The Path-Way to Knowledg, Containing the First Principles of Geometrie Part 1

The Path-Way to Knowledg.

by Robert Record.

TO THE GENTLE READER.

Excvse me, gentle reder if oughte be amisse, straung paths ar not trod? al truly at the first: the way muste needes be comberous, wher none hathe gone before. Where no man hathe geuen light, lighte is it to offend, but when the light is shewed ones, light is it to amende. If my light may so light some other, to espie and marke my faultes, I wish it may so lighten th?, that they may voide offence. Of staggeringe and s...o...b..inge, and vnconstaunt turmoilinge: often offending, and seldome amending, such vices to eschewe, and their fine wittes to shew that they may winne the praise, and I to hold the candle, whilest they their glorious works with eloquence sette forth, so cunningly inuented, so finely indited, that my bokes maie seme worthie to occupie no roome. For neither is mi wit so finelie filed, nother mi learning so largly lettred, nother yet mi laiser so quiet and vncbered, that I maie perform iustlie so learned a laboure or accordinglie to accomplishe so haulte an enforcement, yet maie I thinke thus: This candle did I light: this light haue I kindeled: that learned men maie se, to practise their pennes, their eloquence to aduaunce, to register their names in the booke of memorie I drew the platte rudelie, whereon thei maie builde, whom G.o.d hath indued with learning and liuelihod. For liuing by laboure doth learning so hinder, that learning serueth liuinge, whiche is a peruers trade. Yet as carefull familie shall cease hir cruell callinge, and suffre anie laiser to learninge to repaire, I will not cease from trauaile the pathe so to trade, that finer wittes maie fas.h.i.+on them selues with such glimsinge dull light, a more complete woorke at laiser to finisshe, with inuencion agreable, and aptnes of eloquence.

And this gentle reader I hartelie protest where erroure hathe happened I wisshe it redrest.

TO THE MOST NO- ble and puissaunt prince Edwarde the sixte by the grace of G.o.d, of En- gland Fraunce and Ireland kynge, de- fendour of the faithe, and of the Churche of England and Ire- lande in earth the su- preme head.

It is not vnknowen to youre maiestie, moste soueraigne lorde, what great disceptacion hath been amongest the wyttie men of all nacions, for the exacte knoweledge of true felicitie, bothe what it is, and wherein it consisteth: touchynge whiche thyng, their opinions almoste were as many in numbre, as were the persons of them, that either disputed or wrote thereof. But and if the diuersitie of opinions in the vulgar sort for placyng of their felicitie shall be considered also, the varietie shall be found so great, and the opinions so dissonant, yea plainly monsterouse, that no honest witte would vouchesafe to lose time in hearyng th?, or rather (as I may saie) no witte is of so exact remembrance, that can consider together the monsterouse mult.i.tude of them all. And yet not withstdyng this repugnant diuersitie, in two thynges do they all agree. First all do agre, that felicitie is and ought to be the stop and end of all their doynges, so that he that hath a full desire to any thyng how so euer it be estemed of other m?, yet he estemeth him self happie, if he maie obtain it: and contrary waies vnhappie if he can not attaine it. And therfore do all men put their whole studie to gette that thyng, wherin they haue perswaded them self that felicitie doth consist. Wherfore some whiche put their felicitie in fedyng their bellies, thinke no pain to be hard, nor no dede to be vnhonest, that may be a meanes to fill that foule panche.

Other which put their felicitie in play and ydle pastimes, iudge no time euill spent, that is employed thereabout: nor no fraude vnlawfull that may further their winning. If I should particularly ouerrune but the common sortes of men, which put their felicitie in their desires, it wold make a great boke of it self. Therfore wyl I let them al go, and conclude as I began, That all men employ their whole endeuour to that thing, wherin thei thinke felicitie to stand. whiche thyng who so listeth to mark exactly, shall be able to espie and iudge the natures of al men, whose conuersaci he doth know, though thei vse great dissimulacion to colour their desires, especially wh? they perceiue other men to mislyke that which thei so much desire: For no m wold gladly haue his appet.i.te improued. And herof cmeth that secnde thing wherin al agree, that euery man would most gladly win all other men to his sect, and to make th? of his opinion, and as far as he dare, will dispraise all other mens iudgem?tes, and praise his own waies only, onles it be when he dissimuleth, and that for the furtherce of his own purpose.

