The pathvvay to knowledg containing the first principles of geometrie, as they may moste aptly be applied vnto practise, bothe for vse of instrumentes geometricall, and astronomicall and also for proiection of plattes in euerye kinde, and therefore much necessary for all sortes of men.

Title

The pathvvay to knowledg containing the first principles of geometrie, as they may moste aptly be applied vnto practise, bothe for vse of instrumentes geometricall, and astronomicall and also for proiection of plattes in euerye kinde, and therefore much necessary for all sortes of men.

Author

Record, Robert, 1510?-1558.

Publication

[Imprinted at London :: In Poules churcheyarde, at the signe of the Brasen serpent, by Reynold Wolfe. Cum priuilegio ad imprimendum solum,

Anno Domini. M.D.LI. [1551]]

Rights/Permissions

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Subject terms

Geometry -- Early works to 1800.

Link to this Item

https://name.umdl.umich.edu/A10541.0001.001 content_copy

Cite this Item

"The pathvvay to knowledg containing the first principles of geometrie, as they may moste aptly be applied vnto practise, bothe for vse of instrumentes geometricall, and astronomicall and also for proiection of plattes in euerye kinde, and therefore much necessary for all sortes of men." In the digital collection Early English Books Online 2. https://name.umdl.umich.edu/A10541.0001.001. University of Michigan Library Digital Collections. Accessed May 4, 2024. content_copy

