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
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
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
and C. D be called paralleles also bicause E. and K whiche are .ij. ouer corners are equall and lyke waies L. and M. And so are in lyke maner the nether corners N. and H and G. and F. Nowe to the seconde parte of the theoreme those .ij. lynes A. B. and C. D shall be called paralleles because the ij inner corners As for example those two that bee toward the right hande that is G. and L. are equall by the fyrst parte of this nyntenth theoreme therfore muste G. and L. be equall to two ryght angles The xx Theoreme when a right line is drawen crosse ouer .ij. right gemow lines it maketh .ij. matche corners of the one line equall to two matche corners of the other line and also bothe ouer corners of one hande equall togither and bothe nether corners likewaies and more ouer two inner corners and two vtter corners also towarde one hande equall to two right angles Example Bycause A. B. and C. D in the laste figure are paralleles therefore the two matche corners of the one lyne as E. G. be equall vnto the .ij. matche corners of the other line that is K. F and lykewaies M. N equall to H. L. And also E. and K. bothe ouer corners of the lefte hande equall togyther and so are M. and L the two ouer corners on the ryghte hande in lyke maner N. and H the two nether corners on the lefte hande equall eche to other and G. and F. the two nether angles on the right hande equall togither ¶ Farthermore yet G. and L. the .ij. inner angles on the right hande bee equall to two right angles and so are M. and F. the .ij. vtter angles on the same hande in lyke manner shall you say of N. and K. the two inner corners on the left hand and of E. and H. the two vtter corners on the same hande And thus you see the agreable sentence of these .iij. theoremes to tende to this purpose to declare by the angles how to iudge paralleles and contrary waies howe you may by paralleles iudge the proportion of the angles The xxi Theoreme what so euer lines be paralleles to any other line those same be paralleles togither Example A. B. is a gemow line or a parellele vnto C. D. and E. F lyke waies is a parallele vnto C. D. Wherfore it foloweth that A. B. must nedes bee a parallele vnto E. F. The .xxij. theoreme In euery triangle when any side is drawen forth in length the vtter angle is equall to the ij inner angles that lie againste it And all iij. inner angles of any triangle are equall to ij right angles Example The triangle beeyng A. D. E. and the syde A. E. drawen foorthe vnto B there is made an vtter corner whiche is C and this vtter corner C is equall to bothe the inner corners that lye agaynst it whyche are A. and D. And all thre inner corners that is to say A. D. and E are equall to two ryght corners whereof it foloweth that all the three corners of any one triangle are equall to all the three corners of euerye other triangle For what so euer thynges are equalle to anny one thyrde thynge those same are equalle togitther by the fyrste common sentence so that bycause all the .iij. angles of euery triangle are equall to two ryghte angles and all ryghte angles bee equall togyther by the fourth request therfore must it nedes folow that all the thre corners of euery triangle accomptyng them togyther are equall to iij. corners of any other triangle taken all togyther The .xxiii. theoreme when any ij right lines doth touche and couple .ij. other righte lines whiche are equall in length and paralleles and if those .ij. lines bee drawen towarde one hande then are thei also equall together and paralleles Example A. B. and C. D. are ij ryght lynes and paralleles and equall in length and they ar touched and ioyned togither by ij other lynes A. C. and B. D this beyng so and A. C. and B. D. beyng drawen towarde one syde that is to saye bothe towarde the lefte hande therefore are A C. and B. D. bothe equall and also paralleles The .xxiiij. theoreme In any likeiamme the two contrary sides ar equall togither and so are eche .ij. contrary angles and the bias line that is drawen in it dothe diuide it into two equall portions Example Here ar two likeiammes ioyned togither the one is a longe square A. B. E and the other is a losengelike D. C. E. F. which ij likeiammes ar proued equall togither bycause they haue one ground line that is F. E And are made betwene one payre of gemow lines I meane A. D. and E. H. By this Theoreme may you know the arte of the righte measuringe of likeiammes as in my booke of measuring I wil more plainly declare The xxvi Theoreme All likeiammes that haue equal grounde lines and are drawen betwene one paire of paralleles are equal togither Example Fyrste you muste marke the difference betwene this Theoreme and the laste for the laste Theoreme presupposed to the diuers likeiammes one ground line common to them but this theoreme doth presuppose a diuers ground line for euery like iamme only meaning them to be equal in length though they be diuers in numbbe As for example In the last figure ther are two parallels A. D. and E. H and betwene them are drawen thre likeiammes the firste is A. B. E. F the second is E. C. D. F and the thirde is C. G. H. D. The firste and the seconde haue one ground line that is E. F. and therfore in so muche as they are betwene one paire of paralleles they are equall accordinge to the fiue and twentye Theoreme but the thirde likeiamme that is C. G. H. D. hathe his grounde line G. H seuerall frome the other but yet equall vnto it wherefore the third likeiam is equall to the other two firste likeiammes And for a proofe that G.H. being the groud line of the third likeiamme is equal to E. F whiche is the ground line to both the other likeiams that may be thus declared G.H. is equall to C. D seynge they are the contrary sides of one likeiamme by the foure and twēty theoreme and so are C.D. and E. F. by the same theoreme Therfore seynge both those ground lines E.F. and G. H are equall to one thirde line that is C.D. they must nedes bee equall togyther by the firste common sentence The xxvii Theoreme All triangles hauinge one grounde lyne an standing betwene one paire of parallels ar equall togither Example A.B. and C. F. are twoo gemowe lines betweene which there be made two triangles A. D. E. and D. E. B so that D. E is the common ground line to them bothe wherfore it doth folow that those two triangles A.D.E. and D.E.B. are equall eche to other The xxviij Theoreme All triangles that haue like long ground lines and bee made betweene one paire of gemow lines are