And this propertie also doth geue great light to the full knowledge of mens natures, which as all men ought to obserue, so princes aboue other haue most cause to mark for sundrie occasions which may lye them on, wherof I shall not nede to speke any farther, consideryng not only the greatnes of wit, and exactnes of iudgement whiche G.o.d hath lent vnto your highnes person, but also y^e most graue wisdom and profoud knowledge of your maiesties most honorable coucel, by wh your highnes may so sufficiently vnderstd all thinges conuenient, that lesse shal it nede to vnderstand by priuate readying, but yet not vtterly to refuse to read as often as occasion may serue, for bokes dare speake, when men feare to displease. But to returne agayne to my firste matter, if none other good thing maie be lerned at their maners, which so wrgfully place their felicity, in so miserable a cditi (that while they thinke them selfes happy, their felicitie must nedes seme vnluckie, to be by them so euill placed) yet this may men learn at them, by those .ij. spectacles to espye the secrete natures and dispositions of others, whiche thyng vnto a wise man is muche auailable. And thus will I omit this great tablement of vnhappie hap, and wil come to .iij.

other sortes of a better degre, wherof the one putteth felicitie to consist in power and royaltie. The second sorte vnto power annexeth worldly wisdome, thinkyng him full happie, that could attayn those two, wherby he might not onely haue knowledge in all thynges, but also power to bryng his desires to ende. The thyrd sort estemeth true felicitie to consist in wysdom annexed with vertuouse maners, thinkyng that they can take harme of nothyng, if they can with their wysedome ouercome all vyces. Of the firste of those three sortes there hath been a great numbre in all ages, yea many mightie kinges and great gouernoures which cared not greately howe they myght atchieue their pourpose, so that they dyd preuayle: nor did not take any greatter care for gouernance, then to kepe the people in onely feare of them, Whose common sentence was alwaies this: _Oderint dum metuant_.

And what good successe suche menne had, all histories doe report. Yet haue they not wanted excuses: yea Iulius Caesar (whiche in dede was of the second sorte) maketh a kynde of excuse by his common sentence, for them of that fyrste sorte, for he was euer woonte to saie: ??pe? ??? ?d??e? ???, t??a???d?? pe?? ?????st? ?d??e?, t' ???a d' ??see? ??e?.

Whiche sentence I wysshe had neuer been learned out of Grecia.

But now to speake of the second sort, of whiche there hathe been verye many also, yet for this present time amongest them all, I wyll take the exaumples of kynge Phylippe of Macedonie, and of Alexander his sonne, that valiaunt conquerour. First of kinge Phylip it appeareth by his letter sente vnto Aristotle that famous philosopher, that he more delited in the birthe of his sonne, for the hope of learning and good education, that might happen to him by the said Aristotle, then he didde reioyse in the continuaunce of his succession, for these were his wordes and his whole epistle, worthye to bee remembred and registred euery where.

F???pp?? ???st?t??e? ?a??e?.

?s?e ?? ?e????ta ??. p???? ?? t??s ?e??s ???? ???, ???

??t?s ?p? t? ?e???se? t?? pa?d?s, ?s ?p? t? ?at? t? s? ?????a a?t? ?e?????a? ??p??? ??? a?t? ?p? s?? ??af??ta ?a?

pa?de????ta ???? ?ses?a? ?a? ??? ?a? t?? t? t?a??t?

d?ad???s.

That is thus in sense,

Philip vnto Aristotle sendeth gretyng.

You shall vnderstande, that I haue a sonne borne, for whiche cause I yelde vnto G.o.d moste hartie thankes, not so muche for the byrthe of the childe, as that it was his chaunce to be borne in your tyme. For my trust is, that he shall be so brought vp and instructed by you, that he shall become worthie not only to be named our sonne, but also to be the successour of our affayres.

And his good desire was not all vayne, for it appered that Alexander was neuer so busied with warres (yet was he neuer out of moste terrible battaile) but that in the middes thereof he had in remembraunce his studies, and caused in all countreies as he went, all strange beastes, fowles and fisshes, to be taken and kept for the ayd of that knowledg, which he learned of Aristotle: And also to be had with him alwayes a greate numbre of learned men. And in the moste busye tyme of all his warres against Darius kinge of Persia, when he harde that Aristotle had putte forthe certaine bookes of suche knowledge wherein he hadde before studied, hee was offended with Aristotle, and wrote to hym this letter.

????a?d??? ???s?t??e? e? p??tte?.

??? ????s ?p???sas ??d??s t??s ????aat????s t? ????, t???

??? d???s??? ?e?s t? ????, ?? ?a?' ??s ?pa?de???e? ?????s, ??t?? p??t? ?s??ta?? ??????, ??? d? ????? ? ? ta?s pe?? t?

???sta ?pe???a?s, ? t??s d???es? d?af???. ????s?. that is

Alexander vnto Aristotle sendeth greeting.