Contents

• title page
• The argumentes of the foure bookes
• TO THE GENTLE READER.
• TO THE MOST NO∣ble and puissaunt prince Edwarde the sixte by the grace of God, of En∣gland Fraunce and Ireland kynge, de∣fendour of the faithe, and of the Churche of England and Ire∣lande in earth the su∣preme head.
• THE PREFACE, declaring briefely the commodi∣ties of Geometrye, and the necessitye thereof.
• The definitions of the principles of GEOMETRY.
• THE PRACTIKE WORKINGE OF sondry conclusions Geometrical.
  • THE FYRST CONCLVSION.
    • Example.
  • THE .II. CONCLVSION. If you wil make a twileke or a nouelike triangle on ani cer¦taine line.
  • THE III. CONCL. To diuide an angle of right lines into ij. equal partes.
  • THE IIII. CONCL. To deuide any measurable line into ij. equall partes.
    • Example.
  • THE FIFT CONCLVSION. To make a plumme line or any pricke that you will in any right lyne appointed.
    • Example.
    • Example.
  • THE .VI. CONCLVSION. To drawe a streight line from any pricke that is not in a line, and to make it perpendi∣cular to an other line.
    • Example.
  • THE .VII. CONCLVSION. To make a plumbe lyne or any porcion of a circle, and that on the vtter or inner bughte.
    • Example.
  • THE VIII. CONCLVSYON. How to deuide the arche of a circle into two equall partes, without measuring the arche.
    • Example.
  • THE IX. CONCLVSION.
    • Example.
  • THE X. CONCLVSION. How to do the same thinge an other way yet
    • Example.
  • THE XI. CONCLVSION. when any line is appointed and without it a pricke, whereby a parallel must be drawen howe you shall doo it,
    • Example.
  • THE .XII. CONCLVSION. To make a triangle of any .iij. lines, so that the lines be suche, that any .ij. of them be lon∣ger then the thirde. For this rule is generall, that any two sides of euerie triangle taken to∣gether, are longer then the other side that re∣maineth.
    • Example.
  • THE XIII. CONCLVSION. If you haue a line appointed, and a pointe in it limited, howe you maye make on it a righte lined angle, equall to an other right lined an∣gle, all ready assigned.
    • Example.
  • THE XIIII. CONCLVSION. To make a square quadrate of any righte lyne appoincted.
    • Example.
  • THE .XV. CONCLVSION. To make a likeiāme equall to a triangle ap∣pointed, and that in a right lined āgle limited.
  • THE .XVI. CONCLVSION. To make a likeiamme equall to a triangle appoincted, accordyng to an angle limitted, and on a line also assigned.
    • Example.
  • THE XVII. CONCLVSION. To make a likeiamme equal to any right lined figure, and that on an angle appointed.
    • Example.
  • THE XVIII. CONCLVSION. To parte a line assigned after suche a sorte, that the square that is made of the whole line and one of his parts, shal be equal to the squar that cometh of the other parte alone.
    • Example.
  • THE .XIX. CONCLVSION. To make a square quadrate equall to any right lined figure appoincted.
    • Example.
  • THE .XX. CONCLVSION. when any .ij. square quadrates are set forth, how you maie make one equall to them bothe.
    • Example.
  • THE XXI. CONCLVSION. when any two quadrates be set forth, howe to make a squire about the one quadrate, whi∣che shall be equall to the other quadrate.
    • Example.
  • THE .XXI. CONCLVSION. To find out the cētre of any circle assigned.
    • Example.
  • THE XXIII. CONCLVSION. To find the commen centre belongyng to anye three prickes appointed, if they be not in an ex¦acte right line.
  • THE XXIIII. CONCLVSION. To drawe a touche line vnto a circle, from any poincte assigned.
    • Example.
  • THE XXV. CONCLVSION. when you haue any peece of the circumference of a circle assigned, howe you may make oute the whole circle agreynge therevnto.
    • Example.
  • THE XXVI. CONCLVSION. To finde the centre to any arche of a circle.
    • Example.
  • THE XXVII. CONCLVSION. To drawe a circle within a triangle ap∣poincted.
    • Example.
    • Example.
  • THE XXVIII. CONCLVSION. To drawe a circle about any triāgle assigned.
    • Example.
    • Example.
  • THE XXIX. CONCLVSION. To make a triangle in a circle appoynted whose corners shalbe equall to the corners of any triangle assigned.
    • Example.
  • THE XXX. CONCLVSION. To make a triangle about a circle assigned whiche shall haue corners, equall to the cor∣ners of any triangle appointed.
    • Example.
  • THE XXXI. CONCLVSION. To make a portion of a circle on any right line assigned, whiche shall conteine an angle e∣quall to a right lined angle appointed.
    • Example.
  • THE XXXII. CONCLVSION. To cutte of from any circle appoineed, a portion containyng an angle equall to a right lyned angle assigned.
    • Example.
  • THE XXXIII. CONCLVSION. To make a square quadrate in a circle assigned
    • Example.
  • THE XXXIIII. CONCLVSION. To make a square quadrate aboute annye circle assigned.
    • Example.
  • THE XXXV. CONCLVSION. To drawe a circle in any square quadrate appointed.
    • Example.
  • THE XXXVI. CONCLVSION. To draw a circle about a square quadrate.
    • Example.
  • THE XXXVII. CONCLVSION. To make a twileke triangle, whiche shall haue euery of the ij. angles that lye about the ground line, double to the other corner.
    • Example.
  • THE XXXVII. CONCLVSION. To make a cinkangle of equall sides, and equall corners in any circle appointed.
    • Example.
  • THE XXXIX. CONCLVSION. How to make a cinkangle of equall sides and equall angles about any circle appointed.
    • Example.
  • THE XL. CONCLVSION. To make a circle in any appointed cinke∣angle of equall sides and equall corners.
    • Example.
  • THE XLI. CONCLVSION. To make a circle about any assigned cinke∣angle of equall sides, and equall corners.
    • Example.
  • THE XLII. CONCLVSION. To make a siseangle of equall sides, and equall angles, in any circle assigned.
    • Example.
  • THE XLIII. CONCLVSION.
  • THE XLIIII. CONCLVSION.
  • THE XLV. CONCLVSION.
  • THE XLVI. CONCLVSION.
• title page
• epigraph
• THE PREFACE VNTO the Theoremes.
• The brefe argumentes of suche bokes as ar appoyn¦ted shortly to be set forth by the author herof.
• The Theoremes of Geometry, before WHICHE ARE SET FORTHE certaine grauntable requestes whiche serue for demonstrations Mathematicall.
  • That frō any pricke to one other, there may be drawen a right line.
  • That any right line of measurable length may be drawen forth longer, and straight.
  • That vpon any centre, there

