THE PREFACE,declaring briefely the commodi-tes of Geometrye, and thenecessitye thereof.GEometrye may thinke it selfe to sustaine great iniury, if it shall be inforced other to show her manifold commodities, or els not to prease into the sight of men, and therefore might this wayes answere briefely: Other I am able to do you much good, or els but litle. If I bee able to doo you much good, then be you not your owne friendes, but greatlye your owne enemies to make so little of me, which maye profite you so muche. For if I were as vncurteous as you vnkind, I shuld vtterly refuse to do them any good, which will so curiously put me to the trial and profe of my commodities, or els to suffre exile, and namely sithe I shal only yeld benefites to other, and receaue none againe. But and if you could saye truely, that my benefites be nother many nor yet greate, yet if they bee anye, I doo yelde more to you, then I doo receaue againe of you, and therefore I oughte not to bee repelled of them that loue them selfe, althoughe they loue me not all for my selfe. But as I am in nature a liberall science, so canne I not againste nature contende with your inhumanitye, but muste shewe my selfe liberall euen to myne enemies. Yet this is my comforte againe, that I haue none enemies but them that knowe me not, and therefore may hurte themselues, but can not noye me. Yf they dispraise the thinge that they know not, all wise men will blame them and not credite them, and yf they thinke they knowe me, lette theym shewe one vntruthe and erroure in me, and I wyll geue the victorye.Yetcan no humayne science saie thus, but I onely, that there is no sparke of vntruthe in me: but all my doctrine and workes are without any blemishe of errour that mans reason can discerne. And nexte vnto me in certaintie are my three systers, Arithmetike, Musike, and Astronomie, whiche are also so nere knitte in amitee, that he that loueth the one, can not despise the other, and in especiall Geometrie, of whiche not only these thre, but all other artes do borow great ayde, as partly hereafter shall be shewed. But first will I beginne with the vnlearned sorte, that you maie perceiue how that no arte can stand without me. For if I should declare how many wayes my helpe is vsed, in measuryng of ground, for medow, corne, and wodde: in hedgyng, in dichyng, and in stackes makyng, I thinke the poore Husband man would be more thankefull vnto me, then he is nowe, whyles he thinketh that he hath small benefite by me. Yet this maie he coniecture certainly, that if he kepe not the rules of Geometrie, he can not measure any ground truely. And in dichyng, if he kepe not a proportion of bredth in the mouthe, to the bredthe of the bottome, and iuste slopenesse in the sides agreable to them bothe, the diche shall be faultie many waies. When he doth make stackes for corne, or for heye, he practiseth good Geometrie, els would thei not long stand: So that in some stakes, whiche stand on foure pillers, and yet made round, doe increase greatter and greatter a good height, and then againe turne smaller and smaller vnto the toppe: you maie see so good Geometrie, that it were very difficult to counterfaite the lyke in any kynde of buildyng. As for other infinite waies that he vseth my benefite, I ouerpasse for shortnesse.Carpenters, Karuers, Ioyners, and Masons, doe willingly acknowledge that they can worke nothyng without reason of Geometrie, in so muche that they chalenge me as a peculiare science for them. But in that they should do wrong to all other men, seyng euerie kynde of men haue som benefit by me, not only in buildyng, whiche is but other mennes costes, and the arte of Carpenters, Masons, and the other aforesayd, but in theirowne priuate profession, whereof to auoide tediousnes I make this rehersall.Sith Merchauntes by shippes great riches do winne,I may with good righte at their seate beginne.The Shippes on the sea with Saile and with Ore,were firste founde, and styll made, by Geometries lore.Their Compas, their Carde, their Pulleis, their Ankers,were founde by the skill of witty Geometers.To sette forth the Capstocke, and eche other parte,wold make a greate showe of Geometries arte.Carpenters, Caruers, Ioiners and Masons,Painters and Limners with suche occupations,Broderers, Goldesmithes, if they be cunning,Must yelde to Geometrye thankes for their learning.The Carte and the Plowe, who doth them well marke,Are made by good Geometrye. And so in the warkeOf Tailers and Shoomakers, in all shapes and fashion,The woorke is not praised, if it wante proportion.So weauers by Geometrye hade their foundacion,Their Loome is a frame of straunge imaginacion.The wheele that doth spinne, the stone that doth grind,The Myll that is driuen by water or winde,Are workes of Geometrye straunge in their trade,Fewe could them deuise, if they were vnmade.And all that is wrought by waight or by measure,without proofe of Geometry can neuer be sure.Clockes that be made the times to deuide,The wittiest inuencion that euer was spied,Nowe that they are common they are not regarded,The artes man contemned, the woorke vnrewarded.But if they were scarse, and one for a shewe,Made by Geometrye, then shoulde men know,That neuer was arte so wonderfull witty,So needefull to man, as is good Geometry.The firste findinge out of euery good arte,Seemed then vnto men so godly a parte,That no recompence might satisfye the finder,But to make him a god, and honoure him for euer.So Ceres and Pallas, and Mercury also,Eolus and Neptune, and many other mo,Were honoured as goddes, bicause they did teache,Firste tillage and weuinge and eloquent speache,Or windes to obserue, the seas to saile ouer,They were called goddes for their good indeuour.Then were men more thankefull in that golden age:This yron wolde nowe vngratefull in rage,Wyll yelde the thy reward for trauaile and paine,With sclaunderous reproch, and spitefull disdaine.Yet thoughe other men vnthankfull will be,Suruayers haue cause to make muche of me.And so haue all Lordes, that landes do possesse:But Tennaunted I feare will like me the lesse.Yet do I not wrong but measure all truely,All yelde the full right of euerye man iustely.Proportion Geometricall hath no man opprest,Yf anye bee wronged, I wishe it redrest.But now to procede with learned professions, in Logike and Rhetorike and all partes of phylosophy, there neadeth none other proofe then Aristotle his testimony, whiche without Geometry proueth almost nothinge. In Logike all his good syllogismes and demonstrations, hee declareth by the principles of Geometrye. In philosophye, nether motion, nor time, nor ayrye impressions could hee aptely declare, but by the helpe of Geometrye as his woorkes do witnes. Yea the faculties of the minde dothe hee expresse by similitude to figures of Geometrye. And in morall phylosophy he thought that iustice coulde not wel be taught, nor yet well executed without proportion geometricall. And this estimacion of Geometry he maye seeme to haue learned of his maister Plato, which without Geometrye wolde teache nothinge, nother wold admitte any to heare him, except he were experte in Geometry. And what merualle if he so muche estemed geometrye, seinge his opinion was, that Godde wasalwaiesworkingeby Geometrie? Whiche sentence Plutarche declareth at large. And although Platto do vse the helpe of Geometrye in all the most waighte matter of a common wealth, yet it is so generall in vse, that no small thinges almost can be wel done without it. And therfore saith he: that Geometrye is to be learned, if it were for none other cause, but that all other artes are bothe soner and more surely vnderstand by helpe of it.What greate help it dothe in physike, Galene doth so often and so copiousely declare, that no man whiche hath redde any booke almoste of his, can be ignorant thereof,in so much that he coulde neuer cure well a rounde vlcere, tyll reason geometricall dydde teache it hym. Hippocrates is earnest in admonyshynge that study of geometrie must prepare the way to physike, as well as to all other artes.I shoulde seeme somewhat to tedious, if I shoulde recken vp, howe the diuines also in all their mysteries of scripture doo vse healpe of geometrie: and also that lawyers can neuer vnderstande the hole lawe, no nor yet the firste title therof exactly without Geometrie. For if lawes can not well be established, nor iustice duelie executed without geometricall proportion, as bothe Plato in his Politike bokes, and Aristotle in his Moralles doo largely declare. Yea sithe Lycurgus that cheefe lawmaker amongest the Lacedemonians, is moste praised for that he didde chaunge the state of their common wealthe frome the proportion Arithmeticall to a proportion geometricall, whiche without knowledg of bothe he coulde not dooe, than is it easye to perceaue howe necessarie Geometrie is for the lawe and studentes thereof. And if I shall saie preciselie and freelie as I thinke, he is vtterlie destitute of all abilitee to iudge in anie arte, that is not sommewhat experte in the Theoremes of Geometrie.Andthat caused Galene to say of hym selfe, that he coulde neuer perceaue what a demonstration was, no not so muche, as whether there were any or none, tyll he had by geometrie gotten abilitee to vnderstande it, although he heard the beste teachers that were in his tyme. It shuld be to longe and nedelessealso to declare what helpe all other artes Mathematicall haue by geometrie, sith it is the grounde of all theyr certeintie, and no man studious in them is so doubtful therof, that he shall nede any persuasion to procure credite thereto. For he can not reade .ij. lines almoste in any mathematicall science, but he shall espie the nedefulnes of geometrie. But to auoyde tediousnesse I will make an ende hereof with that famous sentence of auncient Pythagoras, That who so will trauayle by learnyng to attayne wysedome, shall neuer approche to any excellencie without the artes mathematicall, and especially Arithmetike and Geometrie.And yf I shall somewhat speake of noble men, and gouernours of realmes, howe needefull Geometrye maye bee vnto them, then must I repete all that I haue sayde before, sithe in them ought all knowledge to abounde, namely that maye appertaine either to good gouernaunce in time of peace, eyther wittye pollicies in time of warre. For ministration of good lawes in time of peace Lycurgus example with the testimonies of Plato and Aristotle may suffise. And as for warres, I might thinke it sufficient that Vegetius hath written, and after him Valturius in commendation of Geometry, for vse of warres, but all their woordes seeme to saye nothinge, in comparison to the example of Archimedes worthy woorkes make by geometrie, for the defence of his countrey, to reade the wonderfull praise of his wittie deuises, set foorthe by the most famous hystories of Liuius, Plutarche, and Plinie, and all other hystoriographiers, whyche wryte of the stronge siege ofSyracusæmade by that valiant capitayne, and noble warriourMarcellus, whose power was so great, that all men meruayled how that one citee coulde withstande his wonderfull force so longe. But much more woulde they meruaile, if they vnderstode that one man onely dyd withstand all Marcellus strength, and with counter engines destroied his engines to the vtter astonyshment ofMarcellus, and all that were with hym. He had inuented suche balastelas that dyd shoote out a hundred dartes at oneshotte, to the great destruction ofMarcellussouldiours, wherby a fonde tale was spredde abrode, how that in Syracusæ there was a wonderfull gyant, whiche had a hundred handes, and coulde shoote a hundred dartes at ones. And as this fable was spredde of Archimedes, so many other haue been fayned to bee gyantes and monsters, bycause they dyd suche thynges, whiche farre passed the witte of the common people. So dyd they feyne Argus to haue a hundred eies, bicause they herde of his wonderfull circumspection, and thoughte that as it was aboue their capacitee, so it could not be, onlesse he had a hundred eies. So imagined they Ianus to haue two faces, one lokyng forwarde, and an other backwarde, bycause he coulde so wittily compare thynges paste with thynges that were to come, and so duely pondre them, as yf they were all present. Of like reasõ did they feyn Lynceus to haue such sharp syght, that he could see through walles and hylles, bycause peraduenture he dyd by naturall iudgement declare what cõmoditees myght be digged out of the grounde. And an infinite noumbre lyke fables are there, whiche sprange all of lyke reason.For what other thyng meaneth the fable of the great gyant Atlas, whiche was ymagined to beare vp heauen on his shulders? but that he was a man of so high a witte, that it reached vnto the skye, and was so skylfull in Astronomie, and coulde tell before hande of Eclipses, and other like thynges as truely as though he dyd rule the sterres, and gouerne the planettes.So was Eolus accompted god of the wyndes, and to haue theim all in a caue at his pleasure, by reason that he was a wittie man in naturall knowlege, and obserued well the change of wethers, aud was the fyrst that taught the obseruation of the wyndes. And lyke reson is to be geuen of al the old fables.But to retourne agayne to Archimedes, he dyd also by arte perspectiue (whiche is a parte of geometrie) deuise such glasses within the towne of Syracusæ, that dyd bourne their ennemies shyppes a great way from the towne, whyche was a meruaylous politike thynge. And if I shulde repete thevarietees of suche straunge inuentions, as Archimedes and others haue wrought by geometrie, I should not onely excede the order of a preface, but I should also speake of suche thynges as can not well be vnderstande in talke, without somme knowledge in the principles of geometrie.But this will I promyse, that if I may perceaue my paynes to be thankfully taken, I wyll not onely write of suche pleasant inuentions, declaryng what they were, but also wil teache howe a great numbre of them were wroughte, that they may be practised in this tyme also. Wherby shallbe plainly perceaued, that many thynges seme impossible to be done, whiche by arte may very well be wrought. And whan they be wrought, and the reason therof not vnderstande, than say the vulgare people, that those thynges are done bynegromancy. And hereof came it that fryer Bakon was accompted so greate anegromancier, whiche neuer vsed that arte (by any coniecture that I can fynde) but was in geometrie and other mathematicall sciences so experte, that he coulde dooe by theim suche thynges as were wonderfull in the syght of most people.Great talke there is of a glasse that he made in Oxforde, in whiche men myght see thynges that were doon in other places, and that was iudged to be done by power of euyll spirites. But I knowe the reason of it to bee good and naturall, and to be wrought by geometrie (sythe perspectiue is a parte of it) and to stande as well with reason as to see your face in cõmon glasse. But this conclusion and other dyuers of lyke sorte, are more mete for princes, for sundry causes, than for other men, and ought not to bee taught commonly. Yet to repete it, I thought good for this cause, that the worthynes of geometry myght the better be knowen, & partly vnderstanding geuen, what wonderfull thynges may be wrought by it, and so consequently how pleasant it is, and how necessary also.And thus for this tyme I make an end. The reason of som thynges done in this boke, or omitted in the same, you shall fynde in the preface before the Theoremes.*The definitions of the principles ofGEOMETRY.Geometry teacheth the drawyng,Measuringand proporcion offigures. butin as muche as no figure can bee drawen, but it muste haue certayne boũdes and inclosures of lines: and euery lyne also is begon and ended at some certaine prycke, fyrst it shal be meete to know these smaller partes of euery figure, that therby the whole figures may the better bee iudged, and distincte in sonder.A poincte.A Poynt or a Prycke, is named of Geometricians that small and vnsensible shape, whiche hath in it no partes, that is to say: nother length, breadth nor depth. But as their exactnes of definition is more meeter for onlye Theorike speculacion, then for practise and outwarde worke (consideringe that myne intent is to applye all these whole principles to woorke) I thynke meeter for this purpose, to call apoynt or prycke, that small printe of penne, pencyle, or other instrumente, whiche is not moued, nor drawen from his fyrst touche, and therfore hath no notable length nor bredthe: as this example doeth declare.