Chapter V: Part 5

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._

Take the circle to be A.B.C, and the point assigned without it to be D. Now say I, that if there be drawen sundrie lines from D, and crosse the circle, endyng in the circumference on the contrary side, as here you see, D.A, D.E, D.F, and D.B, then of all these lines the longest must needes be D.A, which goeth by the centre of the circle, and the nexte vnto it, that is D.E, is the longest amongest the rest. And contrarie waies, D.B, is the shorteste, because it is farthest distaunt from D.A. And so maie you iudge of D.F, because it is nerer vnto D.A, then is D.B, therefore is it longer then D.B. And likewaies because it is farther of from D.A, then is D.E, therfore is it shorter then D.E. Now for those partes of the lines whiche bee withoute the circle (as you see) D.C, is the shortest. because it is the parte of that line which passeth by the centre, And D.K, is next to it in distance, and therefore also in shortnes, so D.G, is farthest from it in distance, and therfore is the longest of them. Now D.H, beyng nerer then D.G, is also shorter then it, and beynge farther of, then D.K, is longer then it. 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.

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._

For the declaracion of this proposition, it shall not need to vse any other example, then that which is brought for the explication of this laste theoreme, by whiche you may without any teachinge easyly perceaue both the meanyng and also the truth of this proposition.

_The Liiij. Theoreme._

If a point be set forthe in a circle, and from that pointe
vnto the circumference many lines drawen, of which more then
two are equal togither, then is that point the centre of
that circle.

_Example._

The circle is A.B.C, and within it I haue sette fourth for an example three prickes, which are D.E. and F, from euery one of them I haue drawen (at the leaste) iiij. lines vnto the circumference of the circle but frome D, I haue drawen more, yet maye it appear readily vnto your eye, that of all the lines whiche be drawen from E. and F, vnto the circumference, there are but twoo equall, and more can not bee, for G.E. nor E.H. hath none other equal to theim, nor canne not haue any beinge drawen from the same point E. No more can L.F, or F.K, haue anye line equall to either of theim, beinge drawen from the same pointe F. And yet from either of those two poinctes are there drawen twoo lines equall togither, as A.E, is equall to E.B, and B.F, is equall to F.C, but there can no third line be drawen equall to either of these two couples, and that is by reason that they be drawen from a pointe distaunte from the centre of the circle. But from D, althoughe there be seuen lines drawen, to the circumference, yet all bee equall, bicause it is the centre of the circle. And therefore if you drawe neuer so mannye more from it vnto the circumference, all shall be equal, so that this is the priuilege (as it were of the centre) and therfore no other point can haue aboue two equal lines drawen from it vnto the circumference. And from all pointes you maye drawe ij. equall lines to the circumference of the circle, whether that pointe be within the circle or without it.

_The lv. Theoreme._

No circle canne cut an other circle in more pointes then
two.

_Example._

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 example. 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 circles are regulare formes, that is to say, such formes as haue in their protracture a iuste and certaine proportion, so that certain and determinate truths may be affirmed of them, sith they ar vniforme and vnchaungable.

_The lvi. Theoreme._

If two circles be so drawen, that the one be within the
other, and that they touche one an other: If a line bee
drawen by bothe their centres, and so forthe in lengthe,
that line shall runne to that pointe, where the circles do
touche.

_Example._

The one circle, which is the greattest and vttermost is A.B.C, the other circle that is y^e lesser, and is drawen within the firste, is A.D.E. The centre of the greater circle is F, and the centre of the lesser circle is G, the pointe where they touche is A. And now you may see the truthe of the theoreme so plainely, that it needeth no farther declaracion. For you maye see, that drawinge a line from F. to G, and so forth in lengthe, vntill it come to the circumference, it wyll lighte in the very poincte A, where the circles touche one an other.

_The Lvij. Theoreme._

If two circles bee drawen so one withoute an other, that
their edges doo touche and a right line bee drawnenne frome
the centre of the one to the centre of the other, that line
shall passe by the place of their touching.

