circle So that if any angle do not touch the circumference or if the lines that in close that angle doo not ende in the extremities of the corde line but ende other in some other part of the faid corde or in the circumference or that any one of them do so eande then is not that angle accompted to be drawen in the faid cantle of the circle And this promised nowe will I cumme to the meaninge of the theoreme I sette forthe a circle whiche is A. B. C. D and his centre E in this circle I drawe a line D. C whereby there ar made two cantels a more and a lesser The lesser is D. F. C and the geater is D.A.C.C. In this greater cantle I drawe two angles the firste is D. A.C and the second is D. B.C which two angles by reason they are made bothe in one cantle of a circle that is the cantle D.A.B. C therefore are they both equall 〈…〉 Now doth there appere an other triangle whose angle lighteth on the centre of the circle and that triangle is D. E. C whose angle is double to the other angles as is declared in the lxiiij Theoreme whiche maie stande well enough with this Theoreme for it is not made in this cantle of the circle as the other are by reason that his angle doth not light in the circumference of the circle but on the centre of it The .lxvi. theoreme Euerie figure of foure sides drawen in a circle hath his two contrarie angles equall vnto two right angles Example The circle is A. B. C. D and the figure of foure sides in it is made of the sides B. C and C. D and D. A and A.B. Now if you take any two angles that be contrary as the angle by A and the angle by C I saie that those .ij. be equall to .ij. right angles Also if you take the angle by B and the angle by D whiche two are also contrary those two angles are like waies equall to two right angles But if any man will take the angle by A with the angle by B or D they can not be accompted contrary no more is not the angle by C. estemed contrary to the angle by B or yet to the angle by D for they onely be accompted contrary angles whiche haue no one line common to them bothe Suche is the angle by A in respect of the angle by C for there both lynes be distinct where as the angle by A and the angle by D haue one common line A. D and therfore can not be accompted contrary angles So the angle by D and the angle by C haue D. C as a common line and therfore be not contrary angles And this maie you iudge of the residewe by like reason The lxvij Theoreme Vpon one right lyne there can not be made two cantles of circles like and vnequall and drawent towarde one parte Example Cantles of circles be then called like when the angles that are made in them be equall But now for the Theoreme let the right line be A. E.C on whiche I draw a cantle of a circle whiche is A.B.C. Now saieth the Theoreme that it is not possible to draw an other cantle of a circle whiche shall be vnequall vnto this first cantle that is to say other greatter or lesser then it and yet be lyke it also that is to say that the angle in the one shall be equall to the angle in the other For as in this example you see a lesser cantle drawen also that is A. D.C so if an angle were made in it that angle would be greatter then the angle made in the cantle A. B. C and therfore ban not they be called lyke cantess but and if any other cantle were made greatter then the first then would the angle it it be lesser then that in the firste and so nother a lesser nother a greater cantle can be made vpon one line with an other but it will be vnlike to it also The .lxviij. Theoreme Lyke cantelles of circles made on equall righte lynes are equall together Example What is mentby like cantles you haue heard before and it is easie to vnderstand that suche figures are called equall that be of one bygnesse so that the one is nother greater nother lesser then the other And in this kinde of comparison they must so agree that if the one be layed on the other they shall exactly agree in all their boundes so that nother shall excede other Nowe for the example of the Theoreme I haue set forthe diuers varieties of cantles of circles amongest which the first and seconde are made vpō equall lines and ar also both equall and like The third couple ar ioyned in one and be nother equall nother like but expressyng an absurde deformitee whiche would folowe if this Theoreme wer not true And so in the fourth couple you maie see that because they are not equall cantles therfore can not they be like cantles for necessarily it goeth together that all cantles of circles made vpon equall right lines if they be like they must be equall also The lxix Theoreme In equall circles suche angles as be equall are made vpon equall arch lines of the circumference whether the angle light on the circumference or on the centre Example Firste I haue sette for an exaumple twoo equall circles that is A. B. C. D whose centre is K and the second circle E. F. G. H and his centre L and in eche of thē is there made two angles one on the circumference and the other on the centre of eche circle and they be all made on two equall arche lines that is B.C.D. the one and F.G.H. the other Now saieth the