∴Where I haue set .iij. prickes, eche of them hauyng both lẽgth and bredth, thogh it be but smal, and thefore not notable.Nowe of a great numbre of these prickes, is made aLyne, as you may perceiue by this forme ensuyng. ············ where as I haue set a numbre of prickes, so if you with your pen will set in more other prickes betweene euerye two of these,A lyne.then wil it be a lyne, as here you may seeand thislyne, is called of Geometricians,Lengthe withoute breadth.Butas they in theyr theorikes (which ar only mind workes)do precisely vnderstand these definitions, so it shal be sufficient for those men, whiche seke the vse of the same thinges, as sense may duely iudge them, and applye to handy workes if they vnderstand them so to be true, that outwarde sense canne fynde none erroure therein.Of lynes there bee two principall kyndes. The one is called a right or straight lyne, and the other a croked lyne.A streghte lyne.A Straight lyne, is the shortest that maye be drawenne between two prickes.A crokyd lyne.And all other lines, that go not right forth from prick to prick, but boweth any waye,such are calledCroked lynesas in these examples folowyng ye may se, where I haue set but one forme of a straight lyne, for more formes there be not, but of crooked lynes there bee innumerable diuersities, whereof for examples sum I haue sette here.see caption and textA right lyne.Croked lynes.see caption and textCroked lines.see caption and textsee caption and textSo now you must vnderstand, thateuery lyne is drawen betwene twoo prickes, wherof the one is at the beginning, and the other at the ende.see textTherefore when soeuer you do see any formes of lynes to touche at one notable pricke, as in this example, then shall younot call it onecrokedlyne, but rather twoo lynes:an Angle.in as muche as there is a notable and sensible angle by.A.whiche euermore is made by the meetyng of two seuerall lynes. And likewayes shall you iudge of this figure, whiche is made of two lines, and not of one onely.see textSo that whan so euer any suche meetyng of lines doth happen, the place of their metyng is called anAngle or corner.Of angles there be three generall kindes: a sharpe angle, a square angle, and a blunte angle.A righte angle.The square angle, whiche is commonly nameda right corner, is made of twoo lynes meetyng together in fourme of a squire, whiche two lines, if they be drawen forth in length, will crosse one an other: as in the examples folowyng you maie see.Right angles.see caption and textA sharpe corner.A sharpe angleis so called, because it is lesser than is a square angle, and the lines that make it, do not open so wide in their departynge as in a square corner, and if thei be drawen crosse, all fower corners will not be equall.A blunte angle.A blunt or brode corner, is greater then is a square angle, and his lines do parte more in sonder then in a right angle, of whiche all take these examples.And these angles (as you see) are made partly of streght lynes, partly of croken lines, and partly of both together. Howbeit in right angles I haue put none example of croked lines, because it wouldmuche trouble a lerner to iudge them: for their true iudgment doth appertaine to arte perspectiue, and as I may say, rather to reason then to sense.Sharpe angles.see caption and textBlunte or brode angles.see caption and textBut now as of many prickes there is made one line, soof diuerse lines are there made sundry formes, figures, and shapes, whiche all yet be called by one propre name,A platte forme.Platte formes, and thei haue bothelength and bredth, but yet no depenesse.Andthe boundesof euerie platte forme are lines: as by the examples you maie perceiue.Of platte formes some be plain, and some be croked, and some parly plaine, and partlie croked.A plaine platte.A plaine platteis that, whiche is made al equall in height, so that the middle partes nother bulke vp, nother shrink down more then the bothe endes.A crooked platte.For whan the one parte is higher then the other, then is it named aCroked platte.And if it be partlie plaine, and partlie crooked, then is it called aMyxte platte, of all whiche, these are exaumples.A plaine platte.A croked platte.A myxte platte.see caption and textsee caption and textsee caption and textAnd as of many prickes is made a line, and of diuerse lines one platte forme,A bodie.so of manie plattes is madea bodie, whiche conteignethLengthe, bredth, and depenesse.Depenesse.ByDepenesseI vnderstand, not as the common sort doth, the holownesse of any thing, as of a well, a diche, a potte, and suche like, but I meane the massie thicknesseof any bodie, as in exaumple of a potte: the depenesse is after the common name, the space from his brimme to his bottome. But as I take it here, the depenesse of his bodie is his thicknesse in the sides, whiche is an other thyng cleane different from the depenesse of his holownes, that the common people meaneth.Now all bodies haue platte formes for their boundes,Cubike.so in a dye (whiche is calleda cubike bodie) by geomatricians,Asheler.and anashlerof masons, there are .vi. sides, whiche are .vi. platte formes, and are the boundes of the dye.A globe.But in aGlobe, (whiche is a bodie rounde as a bowle) there is but one platte forme, and one bounde, and these are the exaumples of them bothe.A dye or ashler.A globe.see caption and textsee caption and textBut because you shall not muse what I dooe calla bound,A bounde.I mean therby a generall name, betokening the beginning, end and side, of any forme.Forme, Fygure.A forme, figure, or shape, is that thyng that is inclosed within one bond or manie bondes, so that you vnderstand that shape, that the eye doth discerne, and not the substance of the bodie.Offiguresthere be manie sortes, for either thei be made of prickes, lines, or platte formes. Not withstandyng to speake properlie,a figureis euer made by platte formes, and not of bare lines vnclosed, neither yet of prickes.Yet for the lighter forme of teachyng, it shall not be vnsemely to call all suche shapes, formes and figures, whiche yeeye maie discerne distinctly.And first to begin with prickes, there maie be made diuerse formes of them, as partely here doeth folowe.see captionA lynearic numbre.see captionTrianguler numbressee captionLongsquare nũbre.see captionIust square numbressee captiona threcornered spire.see captionA square spire.And so maie there be infinite formes more, whiche I omitte for this time, cõsidering that their knowledg appertaineth more to Arithmetike figurall, than to Geometrie.But yet one name of a pricke, whiche he taketh rather of his place then of his fourme, maie I not ouerpasse. And that is, when a pricke standeth in the middell of a circle (as no circle can be made by cõpasse without it) then is it calleda centre.A centreAnd thereof doe masons, and other worke menne call that patron, acentre, 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.see textLynes make diuerse figures also, though properly thei maie not be called figures, as I said before (vnles the lines do close) but onely for easie maner of teachyng, all shall be calledfigures, that the eye can discerne, of whiche this is one, when one line lyeth flatte (whiche is namedA ground line.theground line) and an other commeth downe on it, and is calledA perpendicular.A plume lyne.aperpendiculerorplũme lyne, as in this example you maysee. where.A.B.is the grounde line, andC.D.the plumbe line.see textAndlike waies in this figure there are three lines, the grounde lyne whiche isA.B.the plumme line that isA.C.and thebias line, whiche goeth from the one of thẽ to the other, and lieth against the right corner in such a figure whiche is here.C.B.Butconsideryng that I shall haue occasion to declare sundry figures anon, I will first shew some certaine varietees of lines that close no figures, but are bare lynes, and of the other lines will I make mencion in the description of the figures.tortuouse paralleles.ParallelysGemowe lynes.Paralleles, orgemowe lynesbe suche lines as be drawen foorth still in one distaunce, 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 calledbought lynes, and loe here exaumples of them bothe.parallelis.bought linessee caption and textsee caption and textparallelis:circular.Concen-trikes.see caption and textI haue added alsoparalleles tortuouse, whiche bowe cõtrarie waies with their two endes: andparalleles circular, whiche be lyke vnperfecte compasses: for if they bee whole circles,Concentrikesthen are they calledcõcentrikes, that is to saie, circles drawẽ on one centre.Heremight I note the error of goodAlbert Durer, which affirmeth that no perpendicular lines can beparalleles. whicherrour doeth spring partlie of ouersight of the difference of a streight line, and partlie of mistakyng certain principles geometrical, which al I wil let passe vntil an other tyme, and wil not blame him, which hath deserued worthyly infinite praise.And to returne to my matter.A twine line.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.And an other sorte of lines is there, that is called aspirall line,A worme line.or aworm line, whiche representeth an apparant forme of many circles, where there is not one in dede: of these .ii. kindes of lines, these be examples.Atwistelyne.see caption and textA spirail lynesee caption and textA touche lyne.A tuch line.A touche lyne, is a line that runneth a long by the edge of a circle, onely touching it, but doth not crosse the circumference of it, as in this exaumple you maie see.A corde,And when that a line doth crosse the edg of the circle, thẽ is it calleda cord, as you shall see anon in the speakynge of circles.Matche cornersIn the meane season must I not omit to declare what angles bee calledmatche corners, that is to saie, suche as stande directly one against the other, when twoo lines be drawen a crosse, as here appereth.Matche corner.WhereA.andB.are matche corners, so areC.andD.but notA.andC.notherD.andA.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.A circleis a figure made and enclosed with one line, and hath in the middell of it a pricke or centre, from whiche all the lines that be drawen to thecircumferenceare equall all in length, as here you see.Circumference.see textAnd the line that encloseth the whole compasse, is called thecircumference.A diameter.And all the lines that bee drawen crosse the circle, and goe by the centre, are nameddiameters, whose halfe, I meane from the center to the circumferenceany waie,Semidiameter.is called thesemidiameter, orhalfe diameter.A cord, or a stringlyne.see textBut and if the line goe crosse the circle, and passe beside the centre, then is it calleda corde, ora stryng line, as I said before, and as this exaumple sheweth: whereA.is the corde. And the compassed line that aunswereth to it,An archlineis calledan arche lyne,A bowline.ora bowe lyne, whiche here marked withB.and the diameter withC.see textsee textBut and if that part be separate from the rest of the circle (as in this exãple you see) then ar both partes called cãtelles,A cantlethe one thegreatter cantleasE.and the other thelesser cantle, asD.And if it be parted iuste by the centre (as you see inF.)A semyecirclethen is it called asemicircle, orhalfe compasse.see caption and textSometimes it happeneth that a cantle is cutte out with two lynes drawen from the centre to the circumference (asG.is)A nooke cantleand then maie it be called anooke cantle, and if it be not parted from the reste of the circle (as you see inH.)A nooke.then is it called anookeplainely without any addicion. And the compassed lyne in it is called anarche lyne, as the exaumple here doeth shewe.An arche.see caption and textNowe haue you heard as touchyng circles, meetely sufficient instruction, so that it should seme nedeles to speake any more of figures in that kynde, saue that there doeth yet remaine ij. formes of an imperfecte circle, for it is lyke a circle that were brused, and thereby did runne out endelong one waie, whiche forme Geometricians dooe call anAn egge fourme.egge forme, because it doeth represent the figure and shape of an egge duely proportioned (as this figure sheweth) hauyng the one ende greate then the other.A tunne or barrel formA tunne forme.see textAn egge formesee caption and textFor if it be lyke the figure of a circle pressed in length, and bothe endes lyke bygge, then is it called atunne forme, orbarrell 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,An axtre or axe lyne.and that is called anaxelyne, oraxtre. Howe be it properly that line that is called anaxtre, whiche gooeth throughe 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 namedpolis, wchyou 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.see textBut to returne to the diuersityes of figures that remayne vndeclared, the most simple of them ar such ones as be made but of two lynes, as are thecantle of a circle, and thehalfe circle, of which I haue spoken allready. Likewyse thehalfe of an egge forme, thecantle of an egge forme, thehalfe of a tunne fourme, and thecantle of a tunne fourme, and besyde these a figure moche like to a tunne fourne, saue that it is sharp couered at both the endes, and therfore doth consist of twoo lynes, where a tunne forme is made of one lyne,An yey fourmeand that figure is named anyey fourme.A trianglesee textsee textThe nexte kynd of figures are those that be made of .iij. lynes other be all right lynes, all crooked lynes, other some right and some crooked. But what fourme so euer they be of, they are named generallytriangles. fora triangleis nothinge els to say, but a figure of three corners. And thys is a generall rule, looke how many lynes any figure hath, so mannye corners it hath also, yf it bee a platte forme, and not a bodye. For a bodye hath dyuers lynes metyng sometime in one corner.Now to geue you example of triangles,there is one whiche is all of croked lynes, and may be taken fur a portiõ of a globe as the figur marked wtA.An other hath two compassed lines and one right lyne, and is as the portiõ of halfe a globe, example ofB.An otherhathbut one compassedlyne, and is the quarter of a circle, named a quadrate, and the ryght lynes make a right corner, as you se inC.Otherlessethen it asyou seD,whose right lines make a sharpe corner, or greater then a quadrate, as isF,and then the right lynes of it do make a blunt corner.see caption and textsee caption and textsee caption and textAlso some triangles haue all righte lynes and they be distincted in sonder by their angles, orcorners. forother their corners bee all sharpe, as you see in the figure,E.other ij. sharpe and one blunt, as is the figureG.other ij. sharp and one blunt as in the figureH.see caption and textThere is also an other distinction of the names of triangles, according to their sides, whiche other be all equal as in the figureE,and that the Greekes doo callIsopleuron,ἰσόπλευρομ.and Latine menæequilaterum: and in english it may be called athrelike triangle, other els two sydes bee equall and the thyrd vnequall, which the Greekes callIsosceles,ισόσκελεσ.the Latine menæquicurio, and in englishtweylekemay they be called, as inG, H,andK.For, they may be of iij. kinds that is to say, with one square angle, as isG,or with a blunte corner asH,or with all in sharpe korners, as you see inK.see caption and textsee caption and textsee caption and textsee caption and textFurther more it may be ytthey haue neuer a one syde equall to an other, and they be in iij kyndes also distinct lyke the twilekes, as you maye perceaue by these examples.M. N,andO.whereM.hath a right angle,N,ablunte angle, andO,all sharpe anglesσκαλενὄμ.these the Greekes and latine men docalscalenaand in englishe theye may be callednouelekes, for thei haue no side equall, or like lõg, to ani other in the same figur. Here it is to be noted, that in a triãgle al the angles bee calledinnerãglesexcept ani side bee drawenne forth in lengthe, for then is that fourthe corner caled anvtter corner, as in this exãple becauseA.B,is drawen in length, therfore the ãgleC,is called an vtter ãgle.see caption and textsee caption and textsee caption and textsee caption and textQuadrãgleAnd thus haue I done with triãguled figures, and nowe folowethquadrangles, which are figures of iiij. corners and of iiij. lines also, of whiche there be diuers kindes, but chiefely v. that is to say,A square quadrate.asquare quadrate, whose sides bee all equall, and al the angles square, as you se here in this figureQ.A longe square.The second kind is called a long square, whose foure corners be all square, but the sides are not equall eche to other, yet is euery side equall to that other that is against it, as you maye perceaue in this figure.R.A losengesee caption and textThe thyrd kind is calledlosengesA diamõd.ordiamondes, whose sides bee all equall, but it hath neuer a square corner, for two of them be sharpe, and the other two be blunt, as appeareth in.S.see caption and textThe iiij. sorte are like vnto losenges, saue that they are longer one waye, and their sides be not equal, yet ther corners are like the corners of a losing, and therfore ar they namedA losenge lyke.losengelikeordiamõdlike, whose figur is noted with T.Here shal you marke that al those squares which haue their sides al equal, may be called also for easy vnderstandinge,likesides, asQ.andS.and those that haue only the contrary sydes equal, asR.andT.haue, those wyll I calllikeiammys, for a difference.see caption and textsee textThe fift sorte doth containe all other fashions of foure cornered figurs, and ar called of the Grekestrapezia, of Latin mẽmensulæand of Arabitians,helmuariphe, they may be called in englisheborde formes,Borde formes.they haue no syde equall to an other as these examples shew, neither keepe they any rate in their corners, and therfore are they countedvnruled formes, and the other foure kindes onely are countedruled formes, in the kynde of quadrangles. Of these vnruled formes ther is no numbre, they are so mannye and so dyuers, yet by arte they may be changed into other kindes of figures, and therby be brought to measure and proportion, as in the thirtene conclusion is partly taught, but more plainly in my booke of measuring you may see it.see caption and textAnd nowe to make an eande of the dyuers kyndes of figures, there dothe folowe now figures of .v. sydes, other .v. corners, which we may callcink-angles, whose sydes partlye are allequallas inA,and those are countedruled cinkeangles, and partlye vnequall, asinB,and they are calledvnruled.see caption and textLikewyse shall you iudge ofsiseangles, which haue sixe corners,septangles, whiche haue seuen angles, and so forth, for as mannye numbres as there maye be of sydes and angles, so manye diuers kindes be there of figures, vnto which yow shall geue names according to the numbre of their sides and angles, of whiche for this tyme I wyll make an ende,A squyre.and wyll sette forthe on example of a syseangle, whiche I had almost forgotten, and that is it, whose vse commeth often in Geometry, and is called asquire, is made of two long squares ioyned togither, as this example sheweth.The globe as is before.see textAnd thus I make an eand to speake of platte formes, and will briefelye saye somwhat touching the figures ofbodeiswhich partly haue one platte forme for their bound, and ytiust roũd as aglobehath, or ended long as in anegge, and atunne 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 atune fourme,hath but one plat forme, and therfore must needs be round at the endes, where as atunnehath thre platte formes, and is flatte at eche end, as partly these pictures do shewe.Bodies of two plattes, are other cantles or halues of those other bodies, that haue but one platte forme, or els they are lyke in foorme to two such cantles ioyned togither as thisA.doth partlyexpresse:A rounde spier.or els it is called arounde spire, orstiple fourme, as in this figure is some what expressed.see caption and textsee caption and textNowe of three plattes there are made certain figures of bodyes, as the cantels and halues of all bodyes that haue but ij. plattys, and also the halues of halfe globys and canteles of a globe. Lykewyse a rounde piller, and a spyre made of a rounde spyre, slytte in ij. partes long ways.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 made an ende for this tyme, of the defi-nitions Geometricall, appertayning to thisparte of practise, and the rest wilI prosecute as cause shallserue.
