Read Ebook: The Path-Way to Knowledg Containing the First Principles of Geometrie by Record Robert

Font size: 10 px 15 px 20 px 25 px 30 px 35 px

Background color: BlackWhiteGrayLight GrayDark GrayDim Gray

Text color: BlackWhiteGrayLight GrayDark GrayDim Gray

Apply/Save settings

Ebook has 436 lines and 51615 words, and 9 pages

The pathway to KNOWLEDG, CONTAI- NING THE FIRST PRIN- ciples of Geometrie, as they may moste aptly be applied vn- to practice, bothe for vse of instrumentes Geome- tricall, and astrono- micall and also for proiection of plattes in euerye kinde, and therfore much ne- cessary for all sortes of men.

Geometries verdicte

All fresshe fine wittes by me are filed, All grosse dull wittes wishe me exiled: Thoughe no mannes witte reiect will I, Yet as they be, I wyll them trye.

The argumentes of the foure bookes

The first booke declareth the definitions of the termes and names vsed in Geometry, with certaine of the chiefe grounds whereon the arte is founded. And then teacheth those conclusions, which may serue diuersely in al workes Geometricall.

The second booke doth sette forth the Theoremes, seruinge for the due knowledge and sure proofe of all conclusions and workes in Geometrye.

The third booke intreateth of diuers formes, and sondry protractions thereto belonging, with the vse of certain conclusions.

The fourth booke teacheth the right order of measuringe all platte formes, and bodies also, by reson Geometricall.

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 stomblinge, 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 vnc?bered, 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 god 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 fashion 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 God, 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.

???????? ??????????? ???????.

???? ??? ???????? ???. ?????? ??? ???? ????? ????? ???, ??? ????? ??? ?? ???????? ??? ??????, ?? ??? ?? ???? ??? ??? ??????? ????? ????????? ?????? ??? ????? ??? ??? ???????? ??? ??????????? ????? ??????? ??? ????? ??? ??? ??? ????????? ????????.

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 God 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 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.

?????????? ?????????? ?? ????????.

??? ????? ???????? ?????? ???? ????????????? ??? ?????, ???? ??? ????????? ????? ??? ?????, ?? ???' ??? ???????????? ??????, ????? ?????? ???????? ??????, ??? ?? ?????? ??? ?? ???? ???? ?? ?????? ??????????, ? ???? ???????? ????????. ??????. that is

Alexander vnto Aristotle sendeth greeting.

You haue not doone well, to put forthe those bookes of secrete phylosophy intituled, ????????????. 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.

+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 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 their owne 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 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 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 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.

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.

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 deuise such glasses within the towne of Syracusae, that dyd bourne their ennemies shyppes a great way from the towne, whyche was a meruaylous politike thynge. And if I shulde repete the varietees 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 by negromancy. And hereof came it that fryer Bakon was accompted so greate a negromancier, whiche neuer vsed that arte 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 and to stande as well with reason as to see your face in c?mon 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.

Geometry teacheth the drawyng, Measuring and proporcion of figures. but in as muche as no figure can bee drawen, but it muste haue certayne bo?des 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.

Where I haue set .iij. prickes, eche of them hauyng both l?gth and bredth, thogh it be but smal, and thefore not notable.

But as they in theyr theorikes 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.

Therefore when soeuer you do see any formes of lynes to touche at one notable pricke, as in this example, then shall you not call it one croked lyne, but rather twoo lynes: 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.

And these angles 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 would muche 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.

Of platte formes some be plain, and some be croked, and some parly plaine, and partlie croked.

Yet for the lighter forme of teachyng, it shall not be vnsemely to call all suche shapes, formes and figures, whiche y^e eye 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, c?sidering that their knowledg appertaineth more to Arithmetike figurall, than to Geometrie.

But consideryng 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.

Where A. and B. are matche corners, so are C. and D. but not A. and C. nother D. and A.

Nowe will I beginne to speak of figures, that be properly so called, of whiche all be made of diuerse lines, except onely a circle, an egge forme, and a tunne forme, which .iij. haue no angle and haue but one line for their bounde, and an eye fourme whiche is made of one lyne, and hath an angle onely.

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.

Share on: WhatsApp @Hash Facebook Tweet