equal to ij sides of the other triangle F. G.H for A. C. is equall to F. G and B.C. is equall to G.H. And also the angle C. contayned beetweene F. G and G. H for both of them answere to the eight parte of a circle Therfore doth it remayne that A. B. whiche is the thirde lyne in the firste triangle doth agre in lengthe with F. H which is the third line in that secōd triāgle the hole triāgle A.B.C. must nedes be equal to the hole triangle F.G.H. And euery corner equall to his match that is to say A. equall to F B. to H and C. to G for those bee called match corners which are inclosed with like sides other els do lye against like sides The second Theoreme In twileke triangles the ij corners that be about the groud line are equal togither And if the sides that be equal be drawē out in lēgth thē wil the corners that are vnder the ground line be equal also togither Example A.B.C. is a twileke triangle for the one side A. C is equal to the other side B.C. And therfore I saye that the inner corners A. and B which are about the ground lines that is A.B. be equall to gither And farther if C. A. and C. B. bee drawen forthe vnto D and E. as you se that I haue drawen them then saye I that the two vtter angles vnder A. and B are equal also togither as the theorem said The profe wherof as of al the rest shal apeare in Euclide whome I intende to set foorth in english with sondry new additions if I may perceaue that it wil be thankfully taken The thirde Theoreme If in annye triangle there bee twoo angles equall togither then shall the sides that lie against those angles be equal also Example This triangle A.B.C. hath two corners equal eche to other that is A. and B as I do by supposition limite wherfore it foloweth that the side A. C is equal to that other side B. C for the side A. C lieth againste the angle B and the side B. C lieth against the angle A. The fourth Theoreme when two lines are drawen frō the endes of anie one line and meet in anie pointe it is not possible to draw two other lines of like lengthe ech to his match that shal begī at the same pointes and end in anie other pointe then the twoo first did Example The first line is A. B on which I haue erected two other lines A. C and B. C that meete in the pricke C wherefore I say it is not possible to draw ij other lines from A. and B. which shal mete in one point as you se A. D. and B.D. mete in D. but that the match lines shal be vnequa◠I mean by match lines the two lines on one side that is the ij on the right hand or the ij on the lefte hand for as youse in this example A. D. is longer thē A. C and B.C. is longer then B.D. And it is not possible that A. C. and A. D. shall bee of one lengthe if B. D. and B.C. bee like longe For if one couple of matche lines be equall as the same example A.E. is equall to A.C. in length then must B. E needes be vnequall to B.C. as you see it is here shorter The fifte Theoreme If two triāgles haue there ij sides equal one to another and their groūd lines equal also then shall their corners whiche are contained betwene like sides be equall one to the other Example Because these two triangles A.B. C and D.E.F. haue two sides equall one to an other For A. C. is equall to D. F and B.C. is equall to E. F and again their groūd lines A.B. and D.E. are lyke in length therfore is eche angle of the one triangle equall to ech angle of the other comparyng together those angles that are contained within lyke sides so is A. equall to D B. to E and C. to F for they are contayned within like sides as before is said The sixt Theoreme when any right line standeth on an other the ij angles that thei make other are both right angles or els equall to .ij. righte angles Example A.B. is a right line and on it there doth light another right line drawen from C. perpendicularly on it therefore saie J that the .ij. angles that thei do make are .ij. right angles as maie be iudged by the definition of a right angle But in the second part of the example where A.B. beyng still the right line on whiche D. standeth inslope wayes the two angles that be made of them are not righte angles but yet they are equall to two righte angles for so muche as the one is to greate more then a righte angle so muche iuste is the other to little so that bothe togither are equall to two right angles as you maye perceiue The seuenth Theoreme If .ij. lines be drawen to any one pricke in an other lyne and those .ij. lines do make with the fyrst lyne two right angles other suche as be equall to two right angles and that towarde one hande than those two lines doo make one streyght lyne Example A.B. is a streyght lyne on which there doth lyght two other lines one frome D and the other frome C but considerynge that they meete in one pricke E and that the angles on one hand be equal to two right corners as the laste theoreme dothe declare therfore maye D.E. and E.C. be counted for one ryght lyne The eight Theoreme when two lines do cut one an other crosse ways they do make their matche angles equall Example What matche angles are I haue tolde you in the definitions of the termes And here A and B. are matche corners in this example as are also C. and D so that the corner A is equall to B and
