superficies ABCD. whose length is taken by the lyne AB or CD and breadth by the lyne AC or BD and by reason of those two dimensions a superficies may be deuided two wayes namely by his length and by hys breadth but not by thicknesse for it hath none For that is attributed onely to a body which is the third kynde of quantitie and hath all three dimensions length breadth and thicknes and may be deuided according to any of them Others define a superficies thus A superficies is the terms or ende of a body As a line is the ende and terme of a superficies 6 Extremes of a superficies are lynes As the endes limites or borders of a lyne are pointes inclosing the line so are lines the limites borders and endes inclosing a superficies As in the figure aforesayde you maye see the superficies inclosed with foure lynes The extremes or limites of a bodye are superficiesles And therfore a superficies is of some thus defined A superficies is that which endeth or incloseâ—h a body as is to be sene in the sides of a die or of any other body 7 A plaine superficies is that which lieth equally betwene his lines A plaine superficies is the shortest extension or drâ—ught from one lyne to an other like as a right lyne is the shortest extension or draught from one point to an other Euclide also leaueth out here to speake of a crooked and hollow superficies because it may easely be vnderstand by the diffinition of a plaine superficies being hys contrary And euen as from one point to an other may be drawen infinite crooked lines but one right line which is the shortest so from one lyne to an other may be drawen infinite croked superficiesses but one plain superficies which is the shortest Here must you consider when there is in Geometry mention made of pointes lines circles triangles or of any other figures ye may not conceyue of them as they be in matter as in woode in mettall in paper or in any such lyke for so is there no lyne but hath some breadth and may be deuidedâ— nor points but that shal haue some partes and may also be deuided and so of others But you must conceiue them in mynde plucking them by imagination from all matter so shall ye vnderstande them truely and perfectly in their owne nature as they are defined As a lyne to be long and not broade and a poynte to be so little that it shall haue no part at all Others otherwyse define a playne superficies A plaine superficies is that which is firmly set betwene his extremes as before was sayd of a right lyne Agayne A plaine superficies is that vnto all whose partes a right line may well be applied Again A plaine superficies is that which is the shortest of al superficies which haue one the self extremes As a rigâ—t line was the shortest line that can be drawen betwene two pointes Againe A playne superficies is that whose middle darkeneth the extremes as was also sayd of a right lyne 8 A plaine angle is an inclination or bowing of two lines the one to the other and the one touching the other and not beyng directly ioyned together As the two lines AB BC incline the one to the other and touch the one the other in the point B in which point by reason of the inclination of the sayd lines is made the angle ABC But if the two lines which touch the one the other be without all inclination of the one to the other and be drawne directly the one to the other then make they not any angle at all as the lines CD and DE touch the one the other in the point D and yet as ye see they make no angle 9 And if the lines which containe the angle be right lynes then is it called a rightlyned angle As the angle ABC in the former figures is a rightlined angle because it is contained of right lines where note that an angle is for the most part described by thre letters of which the second or middle letter representeth the very angle and therfore is set at the angle By the contrary a crooked lyned angle is that which is contained of crooked lines which may be diuersly figured Also a mixt angle is that which is caused of them both namely of a right line and a crooked which may also be diuersly figured as in the figures before set ye may see There are of angles thre kindes a right angle an acute angle and an obtuse angle the definitions of which now follow 10 VVhen a right line standing vpon a right line maketh the angles on either side equallâ— then either of those angles is a right angle And the right lyne which standeth erected is called a perpendiculer line to that vpon which it standeth As vpon the right line CD suppose there do stand an other line AB in such sort that it maketh the angles on either side therof equall namely the angle ABC on the one side equall to the angle ABD on the other side then is eche of the two angles ABC and ABD a right angle and the line AB which standeth erected vpon the line CD without inclination to either part is a perpendicular line commonly called among artificers a plumbe lyne 11 An obtuse angle is that which is greater then a right angle As the angle CBE in the example is an obtuse angle for it is greater then the angle ABC which is a right angle because it contayneth it and containeth moreouer the angle ABE 12 An acute angle is that which is lesse then a right angle As the angle EBD in the figure before put is an acute angle for that it is lesse then the angle ABD which is a right angle for the right angle containeth it and moreouer the angle ABE 13 A limite or terme is the ende of euery thing For as much as of thinges infinite as Plato saith there is no science therefore must magnitude or quantitie wherof Geometry entreateth be finite and haue borders and limites to inclose it which are here defined to be the endes therof As a point is the limite or terme of a line because it is th end therof A line likewise is the limite terme of a superficies and likewise a superficies is the limite and terme of a body as is before declared 14 A figure is that which is contayned vnder one limite or terme or many As the figure A is contained vnder one limit which is the round line also the figure B is contayned vnder three right lines And the figure C vnder foure and so of others which are their limites or termes 15 A circle is a plaine figure conteyned vnder one line which is called a circumference vnto which all lynes drawen from one poynt within the figure and falling vpon
greater perfection then is a line but here in the definitioÌ„ of a solide or body Euclide attributeth vnto it all the three dimensioÌ„s leÌ„gth breadth and thicknes Wherfore a solide is the most perfectest quantitie which wanteth no dimension at all passing a lyne by two dimensions and passing a superâ—icies by one This definition of a solide is without any designation of â—orme or figure easily vnderstanded onely conceiuing in minde or beholding with the eye a piece of timber or stone or what matter so euer els whose dimensionâ— let be equall or vnequall For example let the length therof be 5. inches the breadth 4. and the thicknes 2. if the dimensions were equall the reason is like and all one as it is in a Sphere and in â— cube For in that respect and consideration onely that it is long broade and thicke it beareth the name of a solide or body â—nd hath the nature and properties therof There is added to the endâ— of the definition of a solide that the terme and limite of a solide â—s a superficies Of thinges infinitie there iâ— no Arte or Scienâ—e All quantities therfore in this Arte entreated of are imagined to be finite and to haue their endes and borders as hath bene shewed in the first booke that the limites and endes of a line are pointes and the limites or borders of a superficies are lines so now he saith thaâ— the endes limites or borders of a solideâ— are superficieces As the side of any â—quare piece of timber or of a table or die or any other likâ— are the termes and limites of them 2 A right line is then erected perpendicularly to a pl 〈…〉 erficies wheÌ„ the right line maketh right angles with all the lines 〈…〉 it and are drawen vpon the ground plaine superficies Suppose that vpon the grounde playne superficies CDEF from the pointe B be erected a right line namely â—A so that let the point A be a loâ—e in the ayre Drawe also from the poynte â— in the playne superficies CDBF as many right lines as ye list as the lines BC BD â—â— BF BG HK BH and BL If the erected line BA with all these lines drawen in the superficies CDEF make a right angle so that all those â—ngles Aâ—â— Aâ—D Aâ—E ABFâ— Aâ—G Aâ—K ABH ABL and so of others be right angles then by this definition the line AB iâ— a line â—â—â—cted vpon the superficies CDEF it is also called commonly a perpendicular line or a plumb line vnto or vpon a superficies 3 A plaine superficies is then vpright or erected perpendicularly to a plaine superficies when all the right lines drawen in one of the plaine superficieces vnto the common section of those two plaine superficieces making therwith right angles do also make right angles to the other plaine superficies Inclination or leaning of a right line to a plaine superficies is an acute angle contained vnder a right line falling from a point aboue to the plaine superficies and vnder an other right line from the lower end of the sayd line let downe drawen in the same plaine superficies by a certaine point assigned where a right line from the first point aboue to the same plaine superficies falling perpendicularly toucheth In this third definition are included two definitions the first is of a plaine superficies erected perpendicularly vpon a plaine superficies The second is of the inclination or leaning of a right line vnto a