these latter daies the Germaines especially as Regiomontanus Werner Schoner and Appian have grac'd it But above all other the learned Gemma Phrisius in a severall worke of that argument onely hath illustrated and taught the use of it plainely and fully The Iacobs staffe therefore according to his owne and those Geometricall parts shall here be described The astronomicall distribution wee reserve to his time and place And that done the use of it shall be shewed in the measuring of lines 2 The shankes of the staffe are the Index and the Transome 3 The Index is the double and one tenth part of the transome Or thus The Index is to the transversary double and 1 10 part thereof H. As here thou seest 4 The Transome is that which rideth upon the Index and is to be slid higher or lower at pleasure Or The transversary is to be moved upon the Index sometimes higher sometimes lower H. This proportion in defining and making of the shankes of the instrument is perpetually to be observed as if the transome be 10. parts the Index must be 21. If that be 189. this shall be 90. or if it be 2000. this shall be 4200. Neither doth it skill what the numbers be so this be their proportion More than this That the greater the numbers be that is the lesser that the divisions be the better will it be in the use And because the Index must beare and the transome is to be borne let the index be thicker and the transome the thinner But of what matter each part of the staffe be made whether of brasse or wood it skilleth not so it be firme and will not cast or warpe Notwithstanding the transome will more conveniently be moved up and downe by brasen pipes both by it selfe and upon the Index higher or lower right angle wise so touching one another that the alterne mouth of the one may touch the side of the other The thrid pipe is to be moved or slid up and downe from one end of the transome to the other and therefore it may be called the Cursor The fourth and fifth pipes fixed and immoveable are set upon the ends of the transome are unto the third and second of equall height with â—innes to restraine when neede is the opticke line and as it were with certaine points to define it in the transome The three first pipes may as occasion shall require be fastened or staied with brasen scrues With these pipes therefore the transome may be made as great as need shall require as here thou seest The fabricke or manner of making the instrument hath hitherto beene taught the use thereof followeth unto which in generall is required First a just distance For the sight is not infinite Secondly that one eye be closed For the optick faculty conveighed from both the eyes into one doth aime more certainely and the instrument is more fitly applied and set to the cheeke bone then to any other place For here the eye is as it were the center of the circle into which the transome is inscribed Thirdly the hands must be steady for if they shake the proportion of the Geodesy must needes be troubled and uncertaine Lastly the place of the station is from the midst of the foote 5 If the sight doe passe from the beginning of one shanke it passeth by the end of the other And the one shanke is perpendicular unto the magnitude to be measured the other parallell These common and generall things are premised That the sight is from the beginning of the Index by the end of the transome Or contrariwise From the beginning of the transome unto the end of the Index And that the Index is right that is perpendicular to the line to be measured the transome parallell Or contrariwise Now the perpendicularity of the Index in measurings of lengthts may be tried by a plummet of lead appendentâ— But in heights and breadths the eye must be trusted although a little varying of the plummet can make no sensible errour By the end of the transome understand that which is made by the line visuall whether it be the outmost finne or the Cursour in any other place whatsoever 6 Length and Altitude have a threefold measure The first and second kinde of measure require but one distance and that by granting a dimension of one of them for the third proportionall The third two distances and such onely is the dimension of Latitude Geodesy of right lines is two fold of one distance or of two Geodesy of one distance is when the measurer for the finding of the desired dimension doth not change his place or standing Geodesy of two distances is when the measurer by reason of some impediment lying in the way betweene him and the magnitude to be measured is constrained to change his place and make a double standing Here observe That length and heighth may be joyntly measured both with one and with a double station But breadth may not be measured otherwise than with two 7 If the sight be from the beginning of the Index râ—ght or plumbe unto the length and unto the father end of the same as the segment of the Index is unto the segment of the transome so is the heighth of the measurer unto the length The same manner of measuring shall be used form an higher place as out of y the segment of the Index is 5. parts the segment of the transome 6 and then the height be 10 foote the same Length shall be found to bee 12 foote Neither is it any matter at all whether the length in a plaine or levell underneath Or in an ascent or descent of a mountaine as in the figure under written Thus mayest thou measure the breadths of Rivers Valleys and Ditches For the Length is alwayes after this manner so that one may measure the distance of shippes on the Sea as also Thales Milesius in Proclus at the 26 pj did measure them An example thou hast here Hereafter in the measuring of Longitude and Altitude fight is unto the toppe of the heighth Which here I doe now forewarne thee of least afterward it should in vaine be reitered often The second manner of measuring a Length is thus 8. If the sight be from the beginning of the index parallell to the length to be measured as the segment of the transome is unto the segment of the index so shall the heighth given be to the length As if the segment of the Transome be 120 parts the height given 400-foote The segment of the Index 210 parts The length by the golden rule shall be 700 foote The figure is thus And the demonstration is like unto the former or indeed more easier For the triangles are equiangles as afore Therefore as o u is to u a so is e i to i a. This is the first and second kinde of measuring of a Longitude by one single distance or station The third which is by a double distance doth now
quem Agricola alijex antiquis monumentis tradi derunt Now by any one of these knowne and compared with ours to all English men well knowne the rest may easily be proportioned out 2. The thing proposed to bee measured is a Magnitude Magnitudo a Magnitude or Bignesse is the subject about which Geometry is busied For every Art hath a proper subject about which it doth employ al his rules and precepts And by this especially they doe differ one from another So the subject of Grammar was speech of Logicke reason of Arithmeticke numbers and so now of Geometry it is a magnitude all whose kindes differences and affections are hereafter to be declared 3. A Magnitude is a continuall quantity A Magnitude is quantitas continua a continued or continuall quantity A number is quantitas discreta a disjoined quantity As one two three foure doe consist of one two three foure unities which are disjoyned and severed parts whereas the parts of a Line Surface and Body are contained and continued without any manner of disjunction separation or distinction at all as by and by shall better and more plainely appeare Therefore a Magnitude is here understood to be that whereby every thing to be measured is said to bee great As a Line from hence is said to be long a Surface broade a Body solid Wherefore Length Breadth and solidity are Magnitudes 4. That is continuum continuall whose parts are contained or held together by some common bound This definition of it selfe is somewhat obscure and to be understand onely in a geometricall sense And it dependeth especially of the common bounde For the parts which here are so called are nothing in the whole but in a potentia or powre Neither indeede may the whole magnitude bee conceived but as it is compact of his parts which notwithstanding wee may in all places assume or take as conteined and continued with a common bound which Aristotle nameth a Common limit but Euclide a Common section as in a line is a Point in a surface a Line in a body a Surface 5. A bound is the outmost of a Magnitude Terminus a Terme or Bound is here understood to bee that which doth either bound limite or end actu in deede as in the beginning and end of a magnitude Or potentia in powre or ability as when it is the common bound of the continuall magnitude Neither is the Bound a parte of the bounded magnitude For the thing bounding is one thing and the thing bounded is another For the Bound is one distance dimension or degree inferiour to the thing bounded A Point is the bound of a line and it is lesse then a line by one degree because it cannot bee divided which a line may A Line is the bound of a surface and it is also lesse then a surface by one distance or dimension because it is only length wheras a surface hath both length and breadth A Surface is the bound of a body and it is lesse likewise then it is by one dimension because it is onely length and breadth whereas as a body hath both length breadth and thickenesse Now every Magnitude actu in deede is terminate bounded and finite yet the geometer doth desire some time to have an infinite line granted him but no otherwise infinite or farther to bee drawane out then may serve his turne 6. A Magnitude is both infinitely made and continued and cut or divided by those things wherewith it is