THE PREFACE,declaring briefely the commodi-tes of Geometrye, and thenecessitye thereof.GEometrye may thinke it selfe to sustaine great iniury, if it shall be inforced other to show her manifold commodities, or els not to prease into the sight of men, and therefore might this wayes answere briefely: Other I am able to do you much good, or els but litle. If I bee able to doo you much good, then be you not your owne friendes, but greatlye your owne enemies to make so little of me, which maye profite you so muche. For if I were as vncurteous as you vnkind, I shuld vtterly refuse to do them any good, which will so curiously put me to the trial and profe of my commodities, or els to suffre exile, and namely sithe I shal only yeld benefites to other, and receaue none againe. But and if you could saye truely, that my benefites be nother many nor yet greate, yet if they bee anye, I doo yelde more to you, then I doo receaue againe of you, and therefore I oughte not to bee repelled of them that loue them selfe, althoughe they loue me not all for my selfe. But as I am in nature a liberall science, so canne I not againste nature contende with your inhumanitye, but muste shewe my selfe liberall euen to myne enemies. Yet this is my comforte againe, that I haue none enemies but them that knowe me not, and therefore may hurte themselues, but can not noye me. Yf they dispraise the thinge that they know not, all wise men will blame them and not credite them, and yf they thinke they knowe me, lette theym shewe one vntruthe and erroure in me, and I wyll geue the victorye.Yetcan no humayne science saie thus, but I onely, that there is no sparke of vntruthe in me: but all my doctrine and workes are without any blemishe of errour that mans reason can discerne. And nexte vnto me in certaintie are my three systers, Arithmetike, Musike, and Astronomie, whiche are also so nere knitte in amitee, that he that loueth the one, can not despise the other, and in especiall Geometrie, of whiche not only these thre, but all other artes do borow great ayde, as partly hereafter shall be shewed. But first will I beginne with the vnlearned sorte, that you maie perceiue how that no arte can stand without me. For if I should declare how many wayes my helpe is vsed, in measuryng of ground, for medow, corne, and wodde: in hedgyng, in dichyng, and in stackes makyng, I thinke the poore Husband man would be more thankefull vnto me, then he is nowe, whyles he thinketh that he hath small benefite by me. Yet this maie he coniecture certainly, that if he kepe not the rules of Geometrie, he can not measure any ground truely. And in dichyng, if he kepe not a proportion of bredth in the mouthe, to the bredthe of the bottome, and iuste slopenesse in the sides agreable to them bothe, the diche shall be faultie many waies. When he doth make stackes for corne, or for heye, he practiseth good Geometrie, els would thei not long stand: So that in some stakes, whiche stand on foure pillers, and yet made round, doe increase greatter and greatter a good height, and then againe turne smaller and smaller vnto the toppe: you maie see so good Geometrie, that it were very difficult to counterfaite the lyke in any kynde of buildyng. As for other infinite waies that he vseth my benefite, I ouerpasse for shortnesse.Carpenters, Karuers, Ioyners, and Masons, doe willingly acknowledge that they can worke nothyng without reason of Geometrie, in so muche that they chalenge me as a peculiare science for them. But in that they should do wrong to all other men, seyng euerie kynde of men haue som benefit by me, not only in buildyng, whiche is but other mennes costes, and the arte of Carpenters, Masons, and the other aforesayd, but in theirowne priuate profession, whereof to auoide tediousnes I make this rehersall.Sith Merchauntes by shippes great riches do winne,I may with good righte at their seate beginne.The Shippes on the sea with Saile and with Ore,were firste founde, and styll made, by Geometries lore.Their Compas, their Carde, their Pulleis, their Ankers,were founde by the skill of witty Geometers.To sette forth the Capstocke, and eche other parte,wold make a greate showe of Geometries arte.Carpenters, Caruers, Ioiners and Masons,Painters and Limners with suche occupations,Broderers, Goldesmithes, if they be cunning,Must yelde to Geometrye thankes for their learning.The Carte and the Plowe, who doth them well marke,Are made by good Geometrye. And so in the warkeOf Tailers and Shoomakers, in all shapes and fashion,The woorke is not praised, if it wante proportion.So weauers by Geometrye hade their foundacion,Their Loome is a frame of straunge imaginacion.The wheele that doth spinne, the stone that doth grind,The Myll that is driuen by water or winde,Are workes of Geometrye straunge in their trade,Fewe could them deuise, if they were vnmade.And all that is wrought by waight or by measure,without proofe of Geometry can neuer be sure.Clockes that be made the times to deuide,The wittiest inuencion that euer was spied,Nowe that they are common they are not regarded,The artes man contemned, the woorke vnrewarded.But if they were scarse, and one for a shewe,Made by Geometrye, then shoulde men know,That neuer was arte so wonderfull witty,So needefull to man, as is good Geometry.The firste findinge out of euery good arte,Seemed then vnto men so godly a parte,That no recompence might satisfye the finder,But to make him a god, and honoure him for euer.So Ceres and Pallas, and Mercury also,Eolus and Neptune, and many other mo,Were honoured as goddes, bicause they did teache,Firste tillage and weuinge and eloquent speache,Or windes to obserue, the seas to saile ouer,They were called goddes for their good indeuour.Then were men more thankefull in that golden age:This yron wolde nowe vngratefull in rage,Wyll yelde the thy reward for trauaile and paine,With sclaunderous reproch, and spitefull disdaine.Yet thoughe other men vnthankfull will be,Suruayers haue cause to make muche of me.And so haue all Lordes, that landes do possesse:But Tennaunted I feare will like me the lesse.Yet do I not wrong but measure all truely,All yelde the full right of euerye man iustely.Proportion Geometricall hath no man opprest,Yf anye bee wronged, I wishe it redrest.But now to procede with learned professions, in Logike and Rhetorike and all partes of phylosophy, there neadeth none other proofe then Aristotle his testimony, whiche without Geometry proueth almost nothinge. In Logike all his good syllogismes and demonstrations, hee declareth by the principles of Geometrye. In philosophye, nether motion, nor time, nor ayrye impressions could hee aptely declare, but by the helpe of Geometrye as his woorkes do witnes. Yea the faculties of the minde dothe hee expresse by similitude to figures of Geometrye. And in morall phylosophy he thought that iustice coulde not wel be taught, nor yet well executed without proportion geometricall. And this estimacion of Geometry he maye seeme to haue learned of his maister Plato, which without Geometrye wolde teache nothinge, nother wold admitte any to heare him, except he were experte in Geometry. And what merualle if he so muche estemed geometrye, seinge his opinion was, that Godde wasalwaiesworkingeby Geometrie? Whiche sentence Plutarche declareth at large. And although Platto do vse the helpe of Geometrye in all the most waighte matter of a common wealth, yet it is so generall in vse, that no small thinges almost can be wel done without it. And therfore saith he: that Geometrye is to be learned, if it were for none other cause, but that all other artes are bothe soner and more surely vnderstand by helpe of it.What greate help it dothe in physike, Galene doth so often and so copiousely declare, that no man whiche hath redde any booke almoste of his, can be ignorant thereof,in so much that he coulde neuer cure well a rounde vlcere, tyll reason geometricall dydde teache it hym. Hippocrates is earnest in admonyshynge that study of geometrie must prepare the way to physike, as well as to all other artes.I shoulde seeme somewhat to tedious, if I shoulde recken vp, howe the diuines also in all their mysteries of scripture doo vse healpe of geometrie: and also that lawyers can neuer vnderstande the hole lawe, no nor yet the firste title therof exactly without Geometrie. For if lawes can not well be established, nor iustice duelie executed without geometricall proportion, as bothe Plato in his Politike bokes, and Aristotle in his Moralles doo largely declare. Yea sithe Lycurgus that cheefe lawmaker amongest the Lacedemonians, is moste praised for that he didde chaunge the state of their common wealthe frome the proportion Arithmeticall to a proportion geometricall, whiche without knowledg of bothe he coulde not dooe, than is it easye to perceaue howe necessarie Geometrie is for the lawe and studentes thereof. And if I shall saie preciselie and freelie as I thinke, he is vtterlie destitute of all abilitee to iudge in anie arte, that is not sommewhat experte in the Theoremes of Geometrie.Andthat caused Galene to say of hym selfe, that he coulde neuer perceaue what a demonstration was, no not so muche, as whether there were any or none, tyll he had by geometrie gotten abilitee to vnderstande it, although he heard the beste teachers that were in his tyme. It shuld be to longe and nedelessealso to declare what helpe all other artes Mathematicall haue by geometrie, sith it is the grounde of all theyr certeintie, and no man studious in them is so doubtful therof, that he shall nede any persuasion to procure credite thereto. For he can not reade .ij. lines almoste in any mathematicall science, but he shall espie the nedefulnes of geometrie. But to auoyde tediousnesse I will make an ende hereof with that famous sentence of auncient Pythagoras, That who so will trauayle by learnyng to attayne wysedome, shall neuer approche to any excellencie without the artes mathematicall, and especially Arithmetike and Geometrie.And yf I shall somewhat speake of noble men, and gouernours of realmes, howe needefull Geometrye maye bee vnto them, then must I repete all that I haue sayde before, sithe in them ought all knowledge to abounde, namely that maye appertaine either to good gouernaunce in time of peace, eyther wittye pollicies in time of warre. For ministration of good lawes in time of peace Lycurgus example with the testimonies of Plato and Aristotle may suffise. And as for warres, I might thinke it sufficient that Vegetius hath written, and after him Valturius in commendation of Geometry, for vse of warres, but all their woordes seeme to saye nothinge, in comparison to the example of Archimedes worthy woorkes make by geometrie, for the defence of his countrey, to reade the wonderfull praise of his wittie deuises, set foorthe by the most famous hystories of Liuius, Plutarche, and Plinie, and all other hystoriographiers, whyche wryte of the stronge siege ofSyracusæmade by that valiant capitayne, and noble warriourMarcellus, whose power was so great, that all men meruayled how that one citee coulde withstande his wonderfull force so longe. But much more woulde they meruaile, if they vnderstode that one man onely dyd withstand all Marcellus strength, and with counter engines destroied his engines to the vtter astonyshment ofMarcellus, and all that were with hym. He had inuented suche balastelas that dyd shoote out a hundred dartes at oneshotte, to the great destruction ofMarcellussouldiours, wherby a fonde tale was spredde abrode, how that in Syracusæ there was a wonderfull gyant, whiche had a hundred handes, and coulde shoote a hundred dartes at ones. And as this fable was spredde of Archimedes, so many other haue been fayned to bee gyantes and monsters, bycause they dyd suche thynges, whiche farre passed the witte of the common people. So dyd they feyne Argus to haue a hundred eies, bicause they herde of his wonderfull circumspection, and thoughte that as it was aboue their capacitee, so it could not be, onlesse he had a hundred eies. So imagined they Ianus to haue two faces, one lokyng forwarde, and an other backwarde, bycause he coulde so wittily compare thynges paste with thynges that were to come, and so duely pondre them, as yf they were all present. Of like reasõ did they feyn Lynceus to haue such sharp syght, that he could see through walles and hylles, bycause peraduenture he dyd by naturall iudgement declare what cõmoditees myght be digged out of the grounde. And an infinite noumbre lyke fables are there, whiche sprange all of lyke reason.For what other thyng meaneth the fable of the great gyant Atlas, whiche was ymagined to beare vp heauen on his shulders? but that he was a man of so high a witte, that it reached vnto the skye, and was so skylfull in Astronomie, and coulde tell before hande of Eclipses, and other like thynges as truely as though he dyd rule the sterres, and gouerne the planettes.So was Eolus accompted god of the wyndes, and to haue theim all in a caue at his pleasure, by reason that he was a wittie man in naturall knowlege, and obserued well the change of wethers, aud was the fyrst that taught the obseruation of the wyndes. And lyke reson is to be geuen of al the old fables.But to retourne agayne to Archimedes, he dyd also by arte perspectiue (whiche is a parte of geometrie) deuise such glasses within the towne of Syracusæ, that dyd bourne their ennemies shyppes a great way from the towne, whyche was a meruaylous politike thynge. And if I shulde repete thevarietees of suche straunge inuentions, as Archimedes and others haue wrought by geometrie, I should not onely excede the order of a preface, but I should also speake of suche thynges as can not well be vnderstande in talke, without somme knowledge in the principles of geometrie.But this will I promyse, that if I may perceaue my paynes to be thankfully taken, I wyll not onely write of suche pleasant inuentions, declaryng what they were, but also wil teache howe a great numbre of them were wroughte, that they may be practised in this tyme also. Wherby shallbe plainly perceaued, that many thynges seme impossible to be done, whiche by arte may very well be wrought. And whan they be wrought, and the reason therof not vnderstande, than say the vulgare people, that those thynges are done bynegromancy. And hereof came it that fryer Bakon was accompted so greate anegromancier, whiche neuer vsed that arte (by any coniecture that I can fynde) but was in geometrie and other mathematicall sciences so experte, that he coulde dooe by theim suche thynges as were wonderfull in the syght of most people.Great talke there is of a glasse that he made in Oxforde, in whiche men myght see thynges that were doon in other places, and that was iudged to be done by power of euyll spirites. But I knowe the reason of it to bee good and naturall, and to be wrought by geometrie (sythe perspectiue is a parte of it) and to stande as well with reason as to see your face in cõmon glasse. But this conclusion and other dyuers of lyke sorte, are more mete for princes, for sundry causes, than for other men, and ought not to bee taught commonly. Yet to repete it, I thought good for this cause, that the worthynes of geometry myght the better be knowen, & partly vnderstanding geuen, what wonderfull thynges may be wrought by it, and so consequently how pleasant it is, and how necessary also.And thus for this tyme I make an end. The reason of som thynges done in this boke, or omitted in the same, you shall fynde in the preface before the Theoremes.