Theoreme that if the angle B. A. D be equall to the angle F. E. H then are they made in equall circles and on equall arch lines of their circumference Also if the angle B. K.D be equall to the angle F. L.H then be they made on the centres of equall circles and on equall arche lines so that you muste compare those angles together whiche are made both on the centres or both on the circumference and maie not conferre those angles wherof one is drawen on the circumference and the other on the centre For euermore the angle on the centre in suche sorte shall be double to the angle on the circumference as is declared in the three score and foure Theoreme The .lxx. Theoreme In equall circles those angles whiche bee made on equall arche lynes are euer equall together whether they be made on the centre or on the circumference Example This Theoreme doth but conuert the sentence of the last Theoreme before and therfore is to be vnderstande by the same examples for as that saith that equall angles occupie equall archesynes so this saith that equal arche lines causeth equal angles consideringe all other circumstances as was taughte in the laste theoreme before so that this theoreme dooeth affirming speake of the equalitie of those
equall togither Example Example of this Theoreme you may see in the last figure where as sixe triangles made betwene those two gemowe lines A. B. and C. F the first triangle is A. C. D the seconde is A. D.E the thirde is A. D.B the fourth is A. B. E the fifte is D. E.B and the sixte is B. E.F of which fixe triangles A. D.E. and D. E. B. are equall bicause they haue one common grounde line And so likewise A.B.E. and A.B. D whose commen grounde line is A. B but A.C.D. is equal to B. E.F being both betwene one couple of parallels not bicause thei haue one ground line but bicause they haue their ground lines equall for C.D. is equall to E. F as you may declare thus C. D is equall to A.B. by the foure and twenty Theoreme for theiare two contrary sides of one lykeiamme A. C.D.B and E. F by the same theoreme is equall to A. B for thei ar the two the contrary sides of the likeiamme A. E.F.B wherfore C.D. must needes be equall to E.F. like wise the triangle A. C.D is equal to A. B.E bicause they ar made betwene one paire of parallels and haue their groundlines like I meane C. D. and A. B. Againe A. D.E is equal to eche of them both for his ground line D. E is equall to A. B in so muche as they are the contrary sides of one likeiamme that is the long square A. B. D. E. And thus may you proue the equalnes of all the reste The xxix Theoreme Alequal triangles that are made on one grounde line and rise one waye must needes be betwene one paire of parallels Example Take for example A. D.E and D. E.B which as the xxvij conclusion dooth proue are equall togither and as you see they haue on ground line D.E. And ag aine they rise towarde one side that is to say vp warde toward the line A. B wherfore they must needes be inclosed betweene one paire of parallels which are heere in this example A.B. and D.E. The thirty Theoreme Equal triangles that haue the irground lines equal and be drawē toward one side ar made betwene one paire of paralleles Example The example that declared the last theoreme maye well serue to the declaracion of this also For those ij theoremes do diffre but in this one pointe that the laste theoreme meaneth of triangles that haue one ground line common to them both and this theoreme dothe presuppose the grounde lines to bee diuers but yet of one length as A. C. D and B. E.F as they are ij equall triangles approued by the eighte and twentye Theorem so in the same Theorem it is declared that their groūd lines are equall togither that is C. D and E. F now this beeynge true and considering that they are made towarde one side it foloweth that they are made betwene one paire of parallels when I saye drawen towarde one side I meane that the triangles must be drawen other both vpward frome one parallel other els both downward for if the one be drawen vpward and the other downward then are they drawen betwene two paire of parallels presupposinge one to bee drawen by their ground line and then do they ryse toward contrary sides The xxxi theoreme If a likeiamme haue one ground line with a triangle and be drawen betwene one paire of paralleles then shall the likeiamme be double to the triangle Example A. H. and B.G. are .ij. gemow lines betwene which there is made a triangle B. C G and a lykeiamme A.B.G. C whiche haue a grounde lyne that is to saye B. G. Therfore doth it folow that the lykeiamme A.B.G.C. is double to the triangle B. C. G. For euery halfe of that lykeiamme is equall to the triangle I meane A.B.F.E. other F.E. C.G. as you may coniecture by the .xi. conclusion geometrical And as this Theoreme dothe speake of a triangle and likeiamme that haue one groundelyne so is it true also yf theyr groundelynes bee equall though they bee dyuers so that theibe made betwene one payre of paralleles And hereof may you perceaue the reason why in measuryng the platte of a triangle you must multiply the perpendicular lyne by halfe