Robert Record

“ Theoreme. If a pointe be assigned without a circle, and from that pointe .ij. right lynes be drawen to the circle, so that the one doe crosse the circle, and the other dooe ende at the circumference, and that the longsquare of the line which crosseth [m.i.v] the circle made with the portiõ of the same line beyng without the circle betweene the vtter circumference and the pointe assigned, doe equally agree with the iuste square of that line that endeth at the circumference, then is that lyne so endyng on the circumference a touche line vnto that circle. Example. ”

“ 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 called a cubike bodie) by geomatricians, Asheler. and an ashler of masons, there are .vi. sides, whiche are .vi. platte formes, and are the boundes of the dye.
A globe. But in a Globe, (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. ”

“ To drawe a streight line from any pricke that is not in a line, and to make it perpendicular to an other line.
Open your compas as so wide that it may extend somewhat farther, thẽ from the prick to the line, then sette the one foote of the compas in the pricke, and with the other shall you draw a cõpassed line, that shall crosse that other first line in .ij. places. Now if you deuide that arch line into .ij. equall partes, and from the middell pricke therof vnto the prick without the line you drawe a streight line, it shalbe a plumbe line to that firste lyne, accordyng to the conclusion. ”

“ If a triangle haue twoo sides equalle to the two sides of an other triangle, but yet hathe a longer ground line thẽ that other triangle, then is his angle that lieth betwene the equall sides, greater thẽ the like corner in the other triangle. ”

“ The first circle is A.B.F.E, the second circle is B.C.D.E, and they crosse one an other in B. and in E, and in no more pointes. Nother is it possible that they should, but other figures ther be, which maye cutte a circle in foure partes, as you se in this exãple. Where I haue set forthe one tunne forme, and one eye forme, and eche of them cutteth euery of their two circles into foure partes. But as they be irregulare formes, that is to saye, suche formes as haue no precise measure nother proportion in their draughte, so can there scarcely be made any certaine theorem of them. ”

“ 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. ”

“ The firste circle is A.B.E, and his centre is K, The secõd circle is D.B.C, and his cẽtre is H, the point wher they do touch is B. Nowe doo you se that the line K.H, whiche is drawen i.iij. from K, that is centre of the firste circle, vnto H, beyng centre of the second circle, doth passe (as it must nedes by the pointe B,) whiche is the verye poynte wher they do to touche together.
The .lviij. theoreme. One circle can not touche an other in more pointes then one, whether they touche within or without. ”

“ 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. ”

“ A circle is 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 the circumference are equall all in length, as here you see. ”

“ A righte angle. The square angle, whiche is commonly named a 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.
A sharpe corner. A sharpe angle is 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. ”

“ Fyrste draw a line against the corner assigned, and so is it a triangle, then take heede to the line and the pointe in it assigned, and consider if that line from the pricke to this end bee as long as any of the sides that make the triangle assigned, and if it bee longe enoughe, then prick out there the length of one of the lines, and then woorke with the other two lines, accordinge to the laste conlusion, makynge a triangle of thre like lynes to that assigned triangle. ”

“ Take for an example firste these two right lines A.B. and A.C. whiche meete togither in A, but yet cannot be called a platte forme, bicause there is no bond from B. to C, but if you will drawe a line betwene them twoo, that is frome B. to C, then will it be a platte forme, that is to say, a triangle, but then are there iij. lines, and not only ij. Likewise may you say of D.E. and F.G, whiche doo make a platte forme, nother yet can they make any without helpe of two lines more, whereof the one must be drawen from D. to F, and the other frome E. to G, and then will it be a longe rquare. ”

“ And for humaine knowledge thys wil I boldly say, that who soeuer wyll attain true iudgment therein, must not only trauail in ye 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 sufficiẽtly by gathered by Aristotle not õly in his bookes ⁊.ii. of demonstration (whiche can not be vnderstand without Geometry) but also in all his other workes. ”