    diagram

    may be made a circle of anye quātitee that a man wyll.
  • That all right angles be equall eche to other.
  • Yf one right line do crosse two other right lines, and make ij. inner corners of one side les¦ser thē ij. righte corners, it is certaine, that if those two lines be drawen forth right on that side that the sharpe inner corners be, they wil at lēgth mete togither, and crosse on an other.
  • Two right lines make no platte forme.
  • 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.
    • The seconde common sentence. And if you adde equall portions to thin∣ges that be equall, what so amounteth of them shallbe equall.
    • The thirde common sentence. And if you abate euen portions from things that are equal, those partes that remain shall be equall also.
    • The fourth common sentence. If you abate equalle partes from vnequal thin¦ges, the remainers shall be vnequall.
    • The fifte common sentence. when euen portions are added to vnequalle thinges, those that amounte shalbe vnequall.
    • The syxt common sentence. If two thinges be double to any other, those same two thinges are equal togither.
    • The seuenth common sentence. If any two thinges be the halfes of one other thing, than are thei .ij. equall togither.
    • The eyght common sentence. If any one quantitee be laide on an o∣ther, and thei agree, so that the one excedeth not the other, then are they e∣quall togither.
    • The nynth common sentence. Euery whole thing is greater than any of his partes.
    • The tenthe common sentence. Euery whole thinge is equall to all his partes taken togither.
  • The theoremes of Geometry brieflye declared by shorte examples.
    • The firste Theoreme. When .ij. triangles be so drawen, that the one of thē hath ij. sides equal to ij. sides of the other triangle, and that the angles enclosed with those sides, bee equal also in bothe trian∣gles, then is the thirde side likewise equall in them. And the whole triangles be of one greatnes, and euery angle in the one equall to his matche angle in the other, I meane those angles that be inclosed with like sides.
      • Example.
    • The second Theoreme. In twileke triangles the ij. corners that be about the groud line, are equal togither. And if the sides that be equal, be drawē out in lēgth thē wil the corners that are vnder the ground line, be equal also togither.
      • Example
    • The thirde Theoreme. If in annye triangle there bee twoo angles equall togither, then shall the sides, that lie a∣gainst those angles, be equal also.
      • Example
    • The fourth Theoreme. when two lines are drawen frō the endes of anie one line, and meet in anie pointe, it is not possible to draw two other lines of like lengthe ech to his match that shal begī at the same poin¦tes, and end in anie other pointe then the twoo first did.
      • Example.
    • The fifte Theoreme. If two triāgles haue there ij. sides equal one to another, and their groūd lines equal also, then shall their corners, whiche are contained be∣twene like sides, be equall one to the other.
      • Example.
    • The sixt Theoreme. when any right line standeth on an other, the ij. angles that thei make, other are both right angles, or els equall to .ij. righte angles.
      • Example.
    • The seuenth Theoreme. If .ij. lines be drawen to any one pricke in an other lyne, and those .ij. lines do make with the fyrst lyne, two right angles, other suche as be equall to two right angles, and that towarde one hande, than those two lines doo make one streyght lyne.
      • Example.
    • The eight Theoreme. when two lines do cut one an other crosse ways they do make their matche angles equall.
      • Example.
    • The nynth Theoreme. whan so euer in any triangle the line of one side is drawen forthe in lengthe, that vtter an∣gle is greater than any of the two inner cor∣ners, that ioyne not with it.
      • Example.
    • The tenth Theoreme. In euery triangle any .ij. corners, how so e∣uer you take thē, ar lesse thē ij. right corners.
      • Example.
    • The .xi. Theoreme. In euery triangle, the greattest side lieth against the greattest angle.
      • Example.
    • The twelft Theoreme. In euery triangle the greattest angle lieth against the longest side.
    • The thirtenth theoreme. In euerie triangle anie ij. sides togither how so euer you take them, are longer thē the thirde.
    • The fourtenth theoreme. If there be drawen from the endes of anie side of a triangle .ij. lines metinge within the triangle, those two lines shall be lesse then the other twoo sides of the triangle, but yet the corner that thei make, shall bee greater then that corner of the triangle, whiche standeth ouer it.