the grounde lyne or els the hole grounde lyne by halfe the perpendicular for by any of these bothe waies is there made a lykeiamme equall to halfe suche a one as shulde be made on the same hole grounde lyne with the triangle and betweene one payre of paralleles Therfore as that lykeiamme is double to the triangle so the halfe of it must needes be equall to the triangle Compare the .xv. conclusion with this theoreme The .xxxij. Theoreme In all likeiammes where there are more than one made aboute one bias line the fill squares of euery of them must nedes be equall Example Fyrst before I declare the examples it shal be mete to shew the true vnderstādyngof this heorem Therfore by the Bias line Bias lyne I meane that lyne whiche in any square figure dooth runne from corner to corner And euery square which is diuided by that bias line into equall halues from corner to corner that is to say into .ij. equall triangles those be counted to stande aboute one bias line and the other squares whiche touche that bias line with one of their corners onely those doo I call Fyll squares accordyng to the greke name Fyll squares 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 whiche is anapleromata and called in latin supplementa bycause that they make one generall square includyng and enclosyng the other diuers squares as in this exāple H. C. E. N. is one square likeiamme and L. M. G. C. is an other whiche bothe are made aboute one bias line that is N. M than K. L. H. C. and C. E. F. G. are .ij. syll squares for they doo syll vp the sydes of the .ij. fyrste square sykeiammes in suche sorte that of all them foure is made one greate generall square K. M.F.N. Nowe to the sentence of the theoreme I say that the .ij. fill squres H. K. L. C. and C. E. F. G. are both equall togither as it shall bee declared in the booke of proofes bicause they are the fill squres of two likeiammes made aboute one bias line as the exaumple sheweth Conferre the twelfthe conclusion with this the oreme The xxxiij Theoreme In all right anguled triangles the square of that side whiche lieth against the right angle is equall to the .ij. squares of both the other sides Example A.B.C. is a triangle hauing a ryght angle in B. Wherfore it soloweth that the square of A. C whiche is the side that lyeth agaynst the right angle shall be as muche as the two squares of A. B. and B. C. which are the other .ij. sides ¶ By the square of any syne you muste vnderstande a figure made iuste square hauyng all his iiij sydes equall to that line whereof it is the square so is A. C.F the square of A.C. Lykewais A. B. D.
are equal togither and contrary waies if they be equall togither they ar also equally distant from that least line For the declaracion of this proposition it shall not need to vse any other example then that which is brought for the explication of this laste theoreme by whiche you may without any teachinge casyly perceaue both the meanyng and also the truch of this proposition The L iiij Theoreme If a point be set forthe in a circle and frō that pointe vnto the circumference many lines drawen of which more then two are equal togither then is that point the centre of that circle Example The circle is A. B.C and with init J haue sette fourth for an example three prickes which are D.E. and F and from euery one of them J haue drawē at the leaste iiij lines vnto the circumference of the circle but frome D J haue drawen more yet maye it appear readily vnto your eye that of all the lines whiche be drawen from E. and F vnto the circumference there are but twoo equall and more can not bee for G.E. nor E.H. hath none other equal to theim nor canne not haue any beinge drawen from the fame point E. No more can L. F or F. K haue anye line equall to either of theim beinge drawen from the same pointe F. And yet from either of those two poinctes are there drawen twoo lines equall togither as A. E is equall to E. B and B. F is equall to F. C but there can no third line be drawen equall to either of these two couples and that is by reason that they be drawen from a pointe distaunte from the centre of the circle But from D althoughe there be seuen lines drawen to the circumference yet all bee equall bicause it is the centre of the circle And therefore if you drawe neuer so mannye more from it vnto the circumference all shall be equal so that this is the priuilege as it were of the centre and therfore no other point can haue aboue two equal lines drawen from it vnto the circumference And from all poittes you maye drawe ij equall lines to the circumference of the cirle whether that pointe be within the circle or without it The l v. Theoreme No circle canne cut an other circle in more pointes then two Example The first circle is A. B.F.E the second circle is B. C. D F and they crosse one an other in B. and in E and in no more pointes Nother is it possible that they should but other figures ther be which maye cutte a circle in foure partes as you se in this exāple where I haue set forthe one tunne forme and one eye forme and eche of them cutteth euery of their two circles into foure partes But