Part 9

If a right line be deuided into two equall partes, and one of these .ij. partes diuided agayn into two other partes, as happeneth the longe square that is made of the thyrd or later part of that diuided line, with the residue of the same line, and the square of the mydlemoste parte, are bothe togither equall to the square of halfe the firste line.

_Example._

The line A.B. is diuided into ij. equal partes in C, and that parte C.B. is diuided agayne as hapneth in D. Wherfore saith the Theorem that the long square made of D.B. and A.D, with the square of C.D. (which is the mydle portion) shall bothe be equall to the square of half the lyne A.B, that is to saye, to the square of A.C, or els of C.D, which make all one. The long square F.G.N.O. whiche is the longe square that the theoreme speaketh of, is made of .ij. long squares, wherof the fyrst is F.G.M.K, and the seconde is K.N.O.M. The square of the myddle portion is L.M.O.P. and the square of the halfe of the fyrste lyne is E.K.Q.L. Nowe by the theoreme, that longe square F.G.M.O, with the iuste square L.M.O.P, muste bee equall to the greate square E.K.Q.L, whyche thynge bycause it seemeth somewhat difficult to vnderstande, althoughe I intende not here to make demonstrations of the Theoremes, bycause it is appoynted to be done in the newe edition of Euclide, yet I wyll shew you brefely how the equalitee of the partes doth stande. And fyrst I say, that where the comparyson of equalitee is made betweene the greate square (whiche is made of halfe the line A.B.) and two other, where of the fyrst is the longe square F.G.N.O, and the second is the full square L.M.O.P, which is one portion of the great square all redye, and so is that longe square K.N.M.O, beynge a parcell also of the longe square F.G.N.O, Wherfore as those two partes are common to bothe partes compared in equalitee, and therfore beynge bothe abated from eche parte, if the reste of bothe the other partes bee equall, than were those whole partes equall before: Nowe the reste of the great square, those two lesser squares beyng taken away, is that longe square E.N.P.Q, whyche is equall to the long square F.G.K.M, beyng the rest of the other parte. And that they two be equall, theyr sydes doo declare. For the longest lynes that is F.K and E.Q are equall, and so are the shorter lynes, F.G, and E.N, and so appereth the truthe of the Theoreme.

_The .xl. theoreme._

If a right line be diuided into .ij. euen partes, and an other right line annexed to one ende of that line, so that it make one righte line with the firste. The longe square that is made of this whole line so augmented, and the portion that is added, with the square of halfe the right line, shall be equall to the square of that line, whiche is compounded of halfe the firste line, and the parte newly added.

_Example._

The fyrst lyne propounded is A.B, and it is diuided into ij. equall partes in C, and an other ryght lyne, I meane B.D annexed to one ende of the fyrste lyne.

Nowe say I, that the long square A.D.M.K, is made of the whole lyne so augmented, that is A.D, and the portion annexed, y^t is D.M, for D.M is equall to B.D, wherfore y^t long square A.D.M.K, with the square of halfe the first line, that is E.G.H.L, is equall to the great square E.F.D.C. whiche square is made of the line C.D. that is to saie, of a line compounded of halfe the first line, beyng C.B, and the portion annexed, that is B.D. And it is easyly perceaued, if you consyder that the longe square A.C.L.K. (whiche onely is lefte out of the great square) hath another longe square equall to hym, and to supply his steede in the great square, and that is G.F.M.H. For their sydes be of lyke lines in length.

_The xli. Theoreme._

If a right line bee diuided by chaunce, the square of the same whole line, and the square of one of his partes are iuste equall to the long square of the whole line, and the sayde parte twise taken, and more ouer to the square of the other parte of the sayd line.