prickes that be nighest togither and ther wil appear rightly drawē the figure of fiftene sides and angles equall And so do with any other figure of what numbre of sides so euer it bee FINIS THE SECOND BOOKE OF THE PRINCIPLES of Geometry containing certaine Theoremes whiche may be called Approued truthes And be as it were the moste certaine groundes wheron the practike cōclusions of Geometry ar founded Whervnto are annexed certaine declarations by examples for the right vnderstanding of the same to the ende that the simple reader might not iustly cōplain of hardnes or obscuritee and for the same cause ar the demonstrations and iust profes omitted vntill a more conuient time 1551. If truthe maie trie it selfe By Reasons prudent skyll If reason maie preuayle by right And rule the rage of will I dare the triall byde For truthe that I pretende And though some lyst at me repine Iuste truthe shall me defende THE PREFACE VNTO the Theoremes I Doubt not gentle reader but as my argument is straunge and vnacquainted with the vulgare tougue so shall I of many men be straungly talked of and as straunglye iudged Some men will saye peraduenture I mighte haue better imployed my tyme in some pleasaunte historye comprisinge matter of chiuasrye Some other wolde more haue preised my trauaile if I hadde spente the like time in some morall matter other in decising some controuersy of religion And yet some men as I iudg will not mislike this kind of mater but then will they wishe that I had vsed a more certaine order in placinge bothe the Propositions and Theoremes and also a more exacter proofe of eche of theim bothe by demonstrations mathematicall Some also will mislike my shortenes and simple plainesse as other of other affections diuersely shall espye somwhat that they shall thinke blame worthy and shal misse somewhat that thei wold wish to haue bene here vsed so that euerie manne shall giue his verdicte of me according to his phantasie vnto whome ioinctly I make this my firste answere that as they ar many and in opinions verie diuers so were it scarse possible to please them all with anie one argumente of what kinde so euer it were And for my seconde aunswere I saye thus That if annye one argumente mighte please them all then shoulde thei be thankfull vnto me for this kind of matter For nother is there anie matter more straunge in the englishe tungue then this whereof neuer booke was written before now in that tungue and therefore oughte to delite all them that desire to vnderstand straunge matters as most men commonlie doo And againe the practise is so pleasaunt in vsinge and so profitable in appliynge that who so euer do the delite in anie of bothe ought not of right to mislike this arte And if any manne shall like the arte welle for it selfe but shall mislyke the fourme that I haue vsed in teachyng of it to hym I shall saie Firste that I dooe wishe with hym that some other man whiche coulde better haue doone it hadde shewed his good will and vsed his diligence in suche sorte that I myght haue bene therby occasioned iustely to haue left of my laboure or after my trauaile to haue suppressed my bookes But sithe no manne hath yet attempted the like as far as I canne learne I truste all suche as bee not exercised in the studie of Geometrye shall finde greate ease and furtheraunce by this simple plaine and easie forme of writinge And shall perceaue the exacte woorkes of Theon and others that write on Euclide a greate deale the somner by this blunte delineacion afore hande to them taughte For I dare presuppose of them that thing which I haue sette in my selfe and haue marked in others that is to saye that it is not easie for a man that shall trauaile in a straunge arte to vnderstand at the beginninge bothe the thing that is taught and also the iuste reason whie it is so And by experience of teachinge I haue tried it to bee true for whenne I haue taughte the proposition as it imported in meaninge and annexed the demonstration with all I didde perceaue that it was a greate trouble and a painefull vexacion of mynde to the learner to comprehend bothe those thinges at ones And therfore did I proue firste to make them to vnderstande the sence of the propositions and then afterward did they conceaue the demonstrations muche soner when they hadde the sentence of the propositions first ingrafted in their mindes This thinge caused me in bothe these bookes to omitte the demonstrations and to vse onlye a plaine forme of declaration which might best serue for the firste introduction Whiche example hat beene vsed by other learned menne before nowe for not only Georgius Ioachimus Rheticus but also Boetius that wittye clarke did set forth some whole books of Euclide without any demonstration or any other declaratiō at al. But if I shal hereafter perceaue that it maie be a thankefull trauaile to sette foorth the propositions of geometrie with demonstrations I will not refuse to dooe it and that with sundry varietees of demonstrations bothe pleasaunt and profitable also And then will I in like maner prepare to sette foorth the other bookes whiche now are lefte vnprinted by occasion not so muche of the charges in cuttyng of the figures as for other iuste hynderances whiche I truste hereafter shall bee remedied In the meane season if any man muse why I haue sette the Conclusions beefore the Teoremes seynge many of the Theoremes seeme to include the cause of some of the conclusions and therfore oughte to haue gone before them as the cause goeth before the effecte Herevnto I saie that although the cause doo go beefore the effect in order of nature yet in order of teachyng the effect must be fyrst declared and than the cause therof shewed for so shal men best vnderstād things First to lerne that such thinges ar to be wrought and secondarily what thei ar and what thei do import and thā thirdly what is the cause thereof An other cause why that the theoremes be put after the cōclusions is this whā I wrote these first cūclusions which was iiiij yeres passed I thought not then to haue added any theoremes but next vnto the cōclusiōs to haue taught the order how to haue applied thē to work for drawing of plottes such like vses But afterward cōsidering the great cōmoditie that thei serue for and the light that thei do geue to all sortes of practise geometricall besyde other more notable benefites whiche shall be declared more specially in a place conuenient I thoughte beste to geue you some taste of theym and the pleasaunt contemplation of suche geometrical propositions which might serue diuerselye in other bookes for the demonstrations and proofes of all Geometricall woorkes And in theim as well as in the propositions I haue drawen in the Linearie examples many tymes more lynes than be spoken of in the explication of