bounded A line a surface and a body are made gemetrically by the motion of a point line and surface Item they are conteined continued and cut or divided by a point line and surface But a Line is bounded by a point a surface by a line And a Body by a surface as afterward by their severall kindes shall be understood Now that all magnitudes are cut or divided by the same wherewith they are bounded is conceived out of the definition of Continuum e. 4. For if the common band to containe and couple together the parts of a Line surface Body be a Point Line and Surface it must needes bee that a section or division shall be made by those common bandes And that to bee dissolved which they did containe and knitt together 7. A point is an undivisible signe in a magnitude A Point as here it is defined is not naturall and to bee perceived by sense Because sense onely perceiveth that which is a body And if there be any thing lesse then other to be perceived by sense that is called a Point Wherefore a Point is no Magnitude But it is onely that which in a Magnitude is conceived and imagined to bee undivisible And although it be voide of all bignesse or Magnitude yet is it the beginning of all magnitudes the beginning I meane potentiâ in powre 8. Magnitudes commensurable are those which one and the same measure doth measure contrariwise Magnitudes incommensurable are those which the same measure cannot measure 1 2. d. X. Magnitudes compared betweene themselves in respect of numbers have Symmetry or commensurability and Reason or rationality Of themselves Congruity and Adscription But the measure of a magnitude is onely by supposition and at the discretion of the Geometer to take as pleaseth him whether an ynch an hand breadth foote or any other thing whatsoever for a measure Therefore two magnitudes the one a foote long the other two foote long are commensurable because the magnitude of one foote doth measure them both the first once the second twice But some magnitudes there are which have no common measure as the Diagony of a quadrate and his side 116. p. X. actu in deede are Asymmetra incommensurable And yet they are potentiâ by power symmetra commensurable to witt by their quadrates For the quadrate of the diagony is double to the quadrate of the side 9. Rationall Magnitudes are those whose reason may bee expressed by a number of the measure given Contrariwise they are irrationalls 5. d. X. Ratio Reason Rate or Rationality what it is our Authour and likewise Salignacus have taught us in the first Chapter of the second booke of their Arithmetickes Thither therefore I referre thee Data mensura a Measure given or assigned is of Euclide called Rhetè that is spoken or which may be uttered definite certaine to witt which may bee expressed by some number which is no other then that which as we said was called mensura famosa a knowne or famous measure Therefore Irrationall magnitudes on the contrary are understood to be such whose reason or rate may not bee expressed by a number or a measure assigned As the side of the side of a quadrate of 20. foote unto a magnitude of two foote of which kinde of magnitudes thirteene sorts are mentioned in the tenth booke of Euclides Elements such are the segments of a right line proportionally cutte unto the whole line The Diameter in a circle is rationall But it is irrationall unto the side of
right line But many doe fall out to be in a crooked line And in a Spheare a Cone Cylinderâ— a Ruler may be applyed but it must be a sphearicall Conicall or Cylindraceall But by the example of a right line doth Vitellio 2 p j. demaund that betweene two lines a surface may be extended And so may it seeme in the Elements of many figures both plaine and solids by Euclide to be demanded That a figure may be described at the 7. and 8. e ij Item that a figure may be made vp at the 8. 14. 16. 23.28 p. vj which are of Plaines Item at the 25. 31. 33. 34. 36. 49. p.xj. which are of Solids Yet notwithstanding a plaine surface and a plaine body doe measure their rectitude by a right line so that jus postulandi this right of begging to have a thing granted may seeme primarily to bee in a right plaine line Now the Continuation of a right line is nothing else but the drawing out farther of a line now drawne and that from a point unto a point as we may continue the right line a e. unto i. wherefore the first and second Petitions of Euâ—lde do agree in one And 7. To set at a point assigned a Right line equall to another right line given And from a greater to cut off a part equall to a lesser 2. and 3. pj. Therefore 8. One right line or two cutting one another are in the same plaine out of the 1. and 2. p xj One Right line may bee the common section of two plaines yet all or the whole in the same plaine is one And all the whole is in the same other And so the whole is the same plaine Two Right lines cutting one another may bee in two plaines cutting one of another But then a plainâ— may be drawne by them Therefore both of them shall be in the same plaine And this plaine is geometrically to be conceived Because the same plaine is not alwaies made the ground whereupon one oblique line or two cutting one another are drawne when a periphery is in a sphearicall Neither may all peripheries cutting one another be possibly in one plaine And 9. With a right line given to describe a peripherie Talus the nephew of Daedalus by his sister is said in the viij booke of Ovids Metamorphosis to have beene the inventour of this instrument For there he thus writeth of him and this matter Et ex uno duo ferrea brachia nodo Iunxit ut aequali spatio distantibus ipsis Altera pars staret pars altera duceâ—et orbem Therfore 10. The raiâ—s of the same or of an equall periphery are equall The reason is because the same right line is every where converted or turned about But here by the Ray of the â—eriphery must bee understood the Ray the figure contained within the periphery 11. If two equall peripâ—eries from the ends of equall shankes of an assigned rectilineall angle doe meete before it a right line drawne from the meeting of them unto the toppe or point of the angle shall cut it into two equall parts 9. pj. Hitherto we have spoken of plaine lines Their affection followeth and first in the Bisection or dividing of an Angle into two equall parts 12. If two equall peripheries from the ends of a right line given doe meete on each side of the same a right line drawne from those meetings shall divide the right line given into two equall parts 10. pj. 13. If a right line doe stand perpendicular upon another right line it maketh on each side right angles And contrary wise The Rular for the making of straight lines on a plaine was the first Geometricall instrument The Compasses for the describing of a Circle was the second The Norma or Square for the true â—recting of a right line in the same plaine upon another right line and then of a surface and body upon a surface or body is the third The figure therefore is thus Therefore 14. If a right line do stand upon a right line it maketh the angles on each side equall to two right angles and contrariwise out of the 13. and 14. pj. And 15. If two right lines doe cut one another they doe make the angles at the top equall and all equall to foure right angles 15. pj. And 16. If two right lines cut with one right line doe make the inner angles on the same side greater then two right angles those on the other side against them shall be lesser then two right angles 17. If from â—â—oint assigned of an infinite right line given two equall parts be on each side cut off and then from the points of those sections two equall circles doe meete a right line drawne from their meeting unto the point assigned shall bee perpendicular unto the line given 11. pj. 18. If a part of an infinâ—te right line bee by a periphery from a point given without cut off a right line from the said point cutting in two the said part shall bee perpendicular upon the line given 12. pj. 19. If two right lines drawne at lâ—ngth in the same plaine doe never meete they are parallell◠è 35. dj Therefore 20. If an infinite right line doe cut one of the infinite right parallell lines it shall also cut the other As in the same example u y. cutting a e. it shall also cuâ— i o. Otherwise if it should not cut it it should be parallell unto it by the 18 e. And that against the grant 21. If right lines cut with a right line be pararellells they doe make the inner angles on the same side equall to two right angles And also the alterne angles equall betweene themselves And the outter to the inner opposite to it And contrariwise 29,28,27 p 1. The cause of this threefold propriety is from the perpendicular or plumb-line which falling upon the parallells breedeth and discovereth all this variety As here they are right angles which are the inner on the same part or side Item the alterne angles Item the inner and the outter And therefore they are equall both I meane the two inner to two right angles and the alterne angles between themselvs And the outter to the inner opposite to it If so be that the cutting line be oblique that is fall not upon them plumbe or perpendicularly the same shall on the contrary befall the parallels For by that same obliquaâ—ion or slanting the right lines remaining and the angles unaltered in like manner both one of the inner to wit e u y is made obtuse the other to wiâ— u y o is made acute And the alterne angles are made acute and obtuse As also the outter and inner opposite are likewise made acute and obtuse The same impossibility shall be concluded if they shall be sayd to be lesser than two right anglesâ— The second and third parts may be concluded out of the first The second is thus Twise two angles are equall to two right