_Example._

A.B. is the line diuided in C. And D.E.F.G, is the square of the whole line, D.H.K.M. is the square of the lesser portion (whyche I take for an example) and therfore must bee twise reckened. Nowe I saye that those ij. squares are equall to two longe squares of the whole line A.B, and his sayd portion A.C, and also to the square of the other portion of the sayd first line, whiche portion is C.B, and his square K.N.F.L. In this theoreme there is no difficultie, if you consyder that the litle square D.H.K.M. is .iiij. tymes reckened, that is to say, fyrst of all as a parte of the greatest square, whiche is D.E.F.G. Secondly he is rekned by him selfe. Thirdely he is accompted as parcell of the long square D.E.N.M, And fourthly he is taken as a part of the other long square D.H.L.G, so that in as muche as he is twise reckened in one part of the comparison of equalitee, and twise also in the second parte, there can rise none occasion of errour or doubtfulnes therby.

_The xlij. Theoreme._

If a right line be deuided as chance happeneth the iiij. long squares, that may be made of that whole line and one of his partes with the square of the other part, shall be equall to the square that is made of the whole line and the saide first portion ioyned to him in lengthe as one whole line.

_Example._

The firste line is A.B, and is deuided by C. into two vnequall partes as happeneth. The long square of yt, and his lesser portion A.C, is foure times drawen, the first is E.G.M.K, the seconde is K.M.Q.O, the third is H.K.R.S, and the fourthe is K.L.S.T. And where as it appeareth that one of the little squares (I meane K.L.P.O) is reckened twise, ones as parcell of the second long square and agayne as parte of the thirde long square, to auoide ambiguite, you may place one insteede of it, an other square of equalitee, with it. that is to saye, D.E.K.H, which was at no tyme accompting as parcell of any one of them, and then haue you iiij. long squares distinctly made of the whole line A.B, and his lesser portion A.C. And within them is there a greate full square P.Q.T.V. whiche is the iust square of B.C, beynge the greatter portion of the line A.B. And that those fiue squares doo make iuste as muche as the whole square of that longer line D.G, (whiche is as longe as A.B, and A.C. ioyned togither) it may be iudged easyly by the eye, sith that one greate square doth comprehend in it all the other fiue squares, that is to say, foure long squares (as is before mencioned) and one full square. which is the intent of the Theoreme.

_The xliij. Theoreme._

If a right line be deuided into ij. equal partes first, and one of those parts again into other ij. parts, as chaunce hapeneth, the square that is made of the last part of the line so diuided, and the square of the residue of that whole line, are double to the square of halfe that line, and to the square of the middle portion of the same line.

_Example._

The line to be deuided is A.B, and is parted in C. into two equall partes, and then C.B, is deuided againe into two partes in D, so that the meaninge of the Theoreme, is that the square of D.B. which is the latter parte of the line, and the square of A.D, which is the residue of the whole line. Those two squares, I say, ar double to the square of one halfe of the line, and to the square of C.D, which is the middle portion of those thre diuisions. Which thing that you maye more easilye perceaue, I haue drawen foure squares, whereof the greatest being marked with E. is the square of A.D. The next, which is marked with G, is the square of halfe the line, that is, of A.C, And the other two little squares marked with F. and H, be both of one bignes, by reason that I did diuide C.B. into two equall partes, so that you amy take the square F, for the square of D.B, and the square H, for the square of C.D. Now I thinke you doubt not, but that the square E. and the square F, ar double so much as the square G. and the square H, which thing the easyer is to be vnderstande, bicause that the greate square hath in his side iij. quarters of the firste line, which multiplied by itselfe maketh nyne quarters, and the square F. containeth but one quarter, so that bothe doo make tenne quarters.

Then G. contayneth iiij. quarters, seynge his side containeth twoo, and H. containeth but one quarter, whiche both make but fiue quarters, and that is but halfe of tenne. Whereby you may easylye coniecture, that the meanynge of the theoreme is verified in the figures of this example.

_The xliiij. Theoreme._

If a right line be deuided into ij. partes equally, and an other portion of a righte lyne annexed to that firste line, the square of this whole line so compounded, and the square of the portion that is annexed, ar doule as much as the square of the halfe of the firste line, and the square of the other halfe ioyned in one with the annexed portion, as one whole line.