superficies of the first take this example Suppose ye haue two superâ—icieces ABCD and CDEF Of which let the superficies CDEF be a ground plaine superficies and let the superficies ABCD be erected vnto it and let the line CD be a common terme or intersection to them both that is let it be the end or bound of either of them be drawen in either of them in which line note at pleasure certaine pointes as the point G H. From which pointes vnto the line CD draw perpendicular lines in the superâ—icies ABCD which let be GL and HK which falling vpon the superficies CDEF if they cause right angles with it that is with lines drawen in it from the same pointes G and H as if the angle LGM or the angle LGN contayned vnder the line â—G drawen in the superficies erected and vnder the GM or GN drawen in the ground superficies CDEF lying flat be a right angle then by this definition the superficies ABCD is vpright or erected vpon the superficies CDEF It is also commonly called a superficies perpendicular vpon or vnto a superficies For the second part of this definition which is of the inclination of a right line vnto a plaine superficies take this example Let ABCD be a ground plaine superficies vpon which from a point being a loft namely the point E suppose a right line to fall which let be the line EG touching the plaine superficies ABCD at the poynt G. Againe from the point E being the toppe or higher limite and end of the inclining line EG let a perpendicular line fall vnto the plaine superficies ABCD which let be the line EF and let F be the point where EF toucheth the plaine superficies ABCD. Then from the point of the fall of the line inclining vpon the superficies vnto the point of the falling of the perpendicular line vpon the same superficies that is from the point G to the point F draw a right line GF Now by this definition the acute angle EGF is the inclination of the line EG vnto the superficies ABCD. Because it is contayned of the inclining line and of the right line drawen in the superficies from the point of the fall of the line inclining to the point of the fall of the perpendicular line which angle must of necessitie be an acute angle For the angle EFG is by construction a right angle and three angles in a triangle are equ◠〈…〉 â—ight angles Wherefore the other two angles namely the angles EGF and GEF are equ◠〈…〉 right angle Wherfore either of them is lesse then a right angle Wherfore the angle EGF is an 〈…〉 gle 4 Inclination of a plaine superficies to a plaine superficies is an acute angle contayned vnder the right lines which being drawen in either of the plaine superficieces to one the self same point of the coÌ„mon section make with the section right angles Suppose that there be two superficieces ABCD EFGH and let the superficies ABCD be supposed to be erected not perpendicularly but somewhat leaning and inclining vnto the plaine superficies EFGH as much or as litle as ye will the coÌ„mon terme or section of which two superficieces let be the line CD From some one point aâ— from the point M assigned in the common section of the two superficieces namely in the line CD draw a perpendicular line in either superficies In the ground superficies EFGH draw the line MK and in the superficies ABCD draw the line ML Now
in the middest wherof is a point from which all lines drawen to the circumference therof are equall this definition is essentiall and formall and declareth the very nature of a circle And vnto this definition of a circle is correspondent the deâ—inition of a Sphere geueÌ„ by Theodosius saying that it is a solide oâ— body in the middest whereof there is a point from which all the lines drawen to the circumference are equall So see you the affinitie betwene a circle and a Sphere For what a circle is in a plaine that is a Sphere in a Solide The fulnes and content of a circle is described by the motion of a line moued about but the circumference therof which is the limite and border thereof is described of the end and point of the same line moued about So the fulnes content and body of a Sphere or Globe is described of a semicircle moued about But the Sphericall superficies which is the limite and border of a Sphere is described of the circumference of the same semicircle moued about And this is the superficies ment in the definition when it is sayd that it is contained vnder one superficies which superficies is called of Iohannes de â—acro Busco others the circumference of the Sphere Galene in his booke de diffinitionibus mediciâ— â— geueth yet an other definitioÌ„ of a Sphere by his propertie or coÌ„mon accideÌ„ce of mouing which is thus A Sphere is a figure most apt to all motion as hauing no base whereon th stay This is a very plaine and witty deâ—inition declaring the dignitie thereof aboue all figures generally All other bodyes or solides as Cubes Pyramids and others haue sides bases and angles all which are stayes to rest vpon or impedimentes and lets to motion But the Sphere hauing no side or base to stay one nor angle to let the course thereof but onely in a poynt touching the playne wherein 〈◊〉 standeth moueth freely and fully with out let And for the dignity and worthines thereof this circular and Sphericall motion is attributed to the heauens which are the most worthy bodyes Wherefore there is ascribed vnto them this chiefe kinde of motion This solide or bodely figure is also commonly called a Globe 13 The axe of a Sphere is that right line which abideth fixed about which the semicircle was moued As in the example before geuen in the definition of a Sphere the line AB about which his endes being fixed the semicircle was moued which line also yet remayneth after the motion ended is the axe of the Sphere described of that semicircle Theodosius defineth the axe of a Sphere after this maner The axe of a Sphere is a certayne right line drawen by the centre ending on either side in the superficies of the Sphere about which being fixed the Sphere is turned As the line AB in the former example There nedeth to this definition no other declaration but onely to consider that the whole Sphere turneth vpon that line AB which passeth by the centre D and is extended one either side to the superficies of the Sphere wherefore by this definition of Theodosius it is the axe of the Sphere 14 The centre of a Sphere is that poynt which is also the centre of the semicircle This definition of the centre of a Sphere is geuen as was the other definition of the axe namely hauing a relation to the definition of a Sphere here geuen of Euclide where it was sayd that a Sphere is made by the reuolution of a semicircle whose diameter abideth fixed The diameter of a circle and of a semicrcle is all one And in the diameter either of a circle or of a semicircle is contayned the center of either of them for that they diameter of eche euer passeth by the centre Now sayth Euclide the poynt which is the center of the semicircle by whose motion the Sphere was described is also the centre of the Sphere As in the example there geuen the poynt D is the centre both of the semicircle also of the Sphere Theodosius geueth as other definition of the centre of a Sphere which is thus The centre of a Sphere is a poynt with in the Sphere from which all lines drawen to the superficies of the Sphere are equall As in a circle being a playne figure there is a poynt in the middest from which all lines drawen to the circumfrence are equall which is the centre of the circle so in like maner with in a Sphere which is a solide and bodely figure there must be conceaued a poynt in the middest thereof from which all lines drawen to the superficies thereof are equall And this poynt is the centre of the Sphere by this definition of Theodosius Flussas in defining the centre of a Sphere comprehendeth both those definitions in one after this sort The centre of a Sphere is a poynt assigned in a Sphere from which all the lines drawen to the superficies are equall and it is the same which was also the centre of the semicircle which described the Sphere This definition is superfluous and contayneth more theÌ„ nedeth For either part thereof is a full and sufficient diffinition as before hath bene shewed Or ells had Euclide bene insufficient for leauing out the one part or Theodosius for leauing out the other Paraduenture Flussas did it for the more explication of either that the one part might open the other 15 The diameter of a Sphere is a certayne right line drawen by the ceÌ„tre and one eche side ending at the superficies of the same Sphere This definitioÌ„ also is not hard but may easely be couceaued by the definitioÌ„ of the diameter of a circle For as the diameter of a circle is a right line drawne froÌ„ one side of the circuÌ„frence of a circle to the other passing by the centre of the circle so imagine you a right line to be drawen from one side of the superficies of a Sphere to the other passing by the center of the Sphere and that line is the diameter of the Sphere So it is not all one to say the axe of a Sphere and the diameter of a Sphere Any line in a Sphere drawen from side to side by the centre is a diameter But not euery line so drawen by the centre is the axe of the Sphere but onely one right line about which the Sphere is imagined to be mouedâ— So that the name of a diameter of a Sphere is more general then is the name of an axe For euery axe in a Sphere is a diameter of the same but not euery diameter of a Sphere is an axe of the same And therefore Flussas setteth a diameter in the definition of an axe as a more generall word â—n this maner The axe of a Sphere is that fixed diameter aboue which the Sphere is moued A Sphere as also a circle may haue infinite diameters but it can haue but