angles o y u and e u y by the former part Item a u y and e u y by the 14 e. Therefore they are equall betweene themselves Now from the equall Take away e u y the common angle And the remainders the alterne angles at u and y shall be least equall The third is thus The angles e u y and o y s are equall to the same u y i by the second propriety and by the 15 e. Therefore they are equall betweene themselves If they be oblique angles as here the lines one slanting or liquely crossing one another the angles on one side will grow lesse on the other side greater Therefore they would not be equall to two right angles against the graunt From hence the second and third parts may be concluded The second is thus The alterne angles at u and y are equall to the foresayd inner angles by the 14 e Because both of them are equall to the two right angles And so by the first part the second is concluded The third is therefore by the second demonstrated because the outter o y s is equall to the verticall or opposite angle at the top by the 15 e. Therefore seeing the outter and inner opposite are equall the alterne also are equall Wherefore as Parallelismus parallell-equality argueth a three-fold equality of angels So the threefold equality of angles doth argue the same parallel-equality Therefore 22. If right lines knit together with a right line doe make the inner angles on the same side lesser than two right Angles they being on that side drawne out at length will meete And 23. A right line knitting together parallell right lines is in the same plaine with them 7 p xj And 24. If a right line from a point given doe with a right line given make an angle the other shanke of the angle equalled and alterne to the angle made shall be parallell unto the assigned right line 31 pj. An angle I confesse may bee made equall by the first propriety And so indeed commonly the Architects and Carpenters doe make it by erecting of a perpendicular It may also againe in like manner be made by the outter angle Any man may at his pleasure use which hee shall thinke good But that here taught we take to be the best And 25. The angles of shanks alternly parallell are equall Or Thus The angles whose altenate feete are parallells are equall H. And 26 If parallels doe bound parallels the opposite lines are equall è 34 p.j. Or thus If parallels doe inclose parallels the opposite parallels are equall H. And 27. If right lines doe joyntly bound on the same side equall and parallell lines they are also equall and parallell On the same part or side it is sayd least any man might understand right lines knit together by opposite bounds as here 28. If right lines be cut joyntly by many parallell right lines the segments betweene those lines shall bee proportionall one to another out of the 2 p vj and 17 p x j. Thus much of the Perpendicle and parallell equality of plaine right lines Their Proportion is the last thing to be considered of them If the lines cut be not parallels but doe leane one toward another the portions cut or intercepted betweene them will not be equall yet shall they be proportionall one to another And looke how much greater the line thus cut is so much greater shall the intersegments or portions intercepted be And contrariwise Looke how much lesse so much lesser shall they be The third parallell in the toppe is not expressed yet must it be understood This element is very fruitfull For from hence doe arise and issue First the manner of cutting a line according to any rate or proportion assigned And then the invention or way to finde out both the third and fourth proportionalls 29. If a right line making an angle with another right line be cut according to any reason or proportion assigned parallels drawne from the ends of the segments unto the end of the sayd right line given and unto some contingent point in the same shall cut the line given according to the reason given Schoner hath altered this Consectary and delivereth it thus If a right making an angle with a right line given and 〈◊〉 it unto it with a base be cut according to any rate assigned a parallell to the base from the ends of the segments shall cut the line given according to the rate assigned 9 and 10 p v j. Punctum contingens A contingent point that is falling or lighting in some place at al adventurs not given or assigned This is a marvelous generall consectary serving indifferently for any manner of section of a right line whether it be to be cut into two parts or three parts or into as many patts as you shall thinke good or generally after what manner of way soever thou shalt command or desire a line to be cut or divided Now 〈◊〉 be cut into three parts◠〈◊〉 which the first let it bee the halfe of the second And the second the halfe of the third And the conter minall or right line making an angle with the sayd assigned line let it be cut one part a o Then double this in o u Lastly let u i be taken double to o u and let the whole diagramme be made up with three parallels y◠and os The fourth parallell in the toppe as a fore-sayd shall be understood Therefore that section which was made in the conterminall line by the 28 e shall be in the assigned line Because the segments or portions intercepted are betweene the parallels And 30. If two right lines given making an angle be continued the first equally to the second the second infinitly parallels drawne from the ends of the first continuation unto the beginning of the second and some contingent point in the same shall intercept betweene them the third proportionall 11. p v j. And 31. If of three right lines given the first and the third making an angle be continued the first equally to the second and the third infinitly parallels drawne from the ends of the first continuation unto the beginning of the second and some contingent point the same shall intercept betweene them the fourth proportionall 12. p vj. Let the lines given be these The first a e the second e i the third a o and let the whole diagramme be made up according to the prescript of the consectary Here by 28. e as a e is to e i so is a o to o u. Thus farre Ramus Lazarus Schonerus who about some 25. yeares since did revise and augment this worke of our Authour hath not onely altered the forme of these two next precedent consectaries but he hath also changed their order and that which is here the second is in his edition the third and the third here is in him the second And to the former declaration of them hee addeth these
is a i so is a i unto i e Wherefore by the â— e a e is proportionall cut And the greater segment is a i the same remaine The other propriety of the quintuple doth follow 6 The lesser segment continued to the halfe of the greater is of power quintuple to the same halfe è 3 p x iij. The rate of the triple followeth 7 The whole line and the lesser segment are in power treble unto the greater è 4 p xiij 8 An obliquangled parallelogramme is either a Rhombus or a Rhomboides 9 A Rhombus is an obliquangled equilater parallelogramme 32 dj It is otherwise of some called a Diamond 10 A Rhomboides is an obliquangled parallelogramâ—e not equilater 33. dj And a Rhomboides is so opposed to an oblong as a Rhombus is to a quadrate And the Rhomboides is so called as one would say Rhombuslike although beside the inequality of the angles it hath nothing like to a Rhombus An example of measuring of a Rhombus is thus 11 A Trapezium is a quadrangle not parallelogramme 34. dj The examples both of the figure and of the measure of the same let these be Therefore triangulate quadrangles are of this sort 12 A multangle is a figure that is comprehended of more than foure right lines 23. dj By this generall name all other sorts of right lined figures hereafter following are by Euclide comprehended as are the quinquangle sexangle septangle and such like inumerable taking their names of the number of their angles In every kinde of multangle there is one ordinate as we have in the former signified of which in this place we will say nothing but this one thing of the quinquangle The rest shall be reserved untill we come to Adscription 13 Multangled triangulates doe take their measure also from their triangles 14 If an equilater quinquangle have three sides equall it is equiangled 7 p 13. This of some from the Greeke is called a Pentagon of others a Pentangle by a name partly Greeke partly Latine The fifteenth Booke of Geometry Of the Lines in a Circle AS yet we have had the Geometry of rectilineals The Geometry of Curvilineals of which the Circle is the chiefe doth follow 1. A Circle is a round plaine â— 15 dj The meanes to describe a Circle is the same which was to make a Periphery But with some difference For there was considered no more but the motion the point in the end of the ray describing the periphery Here is considered the motion of the whole ray making the whole plot conteined within the periphery A Circle of all plaines is the most ordinate figure as was before taught at the 10 e iiij 2 Cirâ—les are as the quadrates or squares made of their diameters 2 p. x ij Therefore 3. The Diameters are as their peripheries Pappus 5 l x j and 26 th 18. As here thou seest in a e and i o. 4. Circular Geometry is either in Lines or in the segments of a Circle This partition of the subject matters howsoever is taken for the distinguishing and severing with some light a matter somewhat confused And indeed concerning lines the consideration of secants is here the foremost and first of Inscripts 5. If a right line be bounded by two points in the periphery it shall fall within the Circle 2 p iij. From hence doth follow the Infinite section of which we spake at the 6 e j. This proposition teacheth how a Rightline is to be inscribed in a circle to wit by taking of two points in the periphery 6. If from the end of the diameter and with a ray of it equal to