as they be irregulare formes that is to saye suche formes as haue no precise measure nother proportion in their draughte so can there scarsely be made any certaine theorem of them But circles are regulare formes that is to say such formes as haue in their protracture a iuste and certaine proportion so that certain and determinate truths may be affirmed of them sith they ar vniforme and vnchaungable The lvi Theoreme If two circles be so drawen that the one be within the other and that they touche one an other If a line bee drawen by bothe their centres and so forthe in lengthe that line shall runne to that pointe where the circles do touche Example The one circle which is the greattest and vttermost is A. B. C the other circle that is the lesser and is drawen within the firste is A. D. E. The cētre of the greater circle is F and the centre of the lesser circle is G the pointe where they touche is A. And now you may see the truthe of the theoreme so plainely that it needeth no farther declaracion For you maye see that drawinge a line frome F. to G and so forth in lengthe vntill it come to the circumference it wyll lighte in the very poincte A where the circles touche one an other The Lvij. Theoreme If two circles bee drawen so one withoute and other that their edges doo touche and a right line bee drawenne frome the centre of the oneto the centre of the other that line shall passe by the place of their touching Example The firste circle is A. B.E and his centre is K The secōd circle is D B. C and his cētre is H the point wher they do touch is B. Nowe doo you se that the line K. H whiche is drawen from K that is centre of the firste circle vnto H beyng centre of the second circle doth passe as it must nedes by the pointe B whiche is the verye poynte wher they do to tuche together The .lviij. theoreme One circle can not touche an other in more pointes then one whether they touche within or without Example For the declaration of this Theoreme J haue drawen iiij circles the first is A. B. C and his centre H. the second is A. D. G and his centre F. the third is L. M and his centre K. the .iiij. is D. G.L.M and his centre E. Nowe as you perceiue the second circle A. D.G toucheth the first in the inner side inso much as it is drawen within the other and yet it toucheth him but in one point that is to say in A so lyke waies the third hym as you maie see but in one place And now as for the .iiij. circle it is drawen to declare the diuersitie betwene touchyng and cuttyng or crossyng For one circle maie crosse and cutte a great many other circles yet can be not cutte any one in more places then two as the fiue and fiftie Theoreme affirmeth The .lix. Theoreme In euerie circle those lines are to be counted equall whiche are in lyke distaunce from the centre And contrarie waies they are in lyke distance from the centre whiche be equall Example In this figure you see firste the circle drawen whiche is A. B.C.D and his centre is E. In this circle also there are drawen two lines equally distaunt from the centre for the line A. B and the line D. C are iuste of one distaunce from the centre whiche is E and therfore are they of one length Again thei are of one lengthe as shall be proued in the boke of profes and therefore then distaunce from the centre is all one The .lx. Theoreme In euerie circle the longest line is the diameter and of all the other lines thei are still longest that be nexte vnto the centre and they be the shortest that be farthest distaunt from it Example In this circle A. B.C.D I haue drawen first the diameter whiche is A. D whiche passeth as it must by the centre E Then haue I drawen ij other lines as M. N whiche is neerer the centre and F. G that is farther from the centre The fourth line also on the other side of the diameter that is B. C is neerer to the centre then the line F. G
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
It is to be noted that though euery small arche of a greate circle do seeme to be a right lyne yet in very dede it is not so for euery part of the circumference of al circles is compassed though in litle arches of great circles the eye cannot discerne the crokednes yet reason doeth alwaies declare it therfore iij. prickes in an exact right line can not bee brought into the circumference of a circle But and if they be not in a right line how so euer they stande thus shall you find their cōmon centre Opē your compas so wide that it be some what more then the halfe distance of two of those prickes Then sette the one foote of the compas in the one pricke and with the other foot draw an arche syne toward the other pricke Then againe putte the foot of your compas in the second pricke and with the other foot make an arche line that may crosse the firste arch line in ij places Now as you haue done with those two prickes so do with the middle pricke and the thirde that remayneth Then draw ij lines by the poyntes where those arche lines do crosse and where those two lines do meete there is the centre that you seeke for Example THE XXIIII CONCLVSION To drawe a touche line vnto a circle from any poincte