coÌ„pareth them all with Triangles also together the one with the other In it also is taught how a figure of any forme may be chaunged into a Figure of an other forme And for that it entreateth of these most common and generall thynges thys booke is more vniuersall then is the seconde third or any other and therefore iustly occupieth the first place in order as that without which the other bookes of Eâ—clide which follow and also the workes of others which haue written in Geometry cannot be perceaued nor vnderstanded And forasmuch â—s all the demonstrations and proofes of all the propositions in this whole booke depende of these groundes and principles following which by reason of their playnnes neede no greate declaration yet to remoue all be it neuer so litle obscuritie there are here set certayne shorte and manifesâ— expositions of them Definitions 1. A signe or point is that which hath no part The better to vnderstand what manâ—r of thing a signe or point is ye must note that the nature and propertie of quantitie wherof Geometry entreateth is to be deuided so that whatsoeuer may be deuided into sundâ—y partes is called quantitie But a point although it pertayne to quantitie and hath his beyng in quantitie yet is it no quantitie for that it cannot be deuided Because as the definition saith it hath no partes into which it should be deuided So that a pointe is the least thing that by minde and vnderstanding can be imagined and conceyued then which there can be nothing lesse as the point A in the margent A signe or point is of Pithagoras Scholers after this manner defined A poynt is an vnitie which hath position NuÌ„bers are conceaued in mynde without any forme figure and therfore without matter wheron to 〈◊〉 figure consequently without place and position Wherfore vnitie beyng a part of number hath no position or determinate place Wherby it is manifest that â—umbâ—â— iâ— more simple and pure then is magnitude and also immateriall and so vnity which iâ— the bâ—ginning of number is lesse materiall then a â—igne or poyâ—â— which is the beginnyng of magnitude For a poynt is maâ—eriall and requireth position and place and â—â—â—rby differeth from vnitie â— A line is length â—ithout breadth There pertaine to quantiâ—â—e three dimensions length bredth thicknes or depth and by these thre are all quaÌ„titieâ— measured made known There are also according to these three dimensions three kyndes of continuall quantities a lyne a superficies or plaine and a body The first kynde namely a line is here defined in these wordes A lyne is length without breadth A point for that it is no quantitie nor hath any partes into which it may be deuided but remaineth indiuisible hath not nor can haue any of these three dimensions It neither hath length breadth nor thickenes But to a line which is the first kynde of quantitie is attributed the first dimension namely length and onely that for it hath neither breadth nor thicknes but is conceaued to be drawne in length onely and by it it may be deuided into partes as many as ye list equall or vnequall But as touching breadth it remaineth indiuisible As the lyne AB which is onely drawen in length may be deuided in the pointe C equally or in the point D vnequally and so into as many partes as ye list There are also of diuers other geuen other definitions of a lyne as these which follow A lyne is the mouyng of a poynte as the motion or draught of a pinne or a penne to your sence maketh a lyne Agayne A lyne is a magnitude hauing one onely space or dimension namely length wantyng breadth and thicâ—â—es 3 The endes or limites of a lyne are pointes For a line hath his beginning from a point and likewise endeth in a point so that by this also it is manifest that pointes for their simplicitie and lacke of composition are neither quantitie nor partes of quantitie but only the termes and endes of quantitie As the pointes A B are onely the endes of the line AB and no partes thereof And herein differeth a poynte in quantitie from vnitie in numberâ— for that although vnitie be the beginning of nombers and no number as a point is the beginning of quantitie and no quantitie yet is vnitie a part of number For number is nothyng els but a collection of vnities and therfore may be deuided into them as into his partes But a point is no part of quantitie or of a lyneâ— neither is a lyne composed of pointes as number is of vnities For things indiuisible being neuer so many added together can neuer make a thing diuisible as an instant in time is neither tyme nor part of tyme but only the beginning and end of time and coupleth ioyneth partes of tyme together 4 A right lyne is that which lieth equally betwene his pointes As the whole line AB lyeth straight and equally betwene the poyntes AB without any going vp or comming downe on eyther side A right line is the shortest extension or draught that is or may be from one poynt to an other Archimedes defineth it thus Plato defineth a right line after this maner A right line is that whose middle part shadoweth the exâ—remeâ— As if you put any thyng in the middle of a right lyne you shall not see from the one ende to the other which thyng happeneth not in a crooked lyne The Ecclipse of the Sunne say Astronomers then happeneth when the Sunne the Moone our eye are in one right line For the Moone then being in the midst betwene vs and the Sunne causeth it to be darkened Diuers other define a right line diuersly as followeth A right lyne is that which standeth firme betwene his extremes Agayne A right line is that which with an other line of lyke forme cannot make a figure Agayne A right lyne is that which hath not one part in a plaine superficies and an other erected on high Agayne A right lyne is that all whose partes agree together with all his other partes Agayne A right lyne is that whose extremes abiding cannot be altered Euclide doth not here define a crooked lyne for it neded not It may easely be vnderstand by the definition of a right lyne for euery contrary is well manifested set forth by hys contrary One crooked lyne may be more crooked then an other and from one poynt to an other may be drawen infinite crooked lynes but one right lyne cannot be righter then an other and therfore from one point to an other there may be drawen but one tight lyne As by figure aboue set you may see 5 A superficies is that which hath onely length and breadth A superficies is the second kinde of quantitie and to it are attributed two â—imensions namely length and breadth As in the
be in one and the selfe same plaine superficies wherfore from the one to the other there is some shortest draught whiche is a right line Likewise any two right lines howsoeuer they be set are imagined to be in one superficies and therefore from any one line to any one line may be drawen a superficies 2 To produce a right line finite straight forth continually As to draw in length continually the right line AB who will not graunt For there is no magnitude so great but that there may be a greter nor any so litle but that there may be a lesse And a line is a draught from one point to an other therfore from the point B which is the ende of the line AB may be drawn a line to some other point as to the point C and from that to an otherâ— and so infinitely 3 Vpon any centre and at any distance â—o describe a circle A playne superficies may in compasse be extended infinitely as from any pointe to any pointe may be drawen a right line by reason wherof it commeth to passe that a circle may be described vpon any centre and at any space or distance As vpon the centre A and vpon the space AB ye may describe the circle BC vpon the same centre vpon the distance AD ye may describe the circle DE or vppon the same centre A according to the distaunce AF ye may describe the circle FG and so infinitely extendyng your space 4 All right angles are equall the one to the other This peticion is most plaine and offreth it selfe euen to the sence For as much as a right angle is 〈…〉 riâ—ht lyne falling perpendicularly vppon an other and no one line can fall more perpendicularly vpoÌ„ a line then an otherâ— therfore no one right angle can be greater 〈◊〉 an otherâ— neither dâ— the length or shortenes of the lines alter the greatnes of the angle For in the example the right angle ABC though it be made of much longer lines then the right angle DEF whose lines are much shorter yet is that angle no greater then the other For if ye set the point E â—ust vpon the point B â— then shal the line ED euenly and iustly fall vpon the line AB â— and the line EF shall also fall equally vpon the line BC and so shal the angle DEF be equall to the angle ABC for that the lines which cause them are of like inclination It may euidently also be sene at the centre of a circle For if ye draw in a circle two diameters the one cutting the other in the centre by right angles ye shall deuide the circle into fowre equall partes of which eche contayneth one right angle so are all the foure right angles about the centre of the circle equall 5 VVhen a right line falling vpon â—wo right lines doth make â—n one the selfe same syde the two inwarde angles lesse then two right angles then shal these two right lines beyng produced 〈◊〉 length concurre on that part in which are the two angles lesse then two right angles As if the right line AB fall vpon two right lines namely CD and EF so that it make the two inward angles on the one side as the angles DHI FIH lesse then two right angles