_Example._

The firste circle is A.B.E, and his centre is K, The second circle is D.B.C, and his centre is H, the point wher they do touch is B. Nowe doo you se that the line K.H, whiche is drawen 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.

_Example._

For the declaration of this Theoreme, I haue drawen iiij. circles, the first is A.B.C, and his centre H. the second is A.D.G, and his centre F. the third is L.M, and his centre K. the .iiij. is D.G.L.M, and his centre E. Nowe as you perceiue the second circle A.D.G, toucheth the first in the inner side, in so much as it is drawen within the other, and yet it toucheth him but in one point, that is to say in A, so lykewaies the third circle L.M, is drawen without the firste circle and toucheth hym, as you maie see, but in one place. And now as for the .iiij. circle, it is drawen to declare the diuersitie betwene touchyng and cuttyng, or crossyng. For one circle maie crosse and cutte a great many other circles, yet can be not cutte any one in more places then two, as the fiue and fiftie Theoreme affirmeth.

_The .lix. Theoreme._

In euerie circle those lines are to be counted equall,
whiche are in lyke distaunce from the centre, And contrarie
waies they are in lyke distance from the centre, whiche be
equall.

_Example._

In this figure you see firste the circle drawen, whiche is A.B.C.D, and his centre is E. In this circle also there are drawen two lines equally distaunt from the centre, for the line A.B, and the line D.C, are iuste of one distaunce from the centre, whiche is E, and therfore are they of one length. Again thei are of one lengthe (as shall be proued in the boke of profes) and therefore their distaunce from the centre is all one.

_The lx. Theoreme._

In euerie circle the longest line is the diameter, and of
all the other lines, thei are still longest that be nexte
vnto the centre, and they be the shortest, that be farthest
distaunt from it.

_Example._

In this circle A.B.C.D, I haue drawen first the diameter, whiche is A.D, whiche passeth (as it must) by the centre E, Then haue I drawen ij. other lines as M.N, whiche is neerer the centre, and F.G, that is farther from the centre. The fourth line also on the other side of the diameter, that is B.C, is neerer to the centre then the line F.G, for it is of lyke distance as is the lyne M.N. Nowe saie I, that A.D, beyng the diameter, is the longest of all those lynes, and also of any other that maie be drawen within that circle, And the other line M.N, is longer then F.G. Also the line F.G, is shorter then the line B.C, for because it is farther from the centre then is the lyne B.C. And thus maie you iudge of al lines drawen in any circle, how to know the proportion of their length, by the proportion of their distance, and contrary waies, howe to discerne the proportion of their distance by their lengthes, if you knowe the proportion of their length. And to speake of it by the waie, it is a maruaylouse thyng to consider, that a man maie knowe an exacte proportion betwene two thynges, and yet can not name nor attayne the precise quantitee of those two thynges, As for exaumple, If two squares be sette foorthe, whereof the one containeth in it fiue square feete, and the other contayneth fiue and fortie foote, of like square feete, I am not able to tell, no nor yet anye manne liuyng, what is the precyse measure of the sides of any of those .ij. squares, and yet I can proue by vnfallible reason, that their sides be in a triple proportion, that is to saie, that the side of the greater square (whiche containeth .xlv. foote) is three tymes so long iuste as the side of the lesser square, that includeth but fiue foote. But this seemeth to be spoken out of ceason in this place, therfore I will omitte it now, reseruyng the exacter declaration therof to a more conuenient place and time, and will procede with the residew of the Theoremes appointed for this boke.

_The .lxi. Theoreme._

If a right line be drawen at any end of a diameter in
perpendicular forme, and do make a right angle with the
diameter, that right line shall light without the circle,
and yet so iointly knitte to it, that it is not possible to
draw any other right line betwene that saide line and the
circumference of the circle. And the angle that is made in
the semicircle is greater then any sharpe angle that may be
made of right lines, but the other angle without, is lesser
then any that can be made of right lines.