assigned Here must you vnderstand that the pricke must be without the circle els the conclusion is not possible But the pricke or poinct beyng without the circle thus shall you procede Open your compas so that the one foote of it maie be set in the centre of the circle and the other foote on the pricke appoincted and so draw an other circle of that largenesse about the same centre and it shall gouerne you certainly in makyng the said touche line For if you draw a line frō the pricke appointed vnto the centre of the circle and marke the place where it doeth crosse the lesser circle and from that poincte erect a plumbe line that shall touche the edge of the vtter circle and marke also the place where that plumbe line crosseth that vtter circle and from that place drawe an other line to the centre takyng heede where it crosseth the lesser circle if you drawe a plumbe line from that pricke vnto the edge of the greatter circle that line I say is a touthe line drawen from the point assigned according to the meaning of this conclusion Example Let the circle be called B.C. D and his cētre E and the prick assigned A opē your cōpas now of such widenes that the one foote may be set in E which is the cētre of the circle the other in A which is the pointe assigned so make an other greter circle as here is A. F G thē draw a line from A. vnto E and wher that line doth cross the inner circle which heere is in the prick B. there erect a plūb line vnto the line A.E. and let that plumb line touch the vtter circle as it doth here in the point F so shall B.F. bee that plumbe lyne Then from F. vnto E. drawe an other line whiche shal be F. E and it will cutte the inner circle as it doth here in the point C from which pointe C. if you erect a plumb line vnto A then is that line A. C the touche line whiche you shoulde finde Notwithstandinge that this is a certaine waye to fynde any touche line and a demonstrable forme yet more easyly by many folde may you fynde and make any suche line with a true ruler layinge the edge of the ruler to the edge of the circle and to the pricke and so drawing a right line as this example sheweth where the circle is E the pricke assigned is A. and the ruler C. D. by which the touch line is drawen and that is A B and as this way is light to doo so is it certaine inoughe for any kinde B of workinge THE XXV CONCLVSION when you haue any peece of the circumference of a circle assigned howe you may make oute the whole circle agreynge therevnto First seeke out the centre of that arche according to the doe trine of the seuententh conclusion and then setting one foote of your compas in the centre and extending the other foot vn to the edge of the arche or peece of the circumference it is easy to drawe the whole circle Example A peece of an olde piller was found like informe to thys figure A.D.B. Now to knowe howe muche the cōpasse of the hole piller was seing by this parte it appereth that it was round thus shal you do Make in A table the like draught of that circūference by the self patrō vsing it as it wer a croked ruler Then make .iij. prickes in that arche line as I haue made C. D. and E. And then finde out the common centre to them all as the .xvij. conclusion teacheth And that cētre is here F nowe settyng one foote of your compas in F and the other in C. D other in E and so makyng a compasse you haue youre whole intent THE XXVI CONCLVSION To finde the centre to any arche of a circle If so be it that you desire to find the centre by any other way then by those .iij. prickes consideryng that sometimes you can not haue so muche space in the thyng where the arche is drawen as should serue to make those .iiij. bowe lines then shall you do thus Parte that arche line into two partes equall other vnequall it maketh no force and vnto ech portion draw a corde other a string line And then accordyng as you dyd in one arche in the .xvi. conclusion so doe in bothe those arches here that is to saie deuide the arche in the middle and also the corde and drawe then a line by those two deuisions so then are you sure that that line goeth by the centre Afterward do lykewaies with the other arche and his corde and where those .ij. lines do crosse there is the centre that you seke for Example The arche of the circle is A. B. C vnto whiche I must seke a centre therfore firste I do deuide it into .ij. partes the one of them is A. B and the other is B. C. Then doe I cut euery arche in the middle so is E. the middle of A. B and G. is the middle of B.C. Likewaies I take the middle of their cordes whiche I mark with F. and H settyng F. by E and H. by G. Then drawe I a line from E. to F and from G. to H and they do crosse in D wherefore saie I that D. is the centre that I seke for THE XXVII CONCLVSION To drawe a circle within a triangle appoincted For this conclusion and all other lyke you muste vnderstande that when one figure is named to be within an other that is not other waies to be vnderstande but that eyther euery syde of the inner figure dooeth touche euerie corner of the other other els euery
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.
