Part 11

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._

The circle is D.B.C, and the pointe without the circle is A, from whiche pointe there is drawen one line crosse the circle, and that is A.D.C, and an other lyne is drawn from the said pricke to the marge or edge of the circumference of the circle, and doeth only touche it, that is the line A.B. And of that first line A.D.C, you maie perceiue one part of it, whiche is A.D, to lie without the circle, betweene the vtter circumference of it, and the pointe assigned, whiche was A. Nowe concernyng the meanyng of the Theoreme, if you make a longsquare of the whole line A.C, and of that parte of it that lyeth betwene the circumference and the point, (whiche is A.D,) that longesquare shall be equall to the full square of the touche line A.B, accordyng not onely as this figure sheweth, but also the saied nyneteenth conclusion dooeth proue, if you lyste to examyne the one by the other.

_The lxxvii. 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 the circle made with the portion 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._

In as muche as this Theoreme is nothyng els but the sentence of the last Theoreme before conuerted, therfore it shall not be nedefull to vse any other example then the same, for as in that other Theoreme because the one line is a touche lyne, therfore it maketh a square iust equal with the longsquare made of that whole line, whiche crosseth the circle, and his portion liyng without the same circle. So saith this Theoreme: that if the iust square of the line that endeth on the circumference, be equall to that longsquare whiche is made as for his longer sides of the whole line, which commeth from the pointt assigned, and crosseth the circle, and for his other shorter sides is made of the portion of the same line, liyng betwene the circumference of the circle and the pointe assigned, then is that line whiche endeth on the circumference a right touche line, that is to saie, yf the full square of the right line A.B, be equall to the longsquare made of the whole line A.C, as one of his lines, and of his portion A.D, as his other line, then must it nedes be, that the lyne A.B, is a right touche lyne vnto the circle D.B.C. And thus for this tyme I make an ende of the Theoremes.

+FINIS,+

_IMPRINTED at London in Poules churcheyarde, at the signe of the Bra- senserpent, by Reynold Wolfe._

Cum priuilegio ad imprimen- dum solum.

+ANNO DOMINI+ .M.D.L.I.

* * * * * * * * *

_Errors and Inconsistencies:_

Unless otherwise noted, spelling, punctuation and capitalization are unchanged. Forms were regularized only where there was a very large disparity between the expected form and the apparent errors (for example, a thousand "A.B" against a dozen "A,B"), or a flagrant misprint such as "cnt" for "cut".

The letters u and v follow the conventional "initial v, non-initial u" pattern except in numbers (xv, iv). The lower-case j form occurs only as the last digit of a number (ij, xxj); upper-case I and J share a form, always read as I. Capital and lower-case "w" were often used interchangeably. Words split across line breaks may or may not have a hyphen.

_Language:_

The word "other" is used interchangeably with both "or" and "either"; similarly, "nother" is used in place of "nor" and "neither". The expression "an other" is almost always written as two words.

The spellings "then(ne)" and "than(ne)" are used interchangeably; "than(ne)" is rare. The spelling "liyng", both by itself and as the end of a longer word, is used consistently.

_Illustrations:_

A number of illustrations contain errors such as unmarked or mislabeled points ("circle B.C.D" where only C and D are labeled). Errors of this type are identified in the illustrated HTML version, but not in the present text-only file.

_Greek:_

All Greek is shown and transliterated as printed. Errors or anomalies include missing, misplaced or incorrect diacritics; the non-final form of sigma used at the end of words; and word-final #m# for #n#.

_Meaningful Errors and Anomalies:_

shal be clean extrirped and rooted out [_spelling unchanged: intended form not certain_] new and new causes to pray for your maiestie, perceiuyng [_text "new and new causes" probably intentional_] [Preface] And thus for this tyme I make an end... [Body text] The definitions of the principles of GEOMETRY. [_Pagination as shown by signature numbers demands another leaf (two pages) between the end of the Preface and the beginning of the body text. But no text is missing, and the facsimile has no blank pages._] Otherlesse then it as you se D [_text unchanged: intended wording uncertain_] THE .XXII. CONCLVSION. [.XXI.] THE XXXVIII. CONCLVSION. [XXXVII.] _The xxvi. Theoreme._ There is no xxv. (25) theorem. _The .xxxix. theoreme._ The text of the Example is garbled, and does not fit the illustration. Among other problems, points C and D seem to have been switched, either in the text or in the illustration. 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. [cirle] [_printing of "C.E.D" unclear: looks like "C.F.D" but center of circle is E_] [_lone "t" may be error for ampersand or other punctuation_] [C.F.D,]