as in the example they do the said two lines CD and EF being drawen forth in leÌ„gth on that part wheron the two angles being less◠〈◊〉 â—wo right angleâ— consist shal 〈◊〉 leÌ„gth concurre and meete together as in the point D as it is easie to see For the partes of the lines towardes DF are more enclined the one to the other then the partes of the lines towardes CE are Wherfore the more these parts are produced the more they shall approch neare and neare till at length they shal mete in one point Contrariwise the same lines drawn in leÌ„gth on the other side for that the angles on that side namely the angle CHB and the angle EIA are greater then two right angles so much as the other two angles are lesse thâ—n two right angles shall neuer mete but the further they are drawen the further they shal be distant the one from the other 6 That two right lines include not a superficies If the lines AB and AC being right lines should inclose a superficies they muste of necessitie bee ioyned together at both the endes and the superficies must be betwene theÌ„ Ioyne them on the one side together in the pointe A and imagine the point B to be drawen to the point C so shall the line AB fall on the line AC and couer it and so be all one with it and neuer inclose a space or superficies Common sentences 1 Thinges equall to one and the selfe same thyng are equall also the one to the other After definitions and petitions now are set common sentences which are the third and last kynd of principles Which are certaine general propositioÌ„s commonly known of all men of themselues most manifest cleare therfore are called also dignities not able to be denied of any Peticions also are very manifest but not so fully as are the coÌ„mon sentences and therfore are required or desired to be graunted Peticions also are more peculiar to the arte whereof they are as those before put are proper to Geometry but common sentences are generall to all things wherunto they can be applied Agayne peticions consist in actions or doing of somewhat most easy to be done but common sentences consist in consideration of mynde but yet of such thinges which are most easy to be vnderstanded as is that before set As if the line A be equall to the line B And if the line C be also equall to the line B then of necessitie the lines A and C shal be equal the one to the other So is it in all superâ—iciesses angles numbers in all other things of one kynde that may be compared together 2 And if ye adde equall thinges to equall thinges the whole shal be equall As if the line AB be equal to the line CD to the line AB be added the line BE to the line CD be added also an other line DF being equal to the line BE so that two equal lines namely BE and DF be added to two equall lynes AB CD then shal the whole lyne AE be equall to the whole lyne CF and so of all quantities generally 3 And if from equall thinges ye take away equall thinges the thinges remayning shall be equall As if from the two lines AB and CD being equal ye take away two equall lines namely EB and FD then maye you conclude by this common sentence that the partes remayning namely AE and CF are equall the one to the other and so of all other quantities 4 And if from vnequall thinges ye take away equall thinges the thynges which remayne shall be vnequall As if the lines AB and CD be vnequall
draw by the 11. of the firsâ— a perpeÌ„dicular line AB and let AB be a rationall line and make perfectâ— the parallelogramme BC. Wherefore BG is irrationall by that which was declared and proued in maner of an Assumpt in the end of the demonstration of the 38 and the line that containeth it iâ— power is also irrationall Let the line CD containe in power the superâ—icies BC. Wherefore CD is irrationall not of the selfe same kind with any of those that were before for the square of the line CD applied to a rationall line namely AB maketh the breadth a mediall line namely AC But the square of none of the foresaid lines applied to a rationall line maketh the breadth a mediall line Againe make perfecte the parallelogramme ED. Wherefore the parallelogramme ED is also irrationall by the sayd Assumpt in the end of the 98. his demonstration brieâ—ly proued and the line which containeth it in power is irrationallâ— let the line which containeth it in power be DF. Wherefore DF is irrationall and not of the selfe same kinde with any of the foresaid irrationall lines For the square of none of the foresayd irrationall lines applied vnto a rationall line maketh the breadth the line CD Wherefore of a mediall line are produced infinite irrationall lines of which none is of the selfe same kinde with any of those that were before which was required to be demonstrated ¶ The 92. Theoreme The 116. Proposition Now let vs proue that in square figures the diameter is incommensurable in length to the side SVppose that ABCD be a square and let the diameter therof be AC Then I say that the diameter AC is incommensurable in length to the side AB For if it be possible let it be coÌ„mensurable in leÌ„gth I say that theÌ„ this will follow that one and the selfe same nuÌ„ber shall be both an euen number an odde number It is manifest by the 47. of the first that the square of the line AC is double to the square of the line AB And for that the line AC is commensurable in length to the line AB by supposition therfore the lyne AC hath vnto the line AB that proportion that a number hath to a number by the 5. of the tenth Let the lyne AC haue vnto the line AB that proportion that the number EF hath to the number G. And let EF and G be the least numbers that haue one and the same proportion with them Wherfore EF is not vnitie For if EF be vnitie and it hath to the number G that proportion that the line AC hath to the lyne AB and the line AC is greater then the lyne AB Wherfore vnitie EF is greater then the number G which is impossible Wherfore FE is not vnitie wherfore it is a number And for that as the square of the line AC is to the square of the lyne AB so is the square number of the number EF to the square number of the number G for in eche is the proportion of their sides doubled by the corollary of the 20 of the sixt and 11. of the eight and the proportion of the line AC to the line AB doubled is equal to the proportioÌ„ of the nuÌ„ber EF to the number G doubled for as the line AC is to the line AB so is the nuÌ„ber EF to the number G. But the square of the line AC is double to the square of the line AB Wherfore the square number produced of the number EF is double to the square number produced of the number G. Wherefore the square number produced of EF is an euen number Wherfore EF is also an euen number For if EF were an odde number the square number also produced of it should by the 23. and 29. of the ninth be an odde number For if odde numbers how many soeuer be added together and if the multitude of theÌ„ be odde the whole also shal be odde Wherfore EF is an euen number Deuide the number EF into two equall partes in H. And forasmuch as the numbers EF and G are the lest numbers in that proportion therfore by the 24. of the seuenth they are prime numbers the one to the other And EF is an euen number Wherfore G is an odde number For if G were an euen number the number two should measure both the number EF and the number G for euery euen nuÌ„ber hath an halfe part by the definition but these numbers EF G are prime the one to the other Wherfore it is impossible that they should be measured by two or by any other number besides vnitie Wherfore G is an odde number And forasmuch as the number EF is double to the number EH therfore the square number produced of EF is quadruple to the square number produced of EH And the square number produced of EF is double to the square number produced of G. Wherfore the square number produced of G is double to the square number produced of EH Wherfore the square number produced of G is an euen number Wherfore also by those thinges which haue bene before spoken the number G is an euen number but it is proued that it is an odde number which is impossible Wherefore the line AC is not commensurable in length to the line AB wherfore it is incommensurable An other demonstration We may by an other demonstration proue that the diameter of a square is incommensurable to the side thereof Suppose that there be a square whose diameter let be A and let the side thereof be B. Then I say that the line A is incommensurable in length to the line B. For if it be possible let it be commensurable in length And agayne as the line A is to the line B so let the number EF be to the number G and let them be the least that haue one and the same proportion with them wherefore the numbers EF and G are prime the one to the other First I say that G is not vnitie For if it be possible let it be vnitie And for that the square of the line A is to the square of the line B as the square number produced of EF is to the square number produced of G as it was proued in the â—ormer demonstration but the square of the line A is double to the square of the line B. Wherfore the square nuÌ„ber produced of EF is double to the square number produced of G. And by your supposition G is vnitie Wherefore the square number produced of EF is the number two which is impossible Wherefore G is not vnitie Wherefore it is a number And for that as the square of the line A is to the square of the line B so is the square number produced of EF to the square number produced of G. Wherefore the square number produced of EF is double to the square number produced of G. Wherefore the square number produced of G.