*The definitions of the principles ofGEOMETRY.Geometry teacheth the drawyng,Measuringand proporcion offigures. butin as muche as no figure can bee drawen, but it muste haue certayne boũdes and inclosures of lines: and euery lyne also is begon and ended at some certaine prycke, fyrst it shal be meete to know these smaller partes of euery figure, that therby the whole figures may the better bee iudged, and distincte in sonder.A poincte.A Poynt or a Prycke, is named of Geometricians that small and vnsensible shape, whiche hath in it no partes, that is to say: nother length, breadth nor depth. But as their exactnes of definition is more meeter for onlye Theorike speculacion, then for practise and outwarde worke (consideringe that myne intent is to applye all these whole principles to woorke) I thynke meeter for this purpose, to call apoynt or prycke, that small printe of penne, pencyle, or other instrumente, whiche is not moued, nor drawen from his fyrst touche, and therfore hath no notable length nor bredthe: as this example doeth declare.∴Where I haue set .iij. prickes, eche of them hauyng both lẽgth and bredth, thogh it be but smal, and thefore not notable.Nowe of a great numbre of these prickes, is made aLyne, as you may perceiue by this forme ensuyng. ············ where as I haue set a numbre of prickes, so if you with your pen will set in more other prickes betweene euerye two of these,A lyne.then wil it be a lyne, as here you may seeand thislyne, is called of Geometricians,Lengthe withoute breadth.Butas they in theyr theorikes (which ar only mind workes)do precisely vnderstand these definitions, so it shal be sufficient for those men, whiche seke the vse of the same thinges, as sense may duely iudge them, and applye to handy workes if they vnderstand them so to be true, that outwarde sense canne fynde none erroure therein.Of lynes there bee two principall kyndes. The one is called a right or straight lyne, and the other a croked lyne.A streghte lyne.A Straight lyne, is the shortest that maye be drawenne between two prickes.A crokyd lyne.And all other lines, that go not right forth from prick to prick, but boweth any waye,such are calledCroked lynesas in these examples folowyng ye may se, where I haue set but one forme of a straight lyne, for more formes there be not, but of crooked lynes there bee innumerable diuersities, whereof for examples sum I haue sette here.see caption and textA right lyne.Croked lynes.see caption and textCroked lines.see caption and textsee caption and textSo now you must vnderstand, thateuery lyne is drawen betwene twoo prickes, wherof the one is at the beginning, and the other at the ende.see textTherefore when soeuer you do see any formes of lynes to touche at one notable pricke, as in this example, then shall younot call it onecrokedlyne, but rather twoo lynes:an Angle.in as muche as there is a notable and sensible angle by.A.whiche euermore is made by the meetyng of two seuerall lynes. And likewayes shall you iudge of this figure, whiche is made of two lines, and not of one onely.see textSo that whan so euer any suche meetyng of lines doth happen, the place of their metyng is called anAngle or corner.Of angles there be three generall kindes: a sharpe angle, a square angle, and a blunte angle.A righte angle.The square angle, whiche is commonly nameda right corner, is made of twoo lynes meetyng together in fourme of a squire, whiche two lines, if they be drawen forth in length, will crosse one an other: as in the examples folowyng you maie see.Right angles.see caption and textA sharpe corner.A sharpe angleis so called, because it is lesser than is a square angle, and the lines that make it, do not open so wide in their departynge as in a square corner, and if thei be drawen crosse, all fower corners will not be equall.A blunte angle.A blunt or brode corner, is greater then is a square angle, and his lines do parte more in sonder then in a right angle, of whiche all take these examples.And these angles (as you see) are made partly of streght lynes, partly of croken lines, and partly of both together. Howbeit in right angles I haue put none example of croked lines, because it wouldmuche trouble a lerner to iudge them: for their true iudgment doth appertaine to arte perspectiue, and as I may say, rather to reason then to sense.Sharpe angles.see caption and textBlunte or brode angles.see caption and textBut now as of many prickes there is made one line, soof diuerse lines are there made sundry formes, figures, and shapes, whiche all yet be called by one propre name,A platte forme.Platte formes, and thei haue bothelength and bredth, but yet no depenesse.Andthe boundesof euerie platte forme are lines: as by the examples you maie perceiue.Of platte formes some be plain, and some be croked, and some parly plaine, and partlie croked.A plaine platte.A plaine platteis that, whiche is made al equall in height, so that the middle partes nother bulke vp, nother shrink down more then the bothe endes.A crooked platte.For whan the one parte is higher then the other, then is it named aCroked platte.And if it be partlie plaine, and partlie crooked, then is it called aMyxte platte, of all whiche, these are exaumples.A plaine platte.A croked platte.A myxte platte.see caption and textsee caption and textsee caption and textAnd as of many prickes is made a line, and of diuerse lines one platte forme,A bodie.so of manie plattes is madea bodie, whiche conteignethLengthe, bredth, and depenesse.Depenesse.ByDepenesseI vnderstand, not as the common sort doth, the holownesse of any thing, as of a well, a diche, a potte, and suche like, but I meane the massie thicknesseof any bodie, as in exaumple of a potte: the depenesse is after the common name, the space from his brimme to his bottome. But as I take it here, the depenesse of his bodie is his thicknesse in the sides, whiche is an other thyng cleane different from the depenesse of his holownes, that the common people meaneth.Now all bodies haue platte formes for their boundes,Cubike.so in a dye (whiche is calleda cubike bodie) by geomatricians,Asheler.and anashlerof masons, there are .vi. sides, whiche are .vi. platte formes, and are the boundes of the dye.A globe.But in aGlobe, (whiche is a bodie rounde as a bowle) there is but one platte forme, and one bounde, and these are the exaumples of them bothe.A dye or ashler.A globe.see caption and textsee caption and textBut because you shall not muse what I dooe calla bound,A bounde.I mean therby a generall name, betokening the beginning, end and side, of any forme.Forme, Fygure.A forme, figure, or shape, is that thyng that is inclosed within one bond or manie bondes, so that you vnderstand that shape, that the eye doth discerne, and not the substance of the bodie.Offiguresthere be manie sortes, for either thei be made of prickes, lines, or platte formes. Not withstandyng to speake properlie,a figureis euer made by platte formes, and not of bare lines vnclosed, neither yet of prickes.Yet for the lighter forme of teachyng, it shall not be vnsemely to call all suche shapes, formes and figures, whiche yeeye maie discerne distinctly.And first to begin with prickes, there maie be made diuerse formes of them, as partely here doeth folowe.see captionA lynearic numbre.see captionTrianguler numbressee captionLongsquare nũbre.see captionIust square numbressee captiona threcornered spire.see captionA square spire.And so maie there be infinite formes more, whiche I omitte for this time, cõsidering that their knowledg appertaineth more to Arithmetike figurall, than to Geometrie.But yet one name of a pricke, whiche he taketh rather of his place then of his fourme, maie I not ouerpasse. And that is, when a pricke standeth in the middell of a circle (as no circle can be made by cõpasse without it) then is it calleda centre.A centreAnd thereof doe masons, and other worke menne call that patron, acentre, 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.see textLynes make diuerse figures also, though properly thei maie not be called figures, as I said before (vnles the lines do close) but onely for easie maner of teachyng, all shall be calledfigures, that the eye can discerne, of whiche this is one, when one line lyeth flatte (whiche is namedA ground line.theground line) and an other commeth downe on it, and is calledA perpendicular.A plume lyne.aperpendiculerorplũme lyne, as in this example you maysee. where.A.B.is the grounde line, andC.D.the plumbe line.see textAndlike waies in this figure there are three lines, the grounde lyne whiche isA.B.the plumme line that isA.C.and thebias line, whiche goeth from the one of thẽ to the other, and lieth against the right corner in such a figure whiche is here.C.B.Butconsideryng that I shall haue occasion to declare sundry figures anon, I will first shew some certaine varietees of lines that close no figures, but are bare lynes, and of the other lines will I make mencion in the description of the figures.tortuouse paralleles.ParallelysGemowe lynes.Paralleles, orgemowe lynesbe suche lines as be drawen foorth still in one distaunce, 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 calledbought lynes, and loe here exaumples of them bothe.parallelis.bought linessee caption and textsee caption and textparallelis:circular.Concen-trikes.see caption and textI haue added alsoparalleles tortuouse, whiche bowe cõtrarie waies with their two endes: andparalleles circular, whiche be lyke vnperfecte compasses: for if they bee whole circles,Concentrikesthen are they calledcõcentrikes, that is to saie, circles drawẽ on one centre.Heremight I note the error of goodAlbert Durer, which affirmeth that no perpendicular lines can beparalleles. whicherrour doeth spring partlie of ouersight of the difference of a streight line, and partlie of mistakyng certain principles geometrical, which al I wil let passe vntil an other tyme, and wil not blame him, which hath deserued worthyly infinite praise.And to returne to my matter.A twine line.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.And an other sorte of lines is there, that is called aspirall line,A worme line.or aworm line, whiche representeth an apparant forme of many circles, where there is not one in dede: of these .ii. kindes of lines, these be examples.Atwistelyne.see caption and textA spirail lynesee caption and textA touche lyne.A tuch line.A touche lyne, is a line that runneth a long by the edge of a circle, onely touching it, but doth not crosse the circumference of it, as in this exaumple you maie see.A corde,And when that a line doth crosse the edg of the circle, thẽ is it calleda cord, as you shall see anon in the speakynge of circles.Matche cornersIn the meane season must I not omit to declare what angles bee calledmatche corners, that is to saie, suche as stande directly one against the other, when twoo lines be drawen a crosse, as here appereth.Matche corner.WhereA.andB.are matche corners, so areC.andD.but notA.andC.notherD.andA.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.A circleis a figure made and enclosed with one line, and hath in the middell of it a pricke or centre, from whiche all the lines that be drawen to thecircumferenceare equall all in length, as here you see.Circumference.see textAnd the line that encloseth the whole compasse, is called thecircumference.A diameter.And all the lines that bee drawen crosse the circle, and goe by the centre, are nameddiameters, whose halfe, I meane from the center to the circumferenceany waie,Semidiameter.is called thesemidiameter, orhalfe diameter.A cord, or a stringlyne.see textBut and if the line goe crosse the circle, and passe beside the centre, then is it calleda corde, ora stryng line, as I said before, and as this exaumple sheweth: whereA.is the corde. And the compassed line that aunswereth to it,An archlineis calledan arche lyne,A bowline.ora bowe lyne, whiche here marked withB.and the diameter withC.see textsee textBut and if that part be separate from the rest of the circle (as in this exãple you see) then ar both partes called cãtelles,A cantlethe one thegreatter cantleasE.and the other thelesser cantle, asD.And if it be parted iuste by the centre (as you see inF.)A semyecirclethen is it called asemicircle, orhalfe compasse.see caption and textSometimes it happeneth that a cantle is cutte out with two lynes drawen from the centre to the circumference (asG.is)A nooke cantleand then maie it be called anooke cantle, and if it be not parted from the reste of the circle (as you see inH.)A nooke.then is it called anookeplainely without any addicion. And the compassed lyne in it is called anarche lyne, as the exaumple here doeth shewe.An arche.see caption and textNowe haue you heard as touchyng circles, meetely sufficient instruction, so that it should seme nedeles to speake any more of figures in that kynde, saue that there doeth yet remaine ij. formes of an imperfecte circle, for it is lyke a circle that were brused, and thereby did runne out endelong one waie, whiche forme Geometricians dooe call anAn egge fourme.egge forme, because it doeth represent the figure and shape of an egge duely proportioned (as this figure sheweth) hauyng the one ende greate then the other.A tunne or barrel formA tunne forme.see textAn egge formesee caption and textFor if it be lyke the figure of a circle pressed in length, and bothe endes lyke bygge, then is it called atunne forme, orbarrell 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,An axtre or axe lyne.and that is called anaxelyne, oraxtre. Howe be it properly that line that is called anaxtre, whiche gooeth throughe 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 namedpolis, wchyou 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.see textBut to returne to the diuersityes of figures that remayne vndeclared, the most simple of them ar such ones as be made but of two lynes, as are thecantle of a circle, and thehalfe circle, of which I haue spoken allready. Likewyse thehalfe of an egge forme, thecantle of an egge forme, thehalfe of a tunne fourme, and thecantle of a tunne fourme, and besyde these a figure moche like to a tunne fourne, saue that it is sharp couered at both the endes, and therfore doth consist of twoo lynes, where a tunne forme is made of one lyne,An yey fourmeand that figure is named anyey fourme.A trianglesee textsee textThe nexte kynd of figures are those that be made of .iij. lynes other be all right lynes, all crooked lynes, other some right and some crooked. But what fourme so euer they be of, they are named generallytriangles. fora triangleis nothinge els to say, but a figure of three corners. And thys is a generall rule, looke how many lynes any figure hath, so mannye corners it hath also, yf it bee a platte forme, and not a bodye. For a bodye hath dyuers lynes metyng sometime in one corner.Now to geue you example of triangles,there is one whiche is all of croked lynes, and may be taken