“ For if they be drawen from the centre, then must they of necessitee be all equall, howe many so euer they bee, as the definition of a circle dooeth importe, withoute any regarde how neere so euer they be to the diameter, or how distante from it. And here is to be noted, that in this Theoreme, by neerenesse and distaunce is vnderstand the nereness and distaunce of the extreeme partes of those lynes where they touche the circumference. For at the other end they do all meete and touche. ”

“ A.B.C. is a twileke triangle, for the one side A.C, is equal to the other side B.C. And therfore I saye that the inner corners A. and B, which are about the ground lines, (that is A.B.) be equall togither. And farther if C.A. and C.B. bee drawen forthe vnto D. and E. as you se that I haue drawen them, then saye I that the two vtter angles vnder A. and B, are equal also togither: as the theorem said. The profe wherof, as of al the rest, shal apeare in Euclide, whome I intende to set foorth in english with sondry new additions, if I may perceaue that it wilbe thankfully taken. ”

“ But as god 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 in theim, as well as in the propositions, I haue drawen in the Linearie examples many tymes more lynes, than be spoken of in the explication of them, whiche is doone to this intent, that yf any manne lyst to learne the demonstrations by harte, (as somme [a.iii.v] learned men haue iudged beste to doo) those same men should find the Linearye exaumples to serue for this purpose, and to wante no thyng needefull to the iuste proofe, whereby this booke may bee wel approued to be more complete then many men wolde suppose it. ”

“ And of this Theoreme dothe there folowe an other lyke. whiche you maye calle other a theoreme by it selfe, or else a Corollary vnto this laste theoreme, I passe not so muche for the name. But his sentence is this: when so euer any lynes be drawen frome any pointe, withoute a circle, whether they crosse the circle, or eande in the utter edge of his circumference, those two lines that bee equally distaunt from the least line are equal togither, and contrary waies, if they be equall togither, they ar also equally distant from that least line. ”

“ 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 mẽ, 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 [§.iii.v] 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. ”

“ 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 conuersaciõ he doth know, though thei vse great dissimulacion to colour their desires, especially whẽ they perceiue other men to mislyke that which thei so much desire: For no mã wold gladly haue his appetite improued. ”

“ 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. ”

“ By this theoreme you may vnderstand a redy way to know the syde of any ryght anguled triangle that is vnknowen, so that you knowe the lengthe of any two sydes of it. For by tournynge the two sydes certayne into theyr squares, and so addynge them togyther, other subtractynge the one from the other (accordyng as in the vse of these theoremes I haue sette foorthe) and then fyndynge the roote of the square that remayneth, which roote (I meane the syde of the square) is the iuste length of the vnknowen syde, whyche is sought for. ”

“ Certayn common sentences manifest to sence, and acknowledged of all men. The firste common sentence. What so euer things be equal to one other thinge, those same bee equall betwene them selues.
Examples therof you may take both in greatnes and also in numbre. First (though it pertaine not proprely to geometry, but to helpe the vnderstandinge of the rules, whiche may bee wrought by bothe artes) thus may you perceaue. If the summe of monnye in my purse, and the mony in your purse be equall eche of them to the mony that any other man hathe, then must needes your mony and mine be equall togyther. ”

“ So that for this parte of the theoreme (as I think) you do plainly perceaue the truthe thereof, so the residue hathe no difficulte. For seing that the nearer any line is to D.C, (which ioyneth with the diameter) the shorter it is and the farther of from it, the longer it is. And seyng two lynes can not be of like distaunce beinge bothe on one side, therefore if they shal be of one lengthe, and consequently of one distaunce, they must needes bee on contrary sides of the saide line D.C. And so appeareth the meaning of the whole Theoreme. ”

“ The reason of som thynges done in this boke, or omitted in the same, you shall fynde in the preface before the Theoremes.*
A The definitions of the principles of GEOMETRY. eometry teacheth the drawyng, Measuring and proporcion of figures. but in 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. ”