      • Example.
    • The fiftenth Theoreme. If a triangle haue two sides equall to the two sides of an other triangle, but yet the āgle that is contained betwene those sides, greater then the like angle in the other triangle, then is his grounde line greater then the grounde line of the other triangle.
    • The xvi. Theoreme. If a triangle haue twoo sides equalle to the two sides of an other triangle, but yet hathe a longer ground line thē that other triangle, then is his angle that lieth betwene the equall sides, greater thē the like corner in the other triangle.
      • Example.
    • The seuententh Theoreme. If two triangles be of such sort, that two angles of the one be equal to ij. angles of the o∣ther, and that one side of the one be equal to on side of the other, whether that side do adioyne to one of the equall corners, or els lye againste one of them, then shall the other twoo sides of those triangles bee equalle togither, and the thirde corner also shall be equall in those two triangles.
      • Example.
    • The eightenth Theoreme. when on .ij. right lines ther is drawen a third right line crosse waies, and maketh .ij. matche corners of the one line equall to the like twoo matche corners of the other line, then ar those two lines gemmow lines, or paralleles.
      • Example.
    • The nyntenth Theoreme. when on two right lines there is drawen a thirde right line crosse waies, and maketh the ij. ouer corners towarde one hande equall to∣gither, then ar those .ij. lines paralleles. And in like maner if two inner corners toward one hande, be equall to .ii. right angles.
      • Example.
    • The xx. Theoreme. when a right line is drawen crosse ouer .ij. right gemow lines, it maketh .ij. matche cor∣ners of the one line, equall to two matche cor∣ners of the other line, and also bothe ouer cor¦ners of one hande equall togither, and bothe nether corners likewaies, and more ouer two inner corners, and two vtter corners also to∣warde one hande, equall to two right angles.
      • Example.
    • The xxi. Theoreme. what so euer lines be paralleles to any other line, those same be paralleles togither.
      • Example.
    • The .xxij. theoreme. In euery triangle, when any side is drawen forth in length, the vtter angle is equall to the ij. inner angles that lie againste it. And all iij. inner angles of any triangle are equall to ij. right angles.
      • Example.
    • The .xxiii. theoreme. when any ij. right lines doth touche and cou∣ple .ij. other righte lines, whiche are equall in length and paralleles, and if those .ij. lines bee drawen towarde one hande, then are thei also equall together, and paralleles.
      • Example.
    • The .xxiiij. theoreme. In any likeiamme the two contrary sides ar equall togither, and so are eche .ij. contrary angles, and the bias line that is drawen in it, dothe diuide it into two equall portions.
      • Example.
    • The xxvi. Theoreme. All likeiammes that haue equal grounde lines and are drawen betwene one paire of pa∣ralleles, are equal togither.
      • Example.
    • The xxvii. Theoreme. All triangles hauinge one grounde lyne; an standing betwene one paire of parallels, ar equall togither.
      • Example.
    • The xxviij. Theoreme. All triangles that haue like long ground lines, and bee made betweene one paire of ge∣mow lines, are equall togither.
      • Example.
    • The xxix. Theoreme. Alequal triangles that are made on one grounde line, and rise one waye, must needes be betwene one paire of parallels.
      • Example.
    • The thirty Theoreme. Equal triangles that haue the irground lines equal, and be drawē toward one side, ar made betwene one paire of paralleles.
      • Example.
    • The xxxi, theoreme. If a likeiamme haue one ground line with a triangle, and be drawen betwene one paire of paralleles, then shall the likeiamme be double to the triangle.
      • Example.
    • The .xxxij. Theoreme. In all likeiammes where there are more than one made aboute one bias line, the fill squares of euery of them must nedes be equall.
      • Example.
    • The xxxiij. Theoreme. In all right anguled triangles, the square of that side whiche lieth against the right angle, is equall to the .ij. squares of both the other sides
      • Example.
    • The xxxiiij. Theoreme. If so be it, that in any triangle, the square of the one syde be equall to the .ij. squares of the other ij. sides, than must nedes that corner be a right corner, which is conteined betwene those two lesser sydes.