_Example._

The line is A.B, and is diuided firste into twoo equal partes in C, and then is there annexed to it an other portion whiche is B.D. Now saith the Theoreme, that the square of A.D, and the square of B.D, ar double to the square of A.C, and to the square of C.D. The line A.B. containing four partes, then must needes his halfe containe ij. partes of such partes I suppose B.D. (which is the annexed line) to containe thre, so shal the hole line comprehend vij. parts, and his square xlix. parts, where vnto if you ad y^e square of the annexed lyne, whiche maketh nyne, than those bothe doo yelde, lviij. whyche must be double to the square of the halfe lyne with the annexed portion. The halfe lyne by it selfe conteyneth but .ij. partes, and therfore his square dooth make foure. The halfe lyne with the annexed portion conteyneth fiue, and the square of it is .xxv, now put foure to .xxv, and it maketh iust .xxix, the euen halfe of fifty and eight, wherby appereth the truthe of the theoreme.

_The .xlv. theoreme._

In all triangles that haue a blunt angle, the square of the side that lieth against the blunt angle, is greater than the two squares of the other twoo sydes, by twise as muche as is comprehended of the one of those .ij. sides (inclosyng the blunt corner) and the portion of the same line, beyng drawen foorth in lengthe, which lieth betwene the said blunt corner and a perpendicular line lightyng on it, and drawen from one of the sharpe angles of the foresayd triangle.

_Example._

For the declaration of this theoreme and the next also, whose vse are wonderfull in the practise of Geometrie, and in measuryng especially, it shall be nedefull to declare that euery triangle that hath no ryght angle as those whyche are called (as in the boke of practise is declared) sharp cornered triangles, and blunt cornered triangles, yet may they be brought to haue a ryght angle, eyther by partyng them into two lesser triangles, or els by addyng an other triangle vnto them, whiche may be a great helpe for the ayde of measuryng, as more largely shall be sette foorthe in the boke of measuryng. But for this present place, this forme wyll I vse, (whiche Theon also vseth) to adde one triangle vnto an other, to bryng the blunt cornered triangle into a ryght angled triangle, whereby the proportion of the squares of the sides in suche a blunt cornered triangle may the better bee knowen.

Fyrst therfore I sette foorth the triangle A.B.C, whose corner by C. is a blunt corner as you maye well iudge, than to make an other triangle of yt with a ryght angle, I must drawe forth the side B.C. vnto D, and from the sharp corner by A. I brynge a plumbe lyne or perpendicular on D. And so is there nowe a newe triangle A.B.D. whose angle by D. is a right angle. Nowe accordyng to the meanyng of the Theoreme, I saie, that in the first triangle A.B.C, because it hath a blunt corner at C, the square of the line A.B. whiche lieth against the said blunte corner, is more then the square of the line A.C, and also of the lyne B.C, (whiche inclose the blunte corner) by as muche as will amount twise of the line B.C, and that portion D.C. whiche lieth betwene the blunt angle by C, and the perpendicular line A.D.

The square of the line A.B, is the great square marked with E. The square of A.C, is the meane square marked with F. The square of B.C, is the least square marked with G. And the long square marked with K, is sette in steede of two squares made of B.C, and C.D. For as the shorter side is the iuste lengthe of C.D, so the other longer side is iust twise so longe as B.C, Wherfore I saie now accordyng to the Theoreme, that the greatte square E, is more then the other two squares F. and G, by the quantitee of the longe square K, wherof I reserue the profe to a more conuenient place, where I will also teache the reason howe to fynde the lengthe of all suche perpendicular lynes, and also of the line that is drawen betweene the blunte angle and the perpendicular line, with sundrie other very pleasant conclusions.

_The .xlvi. Theoreme._

In sharpe cornered triangles, the square of anie side that lieth against a sharpe corner, is lesser then the two squares of the other two sides, by as muche as is comprised twise in the long square of that side, on whiche the perpendicular line falleth, and the portion of that same line, liyng betweene the perpendicular, and the foresaid sharpe corner.