the right line given a periphery be described a right line drawne from the said end unto the meeting of the peripheries shall be inscribed into the circle equall to the right line given 1 p iiij And this proposition teacheth How a right line given is to be inscribed into a Circle equall to a line given Moreover of all inscripts the diameter is the chiefe For it sheweth the center and also the reason or proportion of all other inscripts Therefore the invention and making of the diameter of a Circle is first to be taught 7. If an inscript do cut into two equall parts another inscript perpendicularly it is the diamiter of the Circle and the middest of it is the center 1 p iij. The cause is the same which was of the 5 e x j. Because the inscript cut into halfes if for the side of the inscribed rectangle and it doth subtend the periphery cut also into two parts By the which both the Inscript and Periphery also were in like manner cut into two equall parts Therefore the right line thus halfing in the diameter of the rectangle But that the middle of the circle is the center is mâ—nifest out of the 7 e v and 29 e iiij Euclide thought better of Impossibile than he did of the cause And thus he forceth it For if y be not the Center but s the part must be equall to the whole For the Triangle a o s shall be equilater to the triangle e o s. For a o oe are equall by the grant Item s a and s e are the rayes of the circle And s o is common to both the triangles Therefore by the 1 e vij the angles no each side at o are equall And by the 13 e v they are both right angles Therefore s o e is a right angle It is therefore equall by the grant to the right angle y o e that is the part is equall to the whole which is impossible Wherefore y is not the Center The same will fall out of any other points whatsoever â—ut of y. Therefore 8. If two râ—ght lines doe perpendicularly halfe two inscripts the meeting of these two bisecants shall be the Center of the circle è 25 p iij. And one may 9. Draw a periphery by three points which doe not fall in a right line 10. If a diameter doe halfe an inscript that is nâ—t a diameter it doth cut it perpendicularly And contrariwise 3 p iij. 11. If inscripts which are not diameters doe cut one another the segments shall be unequall 4 p iij. But rate hath beene hitherto in the parts of inscripts Proportion in the same parts followeth 12 If two inscripts doe cut one another the rectangle of the segments of the one is equall to the rectangle of the segments of the other 35 p iij. And this is the comparison of the parts inscripts The rate of whole inscripts doth follow the which whole one diameter doth make 13 Inscripts are equall distant from the center unto which the perpendiculars from the center are equall 4 d iij. 14. If inscripts be equall they be equally distant from the center And contrariwise 13 p iij. The diameters in the same circle by the 28 e iiijâ— are equall And they are equally distant from the center seeing they are by the center or rather are no whit at all
an inscribed quinquangle The Diagony of an â—cosahedron and Dodecahedron is irrationall unto the side 10. Congruall or agreeable magnitudes are those whose parts beeing applyed or laid one upon another doe fill an equall place Symmetria Symmetry or Commensurability and Rate were from numbers The next affections of Magnitudes are altogether geometricall Congruentia Congruency Agreeablenesse is of two magnitudes when the first parts of the one doe agree to the first parts of the other the meane to the meane the extreames or ends to the ends and lastly the parts of the one in all respects to the parts of the other so Lines are congruall or agreeable when the bounding points of the one applyed to the bounding points of the other and the whole lengths to the whole lengthes doe occupie or fill the same place So Surfaces doe agree when the bounding lines with the bounding lines And the plots bounded with the plots bounded doe occupie the same place Now bodies if they do agree they do seeme only to agree by their surfaces And by this kind of congruency do we measure the bodies of all both liquid and dry things to witt by filling an equall place Thus also doe the moniers judge the monies and coines to be equall by the equall weight of the plates in filling up of an equall place But here note that there is nothing that is onely a line or a surface onely that is naturall and sensible to the touch but whatsoever is naturall and thus to be discerned is corporeall Therefore 11. Congruall or agreeable Magnitudes are equall 8. ax.j. A lesser right line may agree to a part of a greater but to so much of it it is equall with how much it doth agree Neither is that axiome reciprocall or to be converted For neither in deede are Congruity and Equality reciprocall or convertible For a Triangle may bee equall to a Parallelogramme yet it cannot in all points agree to it And so to a Circle there is sometimes sought an equall quadrate although in congruall or not agreeing with it Because those things which are of the like kinde doe onely agree 12. Magnitudes are described betweene themselves one with another when the bounds of the one are bounded within the boundes of the other That which is within is called the inscript and that which is without the Circumscript Now followeth Adscription whose kindes are Inscription and Circumscription That is when one figure is written or made within another This when it is written or made about another figure Homogenea Homogenealls or figures of the same kinde onely betweene themselves rectitermina or right bounded are properly adscribed betweene themselves and with a round Notwithstanding at the 15. booke of Euclides Elements Heterogenea Heterogenealls or figures of divers kindes are also adscribed to witt the five ordinate plaine bodies betweene themselves And a right line is inscribed within a periphery and a triangle But the use of adscription of a rectilineall and circle shall hereafter manifest singular and notable mysteries by the reason and meanes of adscripts which adscription shall be the key whereby a way is opened unto that most excellent doctrine taught by the subtenses or inscripts of a circle as Ptolomey speakes or Sines as the latter writers call them The second Booke of Geometry Of a Line 1. A Magnitude is either a Line or a Lineate THe Common affections of a magnitude are hitherto declared The Species or kindes doe follow for other then this division our authour could not then meete withall 2. A Line is a Magnitude onely long 3. The bound of a line is a point 4. A Line is either Right or Crooked This division is taken out of the 4 d j. of Euclide where rectitude or straightnes is attributed to a line as if from it both surfaces and bodies were to have it And even so the rectitude of a solid figure here-after shall be understood by a right line perpendicular from the toppe unto the center of the base Wherefore rectitude is propper unto a line And therefore also obliquity or crookednesse from whence a surface is judged to be right or oblique and a body right or oblique 5. A right line is that which lyeth equally betweene his owne bounds A crooked line lieth contrariwise 4. d. j. Therefore 6. A right line is the shortest betweene the same bounds Linea recta a straight or right line is that as Plato defineth it whose middle points do hinder us from seeing both the extremes at once As in the eclipse of the Sunne if a right line should be drawne from the Sunne by the Moone unto our eye the body of the Moone beeing in the midst would hinder our sight and would take away the sight of the Sunne from uâ—â— which is taken from the Opticks in which we are taught that we see by straight beames or rayes Therfore to lye equally betweene the boundes that is by an equall distance to bee the shortest betweene the same bounds And that the middest doth hinder the sight of the extremes is all one 7. A crooked line is touch'd of a right or crooked line when they both doe so meete that being continued or drawne out farther they doe not cut one another Tactus Touching is propper to a crooked line compared either with a right line or crooked as is manifest out of the 2. and 3. d 3. A right line is said to touch a circle which touching the circle and drawne out farther doth not cut the circle 2 d 3. as here a e the right line toucheth the periphery i o u. And a e. doth touch the helix or spirall Circles are said to touch one another when touching they doe not cutte one another 3. d 3. as here the periphery doth a e j. doth touch the periphery o u y. Therefore 8. Touching is but in one point onely è 13. p 3. This Consectary is immediatly conceived out of the definition for otherwise it were a cutting not touching So Aristotle in his Mechanickes saith That a round is easiliest mou'd and most swift Because it is least touch't of the plaine underneath it 9. A crooked line is either a Periphery or an Helix This also is such a division as our Authour could then hitte on 10. A Periphery is a crooked line which is equally distant from the middest of the space comprehended Therefore 11. A Periphery is made by the turning about of a line the one end thereof standing still and the other drawing the line Now the line that is turned about may in a plaine bee either a right line or a crooked line In a sphericall it is onely a crooked line But in a conicall or Cylindraceall it may bee a right line as is the side of a Cone and Cylinder Therefore in the conversion or turning about of a line making a periphery there is considered onely the distance yea two points one in the center the other in the toppe which therefore Aristotle nameth