the wordes of Psellus in his Epitome of Geometrie where he entreateth of the production and constitution of these bodyes His wordes are these All râ—ctiliâ—e figures being erected vpon their playnes or bases by right angles make Prismes Who perceaueth not but that a Pentagon erected vpoÌ„ his base of â—iue sides maketh by his motion a sided Columne of fiue sides Likewise an Hexagon erected at right angles produceth a Columne hauing sixe sides and so of all other rectillne figures All which solides or bodyes so produced whether they be sided Columnes or Parallelipipedons be here in most plaine words of this excelleÌ„t and auncient Greke author Psellus called Prismes Wherfore if the definitioÌ„ of a Prisme geueÌ„ of Euclide should extend it selfe so largely as Flussas imagineth and should enclude such figures or bodyes as he noted he ought not yet for all that so much to be offended and so narowly to haue sought faultes For Euclide in so defining mought haue that meaning sense of a Prisme which Psellus had So ye see that Euclide may be defended either of these two wayes either by that that the definition extendeth not to these figures and so not to be ouer generall nor stretch farther then it ought or ells by that that if it should stretch so far it is not so haynous For that as ye se many haue takâ— it in that sense In deede coÌ„monly a Prisme is taken in that significatioÌ„ and meaning in which Campaâ—â—â— Flussas and others take it In which sense it semeth also that in diuers propositions in these bookes following it ought of necessitie to be taken 12 A Sphere is a figure which is made when the diameter of a semicircle abiding fixed the semicircle is turned round about vntill it returne vnto the selfe same place from whence it began to be moued To the end we may fully and perfectly vnderstand this definition how a Sphere is produced of the motion of a semicircle it shall be expedient to coÌ„sider how quantities Mathematically are by imagination conceaued to be produced by flowing and motion as was somewhat touched in the beginning of the first booke Euer the lesse quantitie by his motion bringeth forâ—h the quaÌ„titie next aboue it As a point mouing flowing or gliding bringeth forth a line which is the first quantitie and next to a point A line mouing produceth a superficies which is the second quantitie and next vnto a line And last of all a superficies mouing bringeth forth a solide or body which is the third last quantitie These thinges well marked it shall not be very hard to attaine to the right vnderstanding of this definition Vpon the line AB being the diameter describe a semicircle ACB whose centre let be D the diameter AB being sixed on his endes orâ— pointes imagine the whole superficies of the semicircle to moue round from some one point assigned till it returne to the same point againe So shall it produce a perfect Sphere or Globe the forme whereof you see in a ball or bowle And it is fully round and solide for that it is described of a semicircle which is perfectly round as our countrey man Iohannes de Sacro Busco in his booke of the Sphere of this definition which he taketh out of Euclide doth well collecte But it is to be noted and taken heede of that none be deceaued by the definition of a Sphere geuen by Iohannes de Sacro Busco A Sphere sayth he is the passage or mouing of the circumference of a semicircle till it returne vnto the place where it beganne which agreeth not with Euclide Euclide plainly sayth that a Sphere is the passage or motion of a semicircle and not the passage or motion of the circumference of a semicircle neither can it be true that the circumference of a semicircle which is a line should describe a body It was before noted that euery quantitie moued describeth and produceth the quantitie next vnto it Wherefore a line moued can not bring forth a body but a superficies onely As if ye imagine a right line fastened at one of his endes to moue about from some one point till it returne to the same againe it shall describe a plaine superficies namely a circle So also if ye likewise conceaue of a crooked line such as is the circumference of a semicircle that his diameter fastened on both the endes it should moue from a point assigned till it returne to the same againe it should describe produce a â—ound superficies onely which is the superficies and limite of the Sphere and should not produce the body and soliditie of the Sphere But the whole semicircle which is a superficies by his motion as is before said produceth a body that is a perfect Sphere So see you the errour of this definition of the author of the Sphere which whether it happened by the author him selfe which I thinke not or that that particle was thrust in by some one after him which is more likely it it not certaine But it is certaine that it is vnaptly put in and maketh an vntrue definition which thing is not here spoken any thing to derogate the author of the booke which assuredly was a man of excellent knowledgeâ— neither to the hindrance or diminishing of the worthines of the booke which vndoubtedly is a very necessary booke then which I know none more meere to be taught and red in scholes touching the groundes and principles of Astronomie and Geographie but onely to admonishe the young and vnskilâ—ull reader of not falling into errour Theodosius in his booke De Sphericis a booke very necessary for all those which will see the groundes and principles of Geometrie and Astronomie which also I haue translated into our vulgare tounge ready to the presse defineth a Sphere after thys maner A Sphere is a solide or body contained vnder one superficies in the midle wherof there is a point froÌ„ which all lines drawen to the circumference are equall This definition of Theodosius is more essentiall and naturall then is the other geuen by Euclide The other did not so much declare the inward nature and substance of a Sphere as it shewed the industry and knowledge of the producing of a Sphere and therfore is a causall definition geuen by the cause efficient or rather a description then a definition But this definition is very esâ—entiall declaring the natuâ—e and substance of a Sphere As if a circle should be thus defined as it well may A circle is the passage or mouing of a line from a point till it returne to the same point againeâ— it is a causall definition shewing the efficient cause wherof a circle is produced namely of the motion of a line And it is a very good description fully shewing what a circle is Such like description is the deâ—inition of a Sphere geuen oâ— Euclide â— by the motion of a semicircle But when a circle is defined to be a plaine superficies
same superficies Wherefore these right lines AB BD and DC are in one and the selfe same superficies and either of these angles ABD and BDC is a right angle by supposition Wherefore by the 28. of the first the line AB is a parallel to the line CD If therefore two right lines be erected perpendicularly to one and the selfe same playne superficies those right lines are parallels the one to the other which was required to be proued Here for the better vnderstanding of this 6. proposition I haue described an other figure as touching which if ye erect the superficies ABD perpendicularly to the superficies BDE and imagine only a line to be drawne from the poynt A to the poynt E if ye will ye may extend a thred from the saide poynt A to the poynt E and so compare it with the demonstration it will make both the proposition and also the demonstration most cleare vnto you ¶ An other demonstration of the sixth proposition by M. Dee Suppose that the two right lines AB CD be perpendicularly erected to one the same playne superficies namely the playne superficies OP Then I say that â—â— and CD are parallels Let the end points of the right lines AB and CD which touch the plaine supâ—â—â—â—cies Oâ— be the poyntes â— and D froÌ„â— to D let a straight line be drawne by the first petition and by the second petition let the straight line â—D be exteÌ„ded as to the poynts M N. Now forasmuch as the right line AB from the poynt â— produced doth cutte the line MN by construction Therefore by the second proposition of this eleuenth booke the right lines AB MN are in one plainâ— superficies Which let be QR cutting the superficies OP in the right line MN By the same meanes may we conclude the right line CD to be in one playne superficies with the right line MN But the right line MN by supposition is in the plaine superficies QR wherefore CD is in the plaine superficies QR And Aâ— the right line was proued to be in the same plaine superficies QR Therfore AB and CD are in one playne superficieâ— namely QR And forasmuch as the lines Aâ— and CD by supposition are perpendicular vpon the playne superficies OP therefore by the second definition of this booke with all the right lines drawne in the superficies OP and touching AB and CD the same perpeÌ„diculars Aâ— and CD do make right angles But by construction MN being drawne in the plaine superficies OP toucheth the perpendiculars AB and CD at the poyntes â— and D. Therefore the perpendiculars Aâ— and CD make with the right line MN two right angles namely ABN and CDM and MN the right line is proued to be in the one and the same playne superficies with the right lines AB CD namely in the playne superficies QR Whâ—refore by the second part of the 28. proposition of the first booke the right lineâ— AB and CD are parallelâ— If therefore two right lines be erected perpendicularly to one and the selfe same playne superficies those right lines are parallels the one to the other which was required to be demonstrated A Corollary added by M. Dee Hereby it is euident that any two right lines perpendicularly erected to one and the selfe same playne superficies are also them selues in one and the same playne superficies which is likewisâ— perpendicularly erected to the same playne superficies vnto which the two right lines are perpendicular The first part hereof is proued by the former construction and demonstration that the right lines AB and CD are in one and the same playne superficies Qâ— The second part is also manifest that is that the playne superficies QR is perpendicularly erected vpon the playne superficies OP for that Aâ— and CD being in the playne superficies QR are by supposition perpendicular to the playne superficies OP wherefore by the third definition of this booke QR is perpendicularly erected to or vpon OP which was required to be proued Io. d ee his aduise vpon the Assumpt of the 6. As concerning