the angle C is equall to D. The nynth Theoreme whan so euer in any triangle the line of one side is drawen forthe in lengthe that vtter angle is greater than any of the two inner corners that ioyne not with it Example The triangle A. D. C hathe hys grounde lyne A. C. drawen forthe in lengthe vnto B so that the vtter corner that it maketh at C is greater then any of the two inner corners that lye againste it and ioyne not wyth it whyche are A. and D for they both are lesser then a ryght angle and be sharpe angles but C. is a blonte angle and therfore greater then a ryght angle The tenth Theoreme In euery triangle any .ij. corners how so euer you take theÌ„ ar lesse theÌ„ ij right corners Example In the firste triangle E whiche is a threlyke and therfore hath all his angles sharpe take anie twoo corners that you will and you shall perceiue that they be lesser then .ij. right corners for in euery triangle that hath all sharpe corners as you see it to be in this example euery corner is lesse then a right corner And therfore also euery two corners must nedes be lesse then two right corners Furthermore in that other triangle marked with M whiche hath .ij. sharpe corners and one right any .ij. of them also are lesse then two right angles For though you take the right corner for one yet the other whiche is a sharpe corner is lesse then a right corner And so it is true in all kindes of triangles as you maie perceiue more plainly by the .xxij. Theoreme The .xi. Theoreme In euery triangle the greattest side lieth against the greattest angle Example As in this triangle A. B. C the greattest angle is C. And A. B. whiche is the side that lieth against it is the greatest and longest side And contrary waies as A. C. is the shortest side so B. whiche is the angle liyng against it is the smallest and sharpest angle for this doth folow also that as the longest side lyeth against the greatest angle so it that foloweth The twelft Theoreme In euery triangle the greattest angle lieth against the longest side For these ij theoremes are one in truthe The thirtenth theoreme In euerie triangle anie ij sides togither how so euer you take them are longer theÌ„ the thirde For example you shal take this triangle A.B.C. which hath a veery blunt corner and therfore one of his sides greater a good deale then any of the other and yet thr ij lesser sides togither ar greate then it And if it bee so in a blunte angeled triangle it must nedes be true in all other for there is no other kinde of triangles that hathe the one side so greate aboue the other sids as thei that haue blunt corners The fourtenth theoreme If there be drawen from the endes of anie side of a triangle .ij. lines metinge within the triangle those two lines shall be lesse then the other twoo sides of the triangle but yet the corner that thei make shall bee greater then that corner of the triangle whiche standeth ouer it Example A.B.C. is a triangle on whose ground line A.B. there is drawen ij lines from the ij endes of it I say from A. and B and they meete within the triangle in the pointe D wherfore I say that as those two lynes A.D. and B. D are lesser then A.C. and B. C so the angle D. is greatter then the angle C which is the angle against it The fiftenth Theoreme If a triangle haue two sides equall to the two sides of an other triangle but yet the aÌ„gle that is contained betwene those sides greater then the like angle in the other triangle then is his grounde line greater then the grounde line of the other triangle Example A.B.C. is a triangle whose sides A.C. and B. C are equall to E.D. and D. F the two sides of the triangle D. E. F but bicause the angle in D is greatter then the angle C. whiche are the ij angles contayned betwene the equal lynes therfore muste the ground line E. F. nedes bee greatter thenne the grounde line A. B as you se plainely The xvi Theoreme If a triangle haue twoo sides equalle to the two sides of an other triangle but yet hathe a longer ground line theÌ„ that other triangle then is his angle that lieth betwene the equall sides greater theÌ„ the like corner in the other triangle Example This Theoreme is nothing els but the sentence of the last Theoreme turned backward and therfore nedeth none other profe nother declaration then the other example The seuententh Theoreme If two triangles be of such sort that two angles of the one be equal to ij angles of the other and that one side of the one be equal to on side of the other whether that side do adioyne to one of the equall corners or els lye againste one of them then shall the other twoo sides of those triangles bee equalle togither and the thirde corner also shall be equall in those two triangles Example Bicause that A. B. C the one triangle hath two corners A. and B equal to D. E that are twoo corners of the other triangle D. E. F. and that they haue one side in theym bothe equall that is A. B which is equaâ—l to D. E therefore shall both the other ij sides be equall one to an other as A C. and B. C. equall to D. F and E. F and also the thirde angle