as is manifest by division The examples are thus And 26. If foure right lines bee proportionall betweene themselves Like figures likelily situate upon them shall be also proportionall betweene themselves And contrariwise out of the 22. pvj. and 37. pxj. The proportion may also here in part bee expressed by numbers And yet a continuall is not required as it was in the former In Plaines let the first example be as followeth The cause of proportionall figures for that twice two figures have the same reason doubled In Solids let this bee the second example And yet here the figures are not proportionall unto the right lines as before figures of equall heighth were unto their basesâ— but they themselves are proportionall one to another And yet are they not proportionall in the same kinde of proportion The cause also is here the same that was before To witt because twice two figures have the same reason trebled 27. Figures filling a place are those which being any way set about the same point doe leave no voide roome This was the definition of the ancient Geometers as appeareth out of Simplicius in his commentaries upon the 8. chapter of Aristotle's iij. booke of Heaven which kinde of figures Aristotle in the same place deemeth to bee onely ordinate and yet not all of that kindâ— But only three among the Plaines to witt a Triangle a Quadrate and a Sexangle amongst Solids two the Pyramis and the Cube But if the filling of a place bee judged by right angles 4. in a Plaine and 8. in a Solid the Oblong of plaines and the Octahedrum of Solids shall as shall appeare in their places fill a place And yet is not this Geometrie of Aristotle accurate enough But right angles doe determine this sentence and so doth Euclide out of the angles demonstrate That there are onely five ordinate solids And so doth Potamon the Geometer as Simplicus testifieth demonstrate the question of figuâ—es filling a place Lastly if figures by laying of their corners together doe make in a Plaine 4. right angles or in a Solid 8. they doe fill a place Of this probleme the ancient geometers have written as we heard even now And of the latter writers Regiomontanus is said to have written accurately And of this argument Maucolycus hath promised a treatise neither of which as yet it hath beene our good hap to see Neither of these are figures of this nature as in their due places shall be proved and demonstrated 28. A round figure is that all whose raies are equall Rotundum a Roundle let it be here used for Rotunda figura a round figure And in deede Thomas Finkius or Finche as we would call him a learned Dane sequestring this argument from the rest of the body of Geometry hath intituled that his worke De Geometria rotundi Of the Geometry of the Round or roundle 29. The diameters of a roundle are cut in two by equall raies The reason is because the halfes of the diameters are the raies Or because the diameter is nothing else but a doubled ray Therefore if thou shalt cut off from the diameter so much as is the radius or ray it followeth that so much shall still remaine as thou hast cutte of to witt one ray which is the other halfe of the diameter Sn. And here observe That Bisecare doth here and in other places following signifie to cutte a thing into two equall parts or portionsâ— And so Bisegmentum to be one such portionâ— And Bisectio such a like cutting or division 30. Rounds of equall diameters are equall Out of the 1. d. iâ—â— Circles and Spheares are equall which have equall diameters For the raies which doe measure the space betweene the Center and Perimeter are equall of which beiâ—g doubled the Diameter doth consist Sn. The fifth Booke of Ramus his Geometry which is of Lines and Angles in a plaine Surface 1. A lineate is either a Surface or a Body LIneatum or Lineamentum a magnitude made of lines as was defined at 1. e. iij. is here divided into two kindes which is easily conceived out of the said definition there in which a line is excluded and a Surface a body are comprehended And from hence arose the division of the arte Metriall into Geometry of a surface and Stereometry of a body after which maner Plato in his vij booke of his Common-wealth and Aristotle in the 7. chapter of the first booke of his Posteriorums doe diâ—tinguish betweene Geometry and Stereometry And yet the name of Geometry is used to signifie the whole arte of measuring in generall 2. A Surface is a lineate only broade 5. dj Epiphania the Greeke word which importeth onely the outter appearance of a thing is here more significant because of a Magnitude there is nothing visible or to bee seene but the surface 3. The bound of a surface is a line 6. dj The matter in Plaines is manifest For a three cornered surface is bounded with 3. lines A foure cornered suâ—face with foure liâ—es and so forth A Circle is bounded with one line But in a Sphearicall surface the matter is not so plaine For it being whole seemeth not to be bounded with a line Yet if the manner of making of a Sphearicall surface by the conversioÌ„ or turning about of a semiperiphery the beginning of it as also the end shal be a line to wit a semiperiphery And as a point doth not only actu or indeede bound and end a line But is potentia or in power the middest of it So also a line boundeth a Surface actu and an innumerable company of lines may be taken or supposed to be throughout the whole surface A Surface therefore is made by the motion of a line as a Line was made by the motion of a point 4. A surface is either Plaine or Bowed The difference of a Surface doth answer to the difference of a Lineâ— in straightnesse and obliquity or crookednesse Obliquum oblique there signified crooked Not righâ— or straight Here uneven or bowed either upward or downeward Sn. 5. A plaine surface is a surface which lyeth â—qually betweene his bounds out of the 7. dj Planum a Plaine is taken and used for a plaine surface as before Rotundum a Round was used for a round figure Therefore 6. From a point unto a point we may in a plaine surface draw a right line 1 and 2. post j. Three things are from the former ground begg'd The first is of a Right line A right line and a periphery were in the ij booke defined But the fabricke or making of them both is here said to bee properly in a plaine Now the Geometricall instrument for the drawing of a right plaine is called Amussis by Petolemey in the 2. chapter of his first booke of his Musicke Regula a Rular such as heere thou seest And from a point unto a point is this justly demanded to be done not unto points For neither doe all points fall in a
words From hence having three lines given is the invention of the fourth proportionall and out of that having two lines given ariseth the invention of the third proportionall 2 Having three right lines given if the first and the third making an angle and knit together with a base be continued the first equally to the second the third infinitly a parallel from the end of the second unto the continuation of the third shall intercept the fourth proportionall 12. pvj. The Diagramme and demonstration is the same with our 31. e or 3 c of Ramus 3 If two right lines given making an angle and knit together with a base be continued the first equally to the second the second infinitly a parallell to the base from the end of the first continuation unto the second shall intercept the third proportionall 11. p v j. The Diagramme here also and demonstration is in all respects the same with our 30 e or 2 c of Ramus Thus farre Ramus And here by the judgement of the learned Finkius two elements of Ptolomey are to be adjoyned 32 If two right lines cutting one another be againe cut with many parallels the parallels are proportionall unto their next segments The same demonstation shall serve if the lines do crosse one another or doe vertically cut one another as in the same diagramme appeareth For if the assigned a i and u s doe cut one another vertically in o let them be cut with the parallels a u and s i the precedent fabricke or figure being made up it shall be by 28. e. as a u is unto a o the segment next unto it so a y that is i s shall be unto o i his next segment The 28. e teacheth how to finde out the third and fourth proportionall This affordeth us a meanes how to find out the continually meane proportionall single or double Thefore 33. If two right lines given be continued into one a perpendicular from the point of continuation unto the angle of the squire including the continued line with the continuation is the meane proportionall betweene the two right lines given A squire Norma Gnomon or Canon is an instrument consisting of two shankes including a right angle Of this we heard before at the 13 e By the meanes of this a meane proportionall unto two lines given is easily found whereupon it may also be called a Mesolabium or Mesographus simplex or single meane finder And 34 If two assigned right lines joyned together by their ends right anglewise be continued vertically a square falling with one of his shankes and another to it parallell and moveable upon the ends of the assigned with the angles upon the continued lines shall cut betweene them from the continued two meanes continually proportionall to the assigned The former consectary was of a single mesolabium this is of a double whose use in making of solids to this or that bignesse desired is notable And thus wee have the composition and use both of the single and double Mesolabium 35. If of foure right lines two doe make an angle the other reflected or turned backe upon themselves from the ends of these doe cut the former the reason of the one unto his owne â—egment or of the segments betweene themselves is made of the reason of the so joyntly bounded that the first of the makers be joyntly bounded with the beginning of the antecedent made the second of this consequent joyntly bounded with the end doe end in the end of the consequent made Let therefore the two right lines