fur a portiõ of a globe as the figur marked wtA.An other hath two compassed lines and one right lyne, and is as the portiõ of halfe a globe, example ofB.An otherhathbut one compassedlyne, and is the quarter of a circle, named a quadrate, and the ryght lynes make a right corner, as you se inC.Otherlessethen it asyou seD,whose right lines make a sharpe corner, or greater then a quadrate, as isF,and then the right lynes of it do make a blunt corner.see caption and textsee caption and textsee caption and textAlso some triangles haue all righte lynes and they be distincted in sonder by their angles, orcorners. forother their corners bee all sharpe, as you see in the figure,E.other ij. sharpe and one blunt, as is the figureG.other ij. sharp and one blunt as in the figureH.see caption and textThere is also an other distinction of the names of triangles, according to their sides, whiche other be all equal as in the figureE,and that the Greekes doo callIsopleuron,ἰσόπλευρομ.and Latine menæequilaterum: and in english it may be called athrelike triangle, other els two sydes bee equall and the thyrd vnequall, which the Greekes callIsosceles,ισόσκελεσ.the Latine menæquicurio, and in englishtweylekemay they be called, as inG, H,andK.For, they may be of iij. kinds that is to say, with one square angle, as isG,or with a blunte corner asH,or with all in sharpe korners, as you see inK.see caption and textsee caption and textsee caption and textsee caption and textFurther more it may be ytthey haue neuer a one syde equall to an other, and they be in iij kyndes also distinct lyke the twilekes, as you maye perceaue by these examples.M. N,andO.whereM.hath a right angle,N,ablunte angle, andO,all sharpe anglesσκαλενὄμ.these the Greekes and latine men docalscalenaand in englishe theye may be callednouelekes, for thei haue no side equall, or like lõg, to ani other in the same figur. Here it is to be noted, that in a triãgle al the angles bee calledinnerãglesexcept ani side bee drawenne forth in lengthe, for then is that fourthe corner caled anvtter corner, as in this exãple becauseA.B,is drawen in length, therfore the ãgleC,is called an vtter ãgle.see caption and textsee caption and textsee caption and textsee caption and textQuadrãgleAnd thus haue I done with triãguled figures, and nowe folowethquadrangles, which are figures of iiij. corners and of iiij. lines also, of whiche there be diuers kindes, but chiefely v. that is to say,A square quadrate.asquare quadrate, whose sides bee all equall, and al the angles square, as you se here in this figureQ.A longe square.The second kind is called a long square, whose foure corners be all square, but the sides are not equall eche to other, yet is euery side equall to that other that is against it, as you maye perceaue in this figure.R.A losengesee caption and textThe thyrd kind is calledlosengesA diamõd.ordiamondes, whose sides bee all equall, but it hath neuer a square corner, for two of them be sharpe, and the other two be blunt, as appeareth in.S.see caption and textThe iiij. sorte are like vnto losenges, saue that they are longer one waye, and their sides be not equal, yet ther corners are like the corners of a losing, and therfore ar they namedA losenge lyke.losengelikeordiamõdlike, whose figur is noted with T.Here shal you marke that al those squares which haue their sides al equal, may be called also for easy vnderstandinge,likesides, asQ.andS.and those that haue only the contrary sydes equal, asR.andT.haue, those wyll I calllikeiammys, for a difference.see caption and textsee textThe fift sorte doth containe all other fashions of foure cornered figurs, and ar called of the Grekestrapezia, of Latin mẽmensulæand of Arabitians,helmuariphe, they may be called in englisheborde formes,Borde formes.they haue no syde equall to an other as these examples shew, neither keepe they any rate in their corners, and therfore are they countedvnruled formes, and the other foure kindes onely are countedruled formes, in the kynde of quadrangles. Of these vnruled formes ther is no numbre, they are so mannye and so dyuers, yet by arte they may be changed into other kindes of figures, and therby be brought to measure and proportion, as in the thirtene conclusion is partly taught, but more plainly in my booke of measuring you may see it.see caption and textAnd nowe to make an eande of the dyuers kyndes of figures, there dothe folowe now figures of .v. sydes, other .v. corners, which we may callcink-angles, whose sydes partlye are allequallas inA,and those are countedruled cinkeangles, and partlye vnequall, asinB,and they are calledvnruled.see caption and textLikewyse shall you iudge ofsiseangles, which haue sixe corners,septangles, whiche haue seuen angles, and so forth, for as mannye numbres as there maye be of sydes and angles, so manye diuers kindes be there of figures, vnto which yow shall geue names according to the numbre of their sides and angles, of whiche for this tyme I wyll make an ende,A squyre.and wyll sette forthe on example of a syseangle, whiche I had almost forgotten, and that is it, whose vse commeth often in Geometry, and is called asquire, is made of two long squares ioyned togither, as this example sheweth.The globe as is before.see textAnd thus I make an eand to speake of platte formes, and will briefelye saye somwhat touching the figures ofbodeiswhich partly haue one platte forme for their bound, and ytiust roũd as aglobehath, or ended long as in anegge, and atunne 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 atune fourme,hath but one plat forme, and therfore must needs be round at the endes, where as atunnehath thre platte formes, and is flatte at eche end, as partly these pictures do shewe.Bodies of two plattes, are other cantles or halues of those other bodies, that haue but one platte forme, or els they are lyke in foorme to two such cantles ioyned togither as thisA.doth partlyexpresse:A rounde spier.or els it is called arounde spire, orstiple fourme, as in this figure is some what expressed.see caption and textsee caption and textNowe of three plattes there are made certain figures of bodyes, as the cantels and halues of all bodyes that haue but ij. plattys, and also the halues of halfe globys and canteles of a globe. Lykewyse a rounde piller, and a spyre made of a rounde spyre, slytte in ij. partes long ways.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 made an ende for this tyme, of the defi-nitions Geometricall, appertayning to thisparte of practise, and the rest wilI prosecute as cause shallserue.
GEometrye may thinke it selfe to sustaine great iniury, if it shall be inforced other to show her manifold commodities, or els not to prease into the sight of men, and therefore might this wayes answere briefely: Other I am able to do you much good, or els but litle. If I bee able to doo you much good, then be you not your owne friendes, but greatlye your owne enemies to make so little of me, which maye profite you so muche. For if I were as vncurteous as you vnkind, I shuld vtterly refuse to do them any good, which will so curiously put me to the trial and profe of my commodities, or els to suffre exile, and namely sithe I shal only yeld benefites to other, and receaue none againe. But and if you could saye truely, that my benefites be nother many nor yet greate, yet if they bee anye, I doo yelde more to you, then I doo receaue againe of you, and therefore I oughte not to bee repelled of them that loue them selfe, althoughe they loue me not all for my selfe. But as I am in nature a liberall science, so canne I not againste nature contende with your inhumanitye, but muste shewe my selfe liberall euen to myne enemies. Yet this is my comforte againe, that I haue none enemies but them that knowe me not, and therefore may hurte themselues, but can not noye me. Yf they dispraise the thinge that they know not, all wise men will blame them and not credite them, and yf they thinke they knowe me, lette theym shewe one vntruthe and erroure in me, and I wyll geue the victorye.
Yetcan no humayne science saie thus, but I onely, that there is no sparke of vntruthe in me: but all my doctrine and workes are without any blemishe of errour that mans reason can discerne. And nexte vnto me in certaintie are my three systers, Arithmetike, Musike, and Astronomie, whiche are also so nere knitte in amitee, that he that loueth the one, can not despise the other, and in especiall Geometrie, of whiche not only these thre, but all other artes do borow great ayde, as partly hereafter shall be shewed. But first will I beginne with the vnlearned sorte, that you maie perceiue how that no arte can stand without me. For if I should declare how many wayes my helpe is vsed, in measuryng of ground, for medow, corne, and wodde: in hedgyng, in dichyng, and in stackes makyng, I thinke the poore Husband man would be more thankefull vnto me, then he is nowe, whyles he thinketh that he hath small benefite by me. Yet this maie he coniecture certainly, that if he kepe not the rules of Geometrie, he can not measure any ground truely. And in dichyng, if he kepe not a proportion of bredth in the mouthe, to the bredthe of the bottome, and iuste slopenesse in the sides agreable to them bothe, the diche shall be faultie many waies. When he doth make stackes for corne, or for heye, he practiseth good Geometrie, els would thei not long stand: So that in some stakes, whiche stand on foure pillers, and yet made round, doe increase greatter and greatter a good height, and then againe turne smaller and smaller vnto the toppe: you maie see so good Geometrie, that it were very difficult to counterfaite the lyke in any kynde of buildyng. As for other infinite waies that he vseth my benefite, I ouerpasse for shortnesse.
Carpenters, Karuers, Ioyners, and Masons, doe willingly acknowledge that they can worke nothyng without reason of Geometrie, in so muche that they chalenge me as a peculiare science for them. But in that they should do wrong to all other men, seyng euerie kynde of men haue som benefit by me, not only in buildyng, whiche is but other mennes costes, and the arte of Carpenters, Masons, and the other aforesayd, but in theirowne priuate profession, whereof to auoide tediousnes I make this rehersall.
Sith Merchauntes by shippes great riches do winne,I may with good righte at their seate beginne.The Shippes on the sea with Saile and with Ore,were firste founde, and styll made, by Geometries lore.Their Compas, their Carde, their Pulleis, their Ankers,were founde by the skill of witty Geometers.To sette forth the Capstocke, and eche other parte,wold make a greate showe of Geometries arte.Carpenters, Caruers, Ioiners and Masons,Painters and Limners with suche occupations,Broderers, Goldesmithes, if they be cunning,Must yelde to Geometrye thankes for their learning.The Carte and the Plowe, who doth them well marke,Are made by good Geometrye. And so in the warkeOf Tailers and Shoomakers, in all shapes and fashion,The woorke is not praised, if it wante proportion.So weauers by Geometrye hade their foundacion,Their Loome is a frame of straunge imaginacion.The wheele that doth spinne, the stone that doth grind,The Myll that is driuen by water or winde,Are workes of Geometrye straunge in their trade,Fewe could them deuise, if they were vnmade.And all that is wrought by waight or by measure,without proofe of Geometry can neuer be sure.Clockes that be made the times to deuide,The wittiest inuencion that euer was spied,Nowe that they are common they are not regarded,The artes man contemned, the woorke vnrewarded.But if they were scarse, and one for a shewe,Made by Geometrye, then shoulde men know,That neuer was arte so wonderfull witty,So needefull to man, as is good Geometry.The firste findinge out of euery good arte,Seemed then vnto men so godly a parte,That no recompence might satisfye the finder,But to make him a god, and honoure him for euer.So Ceres and Pallas, and Mercury also,Eolus and Neptune, and many other mo,Were honoured as goddes, bicause they did teache,Firste tillage and weuinge and eloquent speache,Or windes to obserue, the seas to saile ouer,They were called goddes for their good indeuour.Then were men more thankefull in that golden age:This yron wolde nowe vngratefull in rage,Wyll yelde the thy reward for trauaile and paine,With sclaunderous reproch, and spitefull disdaine.Yet thoughe other men vnthankfull will be,Suruayers haue cause to make muche of me.And so haue all Lordes, that landes do possesse:But Tennaunted I feare will like me the lesse.Yet do I not wrong but measure all truely,All yelde the full right of euerye man iustely.Proportion Geometricall hath no man opprest,Yf anye bee wronged, I wishe it redrest.
Sith Merchauntes by shippes great riches do winne,
I may with good righte at their seate beginne.
The Shippes on the sea with Saile and with Ore,
were firste founde, and styll made, by Geometries lore.
Their Compas, their Carde, their Pulleis, their Ankers,
were founde by the skill of witty Geometers.
To sette forth the Capstocke, and eche other parte,
wold make a greate showe of Geometries arte.
Carpenters, Caruers, Ioiners and Masons,
Painters and Limners with suche occupations,
Broderers, Goldesmithes, if they be cunning,
Must yelde to Geometrye thankes for their learning.
The Carte and the Plowe, who doth them well marke,
Are made by good Geometrye. And so in the warke
Of Tailers and Shoomakers, in all shapes and fashion,
The woorke is not praised, if it wante proportion.
So weauers by Geometrye hade their foundacion,
Their Loome is a frame of straunge imaginacion.
The wheele that doth spinne, the stone that doth grind,
The Myll that is driuen by water or winde,
Are workes of Geometrye straunge in their trade,
Fewe could them deuise, if they were vnmade.
And all that is wrought by waight or by measure,
without proofe of Geometry can neuer be sure.
Clockes that be made the times to deuide,
The wittiest inuencion that euer was spied,
Nowe that they are common they are not regarded,
The artes man contemned, the woorke vnrewarded.
But if they were scarse, and one for a shewe,
Made by Geometrye, then shoulde men know,
That neuer was arte so wonderfull witty,
So needefull to man, as is good Geometry.
The firste findinge out of euery good arte,
Seemed then vnto men so godly a parte,
That no recompence might satisfye the finder,
But to make him a god, and honoure him for euer.
So Ceres and Pallas, and Mercury also,
Eolus and Neptune, and many other mo,
Were honoured as goddes, bicause they did teache,
Firste tillage and weuinge and eloquent speache,
Or windes to obserue, the seas to saile ouer,
They were called goddes for their good indeuour.
Then were men more thankefull in that golden age:
This yron wolde nowe vngratefull in rage,
Wyll yelde the thy reward for trauaile and paine,
With sclaunderous reproch, and spitefull disdaine.
Yet thoughe other men vnthankfull will be,
Suruayers haue cause to make muche of me.
And so haue all Lordes, that landes do possesse:
But Tennaunted I feare will like me the lesse.
Yet do I not wrong but measure all truely,
All yelde the full right of euerye man iustely.
Proportion Geometricall hath no man opprest,
Yf anye bee wronged, I wishe it redrest.