in them both shal be equall that is the angle C. shal be equall to t the angle F. The eightenth Theoreme when on .ij. right lines ther is drawen a third right line crosse waies and maketh .ij. matche corners of the one line equall to the like twoo matche corners of the other line then ar those two lines gemmow lines or paralleles Example The .ij. fyrst lynes are A. B. and C. D the thyrd lyne that crosseth them is E. F. And bycause that E. F. maketh ij matche angles with A. B equall to .ij. other lyke matche angles on C. D that is to say E. G equall to K. F and M. N. equall also to H L. therfore are those ij lynes A. B. and C. D. gemow lynes vnderstand here by lyke matche corners those that go one way as doth E. G and K. F lykeways N. M and H. L for as E. G. and H. L other N. M. and K. F. go not one waie so be not they lyke match corners The nyntenth Theoreme when on two right lines there is drawen a thirde right line crosse waies and maketh the ij ouer corners towarde one hande equall togither then ar those .ij. lines paralleles And in like maner if two inner corners toward one hande be equall to .ii. right angles Example As the Theoreme dothe speake of .ij. ouer angles so muste you vnderstande also of .ij. nether angles for the iudgement is lyke in bothe Take for an example the figure of the last theoreme where A. B
bee moste distaunte in sonder The Li. Theoreme If two circles be so drawen that one of them do touche the other then haue they not one centre Example There are two circles made as you see the one is A. B. C and hath his centre by G the other is B. D. E and his centre is by F so that it is easy enough to perceiue that their centres doe dyffer as muche a sonder as the halfe diameter of the greater circle is lōger then the half diameter of the lesser circle And so must it needes be thought and said of all other circles in lyke kinde The .lij. theoreme If a certaine pointe be assigned in the diameter of a circle distant from the centre of the said circle and from that pointe diuerse lynes drawen to the edge and circumference of the same circle the longest line is that whiche passeth by the centre and the shortest is the residew of the same line And of al the other lines that is euer the greatest that is nighest to the line which passeth by the centre And cōtrary waies that is shortest that is farthest from it And amongest thē all there can be but onely .ij. equall together and they must nedes be so placed that the shortest line shall be in the iust middle betwixte them Example The circle is A. B. C. D. E. H and his centre is F the diameter is A. E in whiche diameter I haue taken a certain point distaunt from the centre and that pointe is G from whiche I haue drawen .iiij. lines to the circumference beside the two partes of the diameter whiche maketh vp vi lynes in all Nowe for the diuersitee in quantitie of these lynes I saie accordyng to the Theoreme that the line whiche goeth by the centre is the longest line that is to saie A. G and the reside we of the same diameter beeyng G. E is the shortest lyne And of all the other that lyne is longest that is neerest vnto that parte of the diameter whiche gooeth by the centre and that is shortest that is farthest distant from it wherefore I saie that G. B is longèr then G. C and therfore muche more longer then G. D sith G. C also is longer then G. D and by this maie you soone perceiue that it is not possible to drawe .ij. lynes on any one side of the diameter whiche might be equall in lengthe together but on the one side of the diameter maie you easylie make one lyne equall to an other on the other side of the same diameter as you see in this example G. H to bee equall to G. B betweene whiche the lyne G. E as the shortest in all the circle doothe stande euen distaunte from eche of them and that is the precise knoweledge of their equalitee if they be equally distaunt from one halfe of the diameter Where as contrary waies if the one be neerer to any one halfe of the diameter then the other is it is not possible that they two may be equall in lengthe namely if they dooe ende bothe in the circumference of the circle and be bothe drawen from one poynte in the diameter so that the saide poynte be as the Theoreme doeth suppose somewhat distaunt from the centre of the said circle For if they be drawen from the centre then must they of necessitee be all equall howe many so euer they bee is the definition of a circle dooeth importe withoute any regarde how neere so euer they be to the diameter or how distante from it And here is to be noted that in this Theoreme by neerenesse and distaunce is vnderstand the nerenesse and distaunce of the extreeme partes of those lynes where they touche the circumference For at the other end they do all meete and touche The .liij. Theoreme If a pointe bee marked without a circle and from it diuerse lines drawen crosse the circle to the circumference on the other side so that one of them passe by the centre then that line whiche passeth by the centre shall be the longest of all them that crosse the circle And of thother lines those are longest that be nexte vnto it