be â— e and a i and from the ends of these other two reflected be i u and e o cutting themselves in y and the two former in u and o. The reason of the particular right lines made shall be as the draught following doth manifest In which the antecedents of the makers are in the upper place the consequents are set under neathe their owne antecedents The businesse is the same in the two other whether you doe crosse the bounds or invert them Here for demonstrations sake we crave no more but that from the beginning of an antecedent made a parallell be drawne to the second consequent of the makers unto one of the assigned infinitely continued then the multiplied proportions shall be The Antecedent the Consequent the Antecedent the Consequent of the second of the makers every way the reason or rate is of Equallity The Antecedent the Consequent of the first of the makers the Parallel the Antecedent of the second of the makers by the 32. e. Therefore by multiplication of proportions the reason of the Parallell unto the Consequent of the second of the makers that is by the fabricke or construction and the 32. e. the reason of the Antecedent of the Product unto the Consequent is made of the reason c. after the manner above written Againe I say that the reason of e y unto y o is compounded of the reason of e u unto u a and of a i unto i â— Theon here draweth a parallell from o unto u i. By the generall fabricke it may be drawne out of e unto o i. Therefore the reason of e n unto i o that is of e y unto y o shall be made of the foresaid reasons Of the segments of divers right linesâ— the Arabians have much under the name of The rule of sixe quantities And the Theoremes of Althinâ—us concerning this matter are in many mens hands And Regiomontanus in his Algorithmus and Maurolycus upon the 1 piij. of Menelaus doe make mention of them but they containe nothing which may not by any man skilfull in Arithmeticke be performed by the multiplication of proportions For all those wayes of theirs are no more but speciall examples of that kinde of multiplication Of Geometry the sixt Booke of a Triangle 1 Like plaines have a double reason of their homâ—logall sides and one proportionall meane out of 20 p vj. and xj and 18. p viij OR thus Like plaines have the proportion of their corespondent proportionall sides doubled one meane proportionall Hitherto wee have spoken of plaine lines and their affections Plaine figures and their kindes doe follow in the next place And first there is premised a common corollary drawne out of the 24. e iiij because in plaines there are but two dimensions 2 A plaine surface is either rectilineall or obliquelineall or rightlined or crookedlined H. Straightnesse and crookednesse was the difference of lines at the 4. e i j. From thence is it here repeated and attributed to a surface which is geometrically made of lines That made of right lines is rectileniall that which is made of crooked lines is Obliquilineall 3. A rectilineall surface is that which is comprehended of right lines 4 A rightilineall doth make all his angles equall to right angles the inner ones generally to paires from two forward the outter alwayes to foure Or thus A right lined plaine maketh his angles equall unto right
angles Namely the inward angles generally are equall unto the even numbers from two forward but the outward angles are equall but to 4. right angles H. 5 A rectilineall is either a Triangle or a Triangulate As before of a line was made a lineate so here in like manner of a triangle is made a triangulate 6 A triangle is a rectilineall figure comprehended of three rightlines 21. dj Therefore 7 A triangle is the prime figure of rectilineals A triangle or threesided figure is the prime or most simple figure of all rectilineals For amongst rectilineall figures there is none of two sides For two right lines cannot inclose a figure What is meant by a prime figure was taught at the 7. e. iiij And 8 If an infinite right line doe cut the angle of a triangle it doth also cut the base of the same Vitell. 29. t j. 9 Any two sides of a triangle are greater than the other Let the triangle be a e i I say the side a i is shorter than the two sides a e and e i because by the 6. e ij a right line is betweene the same bounds the shortest Therefore 10 If of three right lines given any two of them be greater than the other and peripheries described upon the ends of the one at the distances of the other two shall meete the rayes from that meeting unto the said ends shall make a triangle of the lines given And 11 If two equall peripheries from the ends of a right line given and at his distance doe meete liâ—es drawne from the meeting unto the said ends shall make an equilater triangle upon the line given 1 p.j. 12 If a right line in a triangle be parallell to the base it doth cut the shankes proportionally And contrariwise 2 p v j. As here in the triangle a e i let o u be parallell to the base and let a third parallel be understood to be in the toppe a therefore by the 28. e.v. the intersegments are proportionall The converse is forced out of the antecedent because otherwise the whole should be lesse than the part For if o u be not parallell to the base e i then y u is Here by the grant and by the antecedent seeing a o o e a y y e are proportionall and the first a o is lesser than a y the third o e the second must be lesser than y e the fourth that is the whole then the part 13 The three angles of a triangle are equall to two right angles 32. p j. Therefore 14. Any two angles of a triangle are lesse than two right angles For if three angles be equall to two right angles then are two lesser than two right angles And 15 The one side of any triangle being continued or drawne out the outter angle shall be equall to the two inner opposite angles Therefore 16 The said outter angle is greater than either of the inner opposite angles 16. p j. This is a consectary following necessarily upon the next former consectary 17 If a triangle be equicrurall the angles at the base are equall and contrariwise 5. and 6. p.j. Therefore 18 If the equall shankes of a triangle be continued or drawne out the angles under the base shall be equall betweene themselves And 19 If a triangle be an equilater it is also an equiangle And contrariwise And 20 The angle of an equilater triangle doth countervaile two third parts of a right angle Regio 23. p j. For seeing that 3. angles are equall to 2. 1. must needs be equall to â…” And 21 Sixe equilater triangles doe fill a place 22 The greatest side of a triangle subtendeth the greatest angle and the greatest angle is subtended of the greatest side 19. and 18. p j. The converse is manifest by the same figure As let the angle a e i be greater than the angle a i e. Therefore by the same 9 e iij. it is greater in base For what is there spoken of angles in generall are here assumed specially of the angles in a triangle 23 If a right line in a triangle doe cut the angle in two equall parts it shall cut the base according to the reason of the shankes and contrariwise 3. p v j. The mingled proportion of the sides and angles doth now remaine to be handled in the last place The Converse likewise is demonstrated in the same figure For as e a is to a i so is e o to o i And so is e a to a u by the 12 e therefore a i and a u are equall Item the angles e a o and o a i are equall to the angles at u and i by the 21. e vâ— which are equall betweene themselves by the 17. e. Of Geometry the seventh Booke Of the comparison of Triangles 1 Equilater triangles are equiangles 8. p.j. Thus forre of the Geometry or affections and reason of one triangle the comparison of two triangles one with another doth follow And first of their rate or reason out of their sides and angles Whereupon triangles betweene themselves are said to be equilaters and equiangles First out of the equality of the sides is drawne also the equalitie of the angles Triangles therefore are here jointly called equilaters whose sides are severally equall the first to the first the second to the second the third to the third although every severall triangle be inequilaterall Therefore the equality of the sides doth argue the equality of the angles by the 7. e iij. As here 2 If two triangles be equall in angles either the two equicrurals or two of equall either shanke or base of two angles they are equilaters 4. and 26. p j. Oh thus If two triangles be equall in their angles either in two angles contained under equall feet or in two angles whose side or base of both is equall those angles are equilater H. This element hath three parts or it doth conclude two triangles to be equilaters three wayes 1. The first part is apparent thus Let the two triangles be a e i and o u y because the equall angles at a and o are equicrurall therefore they are equall in base by the 7. e iij. 3 The third part is thus forced In the triangles a e i and o u y let the angles at e and i and u and y be equall as afore And a e. the base of the angle at i be equall to o u the base of angle at y I say that the two triangles given are equilaters For if the side e i be greater than the side u y let e s be cut off equall to it and draw the right line a s. Therefore by the antecedent the two triangles a e s and o u y equall in the angle of their equall shankes are equiangle And the angle a s e is equall to the angle o y u which is equall by the grant unto the angle a i e. Therefore a s e is equall to a i e