But now to procede with learned professions, in Logike and Rhetorike and all partes of phylosophy, there neadeth none other proofe then Aristotle his testimony, whiche without Geometry proueth almost nothinge. In Logike all his good syllogismes and demonstrations, hee declareth by the principles of Geometrye. In philosophye, nether motion, nor time, nor ayrye impressions could hee aptely declare, but by the helpe of Geometrye as his woorkes do witnes. Yea the faculties of the minde dothe hee expresse by similitude to figures of Geometrye. And in morall phylosophy he thought that iustice coulde not wel be taught, nor yet well executed without proportion geometricall. And this estimacion of Geometry he maye seeme to haue learned of his maister Plato, which without Geometrye wolde teache nothinge, nother wold admitte any to heare him, except he were experte in Geometry. And what merualle if he so muche estemed geometrye, seinge his opinion was, that Godde wasalwaiesworkingeby Geometrie? Whiche sentence Plutarche declareth at large. And although Platto do vse the helpe of Geometrye in all the most waighte matter of a common wealth, yet it is so generall in vse, that no small thinges almost can be wel done without it. And therfore saith he: that Geometrye is to be learned, if it were for none other cause, but that all other artes are bothe soner and more surely vnderstand by helpe of it.
What greate help it dothe in physike, Galene doth so often and so copiousely declare, that no man whiche hath redde any booke almoste of his, can be ignorant thereof,in so much that he coulde neuer cure well a rounde vlcere, tyll reason geometricall dydde teache it hym. Hippocrates is earnest in admonyshynge that study of geometrie must prepare the way to physike, as well as to all other artes.
I shoulde seeme somewhat to tedious, if I shoulde recken vp, howe the diuines also in all their mysteries of scripture doo vse healpe of geometrie: and also that lawyers can neuer vnderstande the hole lawe, no nor yet the firste title therof exactly without Geometrie. For if lawes can not well be established, nor iustice duelie executed without geometricall proportion, as bothe Plato in his Politike bokes, and Aristotle in his Moralles doo largely declare. Yea sithe Lycurgus that cheefe lawmaker amongest the Lacedemonians, is moste praised for that he didde chaunge the state of their common wealthe frome the proportion Arithmeticall to a proportion geometricall, whiche without knowledg of bothe he coulde not dooe, than is it easye to perceaue howe necessarie Geometrie is for the lawe and studentes thereof. And if I shall saie preciselie and freelie as I thinke, he is vtterlie destitute of all abilitee to iudge in anie arte, that is not sommewhat experte in the Theoremes of Geometrie.
Andthat caused Galene to say of hym selfe, that he coulde neuer perceaue what a demonstration was, no not so muche, as whether there were any or none, tyll he had by geometrie gotten abilitee to vnderstande it, although he heard the beste teachers that were in his tyme. It shuld be to longe and nedelessealso to declare what helpe all other artes Mathematicall haue by geometrie, sith it is the grounde of all theyr certeintie, and no man studious in them is so doubtful therof, that he shall nede any persuasion to procure credite thereto. For he can not reade .ij. lines almoste in any mathematicall science, but he shall espie the nedefulnes of geometrie. But to auoyde tediousnesse I will make an ende hereof with that famous sentence of auncient Pythagoras, That who so will trauayle by learnyng to attayne wysedome, shall neuer approche to any excellencie without the artes mathematicall, and especially Arithmetike and Geometrie.
And yf I shall somewhat speake of noble men, and gouernours of realmes, howe needefull Geometrye maye bee vnto them, then must I repete all that I haue sayde before, sithe in them ought all knowledge to abounde, namely that maye appertaine either to good gouernaunce in time of peace, eyther wittye pollicies in time of warre. For ministration of good lawes in time of peace Lycurgus example with the testimonies of Plato and Aristotle may suffise. And as for warres, I might thinke it sufficient that Vegetius hath written, and after him Valturius in commendation of Geometry, for vse of warres, but all their woordes seeme to saye nothinge, in comparison to the example of Archimedes worthy woorkes make by geometrie, for the defence of his countrey, to reade the wonderfull praise of his wittie deuises, set foorthe by the most famous hystories of Liuius, Plutarche, and Plinie, and all other hystoriographiers, whyche wryte of the stronge siege ofSyracusæmade by that valiant capitayne, and noble warriourMarcellus, whose power was so great, that all men meruayled how that one citee coulde withstande his wonderfull force so longe. But much more woulde they meruaile, if they vnderstode that one man onely dyd withstand all Marcellus strength, and with counter engines destroied his engines to the vtter astonyshment ofMarcellus, and all that were with hym. He had inuented suche balastelas that dyd shoote out a hundred dartes at oneshotte, to the great destruction ofMarcellussouldiours, wherby a fonde tale was spredde abrode, how that in Syracusæ there was a wonderfull gyant, whiche had a hundred handes, and coulde shoote a hundred dartes at ones. And as this fable was spredde of Archimedes, so many other haue been fayned to bee gyantes and monsters, bycause they dyd suche thynges, whiche farre passed the witte of the common people. So dyd they feyne Argus to haue a hundred eies, bicause they herde of his wonderfull circumspection, and thoughte that as it was aboue their capacitee, so it could not be, onlesse he had a hundred eies. So imagined they Ianus to haue two faces, one lokyng forwarde, and an other backwarde, bycause he coulde so wittily compare thynges paste with thynges that were to come, and so duely pondre them, as yf they were all present. Of like reasõ did they feyn Lynceus to haue such sharp syght, that he could see through walles and hylles, bycause peraduenture he dyd by naturall iudgement declare what cõmoditees myght be digged out of the grounde. And an infinite noumbre lyke fables are there, whiche sprange all of lyke reason.
For what other thyng meaneth the fable of the great gyant Atlas, whiche was ymagined to beare vp heauen on his shulders? but that he was a man of so high a witte, that it reached vnto the skye, and was so skylfull in Astronomie, and coulde tell before hande of Eclipses, and other like thynges as truely as though he dyd rule the sterres, and gouerne the planettes.
So was Eolus accompted god of the wyndes, and to haue theim all in a caue at his pleasure, by reason that he was a wittie man in naturall knowlege, and obserued well the change of wethers, aud was the fyrst that taught the obseruation of the wyndes. And lyke reson is to be geuen of al the old fables.
But to retourne agayne to Archimedes, he dyd also by arte perspectiue (whiche is a parte of geometrie) deuise such glasses within the towne of Syracusæ, that dyd bourne their ennemies shyppes a great way from the towne, whyche was a meruaylous politike thynge. And if I shulde repete thevarietees of suche straunge inuentions, as Archimedes and others haue wrought by geometrie, I should not onely excede the order of a preface, but I should also speake of suche thynges as can not well be vnderstande in talke, without somme knowledge in the principles of geometrie.
But this will I promyse, that if I may perceaue my paynes to be thankfully taken, I wyll not onely write of suche pleasant inuentions, declaryng what they were, but also wil teache howe a great numbre of them were wroughte, that they may be practised in this tyme also. Wherby shallbe plainly perceaued, that many thynges seme impossible to be done, whiche by arte may very well be wrought. And whan they be wrought, and the reason therof not vnderstande, than say the vulgare people, that those thynges are done bynegromancy. And hereof came it that fryer Bakon was accompted so greate anegromancier, whiche neuer vsed that arte (by any coniecture that I can fynde) but was in geometrie and other mathematicall sciences so experte, that he coulde dooe by theim suche thynges as were wonderfull in the syght of most people.
Great talke there is of a glasse that he made in Oxforde, in whiche men myght see thynges that were doon in other places, and that was iudged to be done by power of euyll spirites. But I knowe the reason of it to bee good and naturall, and to be wrought by geometrie (sythe perspectiue is a parte of it) and to stande as well with reason as to see your face in cõmon glasse. But this conclusion and other dyuers of lyke sorte, are more mete for princes, for sundry causes, than for other men, and ought not to bee taught commonly. Yet to repete it, I thought good for this cause, that the worthynes of geometry myght the better be knowen, & partly vnderstanding geuen, what wonderfull thynges may be wrought by it, and so consequently how pleasant it is, and how necessary also.
And thus for this tyme I make an end. The reason of som thynges done in this boke, or omitted in the same, you shall fynde in the preface before the Theoremes.*
The definitions of the principles ofGEOMETRY.Geometry teacheth the drawyng,Measuringand proporcion offigures. butin as muche as no figure can bee drawen, but it muste haue certayne boũdes and inclosures of lines: and euery lyne also is begon and ended at some certaine prycke, fyrst it shal be meete to know these smaller partes of euery figure, that therby the whole figures may the better bee iudged, and distincte in sonder.A poincte.A Poynt or a Prycke, is named of Geometricians that small and vnsensible shape, whiche hath in it no partes, that is to say: nother length, breadth nor depth. But as their exactnes of definition is more meeter for onlye Theorike speculacion, then for practise and outwarde worke (consideringe that myne intent is to applye all these whole principles to woorke) I thynke meeter for this purpose, to call apoynt or prycke, that small printe of penne, pencyle, or other instrumente, whiche is not moued, nor drawen from his fyrst touche, and therfore hath no notable length nor bredthe: as this example doeth declare.∴Where I haue set .iij. prickes, eche of them hauyng both lẽgth and bredth, thogh it be but smal, and thefore not notable.Nowe of a great numbre of these prickes, is made aLyne, as you may perceiue by this forme ensuyng. ············ where as I haue set a numbre of prickes, so if you with your pen will set in more other prickes betweene euerye two of these,A lyne.then wil it be a lyne, as here you may seeand thislyne, is called of Geometricians,Lengthe withoute breadth.Butas they in theyr theorikes (which ar only mind workes)do precisely vnderstand these definitions, so it shal be sufficient for those men, whiche seke the vse of the same thinges, as sense may duely iudge them, and applye to handy workes if they vnderstand them so to be true, that outwarde sense canne fynde none erroure therein.Of lynes there bee two principall kyndes. The one is called a right or straight lyne, and the other a croked lyne.A streghte lyne.A Straight lyne, is the shortest that maye be drawenne between two prickes.A crokyd lyne.And all other lines, that go not right forth from prick to prick, but boweth any waye,such are calledCroked lynesas in these examples folowyng ye may se, where I haue set but one forme of a straight lyne, for more formes there be not, but of crooked lynes there bee innumerable diuersities, whereof for examples sum I haue sette here.see caption and textA right lyne.Croked lynes.see caption and textCroked lines.see caption and textsee caption and textSo now you must vnderstand, thateuery lyne is drawen betwene twoo prickes, wherof the one is at the beginning, and the other at the ende.see textTherefore when soeuer you do see any formes of lynes to touche at one notable pricke, as in this example, then shall younot call it onecrokedlyne, but rather twoo lynes:an Angle.in as muche as there is a notable and sensible angle by.A.whiche euermore is made by the meetyng of two seuerall lynes. And likewayes shall you iudge of this figure, whiche is made of two lines, and not of one onely.see textSo that whan so euer any suche meetyng of lines doth happen, the place of their metyng is called anAngle or corner.Of angles there be three generall kindes: a sharpe angle, a square angle, and a blunte angle.A righte angle.The square angle, whiche is commonly nameda right corner, is made of twoo lynes meetyng together in fourme of a squire, whiche two lines, if they be drawen forth in length, will crosse one an other: as in the examples folowyng you maie see.Right angles.see caption and textA sharpe corner.A sharpe angleis so called, because it is lesser than is a square angle, and the lines that make it, do not open so wide in their departynge as in a square corner, and if thei be drawen crosse, all fower corners will not be equall.A blunte angle.A blunt or brode corner, is greater then is a square angle, and his lines do parte more in sonder then in a right angle, of whiche all take these examples.And these angles (as you see) are made partly of streght lynes, partly of croken lines, and partly of both together. Howbeit in right angles I haue put none example of croked lines, because it wouldmuche trouble a lerner to iudge them: for their true iudgment doth appertaine to arte perspectiue, and as I may say, rather to reason then to sense.Sharpe angles.see caption and textBlunte or brode angles.see caption and textBut now as of many prickes there is made one line, soof diuerse lines are there made sundry formes, figures, and shapes, whiche all yet be called by one propre name,A platte forme.Platte formes, and thei haue bothelength and bredth, but yet no depenesse.Andthe boundesof euerie platte forme are lines: as by the examples you maie perceiue.Of platte formes some be plain, and some be croked, and some parly plaine, and partlie croked.A plaine platte.A plaine platteis that, whiche is made al equall in height, so that the middle partes nother bulke vp, nother shrink down more then the bothe endes.A crooked platte.For whan the one parte is higher then the other, then is it named aCroked platte.And if it be partlie plaine, and partlie crooked, then is it called aMyxte platte, of all whiche, these are exaumples.A plaine platte.A croked platte.A myxte platte.see caption and textsee caption and textsee caption and textAnd as of many prickes is made a line, and of diuerse lines one platte forme,A bodie.so of manie plattes is madea bodie, whiche conteignethLengthe, bredth, and depenesse.Depenesse.ByDepenesseI vnderstand, not as the common sort doth, the holownesse of any thing, as of a well, a diche, a potte, and suche like, but I meane the massie thicknesseof any bodie, as in exaumple of a potte: the depenesse is after the common name, the space from his brimme to his bottome. But as I take it here, the depenesse of his bodie is his thicknesse in the sides, whiche is an other thyng cleane different from the depenesse of his holownes, that the common people meaneth.Now all bodies haue platte formes for their boundes,Cubike.so in a dye (whiche is calleda cubike bodie) by geomatricians,Asheler.and anashlerof masons, there are .vi. sides, whiche are .vi. platte formes, and are the boundes of the dye.A globe.But in aGlobe, (whiche is a bodie rounde as a bowle) there is but one platte forme, and one bounde, and these are the exaumples of them bothe.A dye or ashler.A globe.see caption and textsee caption and textBut because you shall not muse what I dooe calla bound,A bounde.I mean therby a generall name, betokening the beginning, end and side, of any forme.Forme, Fygure.A forme, figure, or shape, is that thyng that is inclosed within one bond or manie bondes, so that you vnderstand that shape, that the eye doth discerne, and not the substance of the bodie.Offiguresthere be manie sortes, for either thei be made of prickes, lines, or platte formes. Not withstandyng to speake properlie,a figureis euer made by platte formes, and not of bare lines vnclosed, neither yet of prickes.Yet for the lighter forme of teachyng, it shall not be vnsemely to call all suche shapes, formes and figures, whiche yeeye maie discerne distinctly.And first to begin with prickes, there maie be made diuerse formes of them, as partely here doeth folowe.see captionA lynearic numbre.see captionTrianguler numbressee captionLongsquare nũbre.see captionIust square numbressee captiona threcornered spire.see captionA square spire.And so maie there be infinite formes more, whiche I omitte for this time, cõsidering that their knowledg appertaineth more to Arithmetike figurall, than to Geometrie.But yet one name of a pricke, whiche he taketh rather of his place then of his fourme, maie I not ouerpasse. And that is, when a pricke standeth in the middell of a circle (as no circle can be made by cõpasse without it) then is it calleda centre.A centreAnd thereof doe masons, and other worke menne call that patron, acentre, 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.see textLynes make diuerse figures also, though properly thei maie not be called figures, as I said before (vnles the lines do close) but onely for easie maner of teachyng, all shall be calledfigures, that the eye can discerne, of whiche this is one, when one line lyeth flatte (whiche is namedA ground line.theground line) and an other commeth downe on it, and is calledA perpendicular.A plume lyne.aperpendiculerorplũme lyne, as in this example you maysee. where.A.B.is the grounde line, andC.D.the plumbe line.see textAndlike waies in this figure there are three lines, the grounde lyne whiche isA.B.the plumme line that isA.C.and thebias line, whiche goeth from the one of thẽ to the other, and lieth against the right corner in such a figure whiche is here.C.B.Butconsideryng that I shall haue occasion to declare sundry figures anon, I will first shew some certaine varietees of lines that close no figures, but are bare lynes, and of the other lines will I make mencion in the description of the figures.tortuouse paralleles.ParallelysGemowe lynes.Paralleles, orgemowe lynesbe suche lines as be drawen foorth still in one distaunce, 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 calledbought lynes, and loe here exaumples of them bothe.parallelis.bought linessee caption and textsee caption and textparallelis:circular.Concen-trikes.see caption and textI haue added alsoparalleles tortuouse, whiche bowe cõtrarie waies with their two endes: andparalleles circular, whiche be lyke vnperfecte compasses: for if they bee whole circles,Concentrikesthen are they calledcõcentrikes, that is to saie, circles drawẽ on one centre.Heremight I note the error of goodAlbert Durer, which affirmeth that no perpendicular lines can beparalleles. whicherrour doeth spring partlie of ouersight of the difference of a streight line, and partlie of mistakyng certain principles geometrical, which al I wil let passe vntil an other tyme, and wil not blame him, which hath deserued worthyly infinite praise.And to returne to my matter.A twine line.