two squares made of B. C and C. D. For as the shorter side is the iuste lengthe of C. D so the other longer side is iust twise so longe as B. C Wherfore I saie now accordyng to the Theoreme that the greatte square E is more then the other two squares F. and G by the quantitee of the longe square K wherof I reserue the profe to a more conuenient place where I will also teache the reason howe to fynde the lengthe of all suche perpendicular lynes and also of the line that is drawen betweene the blunte angle and the perpendicular line with sundrie other very pleasant conclusions The .xlvi. Theoreme In sharpe cornered triangles the square of anie side that lieth against a sharpe corner is lesser then the two squares of the other two sides by as muche as is comprised twise in the long square of that side on whiche the perpendicular line falleth and the portion of that same line liyng betweene the perpendicular and the foresaid sharpe corner Example Fyrst I sette foorth the triangle A. B. C and in yt I draw a plūbe line from the angle C. vnto the line A. B and it lighteth in D. Nowe by the theoreme the square of B.C. is not so muche as the square of the other two sydes that of B. A. and of A.C. by as muche as is twise conteyned in the lōg square made of A. B and A. D A. B. beyng the line or syde on which the perpendicular line falleth and A.D. beeyng that portion of the same line whiche doth lye betwene the perpendicular line and the sayd sharpe angle limitted whiche angle is by A. For declaration of the figures the square marked with E. is the square of B. C whiche is the syde that lieth agaynst the sharpe angle the square marked with C. is the square of A. B and the square marked with F. is the square of A. C and the two longe squares marked with H. K are made of the hole line A. B and one of his portions A. D. And truthe it is that the square E. is lesser than the other two squares C. and F. by the quantitee of those two long squares H. and K. Wherby you may consyder agayn an other proportion of equalitee that is to saye that the square E. with the twoo long squares H. K are iuste equall to the other twoo squares C. and F. And so maye you make as it were an other theoreme That in al sharpe cornered triangles where a perpendicular line is drawen frome one angle to the side that lyeth againste it the square of anye one side with the ij longesquares made of that hole line whereon the perpendicular line doth lighte and of that portion of it which ioyneth to that side whose square is all ready taken those thre figures I say are equall to the ij squares of the other ij sides of the triangle In whiche you muste vnderstand that the side on which the perpendiculare falseth is thrise vsed yet is his square but one 's mencioned for twise he is taken for one side of the two long squares And as I haue thus made as it were an other theoreme out of this fourty and sixe theoreme so mighte I out of it and the other that goeth nexte before make as manny as woulde suffice for a whole booke so that when they shall bee applyed to practise and consequently to expresse their benefite no manne that hathe not well wayde their wonderfull commoditee woulde credite the posibilitie of their wonderfull vse and large ayde in knowledge But all this wyll I remitte to a place conuenient The xlvij Theoreme If ij points be marked in the circumferēce of a circle and a right line drawen frome the one to the other that line must needes fal with in the circle Example The circle is A. B.C.D the ij poinctes are A. B the righte line that is drawenne frome the one to the other is the line A. B which as you see must needes lyghte within the circle So if you putte the pointes to be A. D or D. C or A. C other B. C or B. D inany of these cases you see that the line that is drawen from the one pricke to the other dothe euermore run within the edge of the circle els canne it be no right line Howbeit that a croked line especially being more croked then the portion of the circumference maye bee drawen from pointe to pointe withoute the circle But the theoreme speaketh only of right lines and not of croked lines The xlviij Theoreme If a righte line passinge by the centre of a circle doo crosse an other right line within the same circle passinge beside the centre if be deuide the saide line into twoo equal partes then doo they make all their angles righte And contrarie waies if they make all their angles righte then doth the longer line cutte the shorter in twoo partes Example The circle is A. B. C. D the line that passeth by the centre is A. E. C the line that goeth beside the centre is D. B. Nowe saye I that the line A. E. C dothe cutte that other line D. B. into twoo iuste partes and therefore all their four angles ar righte angles And contrarye wayes bicause all their angles are righte angles therfore it muste be true that the greater cutteth the lesser into two equal partes acordinge as the Theoreme would The xlix Theoreme If twoo right lines drawen in a circle doo crosse one an other and doo not passe by the centre euery of them dothe not deuide the other into two equall partions Example The circle is A. B. C. D and the centre is E the one line A. C and the other is B. D which two lines crosse one an other but yet they go not by the centre wherefore accordinge to the woordes of the theoreme eche of theim doth cutte the other into equall portions For as you may easily iudge A C. hath one portiō lōger and an other shorter and so like wise B. D. Howbeit it is not so to be vnderstād but one of them may be diuided into ij euē parts but bothe to bee cutte equally in the middle is not possible onles both passe through the cētre therfore much rather whē bothe go beside the centre it can not be that eche of theym shoulde be iustely parted into ij euen partes The L. Theoreme If two circles crosse and cut one an other then haue not they both one centre Example This theoreme seemeth of it selfe so manifest that it neadeth nother demonstration nother declaraciō Yet for the plaine vnderstanding of it I haue sette forthe a figure here where ij circles be drawē so that one of them doth crosse the other as you see in the pointes B. and G and their centres appear at the firste sighte to bee diuers For the centre of the one is F and the centre of the other is E which diffre as farre a sondre as the edges of the circles where they
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