      • Example.
    • The .xxxv. theoreme. If there be set forth .ij. right lines, and one of them parted into sundry partes, how many or few so euer they be, the square that is made of those ij. right lines proposed, is equal to all the squares, that are made of the vndiuided line, and euery parte of the diuided line.
      • Example.
    • The xxxvi. Theoreme. If a right line be parted into ij. partes, as chaunce may happe, the square that is made of that whole line, is equall to bothe the squares that are made of the same line, and the twoo partes of it seuerally.
      • Example.
    • The xxxvij Theoreme. If a right line be deuided by chaunce, as it maye happen, the square: hat is made of the whole line, and one of the partes of it which so euer it be, shal be equall to that square that is made of the ij. partes ioyned togither, and to another square made of that part, which was before ioyned with the whole line.
      • Example.
    • The xxxviij. Theoreme. If a righte line be deuided by chaunce, into partes, the square that is made of that whole line, is equall to both the squares that ar made of eche parte of the line, and moreouer to two squares made of the one portion of the diuided line ioyned with the other in square.
      • Example.
    • The .xxxix. theoreme. If a right line be deuided into two equall par∣tes, 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 .xl. theoreme. If a right line be diuided into .ij. euen par∣tes, 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 por∣tion 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 xli. Theoreme. If a rightline bee diuided by chaunce, the square of the same whole line, and the square of one of his partes are iuste equall to the lōg 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.
    • The xlij. Theoreme. If a right line be deuided as chance happe∣neth 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 xliij. Theoreme. If a right line be deuided into ij. equal par∣tes first, and one of those parts again into o∣ther ij. parts, as chaūce hapeneth, the square that is made of the last part of the line so di∣uided, 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 por∣tion of the same line.
      • Example.
    • The xliiij. Theoreme. If a right line be deuided into ij. partes e∣qually, 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 double 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 .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 (in∣closyng the blunt corner) and that 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 dra∣wen from one of the sharpe angles of the fore∣sayd triangle.
      • Example.
    • 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 perpen∣dicular line falleth, and the portion of that same line, liyng betweene the perpendicular, and the foresaid sharpe corner.
      • Example.
    • The xlvij Theoreme. If ij. points be marked in the circumferēce of a circle, and a right line drawen frome the one to the other, that line must needes fal with in the circle.
      • Example.
    • 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 be 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 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 o∣ther into two equall partions.
      • Example.
    • The L. Theoreme. If two circles crosse and cut one an other, then haue not they both one centre.
      • Example.
    • The Li. Theoreme. If two circles be so drawen, that one of them do touche the other, then haue they not one centre.
      • Example.
    • The .lij. theoreme. If a certaine pointe be assigned in the dia∣meter 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 pas∣seth by the centre, and the shortest is the resi∣dew 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 cōtra∣ry waies, that is shortest, that is farthest from it. And amongest thē all there can be but one∣ly .ij. equall together, and they must nedes be so placed, that the shortest line shall be in the iust middle betwixte them.
      • Example.
    • 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 longest of all them that crosse the circle. And of thother 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 thē, 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 lēgth▪ and they two muste be on the two contrarie si∣des of the shortest line.
      • Example.
    • The L iiij. Theoreme. If a point be set forthe in a circle, and frō that pointe vnto the circumference many li∣nes drawen, of which more then two are equal togither, then is that point the centre of that circle.
      • Example.