_Example._

Fyrst I sette foorth the triangle A.B.C, and in yt I draw a plumbe line from the angle C. vnto the line A.B, and it lighteth in D. Nowe by the theoreme the square of B.C. is not so muche as the square of the other two sydes, that of B.A. and of A.C. by as muche as is twise conteyned in the long square made of A.B, and A.D, A.B. beyng the line or syde on which the perpendicular line falleth, and A.D. beeyng that portion of the same line whiche doth lye betwene the perpendicular line, and the sayd sharpe angle limitted, whiche angle is by A.

For declaration of the figures, the square marked with E. is the square of B.C, whiche is the syde that lieth agaynst the sharpe angle, the square marked with G. is the square of A.B, and the square marked with F. is the square of A.C, and the two longe squares marked with H.K, are made of the hole line A.B, and one of his portions A.D. And truthe it is that the square E. is lesser than the other two squares C. and F. by the quantitee of those two long squares H. and K. Wherby you may consyder agayn, an other proportion of equalitee, that is to saye, that the square E. with the twoo longsquares H.K, are iuste equall to the other twoo squares C. and F. And so maye you make, as it were an other theoreme. _That in al sharpe cornered triangles, where a perpendicular line is drawen frome one angle to the side that lyeth againste it, the square of anye one side, with the ij. longesquares made at that hole line, whereon the perpendicular line doth lighte, and of that portion of it, which ioyneth to that side whose square is all ready taken, those thre figures, I say, are equall to the ij. squares, of the other ij. sides of the triangle._ In whiche you muste vnderstand, that the side on which the perpendiculare falleth, is thrise vsed, yet is his square but ones mencioned, for twise he is taken for one side of the two long squares. And as I haue thus made as it were an other theoreme out of this fourty and sixe theoreme, so mighte I out of it, and the other that goeth nexte before, make as manny as woulde suffice for a whole booke, so that when they shall bee applyed to practise, and consequently to expresse their benefite, no manne that hathe not well wayde their wonderfull commoditee, would credite the possibilitie of their wonderfull vse, and large ayde in knowledge. But all this wyll I remitte to a place conuenient.

_The xlvij. Theoreme._

If ij. points be marked in the circumference of a circle, and a right line drawen frome the one to the other, that line must needes fal within the circle.

_Example._

The circle is A.B.C.D, the ij. poinctes are A.B, the righte line that is drawenne frome the one to the other, is the line A.B, which as you see, must needes lyghte within the circle. So if you putte the pointes to be A.D, or D.C, or A.C, other B.C, or B.D, in any of these cases you see, that the line that is drawen from the one pricke to the other dothe euermore run within the edge of the circle, els canne it be no right line. How be it, that a croked line, especially being more croked then the portion of the circumference, maye bee drawen from pointe to pointe withoute the circle. But the theoreme speaketh only of right lines, and not of croked lines.

_The xlviij. Theoreme._

If a righte line passinge by the centre of a circle, doo crosse an other right line within the same circle, passinge beside the centre, if he deuide the saide line into twoo equal partes, then doo they make all their angles righte. And contrarie waies, if they make all their angles righte, then doth the longer line cutte the shorter in twoo partes.

_Example._

The circle is A.B.C.D, the line that passeth by the centre, is A.E.C, the line that goeth beside the centre is D.B. Nowe saye I, that the line A.E.C, dothe cutte that other line D.B. into twoo iuste partes, and therefore all their four angles ar righte angles. And contrarye wayes, bicause all their angles are righte angles, therfore it muste be true, that the greater cutteth the lesser into two equal partes, accordinge as the Theoreme would.

_The xlix. Theoreme._

If twoo right lines drawen in a circle doo crosse one an other, and doo not passe by the centre, euery of them dothe not deuide the other into equall partions.

_Example._

The circle is A.B.C.D, and the centre is E, the one line A.C, and the other is B.D, which two lines crosse one an other, but yet they go not by the centre, wherefore accordinge to the woordes of the theoreme, eche of theim doth cuytte the other into equall portions. For as you may easily iudge, A.C. hath one portion longer and an other shorter, and so like wise B.D. Howbeit, it is not so to be vnderstand, but one of them may be deuided into ij. euen parts, but bothe to bee cutte equally in the middle, is not possible, onles both passe through the centre, therfore much rather when bothe go beside the centre, it can not be that eche of theym shoulde be iustely parted into ij. euen partes.