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.And an other sorte of lines is there, that is called aspirall line,A worme line.or aworm line, whiche representeth an apparant forme of many circles, where there is not one in dede: of these .ii. kindes of lines, these be examples.Atwistelyne.see caption and textA spirail lynesee caption and textA touche lyne.A tuch line.A touche lyne, is a line that runneth a long by the edge of a circle, onely touching it, but doth not crosse the circumference of it, as in this exaumple you maie see.A corde,And when that a line doth crosse the edg of the circle, thẽ is it calleda cord, as you shall see anon in the speakynge of circles.Matche cornersIn the meane season must I not omit to declare what angles bee calledmatche corners, that is to saie, suche as stande directly one against the other, when twoo lines be drawen a crosse, as here appereth.Matche corner.WhereA.andB.are matche corners, so areC.andD.but notA.andC.notherD.andA.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.A circleis a figure made and enclosed with one line, and hath in the middell of it a pricke or centre, from whiche all the lines that be drawen to thecircumferenceare equall all in length, as here you see.Circumference.see textAnd the line that encloseth the whole compasse, is called thecircumference.A diameter.And all the lines that bee drawen crosse the circle, and goe by the centre, are nameddiameters, whose halfe, I meane from the center to the circumferenceany waie,Semidiameter.is called thesemidiameter, orhalfe diameter.A cord, or a stringlyne.see textBut and if the line goe crosse the circle, and passe beside the centre, then is it calleda corde, ora stryng line, as I said before, and as this exaumple sheweth: whereA.is the corde. And the compassed line that aunswereth to it,An archlineis calledan arche lyne,A bowline.ora bowe lyne, whiche here marked withB.and the diameter withC.see textsee textBut and if that part be separate from the rest of the circle (as in this exãple you see) then ar both partes called cãtelles,A cantlethe one thegreatter cantleasE.and the other thelesser cantle, asD.And if it be parted iuste by the centre (as you see inF.)A semyecirclethen is it called asemicircle, orhalfe compasse.see caption and textSometimes it happeneth that a cantle is cutte out with two lynes drawen from the centre to the circumference (asG.is)A nooke cantleand then maie it be called anooke cantle, and if it be not parted from the reste of the circle (as you see inH.)A nooke.then is it called anookeplainely without any addicion. And the compassed lyne in it is called anarche lyne, as the exaumple here doeth shewe.An arche.see caption and textNowe haue you heard as touchyng circles, meetely sufficient instruction, so that it should seme nedeles to speake any more of figures in that kynde, saue that there doeth yet remaine ij. formes of an imperfecte circle, for it is lyke a circle that were brused, and thereby did runne out endelong one waie, whiche forme Geometricians dooe call anAn egge fourme.egge forme, because it doeth represent the figure and shape of an egge duely proportioned (as this figure sheweth) hauyng the one ende greate then the other.A tunne or barrel formA tunne forme.see textAn egge formesee caption and textFor if it be lyke the figure of a circle pressed in length, and bothe endes lyke bygge, then is it called atunne forme, orbarrell 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,An axtre or axe lyne.and that is called anaxelyne, oraxtre. Howe be it properly that line that is called anaxtre, whiche gooeth throughe 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 namedpolis, wchyou 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.see textBut to returne to the diuersityes of figures that remayne vndeclared, the most simple of them ar such ones as be made but of two lynes, as are thecantle of a circle, and thehalfe circle, of which I haue spoken allready. Likewyse thehalfe of an egge forme, thecantle of an egge forme, thehalfe of a tunne fourme, and thecantle of a tunne fourme, and besyde these a figure moche like to a tunne fourne, saue that it is sharp couered at both the endes, and therfore doth consist of twoo lynes, where a tunne forme is made of one lyne,An yey fourmeand that figure is named anyey fourme.A trianglesee textsee textThe nexte kynd of figures are those that be made of .iij. lynes other be all right lynes, all crooked lynes, other some right and some crooked. But what fourme so euer they be of, they are named generallytriangles. fora triangleis nothinge els to say, but a figure of three corners. And thys is a generall rule, looke how many lynes any figure hath, so mannye corners it hath also, yf it bee a platte forme, and not a bodye. For a bodye hath dyuers lynes metyng sometime in one corner.Now to geue you example of triangles,there is one whiche is all of croked lynes, and may be taken fur a portiõ of a globe as the figur marked wtA.An other hath two compassed lines and one right lyne, and is as the portiõ of halfe a globe, example ofB.An otherhathbut one compassedlyne, and is the quarter of a circle, named a quadrate, and the ryght lynes make a right corner, as you se inC.Otherlessethen it asyou seD,whose right lines make a sharpe corner, or greater then a quadrate, as isF,and then the right lynes of it do make a blunt corner.see caption and textsee caption and textsee caption and textAlso some triangles haue all righte lynes and they be distincted in sonder by their angles, orcorners. forother their corners bee all sharpe, as you see in the figure,E.other ij. sharpe and one blunt, as is the figureG.other ij. sharp and one blunt as in the figureH.see caption and textThere is also an other distinction of the names of triangles, according to their sides, whiche other be all equal as in the figureE,and that the Greekes doo callIsopleuron,ἰσόπλευρομ.and Latine menæequilaterum: and in english it may be called athrelike triangle, other els two sydes bee equall and the thyrd vnequall, which the Greekes callIsosceles,ισόσκελεσ.the Latine menæquicurio, and in englishtweylekemay they be called, as inG, H,andK.For, they may be of iij. kinds that is to say, with one square angle, as isG,or with a blunte corner asH,or with all in sharpe korners, as you see inK.see caption and textsee caption and textsee caption and textsee caption and textFurther more it may be ytthey haue neuer a one syde equall to an other, and they be in iij kyndes also distinct lyke the twilekes, as you maye perceaue by these examples.M. N,andO.whereM.hath a right angle,N,ablunte angle, andO,all sharpe anglesσκαλενὄμ.these the Greekes and latine men docalscalenaand in englishe theye may be callednouelekes, for thei haue no side equall, or like lõg, to ani other in the same figur. Here it is to be noted, that in a triãgle al the angles bee calledinnerãglesexcept ani side bee drawenne forth in lengthe, for then is that fourthe corner caled anvtter corner, as in this exãple becauseA.B,is drawen in length, therfore the ãgleC,is called an vtter ãgle.see caption and textsee caption and textsee caption and textsee caption and textQuadrãgleAnd thus haue I done with triãguled figures, and nowe folowethquadrangles, which are figures of iiij. corners and of iiij. lines also, of whiche there be diuers kindes, but chiefely v. that is to say,A square quadrate.asquare quadrate, whose sides bee all equall, and al the angles square, as you se here in this figureQ.A longe square.The second kind is called a long square, whose foure corners be all square, but the sides are not equall eche to other, yet is euery side equall to that other that is against it, as you maye perceaue in this figure.R.A losengesee caption and textThe thyrd kind is calledlosengesA diamõd.ordiamondes, whose sides bee all equall, but it hath neuer a square corner, for two of them be sharpe, and the other two be blunt, as appeareth in.S.see caption and textThe iiij. sorte are like vnto losenges, saue that they are longer one waye, and their sides be not equal, yet ther corners are like the corners of a losing, and therfore ar they namedA losenge lyke.losengelikeordiamõdlike, whose figur is noted with T.Here shal you marke that al those squares which haue their sides al equal, may be called also for easy vnderstandinge,likesides, asQ.andS.and those that haue only the contrary sydes equal, asR.andT.haue, those wyll I calllikeiammys, for a difference.see caption and textsee textThe fift sorte doth containe all other fashions of foure cornered figurs, and ar called of the Grekestrapezia, of Latin mẽmensulæand of Arabitians,helmuariphe, they may be called in englisheborde formes,Borde formes.they haue no syde equall to an other as these examples shew, neither keepe they any rate in their corners, and therfore are they countedvnruled formes, and the other foure kindes onely are countedruled formes, in the kynde of quadrangles. Of these vnruled formes ther is no numbre, they are so mannye and so dyuers, yet by arte they may be changed into other kindes of figures, and therby be brought to measure and proportion, as in the thirtene conclusion is partly taught, but more plainly in my booke of measuring you may see it.see caption and textAnd nowe to make an eande of the dyuers kyndes of figures, there dothe folowe now figures of .v. sydes, other .v. corners, which we may callcink-angles, whose sydes partlye are allequallas inA,and those are countedruled cinkeangles, and partlye vnequall, asinB,and they are calledvnruled.see caption and textLikewyse shall you iudge ofsiseangles, which haue sixe corners,septangles, whiche haue seuen angles, and so forth, for as mannye numbres as there maye be of sydes and angles, so manye diuers kindes be there of figures, vnto which yow shall geue names according to the numbre of their sides and angles, of whiche for this tyme I wyll make an ende,A squyre.and wyll sette forthe on example of a syseangle, whiche I had almost forgotten, and that is it, whose vse commeth often in Geometry, and is called asquire, is made of two long squares ioyned togither, as this example sheweth.The globe as is before.see textAnd thus I make an eand to speake of platte formes, and will briefelye saye somwhat touching the figures ofbodeiswhich partly haue one platte forme for their bound, and ytiust roũd as aglobehath, or ended long as in anegge, and atunne 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 atune fourme,hath but one plat forme, and therfore must needs be round at the endes, where as atunnehath thre platte formes, and is flatte at eche end, as partly these pictures do shewe.Bodies of two plattes, are other cantles or halues of those other bodies, that haue but one platte forme, or els they are lyke in foorme to two such cantles ioyned togither as thisA.doth partlyexpresse:A rounde spier.or els it is called arounde spire, orstiple fourme, as in this figure is some what expressed.see caption and textsee caption and textNowe of three plattes there are made certain figures of bodyes, as the cantels and halues of all bodyes that haue but ij. plattys, and also the halues of halfe globys and canteles of a globe. Lykewyse a rounde piller, and a spyre made of a rounde spyre, slytte in ij. partes long ways.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 made an ende for this tyme, of the defi-nitions Geometricall, appertayning to thisparte of practise, and the rest wilI prosecute as cause shallserue.
Geometry teacheth the drawyng,Measuringand proporcion offigures. butin as muche as no figure can bee drawen, but it muste haue certayne boũdes and inclosures of lines: and euery lyne also is begon and ended at some certaine prycke, fyrst it shal be meete to know these smaller partes of euery figure, that therby the whole figures may the better bee iudged, and distincte in sonder.
A poincte.A Poynt or a Prycke, is named of Geometricians that small and vnsensible shape, whiche hath in it no partes, that is to say: nother length, breadth nor depth. But as their exactnes of definition is more meeter for onlye Theorike speculacion, then for practise and outwarde worke (consideringe that myne intent is to applye all these whole principles to woorke) I thynke meeter for this purpose, to call apoynt or prycke, that small printe of penne, pencyle, or other instrumente, whiche is not moued, nor drawen from his fyrst touche, and therfore hath no notable length nor bredthe: as this example doeth declare.∴
Where I haue set .iij. prickes, eche of them hauyng both lẽgth and bredth, thogh it be but smal, and thefore not notable.
Nowe of a great numbre of these prickes, is made aLyne, as you may perceiue by this forme ensuyng. ············ where as I haue set a numbre of prickes, so if you with your pen will set in more other prickes betweene euerye two of these,A lyne.then wil it be a lyne, as here you may seeand thislyne, is called of Geometricians,Lengthe withoute breadth.
Butas they in theyr theorikes (which ar only mind workes)do precisely vnderstand these definitions, so it shal be sufficient for those men, whiche seke the vse of the same thinges, as sense may duely iudge them, and applye to handy workes if they vnderstand them so to be true, that outwarde sense canne fynde none erroure therein.
Of lynes there bee two principall kyndes. The one is called a right or straight lyne, and the other a croked lyne.
A streghte lyne.A Straight lyne, is the shortest that maye be drawenne between two prickes.
A crokyd lyne.And all other lines, that go not right forth from prick to prick, but boweth any waye,such are calledCroked lynesas in these examples folowyng ye may se, where I haue set but one forme of a straight lyne, for more formes there be not, but of crooked lynes there bee innumerable diuersities, whereof for examples sum I haue sette here.
see caption and textA right lyne.
So now you must vnderstand, thateuery lyne is drawen betwene twoo prickes, wherof the one is at the beginning, and the other at the ende.
see text
Therefore when soeuer you do see any formes of lynes to touche at one notable pricke, as in this example, then shall younot call it onecrokedlyne, but rather twoo lynes:an Angle.in as muche as there is a notable and sensible angle by.A.whiche euermore is made by the meetyng of two seuerall lynes. And likewayes shall you iudge of this figure, whiche is made of two lines, and not of one onely.
see text
So that whan so euer any suche meetyng of lines doth happen, the place of their metyng is called anAngle or corner.
Of angles there be three generall kindes: a sharpe angle, a square angle, and a blunte angle.A righte angle.The square angle, whiche is commonly nameda right corner, is made of twoo lynes meetyng together in fourme of a squire, whiche two lines, if they be drawen forth in length, will crosse one an other: as in the examples folowyng you maie see.