distant from it Other inscripts are judged to be equall greater or lesser one than another by the diameter or by the diameters center Euclide doth demonstrate this proposition thus Let first a e and i o be equall I say they are equidistant from the center For let u y and u y be perpendiculars They shall cut the assigned a e i o into halfes by the 5 e xj And y a and s i aâ—e equall because they are the halfes of equals Now let the raies of the circle be u a aund u i Their quadrates by the 9 e xij are equall to the paire of quadrates of the shankes which paires are therefore equall betweene themselves Take from equalls the quadrates y a and s i there shall remaine y u and u s equalls and therefore the sides are equall by the 4 e 12. The converse likewise is manifest For the perpendiculars given do halfe them And the halfes as before are equall 15 Of unequall inscripts the diameter is the greatest And that which is next to the diameter is greater than that which is farther off from it That which is farthest off from it is the least And that which is next to the least is lesser than that which is farther off And those two onely which are on each side of the diameter are equall è 15 e iij. This proposition consisteth of five members The first is The diameter is the greatest iuscript The second That which is next to the diameter is greater than that which is farther off The third That which is farthest off from the diameter is the least The fourth That next to the least is lesser than that farther off The fifth That two onely on each side of the diameter are equall betweene themselves All which are manifest out of that same argument of equalitie that is the center the beginning of decreasing and the end of increasing For looke how much farther off you goe from the center or how much nearer you come unto it so much lesâ—er or greater doe you make the inscript But Euclides conclusion is by triangles of two sides greater than the other and of the greater angle The first part is plaine thus Because the diameter a e is equall to i l and l o viz. to the raies And to those which are greater than i o the base by the 9. e v j c. The second part of the nearer is manifest by the 5 e vij because of the triangle i l o equicrurall to the triangle u l y is greater in angle And therefore it is also greater in base The third and fourth are consectaries of the first The fifth part is manifest by the second For if beside i o and s r there be supposed a third equall the same also shall be unequall because it shall be both nearer and farther off from the diameter 16 Of right lines drawne from a point in the diameter which is not the center unto the periphery that which passeth by the center is the greatest And that which is nearer to the greatest is greater than that which is farther off The other part of the greatest is the lest And that which is nearest to the least is lesser than that which is farther off And two on each side of the greater or least are only equall 7 p iij. The third that a y is lesser than a u because s y which is equall to s u is lesser than the right lines s a and a u by the 9 e v j And the common s a being taken away a y shall be left lesser than a u. The fourth part followeth of the third The fifth let it be thus s r making the angle a s r equall to the angle a s u the bases a u and a r shall be equall by the 2 e v ij To these if the third be supposed to be equall as a l it would follow by the 1 e v ij that the whole angle s a should be equall to r s a the particular angle which is impossible And out of this fifth part issueth this Consectary Therefore 17 If a point in a circle be the bound of three equall right lines determined in the periphery it is the center of the circle 9 p iij. Let the point a in a circle be the common bound of three right lines ending in the periphery and equall betweene themselves be a e a i a u. I say this point is the center of the Circle 18 Of right lines drawne from a point assigned without the periphery unto the concavity or hollow of the same that which is by the center is the greatest And that next to the greatest is greater than that which is farther off But of those which fall upon the convexitiâ— of the circumference the segment of the greatest is leastâ— And that which is next unto the least is lesser than that is farther off And two on each side of the greatest or least are onely equall 8 piij. 19 If a right line be perpendicular unto the end of the diameter it doth touch the periphery And contrariwise è 16 p iij. As for example Let the circle given a e be perpendicular to the end of the diameter or the end of the ray in the end a as suppose the ray be i a I say that e a doth touch not cut the periphery in the common bound a. Therefore 20 If a right line doe passe by the center and touch-point it is perpendicular to the tangent or touch-line 18 p iij. And Or thus as Schoner amendeth it If a right line be the diameter by the touch point it is perpendicular to the tangent 21 If a right line be perpendicular unto the tangent it doth passe by the center and touch-point 19. piij. Or thus if it be perpendicular to the tangent it is a diameter by the touch point Schoner For a right line either from the center unto the touch-point or from the touch point unto the center is radius or semidiameter And 22 The touch-point is that into which the perpendicular from the center doth fall upon the touch line 23 A tangent on the same side is onely one Or touch line is but one upon one and the same side H. Or. A tangent is but one onely in that point of the periphery Schoner Euclide propoundeth this more specially thus that no other right line may possibly fall betweene the periphery and the tangent And 24 A touch-angle is lesser than any rectilineall aâ—ute angle è 16 p ij Angulus contractus A touch angle is an angle of a straight touch-line and a periphery It is commonly called Angulus contingentiae Of Proclus it is named Cornicularis an horne-like cornerâ— because it is made of a right line and periphery like unto a horne It is lesse therefore than any acute or sharpe right-lined angle Because if it were not lesser a right line might fall between the periphery and the
unto the remainder which is thus found 5. If a right line be cut proportionally the base of that triangle whose shankes shall be equall to the whole line cut and the base to the greater segment of the same shall have each of the angles at the base double to the remainder And the base shall be the side of the quinquangle inscribed with the triangle into a circle 10 and 11. p i i i j. 6 If two right lines doe subtend on each side two angles of an inscript quinquangle they are cut proportionally and the greater segments are the sides of the said inscript è 8 p x iij. And from hence the fabricke or construction of an ordinate quinquangle upon a right line given is manifest Therefore 7 If a right line given cut proportionall be continued at each end with the greater segment and sixe peripheries at the distance of the line given shall meete two on each side from the ends of the line given and the continued two others from their meetings right lines drawne from their meetings the ends of the assigned shall make an ordinate quinquangle upon the assigned 8 If the diameter of a circle circumscribed about a quinquangle be rationall it is irrationall unto the side of the inscribed quinquangle è 11. p xiij So before the segments of a right line proportionally cut were irrationall The other triangulates hereafter multiplied from the ternary quaternary or quinary of the sides may be inscribed into a circle by an inscript triangle quadrate or quinquangle Therefore by a triangle there may be inscribed a triangulate of 6. 12,24,46 angles By a quadrate a triangulate of 8. 16,32,64 angles By a quinquangle a triangulate of 10 20. 40,80 angles c. 9 The ray of a circle is the side of the inscript sexangle è 15 p iiij Therefore 10 Three ordinate sexangles doe fill up a place Furthermore also no one figure amongst the plaines doth fill up a place A Quinquangle doth not For three angles a quinquangle may make only 3 ◠5 angles which is too little And foure would make 4 ◠5. which is as much too great The angles of a septangle would make onely two rightangles and 6 7 of one Three would make 3 and 9 7 that is in the whole 4. 2 7 which is too much c. to him that by induction shall thus make triall it will appeare That a plaine place may be filled up by three sorts of ordinate plaines onely And 11 If right lines from one angle of an inscript sexangle unto the third angle on each side be knit together they shall inscribe an equilater triangle into the circle given 12 The side of an inscribed equilater triangle hath a treble power unto the ray of the circle 12. p xiij 13 If the side of a sexangle be cut proportionally the greater segment shall be the side of the decangle Therefore 14 If a decangle and a sexangle be inscribed in the same circle a right line continued and made of both sides shall be cut proportionally and the greater segment shall be the side of a sexangle and if the greater segment of a right line cut proportionally be the side of an hexagon the rest shall be the side of a decagon 9. p xiij The comparison of the decangle and sexangle with the quinangle followeth 15 If a decangle a sexangle and a pentangle be inscribed into the same circle the side of the pentangle shall in power countervaile the sides of the others And if a right line inscribed do countervaile the sides of the sexangle and decangle it is the side of the pentangle 10. p xiiij Let the proportion of this syllogisme be demonstrated For this part onely remaineth doubtfull Therefore two triangles a e i and y e i are equiangles having one common angle at e And also two equall ones a e i and e i y the halfes to wit of the same e i s Because that is by the 17 e vj one of the two equalls unto the which e i s the out angle is equall by the 15 e. vj. And this doth insist upon a halfe periphery For the halfe periphery a l s is equall to the halfe periphery a r s and also a l is equall to a r. Therefore the remnant l s is equall to the remnant r s And the