that passeth by the centre And those ar shortest that be farthest distant from it But among those partes of those lines whiche ende in the outewarde circumference that is most shortest whiche is parte of the line that passeth by the centre and amongeste the othere eche of thē the nerer they are vnto it the shorter they are and the farther from it the longer they be And amongest them all there can not be more then .ij. of any one lēgth ▪ and they two muste be on the two contrarie sides of the shortest line Example Take the circle to be A. B. C and the point assigned without it to be D. Now say J that if there be drawen sundrie lines from D and crosse the circle endyng in the circumference on the cōtrary side as here you see D. A D. E D. F and D. B then of all these lines the longest must needes be D. A which goeth by the centre of the circle and the nexte vnto it that is D. E is the longest amongest the rest And contrarie waies D. B is the shorteste because it is farthest distaunt from D.A. And so maie you iudge of D. F because it is nerer vnto D. A then is D. B therefore is it longer then D. B. And likewaies because it is farther of from D. A then is D. E therfore is it shorter then D.E. Now for those partes of the lines whiche bee withoute the circle as you see D. C is the shortest because it is the parte of that line which passeth by the centre And D. K is next to it in distance and therfore also in shortnes so D. G is farthest from it in distance and thērfore is the longest of them Now D. H beyng nerer then D. G is also shorter then it and beynge farther of then D. K is longer then it ▪ So that for this parte of the theoreme as J think you do plain ly perceaue the truthe thereof so the residue hathe no difficulte For seing that the nearer any line is to D. C which ioyneth with the diameter the shorter it is and the farther of from it the longer it is And seyng two lynes can not be of like distaunce beinge bothe on one side therefore if they shal be of one lengthe and consequently of one distaunce they must needes bee on contrary sides of the saide line D. C. And so appeareth the meaning of the whole Theoreme And of this Theoreme dothe there folo we an other lyke whiche you maye calle other a theoreme by it selfe or else a Corollary vnto this laste theoreme J passe not so muche for the name But his sentence is this when so euer any lynes be drawen frome any pointe withoute a circle whether they crosse the circle or cande in the utter edge of his circumference those two lines that bee equally distaunt from the least line
circle thē must it neds be other a blūt āgle or els a sharpe angle and in no wise a righte angle For if the cantle wherein the angle is made be greater then the halfe circle then is that angle a sharpe angle And generally the greater the cātle is the lesser is the angle comprised in that cantle and contrary waies the lesser any cantle is the greater is the angle that is made in it Wherfore it must nedes folowe that the angle made in a cantle lesse then a semicircle must nedes be greater then a right angle So the angle by B beyng made of the right line A. B and the righte line B. D is a iuste righte angle because it is made in a semicircle But the angle made by A which is made of the right line A. B and of the right line A. D is lesser then a righte angle and is named a sharpe angle for as muche as it is made in a cantle of a circle greater then a semicircle And contrary waies the angle by C beyng made of the righte line B. C and of the right line C. D is greater then a right angle and is named a blunte angle because it is made in a cantle of a circle lesser then a semicircle But now touchyng the other angles of the cantles J saie accordyng to the Theoreme that the .ij. angles of the greater cantle which are by B. and D as is before declared are greatter eche of them then a right angle And the angles of the lesser cantle whiche are by the same letters B and D but be on the other side of the corde are lesser eche of them then a right angle and be therfore sharpe corners The lxxiiij Theoreme If a right line do touche a circle and from the pointe where they touche a righte lyne be drawen crosse the circle and deuide it the angles that the saied lyne dooeth make with the touche line are equall to the angles whiche are made in the cantles of the same circle on the contrarie sides of the lyne aforesaid Example The circle is A. B.C.D and the touche line is E. F. The pointe of the touchyng is D from which point J suppose the line D. B to be drawen crosse the circle and to deuide it into .ij. cantles wherof the greater is B. A.D and the lesser is B. C.D and in ech of them an angle drawen for in the greater cantle the angle is by A and is made of the right lines B. A and A. D in the lesser cantle the angle is by C and is made of the right lines B. C and C.D. Now saith the Theoreme that the angle B. D. F is equall to the angle made in the cantle on the other side of the said line that is to saie in the cantle B. A.D so that the angle B. D.F is equall to the angle B. A.D because the angle B. D.F is on the one side of the line B. D