    • The l v. Theoreme. No circle canne cut an other circle in more pointes then two.
      • Example.
    • 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 Lvij. Theoreme. If two circles bee drawen so one withoute and other, that their edges doo touche and a right line bee drawenne frome the centre of the oneto the centre of the other, that line shall passe by the place of their touching.
      • Example.
    • The .lviij. theoreme. One circle can not touche an other in more pointes then one, whether they touche within or without.
      • Example.
    • 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.
    • The .lx. Theoreme. In euerie circle the longest line is the diame∣ter, and of all the other lines, thei are still lon∣gest that be nexte vnto the centre, and they be the shortest, that be farthest distaunt from it.
      • Example.
    • The .lxi. Theoreme. If a right line be drawen at any end of a di∣ameter in perpendicular forme, and do make a right angle with the diameter, that right line shall light without the circle, and yet so ioint∣ly knitte to it, that it is not possible to draw a∣ny other right line betwene that saide line and the circumferēce of the circle And the angle that is made in the semicircle is greater then any sharpe angle that may be made of right li∣nes, but the other angle without, is lesser then any that can be made of right lines.
      • Example.
    • The lxij. Theoreme. If a right line doo touche a circle, and an other right line drawen frome the centre of tge circle to the point where they touch, that line whiche is drawenne frome the centre, shall be a perpendicular line to the touch line.
      • Example.
    • 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 cen¦tre of the circle bee in that same line, so dra∣wen.
      • Example.
    • The lxiiij. Theoreme. If an angle be made on the centre of a cir¦cle, and an other angle made on the circumfe¦rence of the same circle, and their grounde line be one common portion of the circumfe∣rence, then is the angle on the centre twise so great as the other angle on the circūferēce
      • Example.
    • The lxv. Theoreme. Those angles whiche be made in one cantle of a circle, must needes be equal togither.
      • Example.
    • The .lxvi. theoreme. Euerie figure of foure sides, drawen in a circle, hath his two contrarie angles equall vnto two right angles.
      • Example.
    • The lxvij. Theoreme. Vpon one right lyne there can not be made two cantles of circles, like and vnequall, and drawent towarde one parte.
      • Example.
    • The .lxviij. Theoreme. Lyke cantelles of circles made on equall righte lynes, are equall together.
      • Example.
    • The lxix. Theoreme. In equall circles, suche angles as be equall are made vpon equall arch lines of the circum∣ference, whether the angle light on the cir∣cumference, or on the centre.
      • Example.
    • The .lxx. Theoreme. In equall circles, those angles whiche bee made on equall arche lynes, are euer equall to∣gether, whether they be made on the centre, or on the circumference.
      • Example.
    • The lxxi. Theoreme. In equal circles, equall right lines beinge drawen, doo cutte awaye equalle arche lines frome their circumference, so that the grea∣ter arche line of the one is equall to the grea∣ter arche line of the other, and the lesser to the lesser.
      • Example.
    • The lxxij. Theoreme. In equall circles, vnder equall arche lines the right lines that bee drawen are equall to∣gither.
      • Example.
    • 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 an∣gle, 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.
    • 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 an∣gles 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 .lxxv. Theoreme. In any circle when .ij. right lines do crosse one an other, the likeiamme that is made of the por¦tions of the one line, shall be equall to the lyke∣iamme made of the partes of the other lyne.
    • 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 longe square that is made of that whole lyne whiche crosseth the circle, and the portion of it, that lyeth betwene the vtter cir∣cumference of the circle and the pointe, shall be equall to the full square of the other lyne, that onely toucheth the circle.
      • Example.
    • The .lxxvij. 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 cros∣seth the circle made with the portiō of the same line beyng without the circle betweene the vt∣ter 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.
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