You haue not doone well, to put forthe those bookes of secrete phylosophy int.i.tuled, a???aat????. For wherin shall we excell other, yf that knowledge that wee haue studied, shall be made commen to all other men, namely sithe our desire is to excelle other men in experience and knowledge, rather then in power and strength. Farewell.

By whyche lettre it appeareth that hee estemed learninge and knowledge aboue power of men. And the like iudgement did he vtter, when he beheld the state of Diogenes Linicus, adiudginge it the beste state next to his owne, so that he said: If I were not Alexander, I wolde wishe to be Diogenes. Whereby apeareth, how he esteemed learning, and what felicity he putte therin, reputing al the worlde saue him selfe to be inferiour to Diogenes. And bi al coniecturs, Alexander did esteme Diogenes one of them whiche contemned the vaine estimation of the disceitfull world, and put his whole felicity in knowledg of vertue, and practise of the same, though some reporte that he knew more vertue then he folowed: But whatso euer he was, it appeareth that Socrates and Plato and many other did forsake their liuings and sel away their patrimony, to the intent to seeke and trauaile for learning, which examples I shall not need to repete to your Maiesty, partly for that your highnes doth often reade them and other lyke, and partly sith your maiesty hath at hand such learned schoolemaysters, which can much better th? I, declare them vnto your highnes, and that more largely also then the shortenes of thys epistle will permit. But thys may I yet adde, that King Salomon whose renoume spred so farre abroad, was very greatlye estemed for his wonderfull power and exceading treasure, but yet much more was he estemed for his wisdom. And him selfe doth bear witnes, that wisedom is better then pretious stones . yea all thinges that can be desired ar not to be compared to it. But what needeth to alledge one sentence of him, whose bookes altogither do none other thing, then set forth the praise of wisedom & knowledg? And his father king Dauid ioyneth uertuous conuersacion and knowledg togither, as the summe of perfection and chief felicity. Wherfore I maye iustelye conclude, that true felicity doth consist in wisdome and vertu. Then if wisdome be as Cicero defineth it, _Diuinarum atq; humanarum rerum scientia_, then ought all men to trauail for knowledg in matters both of religion and humaine docrine, if he shall be counted wyse, and able to attaine true felicitie: But as the study of religious matters is most princ.i.p.all, so I leue it for this time to them that better can write of it then I can. And for humaine knowledge thys wil I boldly say, that who soeuer wyll attain true iudgment therein, must not only trauail in y^e knowledg of the tungs, but must also before al other arts, taste of the mathematical sciences, specially Arithmetike and Geometry, without which it is not possible to attayn full knowledg in any art. Which may suffici?tly by gathered by Aristotle not ly in his bookes of demonstration (whiche can not be vnderstand without Geometry) but also in all his other workes. And before him Plato his maister wrote this sentence on his schole house dore. ??e??t??t?s ??d??s ??s?t?. Let no man entre here (saith he) without knowledg in Geometry. Wherfore moste mighty prince, as your most excellent Maiesty appeareth to be borne vnto most perfect felicity, not only by reas that G.o.d moued with the long prayers of this realme, did send your highnes as moste comfortable inheritour to the same, but also in that your Maiesty was borne in the time of such skilful schoolmaisters & learned techers, as your highnes doth not a little reioyse in, and profite by them in all kind of vertu & knowledg. Amgst which is that heau?ly knowledg most worthely to be praised, wherbi the blindnes of errour & superst.i.tion is exiled, & good hope cceiued that al the sedes & fruts therof, with all kindes of vice & iniquite, wherby vertu is hindered, & iustice defaced, shal be clean extrirped and rooted out of this realm, which hope shal increase more and more, if it may appear that learning be estemed & florish within this realm. And al be it the chief learnig be the diuine scriptures, which instruct the mind princ.i.p.ally, & nexte therto the lawes politike, which most specially def?d the right of goodes, yet is it not possible that those two can long be wel vsed, if that ayde want that gouerneth health and expelleth sicknes, which thing is done by Physik, & these require the help of the vij. liberall sciences, but of none more then of Arithmetik and Geometry, by which not only great thinges ar wrought touchig accptes in al kinds, & in suruaiyng & measuring of ldes, but also al arts depend partly of th?, & building which is most necessary can not be w^{t}out them, which thing csidering, moued me to help to serue your maiesty in this point as wel as other wais, & to do what mai be in me, y^t not ly thei which studi pric.i.p.alli for lernig, mai haue furderce bi mi poore help, but also those whiche haue no tyme to trauaile for exacter knowlege, may haue some helpe to vnderstand in those Mathematicall artes, in whiche as I haue all readye set forth sumwhat of Arithmetike, so G.o.d willing I intend shortly to setforth a more exacter worke therof. And in the meane ceason for a taste of Geometry, I haue sette forthe this small introduction, desiring your grace not so muche to beholde