_Example._

In this circle A.B.C, the diameter is A.C, the perpendicular line, which maketh a right angle with the diameter, is C.A, whiche line falleth without the circle, and yet ioyneth so exactly vnto it, that it is not possible to draw an other right line betwene the circumference of the circle and it, whiche thyng is so plainly seene of the eye, that it needeth no farther declaracion. For euery man wil easily consent, that betwene the croked line A.F, (whiche is a parte of the circumference of the circle) and A.E (which is the said perpendicular line) there can none other line bee drawen in that place where they make the angle. Nowe for the residue of the theoreme. The angle D.A.B, which is made in the semicircle, is greater then anye sharpe angle that may bee made of ryghte lines. and yet is it a sharpe angle also, in as much as it is lesser then a right angle, which is the angle E.A.D, and the residue of that right angle, which lieth without the circle, that is to saye, E.A.B, is lesser then any sharpe angle that can be made of right lines also. For as it was before rehersed, there canne no right line be drawen to the angle, betwene the circumference and the right line E.A. Then must it needes folow, that there can be made no lesser angle of righte lines. And againe, if ther canne be no lesser then the one, then doth it sone appear, that there canne be no greater then the other, for they twoo doo make the whole right angle, so that if anye corner coulde be made greater then the one parte, then shoulde the residue bee lesser then the other parte, so that other bothe partes muste be false, or els bothe graunted to be true.

_The lxij. Theoreme._

If a right line doo touche a circle, and an other right line
drawen frome the centre of the circle to the pointe where
they touche, that line whiche is drawenne frome the centre,
shall be a perpendicular line to the touch line.

_Example._

The circle is A.B.C, and his centre is F. The touche line is D.E, and the point wher they touch is C. Now by reason that a right line is drawen frome the centre F. vnto C, which is the point of the touche, therefore saith the theoreme, that the sayde line F.C, muste needes bee a perpendicular line vnto the touche line D.E.

_The lxiij. Theoreme._

If a righte line doo touche a circle, and an other right
line be drawen from the pointe of their touchinge, so that
it doo make righte corners with the touche line, then shal
the centre of the circle bee in that same line, so drawen.

_Example._

The circle is A.B.C, and the centre of it is G. The touche line is D.C.E, and the pointe where it toucheth, is C. Nowe it appeareth manifest, that if a righte line be drawen from the pointe where the touch line doth ioine with the circle, and that the said lyne doo make righte corners with the touche line, then muste it needes go by the centre of the circle, and then consequently it must haue the sayde centre in him. For if the saide line shoulde go beside the centre, as F.C. doth, then dothe it not make righte angles with the touche line, which in the theoreme is supposed.

_The lxiiij. Theoreme._

If an angle be made on the centre of a circle, and an other
angle made on the circumference of the same circle, and
their grounde line be one common portion of the
circumference, then is the angle on the centre twise so
great as the other angle on the circumference.

_Example._

The circle is A.B.C.D, and his centre is E: the angle on the centre is C.E.D, and the angle on the circumference is C.A.D t their commen ground line, is C.F.D. Now say I that the angle C.E.D, whiche is on the centre, is twise so greate as the angle C.A.D, which is on the circumference.

_The lxv. Theoreme._

Those angles whiche be made in one cantle of a circle, must
needes be equal togither.