the making of the line DE equall to the right line AB verely the second of the first without some farther consideration is not properly enough alledged And no wonder it is for that in the former bookeâ— whatsoeâ—â—â—â—aâ—h of lines bene spoken the same hath alwayâ—s bene imagined to be in one onely playne superficies considered or executed But here the perpendicular line AB is not in the same playnâ— superficies that the right line DB is Therfore some other helpe must be put into the handes of young beginners how to bring this probleme to execution which is this most playne and briefe Vnderstand that BD the right line is the common section of the playne superficies wherein the perpendiculars AB and CD are of the other playne superficies to which they are perpendiculars The first of these in my former demonstration of the 6 â— I noted by the playne superficies QR and the other I noted by the plaine superficies OP Wherfore BD being a right line common to both the playne supâ—rficieces QR OP therby the ponits B and D are coÌ„mon to the playnes QR and OP Now from BD sufficiently extended cutte a right line equall to AB which suppose to be BF by the third of the first and orderly to BF make DE equall by the 3. oâ— the first if DE be greater then BF Which alwayes you may cause so to be by producing of DE sufficiently Now forasmuch as BF by construction is cutte equall to AB and DE also by construction put equâ—ll to BF therefore by the 1. common sentence DE is put equall to AB which was required to be done In like sort if DE were a line geuen to whome AB were to be cutte and made equall first out of the line DB suâ—â—iciently produced cutting of DG equall to DE by the third of the first and by the same 3. cutting from BA sufficiently produced BA equall to DG then is it euideÌ„t that to the right line DE the perpeÌ„dicular line AB is put equall And though this be easy to conceaue yet I haue designed the figure accordingly wherby you may instruct your imagination Many such helpes are in this booke requisite as well to enforme the young studentes therewith as also to master the froward gaynesayer of our conclusion or interrupter of our demonstrations course ¶ The 7. Theoreme The 7. Proposition If there be two parallel right lines and in either of them be taken a point at all aduentures a right line drawen by the said pointes is in the self same superficies with the parallel right lines SVppose that these two right lines AB and CD be parallels and in either of theÌ„ take a point at all aduentures namely E and F. Then I say that a right line drawen from the point E to the point F is in the selfe same plaine superficies that the
at all aduentures namely D V G S and a right line is drawen from the point D to the point G and an other from the point V to the point S. Wherefore by the 7. of the eleuenth the lines DG and VS are in one and the selfe same plaine superficies And forasmuch as the line DE is a parallel to the line BG therefore by the 24. of the first the angle EDT is equall to the angle BGT for they are alternate angles and likewise the angle DTV is equall to the angle GTS Now then there are two triangles that is DTV and GTS hauing two angles of the one equall to two angles of the other and one side of the one equall to one side of the other namely the side which subtendeth the equall angles that is the side DV to the side GS for they are the halfes of the lines DE and BG Wherefore the sides remayning are equall to the sides remayning Wherfore the line DT is equall to the line TG and the line VT to the line T S If therefore the opposite sides of a Parallelipipedon be deuided into two equall partes and by their sections be extended plaine superficieces the common section of those plaine superficieces and the diameter of the Parallelipipedon do deuide the one the other into two equall partes which was required to be demonstrated A Corollary added by Flussas Euery playne superficies extended by the center of a parallelipipedon diuideth that solide into two equall partes and so doth not any other playne superficies not extended by the center For euery playne extended by the center cutteth the diameter of the parallelipipedon in the center into two equall partes For it is proued that playne superficieces which cutte the solide into two equall partes do cut the dimetient into two equall partes in the center Wherefore all the lines drawen by the center in that playne superficies shall make angles with the dimetient And forasmuch as the diameter falleth vpon the parallel right lines of the solide which describe the opposite sides of the sayde solide or vpon the parallel playne superficieces of the solide which make angels at the endes of the diameter the triangles contayned vnder the diameter and the right line extended in that playne by the center and the right line which being drawen in the opposite superficieces of the solide ioyneth together the endes of the foresayde right lines namely the ende of the diameter and the ende of the line drawen by the center in the superficies extended by the center shall alwayes be equall and equiangle by the 26. of the first For the opposite right lines drawen by the opposite playne superficieces of the solide do make equall angles with the diameter forasmuch as they are parallel lines by the 16. of this booke But the angles at the ceÌ„ter are equall by the 15. of the first for they are head angles one side is equall to one side namely halfe the dimetient Wherefore the triangles contayned vnder euery right line drawen by the center of the parallelipipedon in the superficies which is extended also by the sayd center and the diameter thereof whose endes are the angles of the solide are equall equilater equiangle by the 26. of the first Wherfore it followeth that the playne superficies which cutteth the parallelipipedon doth make the partes of the bases on the opposite side equall and equiangle and therefore like and equall both in multitude and in magnitude wherefore the two solide sections of that solide shal be equall and like by the 8. diffinition of this booke And now that no other playne superficies besides that which is extended by the center deuideth the parallelipipedon into two equall partes it is manifest if vnto the playne superficies which is not extended by the center we extend by the center a parallel playne superficies by the Corollary of the 15. of this booke For forasmuch as that superficies which is extended by the center doth deuide the parallelipipedoÌ„ into two equall parâ—â— it is manifest that the other playne superficies which is parallel to the superficies which deuideth the solide into two equall partes is in one of the equall partes of the solide wherefore seing that the whole is euer greater then his partes it must nedes be that one of these sections is lesse then the halfe of the solide and therefore the other is greater For the better vnderstanding of this former proposition also of this Corollary added by Flussas it shal be very nedefull for you to describe of pasted paper or such like matter a parallelipipedoÌ„ or a Cube and to deuide all the parallelograÌ„mes therof into two equall parts by drawing by the câ—Ì„ters of the sayd parallelogrammes which centers are the poynts made by the cutting of diagonall lines drawen froÌ„ thâ— opposite angles of the sayd parallelograÌ„mes lines parallels to the sides of the parallelograÌ„mes as in the former figure described in a plaine ye may see are the sixe parallelograÌ„mes DE EH HA AD DH and CG whom these parallel lines drawen by the ceÌ„ters of the sayd parallelograÌ„mes namely XO OR PR and PX do deuide into two equall parts by which fower lines ye must imagine a playne superficies to be extended also these parallel lynes KL LN NM and Mâ— by which fower lines likewise yâ— must imagine a playne superficies to be extended ye may if ye will put within your body made thus of pasted paper two superficieces made also of the sayd paper hauing to their limites lines equall to the foresayde parallel lines which superficieces must also be deuided into two equall partes by parallel lines drawen by their centers and must cut the one the other by these parallel lines And for the diameter of this body exteÌ„d a thred from one angle in the base of the solide to his opposite angle which shall passe by the center of the parallelipipedon as doth the line DG in the figure before described in the playne And draw in the base and the opposite superficies vnto it Diagonall lines from the angles from which is extended the diameter of the solide as in the former description are the lines BG and DE. And when you haue thus described this body compare it with the former demonstration and it will make it very playne vnto you so your letters agree with the letters of the figure described in the booke And this description will playnely set forth vnto you the corollary following that proposition For where as to the vnderstanding of the demonstration of the proposition the superficieces put within the body were extended by parallel lynes drawen by the ceÌ„ters of the bases of the parallelipipedon to the vnderstanding of the sayd Corollary ye may extende a superficies by any other lines drawen in the sayd bases so that yet it passe through the middest of the thred which is supposed to be the center of the parallelipipedon ¶ The 35. Theoreme The 40. Proposition If there be