Right angles.see caption and text
A sharpe corner.A sharpe angleis so called, because it is lesser than is a square angle, and the lines that make it, do not open so wide in their departynge as in a square corner, and if thei be drawen crosse, all fower corners will not be equall.
A blunte angle.A blunt or brode corner, is greater then is a square angle, and his lines do parte more in sonder then in a right angle, of whiche all take these examples.
And these angles (as you see) are made partly of streght lynes, partly of croken lines, and partly of both together. Howbeit in right angles I haue put none example of croked lines, because it wouldmuche trouble a lerner to iudge them: for their true iudgment doth appertaine to arte perspectiue, and as I may say, rather to reason then to sense.
But now as of many prickes there is made one line, soof diuerse lines are there made sundry formes, figures, and shapes, whiche all yet be called by one propre name,A platte forme.Platte formes, and thei haue bothelength and bredth, but yet no depenesse.
Andthe boundesof euerie platte forme are lines: as by the examples you maie perceiue.
Of platte formes some be plain, and some be croked, and some parly plaine, and partlie croked.
A plaine platte.A plaine platteis that, whiche is made al equall in height, so that the middle partes nother bulke vp, nother shrink down more then the bothe endes.
A crooked platte.For whan the one parte is higher then the other, then is it named aCroked platte.
And if it be partlie plaine, and partlie crooked, then is it called aMyxte platte, of all whiche, these are exaumples.
And as of many prickes is made a line, and of diuerse lines one platte forme,A bodie.so of manie plattes is madea bodie, whiche conteignethLengthe, bredth, and depenesse.Depenesse.ByDepenesseI vnderstand, not as the common sort doth, the holownesse of any thing, as of a well, a diche, a potte, and suche like, but I meane the massie thicknesseof any bodie, as in exaumple of a potte: the depenesse is after the common name, the space from his brimme to his bottome. But as I take it here, the depenesse of his bodie is his thicknesse in the sides, whiche is an other thyng cleane different from the depenesse of his holownes, that the common people meaneth.
Now all bodies haue platte formes for their boundes,Cubike.so in a dye (whiche is calleda cubike bodie) by geomatricians,Asheler.and anashlerof masons, there are .vi. sides, whiche are .vi. platte formes, and are the boundes of the dye.
A globe.But in aGlobe, (whiche is a bodie rounde as a bowle) there is but one platte forme, and one bounde, and these are the exaumples of them bothe.
But because you shall not muse what I dooe calla bound,A bounde.I mean therby a generall name, betokening the beginning, end and side, of any forme.
Forme, Fygure.A forme, figure, or shape, is that thyng that is inclosed within one bond or manie bondes, so that you vnderstand that shape, that the eye doth discerne, and not the substance of the bodie.
Offiguresthere be manie sortes, for either thei be made of prickes, lines, or platte formes. Not withstandyng to speake properlie,a figureis euer made by platte formes, and not of bare lines vnclosed, neither yet of prickes.
Yet for the lighter forme of teachyng, it shall not be vnsemely to call all suche shapes, formes and figures, whiche yeeye maie discerne distinctly.
And first to begin with prickes, there maie be made diuerse formes of them, as partely here doeth folowe.
And so maie there be infinite formes more, whiche I omitte for this time, cõsidering that their knowledg appertaineth more to Arithmetike figurall, than to Geometrie.
But yet one name of a pricke, whiche he taketh rather of his place then of his fourme, maie I not ouerpasse. And that is, when a pricke standeth in the middell of a circle (as no circle can be made by cõpasse without it) then is it calleda centre.A centreAnd thereof doe masons, and other worke menne call that patron, acentre, 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.
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Lynes make diuerse figures also, though properly thei maie not be called figures, as I said before (vnles the lines do close) but onely for easie maner of teachyng, all shall be calledfigures, that the eye can discerne, of whiche this is one, when one line lyeth flatte (whiche is namedA ground line.theground line) and an other commeth downe on it, and is calledA perpendicular.A plume lyne.aperpendiculerorplũme lyne, as in this example you maysee. where.A.B.is the grounde line, andC.D.the plumbe line.
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Andlike waies in this figure there are three lines, the grounde lyne whiche isA.B.the plumme line that isA.C.and thebias line, whiche goeth from the one of thẽ to the other, and lieth against the right corner in such a figure whiche is here.C.B.
Butconsideryng that I shall haue occasion to declare sundry figures anon, I will first shew some certaine varietees of lines that close no figures, but are bare lynes, and of the other lines will I make mencion in the description of the figures.
tortuouse paralleles.
ParallelysGemowe lynes.Paralleles, orgemowe lynesbe suche lines as be drawen foorth still in one distaunce, 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 calledbought lynes, and loe here exaumples of them bothe.
I haue added alsoparalleles tortuouse, whiche bowe cõtrarie waies with their two endes: andparalleles circular, whiche be lyke vnperfecte compasses: for if they bee whole circles,Concentrikesthen are they calledcõcentrikes, that is to saie, circles drawẽ on one centre.
Heremight I note the error of goodAlbert Durer, which affirmeth that no perpendicular lines can beparalleles. whicherrour doeth spring partlie of ouersight of the difference of a streight line, and partlie of mistakyng certain principles geometrical, which al I wil let passe vntil an other tyme, and wil not blame him, which hath deserued worthyly infinite praise.
And to returne to my matter.A twine line.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.And an other sorte of lines is there, that is called aspirall line,A worme line.or aworm line, whiche representeth an apparant forme of many circles, where there is not one in dede: of these .ii. kindes of lines, these be examples.
A touche lyne.
A touche lyne, is a line that runneth a long by the edge of a circle, onely touching it, but doth not crosse the circumference of it, as in this exaumple you maie see.
And when that a line doth crosse the edg of the circle, thẽ is it calleda cord, as you shall see anon in the speakynge of circles.
In the meane season must I not omit to declare what angles bee calledmatche corners, that is to saie, suche as stande directly one against the other, when twoo lines be drawen a crosse, as here appereth.
Matche corner.
WhereA.andB.are matche corners, so areC.andD.but notA.andC.notherD.andA.
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 circleis a figure made and enclosed with one line, and hath in the middell of it a pricke or centre, from whiche all the lines that be drawen to thecircumferenceare equall all in length, as here you see.
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And the line that encloseth the whole compasse, is called thecircumference.
A diameter.And all the lines that bee drawen crosse the circle, and goe by the centre, are nameddiameters, whose halfe, I meane from the center to the circumferenceany waie,Semidiameter.is called thesemidiameter, orhalfe diameter.
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But and if the line goe crosse the circle, and passe beside the centre, then is it calleda corde, ora stryng line, as I said before, and as this exaumple sheweth: whereA.is the corde. And the compassed line that aunswereth to it,An archlineis calledan arche lyne,A bowline.ora bowe lyne, whiche here marked withB.and the diameter withC.
But and if that part be separate from the rest of the circle (as in this exãple you see) then ar both partes called cãtelles,A cantlethe one thegreatter cantleasE.and the other thelesser cantle, asD.And if it be parted iuste by the centre (as you see inF.)A semyecirclethen is it called asemicircle, orhalfe compasse.
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Sometimes it happeneth that a cantle is cutte out with two lynes drawen from the centre to the circumference (asG.is)A nooke cantleand then maie it be called anooke cantle, and if it be not parted from the reste of the circle (as you see inH.)A nooke.then is it called anookeplainely without any addicion. And the compassed lyne in it is called anarche lyne, as the exaumple here doeth shewe.
An arche.see caption and text
Nowe haue you heard as touchyng circles, meetely sufficient instruction, so that it should seme nedeles to speake any more of figures in that kynde, saue that there doeth yet remaine ij. formes of an imperfecte circle, for it is lyke a circle that were brused, and thereby did runne out endelong one waie, whiche forme Geometricians dooe call anAn egge fourme.egge forme, because it doeth represent the figure and shape of an egge duely proportioned (as this figure sheweth) hauyng the one ende greate then the other.
A tunne forme.see text
An egge formesee caption and text
For if it be lyke the figure of a circle pressed in length, and bothe endes lyke bygge, then is it called atunne forme, orbarrell 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,An axtre or axe lyne.and that is called anaxelyne, oraxtre. Howe be it properly that line that is called anaxtre, whiche gooeth throughe 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 namedpolis, wchyou 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.
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But to returne to the diuersityes of figures that remayne vndeclared, the most simple of them ar such ones as be made but of two lynes, as are thecantle of a circle, and thehalfe circle, of which I haue spoken allready. Likewyse thehalfe of an egge forme, thecantle of an egge forme, thehalfe of a tunne fourme, and thecantle of a tunne fourme, and besyde these a figure moche like to a tunne fourne, saue that it is sharp couered at both the endes, and therfore doth consist of twoo lynes, where a tunne forme is made of one lyne,An yey fourmeand that figure is named anyey fourme.
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The nexte kynd of figures are those that be made of .iij. lynes other be all right lynes, all crooked lynes, other some right and some crooked. But what fourme so euer they be of, they are named generallytriangles. fora triangleis nothinge els to say, but a figure of three corners. And thys is a generall rule, looke how many lynes any figure hath, so mannye corners it hath also, yf it bee a platte forme, and not a bodye. For a bodye hath dyuers lynes metyng sometime in one corner.
Now to geue you example of triangles,there is one whiche is all of croked lynes, and may be taken fur a portiõ of a globe as the figur marked wtA.
An other hath two compassed lines and one right lyne, and is as the portiõ of halfe a globe, example ofB.
An otherhathbut one compassedlyne, and is the quarter of a circle, named a quadrate, and the ryght lynes make a right corner, as you se inC.Otherlessethen it asyou seD,whose right lines make a sharpe corner, or greater then a quadrate, as isF,and then the right lynes of it do make a blunt corner.
Also some triangles haue all righte lynes and they be distincted in sonder by their angles, orcorners. forother their corners bee all sharpe, as you see in the figure,E.other ij. sharpe and one blunt, as is the figureG.other ij. sharp and one blunt as in the figureH.
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There is also an other distinction of the names of triangles, according to their sides, whiche other be all equal as in the figureE,and that the Greekes doo callIsopleuron,ἰσόπλευρομ.and Latine menæequilaterum: and in english it may be called athrelike triangle, other els two sydes bee equall and the thyrd vnequall, which the Greekes callIsosceles,ισόσκελεσ.the Latine menæquicurio, and in englishtweylekemay they be called, as inG, H,andK.For, they may be of iij. kinds that is to say, with one square angle, as isG,or with a blunte corner asH,or with all in sharpe korners, as you see inK.
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Further more it may be ytthey haue neuer a one syde equall to an other, and they be in iij kyndes also distinct lyke the twilekes, as you maye perceaue by these examples.M. N,andO.whereM.hath a right angle,N,ablunte angle, andO,all sharpe anglesσκαλενὄμ.these the Greekes and latine men docalscalenaand in englishe theye may be callednouelekes, for thei haue no side equall, or like lõg, to ani other in the same figur. Here it is to be noted, that in a triãgle al the angles bee calledinnerãglesexcept ani side bee drawenne forth in lengthe, for then is that fourthe corner caled anvtter corner, as in this exãple becauseA.B,is drawen in length, therfore the ãgleC,is called an vtter ãgle.
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QuadrãgleAnd thus haue I done with triãguled figures, and nowe folowethquadrangles, which are figures of iiij. corners and of iiij. lines also, of whiche there be diuers kindes, but chiefely v. that is to say,A square quadrate.asquare quadrate, whose sides bee all equall, and al the angles square, as you se here in this figureQ.A longe square.The second kind is called a long square, whose foure corners be all square, but the sides are not equall eche to other, yet is euery side equall to that other that is against it, as you maye perceaue in this figure.R.
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The thyrd kind is calledlosengesA diamõd.ordiamondes, whose sides bee all equall, but it hath neuer a square corner, for two of them be sharpe, and the other two be blunt, as appeareth in.S.
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The iiij. sorte are like vnto losenges, saue that they are longer one waye, and their sides be not equal, yet ther corners are like the corners of a losing, and therfore ar they namedA losenge lyke.losengelikeordiamõdlike, whose figur is noted with T.Here shal you marke that al those squares which haue their sides al equal, may be called also for easy vnderstandinge,likesides, asQ.andS.and those that haue only the contrary sydes equal, asR.andT.haue, those wyll I calllikeiammys, for a difference.
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The fift sorte doth containe all other fashions of foure cornered figurs, and ar called of the Grekestrapezia, of Latin mẽmensulæand of Arabitians,helmuariphe, they may be called in englisheborde formes,Borde formes.they haue no syde equall to an other as these examples shew, neither keepe they any rate in their corners, and therfore are they countedvnruled formes, and the other foure kindes onely are countedruled formes, in the kynde of quadrangles. Of these vnruled formes ther is no numbre, they are so mannye and so dyuers, yet by arte they may be changed into other kindes of figures, and therby be brought to measure and proportion, as in the thirtene conclusion is partly taught, but more plainly in my booke of measuring you may see it.
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And nowe to make an eande of the dyuers kyndes of figures, there dothe folowe now figures of .v. sydes, other .v. corners, which we may callcink-angles, whose sydes partlye are allequallas inA,and those are countedruled cinkeangles, and partlye vnequall, asinB,and they are calledvnruled.
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Likewyse shall you iudge ofsiseangles, which haue sixe corners,septangles, whiche haue seuen angles, and so forth, for as mannye numbres as there maye be of sydes and angles, so manye diuers kindes be there of figures, vnto which yow shall geue names according to the numbre of their sides and angles, of whiche for this tyme I wyll make an ende,A squyre.and wyll sette forthe on example of a syseangle, whiche I had almost forgotten, and that is it, whose vse commeth often in Geometry, and is called asquire, is made of two long squares ioyned togither, as this example sheweth.
The globe as is before.see text
And thus I make an eand to speake of platte formes, and will briefelye saye somwhat touching the figures ofbodeiswhich partly haue one platte forme for their bound, and ytiust roũd as aglobehath, or ended long as in anegge, and atunne 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 atune fourme,hath but one plat forme, and therfore must needs be round at the endes, where as atunnehath thre platte formes, and is flatte at eche end, as partly these pictures do shewe.
Bodies of two plattes, are other cantles or halues of those other bodies, that haue but one platte forme, or els they are lyke in foorme to two such cantles ioyned togither as thisA.doth partlyexpresse:A rounde spier.or els it is called arounde spire, orstiple fourme, as in this figure is some what expressed.
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Nowe of three plattes there are made certain figures of bodyes, as the cantels and halues of all bodyes that haue but ij. plattys, and also the halues of halfe globys and canteles of a globe. Lykewyse a rounde piller, and a spyre made of a rounde spyre, slytte in ij. partes long ways.
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 made an ende for this tyme, of the defi-nitions Geometricall, appertayning to thisparte of practise, and the rest wilI prosecute as cause shallserue.