_The L. Theoreme._

If two circles crosse and cut one an other, then haue not they both one centre.

_Example._

This theoreme seemeth of it selfe so manifest, that it neadeth nother demonstration nother declaracion. Yet for the plaine vnderstanding of it, I haue sette forthe a figure here, where ij. circles be drawen, so that one of them doth crosse the other (as you see) in the pointes B. and G, and their centres appear at the firste sighte to bee diuers. For the centre of the one is F, and the centre of the other is E, which diffre as farre asondre as the edges of the circles, where they bee most distaunte in sonder.

_The Li. Theoreme._

If two circles be so drawen, that one of them do touche the other, then haue they not one centre.

_Example._

There are two circles made, as you see, the one is A.B.C, and hath his centre by G, the other is B.D.E, and his centre is by F, so that it is easy enough to perceaue that their centres doe dyffer as muche a sonder, as the halfe diameter of the greater circle is longer then the half diameter of the lesser circle. And so must it needes be thought and said of all other circles in lyke kinde.

_The .lij. theoreme._

If a certaine pointe be assigned in the diameter of a circle, distant from the centre of the said circle, and from that pointe diuerse lynes drawen to the edge and circumference of the same circle, the longest line is that whiche passeth by the centre, and the shortest is the residew of the same line. And of al the other lines that is euer the greatest, that is nighest to the line, which passeth by the centre. And contrary waies, that is the shortest, that is farthest from it. And amongest them all there can be but onely .ij. equall together, and they must nedes be so placed, that the shortest line shall be in the iust middle betwixte them.

_Example._

The circle is A.B.C.D.E.H, and his centre is F, the diameter is A.E, in whiche diameter I haue taken a certain point distaunt from the centre, and that pointe is G, from which I haue drawen .iiij. lines to the circumference, beside the two partes of the diameter, whiche maketh vp vi. lynes in all. Nowe for the diuersitee in quantitie of these lynes, I saie accordyng to the Theoreme, that the line whiche goeth by the centre is the longest line, that is to saie, A.G, and the residewe of the same diameter beeyng G.E, is the shortest lyne. And of all the other that lyne is longest, that is neerest vnto that parte of the diameter whiche gooeth by the centre, and that is shortest, that is farthest distant from it, wherefore I saie, that G.B, is longer then G.C, and therfore muche more longer then G.D, sith G.C, also is longer then G.D, and by this maie you soone perceiue, that it is not possible to drawe .ij. lynes on any one side of the diameter, whiche might be equall in lengthe together, but on the one side of the diameter maie you easylie make one lyne equall to an other, on the other side of the same diameter, as you see in this example G.H, to bee equall to G.D, betweene whiche the lyne G.E, (as the shortest in all the circle) doothe stande euen distaunte from eche of them, and it is the precise knoweledge of their equalitee, if they be equally distaunt from one halfe of the diameter. Where as contrary waies if the one be neerer to any one halfe of the diameter then the other is, it is not possible that they two may be equall in lengthe, namely if they dooe ende bothe in the circumference of the circle, and be bothe drawen from one poynte in the diameter, so that the saide poynte be (as the Theoreme doeth suppose) somewhat distaunt from the centre of the said circle. 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.

_The .liij. Theoreme._

If a pointe bee marked without a circle, and from it diuerse lines drawen crosse the circle, to the circumference on the other side, so that one of them passe by the centre, then that line whiche passeth by the centre shall be the loongest of them all that crosse the circle. And of the other lines those are longest, that be nexte vnto it that passeth by the centre. And those ar shortest, that be farthest distant from it. But among those partes of those lines, whiche ende in the outewarde circumference, that is most shortest, whiche is parte of the line that passeth by the centre, and amongeste the othere eche, of them, the nerer they are vnto it, the shorter they are, and the farther from it, the longer they be. And amongest them all there can not be more then .ij. of any one length, and they two muste be on the two contrarie sides of the shortest line.

_Example._