_Misprints:_

Aristotle had putte forthe certaine bookes [kookes] And his father king Dauid ioyneth [Danid] those thynges are done by negromancy. And hereof came it that fryer Bakon was accompted so greate a negromancier [_spelling unchanged_] For vndoubtedly if they mean [vndoudtedly if the] that Godde was alwaies workinge [alwaaies] all the lines that be drawen to the circumference [circumfernece] An other hath but one compassed [hatht] whose sydes partlye are all equall as in A [eqnall] as this A. doth partly expresse [A doth partly eppresse] To make a threlike triangle on any lyne measurable. [or any] and it shall cut the first line in two equall portions. [cnt] then open I the compass as wide as .iiij. partes [compaas] To make a plumbe lyne on any porcion of a circle [or any] for euery triangle together an equal likeiamme [_text has "to/gether" at line break: may be meant for two words_] as you mai se by C and D, for ij. sides of both the trianngles ar parallels. [you maise ... triangls] then will the square of that greater portion [portior] that line I say is a touche line [touthe] (that is to saie M. G,) [_error for "in G"?_] one perpendicular from G. vnto the side side B.C [A.C] then must I first draw a tuche lyne [wust] anye of the two lynes contayninge the angle appointed. [nye of] draw thence two lines, one to D [one to A] which is made accordinge to the conclusion. [ancordinge] you shal perceaue that there will be [peceaue] a / more conueni/ent time. [_text has "conui/ent" at line break_] other in deciding some controuersy [decising] a great deale the soner [somner] Whiche example hath beene vsede [hat] whan I wrote these first connclusions [cunclusions] suche bokes as ar appoynted [as at] Will I refuse. [Willl] all right angles be equall [eqnall] whiche thinge the better to perceaue [peceaue] whiche I do only by examples declare [declae] about the ground line, are equal togither [groud] you shal take this triangle A.B.C. which hath a very blunt corner [_word "this" illegible_] [veery blunt] the one is a longe square A.B.E [_printed as shown: expected form is "A.B.F.E"_] though they be diuers in numbre. [numbhe] proofe that G.H. being the ground line [groud] and standing betwene one paire of parallels [an] for thei ar the two y^e contrary sides [_word "y^e" superfluous?_] they haue one ground line D.E. [on] By the square of any lyne [sqnare] squares that are made of the same line [sane] The fyrst lyne propounded is A.B [propouned] hath another longe square equall to hym [_text has anomalous "a / nother" at line break_] one of those parts again into other ij. parts [iuto] which thing the easyer is to be vnderstande [eayser] and blunt cornered triangles [couered] the square marked with G. is the square of A.B [with C.] And from all pointes you maye drawe ij. equall lines [poittes] If two circles bee drawen so one withoute an other [and other] drawen frome the centre of the circle to the pointe [tge] which in the theoreme is supposed. [the .heoreme] and therfore can not they be called lyke cantels [ban] then would the angle in it be lesser [it] being cutte frome his circumference, by the right line F.H. [circumforence] And in lyke sorte shall you iudg [And n lyke]

_Punctuation, Spacing, Capitalization:_

Phrase breaks where a comma is followed by a capital letter, or a period by a lower-case letter, are not individually noted.

Number forms such as "those. ij. last" or "line. A.B." were silently regularized to "those .ij. last" and "line .A.B." Missing sentence-final periods at the end of a printed line were silently supplied.

_Initial u or medial v unchanged:_ ioyneth uertuous conuersacion XXV. CONCLVSION.... the whole circle agreynge therevnto. or eande in the utter edge of his circumference onles the uery meaning of the wordes be firste vnderstand

Geometry teacheth the drawyng, Measuring and proporcion [_capital M in original_]