_Example._

Before I declare this theoreme by example, it shall bee needefull to declare, what is it to be vnderstande by the wordes in this theoreme. For the sentence canne not be knowen, onles the uery meaning of the wordes be firste vnderstand. Therefore when it speaketh of angles made in one cantle of a circle, it is this to be vnderstand, that the angle muste touch the circumference: and the lines that doo inclose that angle, muste be drawen to the extremities of that line, which maketh the cantle of the circle. So that if any angle do not touch the circumference, or if the lines that inclose that angle, doo not ende in the extremities of the corde line, but ende other in some other part of the said corde, or in the circumference, or that any one of them do so eande, then is not that angle accompted to be drawen in the said cantle of the circle. And this promised, nowe will I cumme to the meaninge of the theoreme. I sette forthe a circle whiche is A.B.C.D, and his centre E, in this circle I drawe a line D.C, whereby there ar made two cantels, a more and a lesser. The lesser is D.E.C, and the geater is D.A.B.C. In this greater cantle I drawe two angles, the firste is D.A.C, and the second is D.B.C which two angles by reason they are made bothe in one cantle of a circle (that is the cantle D.A.B.C) therefore are they both equall. Now doth there appere an other triangle, whose angle lighteth on the centre of the circle, and that triangle is D.E.C, whose angle is double to the other angles, as is declared in the lxiiij. Theoreme, whiche maie stande well enough with this Theoreme, for it is not made in this cantle of the circle, as the other are, by reason that his angle doth not light in the circumference of the circle, but on the centre of it.

_The .lxvi. theoreme._

Euerie figure of foure sides, drawen in a circle, hath his
two contrarie angles equall vnto two right angles.

_Example._

The circle is A.B.C.D, and the figure of foure sides in it, is made of the sides B.C, and C.D, and D.A, and A.B. Now if you take any two angles that be contrary, as the angle by A, and the angle by C, I saie that those .ij. be equall to .ij. right angles. Also if you take the angle by B, and the angle by D, whiche two are also contray, those two angles are like waies equall to two right angles. But if any man will take the angle by A, with the angle by B, or D, they can not be accompted contrary, no more is not the angle by C. estemed contray to the angle by B, or yet to the angle by D, for they onely be accompted _contrary angles_, whiche haue no one line common to them bothe. Suche is the angle by A, in respect of the angle by C, for there both lynes be distinct, where as the angle by A, and the angle by D, haue one common line A.D, and therfore can not be accompted contrary angles, So the angle by D, and the angle by C, haue D.C, as a common line, and therefore be not contrary angles. And this maie you iudge of the residewe, by like reason.

_The lxvij. Theoreme._

Vpon one right lyne there can not be made two cantles of
circles, like and vnequall, and drawen towarde one parte.

_Example._

Cantles of circles be then called like, when the angles that are made in them be equall. But now for the Theoreme, let the right line be A.E.C, on whiche I draw a cantle of a circle, whiche is A.B.C. Now saieth the Theoreme, that it is not possible to draw an other cantle of a circle, whiche shall be vnequall vnto this first cantle, that is to say, other greatter or lesser then it, and yet be lyke it also, that is to say, that the angle in the one shall be equall to the angle in the other. For as in this example you see a lesser cantle drawen also, that is A.D.C, so if an angle were made in it, that angle would be greatter then the angle made in the cantle A.B.C, and therfore can not they be called lyke cantels, but and if any other cantle were made greater then the first, then would the angle in it be lesser then that in the firste, and so nother a lesser nother a greater cantle can be made vpon one line with an other, but it will be vnlike to it also.

_The .lxviij. Theoreme._

Lyke cantelles of circles made on equal righte lynes, are
equall together.

_Example._

What is ment by like cantles you haue heard before. and it is easie to vnderstand, that suche figures a called equall, that be of one bygnesse, so that the one is nother greater nother lesser then the other. And in this kinde of comparison, they must so agree, that if the one be layed on the other, they shall exactly agree in all their boundes, so that nother shall excede other.

Nowe for the example of the Theoreme, I haue set forthe diuers varieties of cantles of circles, amongest which the first and seconde are made vpon equall lines, and ar also both equall and like. The third couple ar ioyned in one, and be nother equall, nother like, but expressyng an absurde deformitee, whiche would folowe if this Theoreme wer not true. And so in the fourth couple you maie see, that because they are not equall cantles, therfore can not they be like cantles, for necessarily it goeth together, that all cantles of circles made vpon equall right lines, if they be like they must be equall also.