superficies or soliditie in the hole or in partâ— such certaine knowledge demonstratiue may arise and such mechanical exercise thereby be deuised that sure I am to the sincere true student great light ayde and comfortable courage farther to wade will enter into his hart and to the Mechanicall witty and industrous deuiser new maner of inuentions executions in his workes will with small trauayle for fete application come to his perceiueraunce and vnderstanding Therefore euen a manifolde speculations practises may be had with the circle his quantitie being not knowne in any kinde of smallest certayne measure So likewise of the sphere many Problemes may be executed and his precise quantitie in certaine measure not determined or knowne yet because both one of the first humane occasioÌ„s of inuenting and stablishing this Arte was measuring of the earth and therfore called Geometria that is Earthmeasuring and also the chiefe and generall end in deede is measure and measure requireth a determination of quantitie in a certayne measure by nuÌ„ber expressed It was nedefull for Mechanicall earthmeasures not to be ignorant of the measure and contents of the circle neither of the sphere his measure and quantitie as neere as sense can imagine or wish And in very deede the quantitie and measure of the circle being knowne maketh not onely the cone and cylinder but also the sphere his quantitie to be as precisely knowne and certayne Therefore seing in respect of the circles quantitie by Archimedes specified this Theoreme is noted vnto you I wil by order vpon that as a supposition inferre the conclusion of this our Theoremes Note 1. Wherfore if you deuide the one side as TQ of the cube TX into 21. equall partes and where 11. partes do end reckening from T suppose the point P and by that point P imagine a plaine passing parallel to the opposite bases to cut the cube TX and therby the cube TX to be deuided into two rectangle parallelipipedons namely TN and PX It is manifest TN to be equall to the Sphere A by construction and the 7. of the fift Note 2. Secondly the whole quantitie of the Sphere A being coÌ„tayned in the rectangle parallelipipedon TN you may easilie transforme the same quantitie into other parallelipipedons rectangles of what height and of what parallelogramme base you list by my first and second Problemes vpon the 34. of this booke And the like may you do to any assigned part of the Sphere A by the like meanes deuiding the parallelipipedon TN as the part assigned doth require As if a third fourth fifth or sixth part of the Sphere A were to be had in a parallelipipedon of any parallelograâ—â—e base assigned or of any heith assigned then deuiding TP into so many partes as into 4. if a fourth part be to be transformed or into fiue if a fifth part be to be transformed c. and then proceede â—s you did with cutting of TN from TX And that I say of parallelipipedons may in like sort by my â—â—yd two problemes added to the 34. of this booke be done in any sided columnes pyramids and prismeâ— so thâ—â— in pyramids and some prismes you vse the cautions necessary in respect of their quan 〈…〉 odyes hauing parallel equall and opposite bases whose partes 〈…〉 re in their propositions is by Euclide demonstrated And finally 〈…〉 additions you haue the wayes and orders how to geue to a Sphere or any segmeâ—â— oâ— the same Cones or Cylinders equall or in any proportion betwene two right lines geuen with many other most necessary speculations and practises about the Sphere I trust that I haue sufficiently â—raughted your imagination for your honest and profitable studie herein and also geuen you reaâ—â— â—â—tter wheâ—â— with to sâ—â—p the mouthes of the malycious ignorant and arrogant despisers of the most excellent discourses trauayles and inuentions mathematicall Sting aswel the heauenly spheres sterres their sphericall soliditie with their conueâ—e spherical superficies to the earth at all times respecting and their distances from the earth as also the whole earthly Sphere and globe it selfe and infinite other cases concerning Spheres or globes may hereby with as much ease and certainety be determined of as of the quantitie of any bowle ball or bullet which we may gripe in our handes reason and experience being our witnesses and without these aydes such thinges of importance neuer hable of vs certainely to be knowne or attayned vnto Here ende M. Iohn d ee his additions vpon the last proposition of the twelfth booke A proposition added by Flussas If a Sphere touche a playne superficiesâ— a right line drawne from the center to the touche shall be erected perpendicularly to the playne superficies Suppose that there be a Sphere BCDL whose centre let be the poynt A. And let the playne superficies GCI touch the Spere in the poynt C and extend a right line from the centre A to the poynt C. Then I say that the line AC is erected perpendicularly to tâ—e playne GIC. Let the sphere be cutte by playne superficieces passing by the right line LAC which playnes let be ABCDL and ACEL which let cut the playne GCI by the right lines GCH and KCI Now it is manifest by the assumpt put before the 17. of this booke that the two sections of the sphere shall be circles hauing to their diameter the line LAC which is also the diameter of the sphere Wherefore the right lines GCH and KCI which are drawne in the playne GCI do at the poynt C fall without the circles BCDL and ECL. Wherefore they touch the circles in the poynt C by the second definition of the third Wherefore the right line LAC maketh right angles with the lines GCH and KCI by the 16. of the third Wherefore by the 4. of the eleuenth the right line AC is erected perpendicularly to to the playne superficies GCI wherein are drawne the lines GCH and KCI If therefore a Sphere touch a playne superficies a right line drawne from the centre to the touche shall be erected perpendicularly to the playne superficies which was required to be proued The ende of the twelfth booke of Euclides Elementes ¶ The thirtenth booke of Euclides Elementes IN THIS THIRTENTH BOOKE are set forth certayne most wonderfull and excellent passions of a lyne deuided by an extreme and meane proportion a matter vndoubtedly of great and infinite vse in Geometry as ye shall both in thys booke and in the other bookes following most euidently perceaue It teacheth moreouer the composition of the fiue regular solides and how to inscribe them in a Sphere geuen and also setteth forth certayne comparisons of the sayd bodyes both the one to the other and also to the Sphere wherein they are described The 1. Theoreme The 1. Proposition If a right line be deuided by an extreme and meane proportion and to the greater segment be added the halfe of the whole line the square made of those two
EG the sides which include the equall angles are proportionall Wherefore the parallelogramme ABCD is by the first definition of the sixth like vnto the parallelogramme EG And by the same reason also the parallelogramme ABCD is like to the parallelogramme KH wherefore either of these parallelogrammes EG and KH is like vnto the parallelogramme ABCD. But rectiline figures which are like to one and the same rectiline figure are also by the 21. of the sixth like the one to the other Wherefore the parallelogramme EG is like to the parallelogramme HK Wherfore in euery parallelogramme the parallelogrammes about the dimecient are like vnto the whole and also like the one to the other Which was required to be proued ¶ An other more briefe demonstration after Flussates Suppose that there be a parallelograÌ„me ABCD whose dimeâ—ient let bâ— Aâ— about which let consist these parallelogrammes EK and TI hauing the angles at the pointes â— and 〈…〉 with the whole parallelogramme ABCD. Then I say that those parallelogrammes EK and TI are like to the whole parallelogramme DB and also alâ— like the one to the other For forasmuch as BD EK and TI are parallelogrammes therefore the right line AZG falling vpon these parallell lines AEB KZT and DI G or vpon these parallell lines AKD EZI and BTG maketh these angles equall the one to the other namely the angle EAZ to the angle KZA the angle EZA to the angle KAZ and the angle TZG to the angle ZGI and the angle TGZ to the angle IZG and the angle BAG to the angle AGD and finally the angle BGA to the angle DAG Wherefore by the first Corollary of the 32. of the first and by the 34. of the first the angles remayning are equall the one to the other namely the angle B to the angle D and the angle E to the angle K and the angle T to the angle I. Wherefore these triangles are equiangle and therefore like the one to the other namely the triangle ABG to the triangle GDA and the triangle AEZ to the triangle ZKA the triangle ZTG to the triangle GIZ. Wherefore as the side AB is to the side BG so is the side AE to the side EZ and the side ZT to the side TG Wherefore the parallelogrammes contayned vnder those right lines namely the parallelogrammes ABGD EK TI are like the one to the other by the first definition of this booke Wherefore in euery parallelogramme the parallelogrammes c. as before which was required to be demonstrated ¶ A Probleme added by Pelitarius Two equiangle Parallelogrammes being geuen so that they be not like to cut of from one of them a parallelogramme like vnto the other Suppose that the two equiangle parallelogrammes be ABCD and CEFG which let not be like the one to the other It is required from the Parallelogramme ABCD to cut of a parallelogramme like vnto the parallelogramme CEFG Let the angle C of the one be equall to the angle C of the other And let the two parallelogrammes be so 〈◊〉 that the lines BC CG may make both one right line namely BG Wherefore also the right lines DC and CE shall both make one right line namely DE. And drawe a line from the poynt F to the poynt C and produce the line FC till it coÌ„curre with the line AD in the poynt H. And draw the line HK parallell to the line CD by the 31. of the first Then I say that from the parallelogramme AC is cut of the parallelograÌ„me CDHK like vnto the parallelograÌ„me EG Which thing is manifest by thys 24. Proposition For that both the sayd parallelogrammes are described about one the selfe same dimetient And to the end it might the more plainly be seene I haue made complete the Parallelogramme ABGL ¶ An other Probleme added by Pelitarius Betwene two rectiline Superficieces to finde out a meane superficies proportionall Suppose that the two superficieces be A and B betwene which it is required to place a meane superficies proportionall Reduce the sayd two rectiline figures A and B vnto two like parallelograÌ„mes by the 18. of this booke or if you thinke good reduce eyther of them to a square by the last of the second And let the said two parallelogrammes like the one to the other and equall to the superficieces A and B be CDEF and FGHK And let the angles F in either of them be equall which two angles let be placed in such sort that the two parallelogrammes ED and HG may be about one and the selfe same dimetient CK which is done by putting the right lines EF and FG in such sort that they both make one right line namely EG And make coÌ„plete the parallelograÌ„me CLK M. Then I say that either of the supplements FL FM is a meane proportionall betwene the superficieces CF FK that is betwene the superficieces A and B namely as the superficies HG is to the superficies FL so is the same superficies FL to the superficies ED. For by this 24. Proposition the line HF is to the line FD as the line GF is to the line FE But by the first of this booke as the line HF is to the line FD so is the superficies HG to the superficies FL and as the line GF is to the line FE so also by the same is the superficies FL to the superficies ED. Wherfore by the 11. of the fift as the superficies HG is to the superficies FL so is the same superficies FL to the superficies ED which was required to be done The 7. Probleme The 25. Proposition Vnto a rectiline figure geuen to describe an other figure lyke which shal also be equall vnto an other rectiline figure geuen SVppose that the rectiline figure geueÌ„ wherunto is required an other to be made like be ABC and let the other rectiline figure whereunto the same is required to be made equal be D. Now it is required to describe a rectiline figure like vnto the figure ABC and equall vnto the figure D. Vppon the line BC describe by the 44. of the first a parallelogramme BE equall vnto the triangle ABC and by the same vpon the line CE describe the parallelogramme C M equall vnto the rectiline figure D and in the said parallelogramme let the angle FCE be equall vnto the angle CBL And forasmuch as the angle FCE is by construction equall to the angle CBL adde the angle BCE common to them both Wherefore the angles LBC and BCE are equall vnto the angles BCE and ECF but the angles LBC and BCE are equall to two right angles by the 29. of the first wherfore also the angles BCE and ECF are equall to two right angles Wherfore the lines BC and CF by the 14. of the first make both one right line namely BF and in like sort do the lines LE and EM make both one right line namely LM Then by the