whiche is accordyng to the supposition of the Theoreme drawen crosse the circle and the angle B. A.D is in the cātle on the other side Likewaies the angle B. D.E beyng on the one side of the line B. D must be equall to the angle B. C.D that is the āgle by C whiche is made in the cātle on the other side of the right line B.D. The profe of all these J do reserue as J haue often saide to a conuenient boke wherein they shall be all set at large The .lxxv. Theoreme In any circle when .ij. right lines do crosse one an other the likeiamme that is made of the portions of the one line shall be equall to the lykeiamme made of the partes of the other lyne Because this Theoreme doth serue to many vses and wold be wel vnderstande J haue set forth .ij. examples of it Jn the firste the lines by their crossyng do make their portions somewhat toward an equalitie Jn the second the portiōs of the lynes be very far frō an equalitie and yet in bothe these and in all other the Theoreme is true Jn the first exāple the circle is A. B.C.D in which th one line A. C doth crosse thother line B. D in the point E. Now if you do make one likeiāme or lōgsquare of D. E E. B being the .ij. portions of the line D. B that longsquare shall be equall to the other longsquare made of A. E and E. C beyng the portions of the other line A.C. Lykewaies in the second example the circle is F. G.H.K in whiche the line F. H doth crosse the other line G. K in the pointe L. Wherfore if you make a lykeiamme or longsquare of the two partes of the line F. H that is to saye of F. L and L. H that longsquare will be equall to an other longsquare made of the two partes of the line G. K. which partes are G. L and L.K. Those longsquares haue J set foorth vnder the circles containyng their sides that you maie somewhat whet your own wit in practisyng this Theoreme accordyng to the doctrine of the nineteenth conclusion The .lxxvi. Theoreme If a pointe be marked without a circle and from that pointe two right lines drawen to the circle so that the one of them doe runne crosse the circle and the other doe touche the circle onely the longe square that is made of that whole lyne whiche crosseth the circle and the portion of it that lyeth betwene the vtter circumference of the circle and the pointe shall be equall to the full square of the other lyne that onely toucheth the circle Example The circle is D. B.C and the pointe without the circle is A from whiche pointe there is drawen one line crosse the circle and that is A. D.C and an other lyne is drawn from the said pricke to the marge or edge of the circumference of the circle and doeth only touche it that is the line A.B. And of that first line A. D.C you maie perceiue one part of it whiche is A. D to lie without the circle betweene the vtter circumference of it and the pointe assigned whiche was A. No we concernyng the meanyng of the Theoreme if you make a longsquare of the whole line A. C and of that parte of it that lyeth betwene the circumference and the point whiche is A. D that longe square shall be equall to the full square of the touche line A. B accordyng not onely as this figure she weth but also the saied nyneteenth conclusion dooeth proue if you lyste to examyne the one by the other The .lxxvij. Theoreme If a pointe be assigned without a circle and from that pointe .ij. right lynes be drawen to the circle so that the one doe crosse the circle and the other dooe ende at the circumference and that the longsquare of the line which crosseth the circle made with the portiō of the same line beyng without the circle betweene the vtter circumference and the pointe assigned doe equally agree with the iuste square of that line that endeth at the circumference then is that lyne so endyng on the circumference a touche line vnto that circle Example Jn as muche as this Theoreme is nothyng els but the sentence of the last Theoreme before conuerted therfore it shall not be nedefull to vse any other example then the same for as in that other Theoreme because the one line is a touche lyne therfore it maketh a square iust equal with the longsquare made of that whole line whiche crosseth the circle and his portion liyng without the same circle So saith this Theoreme that if the iust square of the line that endeth on the circumference be equall to that longsquare whiche is made as for his longer sides of the whole line which commeth from the point assigned and crosseth the circle and for his other shorter sides is made of the portion of the same line liyng betwene the circumference of the circle and the pointe assigned then is that line whiche endeth on the circumference a right touche line that is to saie yf the full square of the right line A. B be equall to the longsquare made of the whole line A. C as one of his lines and of his portion A. D as his other line then must it nedes be that the lyne A. B is a right touche lyne vnto the circle D. B.C. And thus for this tyme J make an ende of the Theoremes FINIS IMPRINTED at London in Poules churcheyarde at the signe of the Brasen serpent by Reynold wolfe Cum priuilegio ad imprimendum solum ANNO DOMINI M.D.LI.