_The lxix. Theoreme._

In equall circles, suche angles as be equall are made vpon
equall arch lines of the circumference, whether the angle
light on the circumference, or on the centre.

_Example._

Firste I haue sette for an exaumple twoo equall circles, that is A.B.C.D, whose centre is K, and the second circle E.F.G.H, and his centre L, and in eche of them is there made two angles, one on the circumference, and the other on the centre of eche circle, and they be all made on two equall arche lines, that is B.C.D. the one, and F.G.H. the other. Now saieth the Theoreme, that if the angle B.A.D, be equall to the angle F.E.H, then are they made in equall circles, and on equall arch lines of their circumference. Also if the angle B.K.D, be equal to the angle F.L.H, then be they made on the centres of equall circles, and on equall arche lines, so that you muste compare those angles together, whiche are made both on the centres, or both on the circumference, and maie not conferre those angles, wherof one is drawen on the circumference, and the other on the centre. For euermore the angle on the centre in suche sorte shall be double to the angle on the circumference, as is declared in the three score and foure Theoreme.

_The .lxx. Theoreme._

In equall circles, those angles whiche bee made on equall
arche lynes, are euer equall together, whether they be made
on the centre, or on the circumference.

_Example._

This Theoreme doth but conuert the sentence of the last Theoreme before, and therfore is to be vnderstande by the same examples, for as that saith, that equall angles occupie equall archelynes, so this saith, that equal arche lines causeth equal angles, consideringe all other circumstances, as was taughte in the laste theoreme before, so that this theoreme dooeth affirming speake of the equalitie of those angles, of which the laste theoreme spake conditionally. And where the laste theoreme spake affirmatiuely of the arche lines, this theoreme speaketh conditionally of them, as thus: If the arche line B.C.D. be equall to the other arche line F.G.H, then is that angle B.A.D. equall to the other angle F.E.H. Or els thus may you declare it causally: Bicause the arche line B.C.D, is equal to the other arche line F.G.H, therefore is the angle B.K.D. equall to the angle F.L.H, consideringe that they are made on the centres of equall circles. And so of the other angles, bicause those two arche lines aforesaid ar equal, therfore the angle D.A.B, is equall to the angle F.E.H, for as muche as they are made on those equall arche lines, and also on the circumference of equall circles. And thus these theoremes doo one declare an other, and one verifie the other.

_The lxxi. Theoreme._

In equal circles, equall right lines beinge drawen, doo
cutte awaye equalle arche lines frome their circumferences,
so that the greater arche line of the one is equall to the
greater arche line of the other, and the lesser to the
lesser.

_Example._

The circle A.B.C.D, is made equall to the circle E.F.G.H, and the right line B.D. is equal to the righte line F.H, wherfore it foloweth, that the ij. arche lines of the circle A.B.D, whiche are cut from his circumference by the right line B.D, are equall to two other arche lines of the circle E.F.H, being cutte frome his circumference, by the right line F.H. that is to saye, that the arche line B.A.D, beinge the greater arch line of the firste circle, is equall to the arche line F.E.H, beynge the greater arche line of the other circle. And so in like manner the lesser arche line of the firste circle, beynge B.C.D, is equal to the lesser arche line of the seconde circle, that is F.G.H.

_The lxxij. Theoreme._

In equall circles, vnder equall arche lines the right lines
that bee drawen are equall togither.

_Example._

This Theoreme is none other, but the conuersion of the laste Theoreme beefore, and therefore needeth none other example. For as that did declare the equalitie of the arche lines, by the equalitie of the righte lines, so dothe this Theoreme declare the equalnes of the right lines to ensue of the equalnes of the arche lines, and therefore declareth that right lyne B.D, to be equal to the other right line F.H, bicause they both are drawen vnder equall arche lines, that is to saye, the one vnder B.A.D, and thother vnder F.E.H, and those two arch lines are estimed equall by the theoreme laste before, and shal be proued in the booke of proofes.