whole r l is the double of the same r s And therefore e r is the double of e o And r s the double of o u. For the bisegments are manifest by the 10 e xv and the 11 e xvj Therefore the periphery e r s is the double of the periphery e o u And therefore the angle e i u is the halfe of the angle e i s by the 7 e xvj Therefore two angles of two triangles are equall Wherefore the remainder by the 4 e vij is equall to the remainder Wherefore by the 12 e vij as the side a e is to e i so is e i to e y. Therefore by the 8 e xij the oblong of the extreames is equall to the quadrate of the meane Now let o y be knit together with a straight Here againe the two triangles a o e and a o y are equiangles having one common angle at a And a o y and o e a therefore also equall Because both are equall to the angle at a That by the 17 e vj This by the 2 e vij Because the perpendicular halfing the side of the decangle doth make two triangles equicrurall and equall by the right angle of their shankes And therefore they are equiangles Therefore as e a is to a o so is e a to a y. Wherefore by the 8 e xij the oblong of the two extremes is equall to the quadrate of the meane And the proposition of the syllogisme which was to be demonstrated The converse from hence as manifest Euclide doth use at the 16 p xiij 16. If a triangle and a quinquangle be inscribed into the same Circle at the same point the right line inscribed betweene the bases of the both opposite to the sayd point shall be the side of the inscribed quindecangle 16. p. iiij Therefore 17. If a quinquangle and a sexangle be inscribed into the same circle at the same point the periphery intercepted betweene both their sides shall be the thirtieth part of the whole periphery Of Geometry the ninteenth Booke Of the Measuring of ordinate Multangle and of a Circle OVt of the Adscription of a Circle and a Rectilineall is drawne the Geodesy of ordinate Multangles and first of the Circle it selfe For the meeting of two right lines equally dividing two angles is the center of the circumscribed Circle From the center unto the angle is the ray And then if the quadrate of halfe the side be taken out of the quadrate of the ray the side of the remainder shall be the perpendicular by the 9 e xij Therefore a speciall theoreme is here thus make 1. A plaine made of the
by this meanes 14 If a right line equall to the axis of the sphearicall and to it from the end of the perpendicular be knit unto the center a right line drawne from the cutting of the periphery unto the said end shall be the side of the Icosahedrum 15 Of the five ordinate bodies inscribed into the same spheare the tetrahedrum in respect of the greatnesse oâ— his side is first the Octahedrum the second the Cube the third the Icosahedrum the fourth and the Dodecahedrum the fifth The latter Euclide doth demonstrate with a greater circumstance Therefore out of the former figures and demonstrations let here be repeated The sections of the axis first into a double reason in s And the side of the sexangle r l And the side of the Decangle a r inscribed into the same circle circumscribing the quinquangle of an icosahedrum And the perpendiculars i s and u l. Here the two triangles a i e and i e s are by the 8 e viij alike And as s e is unto e i So is i e unto e a And by 25 e iiij as s e is to e a so is the quadrate of s e to the quadrate of e i And inversly or backward as a e is to s e so is the quadrate of i e to the quadrate of s e. But a e is the triple of s e. Therefore the quadrate of i e is the triple of s e. But the quadrate of a s by the grant and 14 e xij is the quadruple of the quadrate of s e. Therefore also it is greater than the quadrate of i e And the right line a s is greater than i e and a l therefore is much greater But a l is by the grant compounded of the sides of the sexangle and decangle r l and a r. Therefore by the 1 c. 5 e 18. it is cut proportionally And the greater segment is the side of the sexangle to wit r l And the greater segment of i e proportionally also cut is y e. Therefore the said r l is greeter than y e And even now it was shewed u l was equall to r l. Therefore u l. is greater than y e But u e the side of the Icosahedrum by 22. e vj. is greater than u l. Therefore the side of the Icosahedrum is much greater then the side of the dodecahedrum Of Geometry the twenty seventh Book Of the Cone and Cylinder 1 A mingled solid is that which is comprehended of a variable surface and of a base FOr here the base is to be added to the variable surface 2 If variable solids have their axes proportionall to their bases they are alike 24. d xj It is a Consectary out of the 19 e iiij For here the axes and diameters are as it were the shankes of equall angles to wit of right angles in the base and perpendicular axis 3 A mingled body is a Cone or a Cylinder The cause of this division of a varied or mingled body is to be conceived from the division of surfaces 4 A Cone is that which is comprehended of a conicall and a base Therefore 5 It is made by the turning about of a rightangled triangle the one shanke standing still As it appeareth out of the definition of a variable body And 6 A Cone is rightangled if the shanke standing still be equall to that turned about It is Obtusangeld if it be lesse and acutangled if it be greater ê 18 d xj And 7 A Cone is the first of all variable For a Cone is so the first in variable solids as a triangle is in rectilineall plaines As a Pyramis is in solid plaines For neither may it indeed be divided into any other variable solids more simple And 8 Cones of equall heighth are as their bases are 11. p xij As here you see And 9 They which are reciprocall in base and heighth are equall 15 p x ij These are consectaries drawne out of the 12 and 13 e iiij As here you see 10 A Cylinder is that which is comprehended of a cyliudricall surface and the opposite bases Therefore 11 It is made by the turning about of a right angled parallelogramme the one side standing still 21. dxj. As is apparant out the same definition of a varium 12. A plaine made of the base and heighth is the solidity of a Cylinder This manner of measuring doth answeare I say to the manner of measuring of a prisma and in all respects to the geodesy of a right angled parallelogramme If the cylinder in the opposite bases be oblique then if what thou cuttest off from one base thou doest adde unto the other thou shalt have the measure of the whole as here thou seest in these cylinders a and b. As here the diameter of the inner Circle is 6 foote The periphery is 18 6 7 Therefore the plot or content of the circle is 28 2 7 Of which and the heighth 10 the plaine is 282 6 7 for the capacity of the vessell Thus therefore shalt thou judge as afore how much liquour or any thing esle conteined a cubicall foote may hold 13. A Cylinder is the triple of a cone equall to it in base and heighth 10 p xij The demonstration of this proposition hath much troubled the interpreters The reason of a Cylinder unto a Cone may more easily be assumed from the reason of a Prisme unto a Pyramis For a Cylinder doth as much resemble a Prisme as the Cone doth a Pyramis Yea and within the same sides may a Prisme and a Cylinder a Pyramis and a Cone be conteined And if a Prisme and a Pyramis have a very multangled base the Prisme and Clinder as also the Pyramis and Cone do seeme to be the same figure Lastly within the same sides as the Cones and Cylinders so the Prisma and Pyramides from their axeletrees and diameters may have the similitude of their bases And with as great reason may the Geometer demand to have it granted him That the Cylinder is the treble of a Coneâ— As it was demanded and granted him That Cylinders and Cones are alike whose axletees are proportionall to the diameters of their bases Therefore 14. A plaine made of the base and third part of the height is the solidity of the cone of equall base height Of two cones of one common base is made Archimede's Rhombus as here whose geodaesy shall be cut of two cones And 15. Cylinder of equall heighth are as their bases are 11 p xij And 16 Cylinders reciprocall in base and heighth are eequall 15 p xij Both these affections are in common attributed to the equally manifold of first figures And 17. If a Cylinder be cut with a plaine surface parallell to his opposite bases the segments are as their axes are 13 p xij The unequall sections of a spheare we have reserved for this place Because they are â—omprehended of a surface both sphearicall and conicall as is the sectour As also of a plaine and sphearicall as is the section And in both like as in a Circle there is but a greater and lesser segment And the sectour as before is considered in the center 18. The sectour of a spheare is a segment of a spheare which without is comprehended of a sphearicallâ— within of a conicall bounded in the center the greater of a concave the lesser of a convex Archimides maketh mention of such kinde of Sectours in his 1 booke of the Spheare From hence also is the geodesy following drawne And here also is there a certaine analogy with a circular sectour 19. A plaine made of the diameter and sixth part of the greater or lesser sphearicall is the greater or lesser sector And from hence lastly doth arise the solidity of the section by addition and subduction 20. If the greater sectour be increased with the internall cone the whole shall be the greater section If the lesser be diminished by it the remaine shall be the lesser section As here the inner cone measured is 126 4 63. The greater sectour by the former was 1026 â…” And 126 â…” 126 4 63 doe make 1152 46 63. Againe the lesser sectour by the next precedent was 410 â…” And here the inner cone is 126 4 63 And therefore 410 2 â— 126 4 63 that is 284 38 63 is the lesser section FINIS Or thus Or thus Or thus