the simplenes of the woorke, in comparison to your Maiesties excellencye, as to fauour the edition thereof, for the ayde of your humble subiectes, which shal thinke them selues more and more dayly bounden to your highnes, if when they shall perceaue your graces desyre to haue theym profited in all knowledge and vertue. And I for my poore ability considering your Maiesties studye for the increase of learning generally through all your highenes dominions, and namely in the vniuersities of Oxforde and Camebridge, as I haue an earnest good will as far as my simple seruice and small knowledg will suffice, to helpe toward the satisfiyng of your graces desire, so if I shall perceaue that my seruice may be to your maiesties cont?tacion, I wil not only put forth the other two books, whiche shoulde haue beene sette forth with these two, yf misfortune had not hindered it, but also I wil set forth other bookes of more exacter arte, bothe in the Latine tongue and also in the Englyshe, whereof parte bee all readye written, and newe instrumentes to theym deuised, and the residue shall bee eanded with all possible speede. I was boldened to dedicate this booke of Geometrye vnto your Maiestye, not so muche bycause it is the firste that euer was sette forthe in Englishe, and therefore for the noueltye a straunge presente, but for that I was perswaded, that suche a wyse prince doothe desire to haue a wise sorte of subiectes. For it is a kynges chiefe reioysinge and glory, if his subiectes be riche in substaunce, and wytty in knowledge: and contrarye waies nothyng can bee more greuouse to a n.o.ble kyng, then that his realme should be other beggerly or full of ignoraunce: But as G.o.d hath geuen your grace a realme bothe riche in commodities and also full of wyttie men, so I truste by the readyng of wyttie artes (whiche be as the whette stones of witte) they muste needes increase more and more in wysedome, and peraduenture fynde some thynge towarde the ayde of their substaunce, whereby your grace shall haue newe occasion to reioyce, seyng your subiectes to increase in substance or wisdom, or in both. And thei again shal haue new and new causes to pray for your maiestie, perceiuyng so graciouse a mind towarde their benefite. And I truste (as I desire) that a great numbre of gentlemen, especially about the courte, whiche vnderstand not the latin tong, or els for the hardnesse of the mater could not away with other mens writyng, will fall in trade with this easie forme of teachyng in their vulgar tong, and so employe some of their tyme in honest studie, whiche were wont to bestowe most part of their time in triflyng pastime: For vndoubtedly if they mean other your maiesties seruice, other their own wisdome, they will be content to employ some tyme aboute this honest and wittie exercise. For whose encouragem?t to the intent they maie perceiue what shall be the vse of this science, I haue not onely written somewhat of the vse of Geometrie, but also I haue annexed to this boke the names and brefe argumentes of those other bokes whiche I will set forth hereafter, and that as shortly as it shall appeare vnto your maiestie by coniecture of their diligent vsyng of this first boke, that they wyll vse well the other bokes also. In the meane ceason, and at all times I wil be a continuall pet.i.tioner, that G.o.d may work in all english hartes an ernest mynde to all honest exercises, wherby thei may serue the better your maiestie and the realm. And for your highnes I besech the most mercifull G.o.d, as he hath most fauourably sent you vnto vs, as our chefe comforter in earthe, so that he will increase your maiestie daiely in all vertue and honor with moste prosperouse successe, and augment in vs your most humble subiectes, true loue to G.o.dward, and iust obedience toward your highnes with all reuerence and subiection.

At London the .xxviij. daie of Ianuarie. M. D. L I.

_Your maiesties moste humble seruant and obedient subiect, Robert Recorde._

+THE PREFACE,+ declaring briefely the commodi- tes of Geometrye, and the necessitye 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.

Yet can 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 ouerpa.s.se 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 their owne priuate profession, whereof to auoide tediousnes I make this rehersall.

Sith Merchauntes by s.h.i.+ppes great riches do winne, I may with good righte at their seate beginne.

The s.h.i.+ppes 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 fas.h.i.+on, 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 sca.r.s.e, 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 G.o.dly a parte, That no recompence might satisfye the finder, But to make him a G.o.d, and honoure him for euer.

So Ceres and Pallas, and Mercury also, Eolus and Neptune, and many other mo, Were honoured as G.o.ddes, bicause they did teache, Firste tillage and weuinge and eloquent speache, Or windes to obserue, the seas to saile ouer, They were called G.o.ddes 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 G.o.dde was alwaies workinge by 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 t.i.tle 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 dest.i.tute of all abilitee to iudge in anie arte, that is not sommewhat experte in the Theoremes of Geometrie.

And that 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 nedelesse also 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.