_The lxxiij. Theoreme._

In euery circle, the angle that is made in the halfe circle,
is a iuste righte angle, and the angle that is made in a
cantle greater then the halfe circle, is lesser thanne a
righte angle, but that angle that is made in a cantle,
lesser then the halfe circle, is greatter then a right
angle. And moreouer the angle of the greater cantle is
greater then a righte angle and the angle of the lesser
cantle is lesser then a right angle.

_Example._

In this proposition, it shal be meete to note, that there is a greate diuersite betwene an angle of a cantle, and an angle made in a cantle, and also betwene the angle of a semicircle, and y^e angle made in a semicircle. Also it is meet to note y^t al angles that be made in y^e part of a circle, ar made other in a semicircle, (which is the iuste half circle) or els in a cantle of the circle, which cantle is other greater or lesser then the semicircle is, as in this figure annexed you maye perceaue euerye one of the thinges seuerallye.

Firste the circle is, as you see, A.B.C.D, and his centre E, his diameter is A.D, Then is ther a line drawen from A. to B, and so forth vnto F, which is without the circle: and an other line also frome B. to D, whiche maketh two cantles of the whole circle. The greater cantle is D.A.B, and the lesser cantle is B.C.D, In whiche lesser cantle also there are two lines that make an angle, the one line is B.C, and the other line is C.D. Now to showe the difference of an angle in a cantle, and an angle of a cantle, first for an example I take the greter cantle B.A.D, in which is but one angle made, and that is the angle by A, which is made of a line A.B, and the line A.D, And this angle is therfore called an angle in a cantle. But now the same cantle hathe two other angles, which be called the angles of that cantle, so the twoo angles made of the righte line D.B, and the arche line D.A.B, are the twoo angles of this cantle, whereof the one is by D, and the other is by B. Wher you must remembre, that the angle by D. is made of the right line B.D, and the arche line D.A. And this angle is diuided by an other right line A.E.D, which in this case must be omitted as no line. Also the angle by B. is made of the right line D.B, and of the arch line .B.A, & although it be deuided with ij. other right lines, of w^{ch} the one is the right line B.A, & thother the right line B.E, yet in this case they ar not to be considered. And by this may you perceaue also which be the angles of the lesser cantle, the first of them is made of y^e right line B.D, & of y^e arch line B.C, the second is made of the right line .D.B, & of the arch line D.C. Then ar ther ij. other lines, w^{ch} deuide those ij. corners, y^t is the line B.C, & the line C.D, w^{ch} ij. lines do meet in the poynte C, and there make an angle, whiche is called an angle made in that lesser cantle, but yet is not any angle of that cantle. And so haue you heard the difference betweene an angle in a cantle, and an angle of a cantle. And in lyke sorte shall you iudg of the angle made in a semicircle, whiche is distinct from the angles of the semicircle. For in this figure, the angles of the semicircle are those angles which be by A. and D, and be made of the right line A.D, beeyng the diameter, and of the halfe circumference of the circle, but by the angle made in the semicircle is that angle by B, whiche is made of the righte line A.B, and that other right line B.D, whiche as they mete in the circumference, and make an angle, so they ende with their other extremities at the endes of the diameter. These thynges premised, now saie I touchyng the Theoreme, that euerye angle that is made in a semicircle, is a right angle, and if it be made in any cantle of a circle, then must it neds be other a blunt angle, or els a sharpe angle, and in no wise a righte angle. For if the cantle wherein the angle is made, be greater then the halfe circle, then is that angle a sharpe angle. And generally the greater the cantle is, the lesser is the angle comprised in that cantle: and contrary waies, the lesser any cantle is, the greater is the angle that is made in it. Wherfore it must nedes folowe, that the angle made in a cantle lesse then a semicircle, must nedes be greater then a right angle. So the angle by B, beyng made at the right line A.B, and the righte line B.D, is a iuste righte angle, because it is made in a semicircle. But the angle made by A, which is made of the right line A.B, and of the right line A.D, is lesser then a righte angle, and is named a sharpe angle, for as muche as it is made in a cantle of a circle, greater then a semicircle. And contrary waies, the angle by C, beyng made of the righte line B.C, and of the right line C.D, is greater then a right angle, and is named a blunte angle, because it is made in a cantle of a circle, lesser then a semicircle. But now touchyng the other angles of the cantles, I saie accordyng to the Theoreme, that the .ij. angles of the greater cantle, which are by B. and D, as is before declared, are greatter eche of them then a right angle. And the angles of the lesser cantle, whiche are by the same letters B, and D, but be on the other side of the corde, are lesser eche of them then a right angle, and be therfore sharpe corners.

_The lxxiiij. Theoreme._

If a right line do touche a circle, and from the pointe
where they touche, a righte lyne be drawen crosse the
circle, and deuide it, the angles that the saied lyne dooeth
make with the touche line, are equall to the angles whiche
are made in the cantles of the same circle, on the contrarie
sides of the lyne aforesaid.

_Example._

The circle is A.B.C.D, and the touche line is E.F. The pointe of the touchyng is D, from which point I suppose the line D.B, to be drawen crosse the circle, and to diuide it into .ij. cantles, wherof the greater is B.A.D, and the lesser is B.C.D, and in ech of them an angle is drawen, for in the greater cantle the angle is by A, and is made of the right lines B.A, and A.D, in the lesser cantle the angle is by C, and is made of y^e right lines B.C, and C.D. Now saith the Theoreme that the angle B.D.F, is equall to the angle made in the cantle on the other side of the said line, that is to saie, in the cantle B.A.D, so that the angle B.D.F, is equall to the angle B.A.D, because the angle B.D.F, is on the one side of the line B.D, (whiche is according to the supposition of the Theoreme drawen crosse the circle) and the angle B.A.D, is in the cantle on the other side. Likewaies the angle B.D.E, beyng on the one side of the line B.D, must be equall to the angle B.C.D, (that is the angle by C,) whiche is made in the cantle on the other side of the right line B.D. The profe of all these I do reserue, as I haue often saide, to a conuenient boke, wherein they shall be all set at large.

_The .lxxv. Theoreme._

In any circle when .ij. right lines do crosse one an other,
the likeiamme that is made of the portions of the one line,
shall be equall to the lykeiamme made of the partes of the
other lyne.

Because this Theoreme doth serue to many vses, and wold be wel vnderstande, I haue set forth .ij. examples of it. In the firste, the lines by their crossyng do make their portions somewhat toward an equalitie. In the second the portions of the lynes be very far from an equalitie, and yet in bothe these and in all other y^e Theoreme is true. In the first example the circle is A.B.C.D, in which thone line A.C, doth crosse thother line B.D, in y^e point E. Now if you do make one likeiamme or longsquare of D.E, & E.B, being y^e .ij. portions of the line D.B, that longsquare shall be equall to the other longsquare made of A.E, and E.C, beyng the portions of the other line A.C. Lykewaies in the second example, the circle is F.G.H.K, in whiche the line F.H, doth crosse the other line G.K, in the pointe L. Wherfore if you make a lykeiamme or longsquare of the two partes of the line F.H, that is to saye, of F.L, and L.H, that longsquare will be equall to an other longsquare made of the two partes of the line G.K. which partes are G.L, and L.K. Those longsquares haue I set foorth vnder the circles containyng their sides, that you maie somewhat whet your own wit in practisyng this Theoreme, accordyng to the doctrine of the nineteenth conclusion.

_The .lxxvi. Theoreme._

If a pointe be marked without a circle, and from that pointe
two right lines drawen to the circle, so that the one of
them doe runne crosse the circle, and the other doe touche
the circle onely, the long square that is made of that whole
lyne which crosseth the circle, and the portion of it, that
lyeth betwene the vtter circumference of the circle and the
pointe, shall be equall to the full square of the other
lyne, that onely toucheth the circle.

_Example._