two other propositions going next before it so farre misplaced that where they are word for word before duâ—ly placed being the 105. and 106. yet here after the booke ended they are repeated with the numbers of 116. and 117. proposition Zambert therein was more faythfull to follow as he found in his greke example than he was skilfull or carefull to doe what was necessary Yea and some greke written auncient copyes haue them not so Though in deede they be well demonstrated yet truth disorded is halfe disgracedâ— especially where the patterne of good order by profession is auouched to be But through ignoraunce arrogancy and â—emerltie of vnskilfull Methode Masters many thinges remayne yet in these Geometricall Elementes vnduely tumbled in though true yet with disgrace which by helpe of so many wittes and habilitie of such as now may haue good cause to be skilfull herein will I hope ere long be taken away and thinges of importance wanting supplied The end of the tenth booke of Euclides Elementes ¶ The eleuenth booke of Euclides Elementes HITHERTO HATH â—VCLIDâ— IN THâ—Sâ— former bookes with a wonderfull Methode and order entreated of such kindes of figures superficial which are or may be described in a superficies or plaine And hath taught and set forth their properties natures generations and productions euen from the first roote ground and beginning of them namely from a point which although it be indiuisible yet is it the beginning of all quantitie and of it and of the motion and slowing therof is produced a line and consequently all quantitie coÌ„tinuall as all figures playne and solide what so euer Euclide therefore in his first booke began with it and from thence went he to a line as to a thing most simple next vnto a point then to a superficies and to angles and so through the whole first booke he intreated of these most simple and plaine groundes In the second booke he entreated further and went vnto more harder matter and taught of diuisions of lines and of the multiplication of lines and of their partes and of their passions and properties And for that rightlined â—igures are far distant in nature and propertie from round and circular figures in the third booke he instructeth the reader of the nature and conditioÌ„ of circles In the fourth booke he compareth figures of right lines and circles together and teacheth how to describe a figure of right lines with in or about a circle and contraâ—iwiâ—e a circle with in or about a rectiline figure In the fifth booke he searcheth out the nature of proportion a matter of wonderfull vse and deepe consideration for that otherwise he could not compare â—igure with figure or the sides of figures together For whatsoeuer is compared to any other thing is compared vnto it vndoubtedly vnder some kinde of proportion Wherefore in the sixth booke he compareth figures together one to an other likewise their sides And for that the nature of proportion can not be fully and clearely sene without the knowledge of number wherein it is first and chiefely found in the seuenth eight and ninth bookes he entreatâ—th of number of the kindes and properties thereof And because that the sides of solide bodyes for the most part are of such sort that compared together they haue such proportion the one to the other which can not be expresâ—ed by any number certayne and therefore are called irrational lines he in the teÌ„th boke hath writteÌ„ taught which lineâ— are coÌ„meÌ„surable or incoÌ„meÌ„surable the one to the other and of the diuersitie of kindes of irrationall lines with all the conditions proprieties of them And thus hath Euclide in these ten foresayd bokes fully most pleÌ„teously in a meruelous order taught whatsoeuer semed necessary and requisite to the knowledge of all superficiall figures of what sort forme so euer they be Now in these bookes following he entreateth of figures of an other kinde namely of bodely figures as of Cubes Piramids Cones Columnes Cilinders Parallelipipedons Spheres and such othersâ— and sheweth the diuersitie of theÌ„ the generation and production of them and demonstrateth with great and wonderfull art their proprieties and passions with all their natures and conditions He also compareth one oâ— them to an other whereby to know the reason and proportion of the one to the other chiefely of the fiue bodyes which are called regular bodyes And these are the thinges of all other entreated of in Geometrie most worthy and of greatest dignitie and as it were the end and finall entent of the whole are of Geometrie and for whose cause hath bene written and spoken whatsoeuer hath hitherto in the former bookes bene sayd or written As the first booke was a ground and a necessary entrye to all the râ—st â—ollowing so is this eleuenth booke a necessary entrie and ground to the rest which follow And as that contayned the declaration of wordes and definitions of thingeâ— requisite to the knowledge of superficiall figures and entreated of lines and of their diuisions and sections which are the termes and limites of superficiall figures so in this booke is set forth the declaration of wordes and definitions of thinges pertayning to solide and corporall figures and also of superficieces which are the termes limites of solides moreouer of the diuision and intersection of them and diuers other thinges without which the knowledge of bodely and solide formes can not be attayned vnto And first is set the definitions as followâ—th Definitions A solide or body is that which hath length breadth and thicknes and the terme or limite of a solide is a superficies There are three kindes of continuall quantitie a line a superficies and a solide or body the beginning of all which as before hath bene sayd is a poynt which is indiuisible Two of these quantities namely a line and a superficies were defined of Euclide before in his first booke But the third kinde namely a solide or body he there defined not as a thing which pertayned not then to his purpose but here in this place he setteth the definitioÌ„ therof as that which chiefely now pertayneth to his purpose and without which nothing in these thinges can profitably be taught A solide sayth he is that which hath leÌ„gth breadth and thicknes or depth There are as before hath bene taught three reasons or meanes of measuring which are called coÌ„monly dimensions namely lâ—ngth breadth and thicknes These dimensions are ascribed vnto quantities onely By these are all kindes of quantitie deâ—ined â—â— are counted perfect or imperfect according as they are pertaker of fewer or more of them As Euclide defined a line ascribing vnto it onely one of these dimensions namely length Wherefore a line is the imperfectest kinde of quantitie In defining of a superficies he ascribed vnto it two dimensions namely length and breadth whereby a superficies is a quantitie of
the 7. of the fift our conclusion is inferred the superficies Sphericall of the segment CAE to be to the superficies Sphericall of the segment FGH as AD is to GI A Theoreme 6. To any solide sector of a Sphere that vpright Câ—â—e is equall whose base is equall to the câ—nnex Sphericall superficies of that sector and heith equall to the semidiameter of the same Sphere Hereof the demonstration in respect of the premises and the common argument of inscriptioâ— and circumscription of figures is easy and neuerthelesse if your owne write will not helpe you sufficiently you may take helpe at Archimedes hand in his first booke last proposition of the sphere and cylinder Whether if ye haue recourse you shalâ— perceaue how your Theoreme here amendeth the common translation there and also our delinâ—ation geueth more sâ—uâ—y shew of the chiefe circumstances necessary to the construction then there you shall finde Of the sphere here imagined to be A we note a solide sector by the letterâ— PQRO. So that PQR doth signifie the sphericall superficies to that solide sector belonging which is also common to the segment of the same sphere PRQ and therefore a line drawne from the toppe of that segmentâ— which toppe suppose to be Q is the semidiameter of the circle which is equall to the sphericall superficies of the sayd solide sector or segmentâ— as before is taught Let that line be QP By Q draw a line contingent which let be SQT. At the poynt Q from the line QS cut a line equall to PQ which let be SQ And vnto SQ make QT equall then draw the right lines OSOâ— and OQ About which OQ as an axe faâ—â—ened if you imagine the triangle OST to make an halfe circular reuolution you shall haue the vpright cone OST whose heith is OQ the semidiameter of the sphere and base the circle whose diameter is ST equall to the solide sector PQRO. A Theoreme 7. To any segment or portion of a Sphere that cone iâ— equall which hath that circle to his base which is the bâ—se of the segmeÌ„t and heith a right line which vnto the heith of the segmeÌ„t hath that proportioÌ„ which the semidiameter of the Sphere together with the heith of the other segment remayâ—â—â—g hath to the heith of the same other segment remaynâ—ng This is well demonstrated by Archiâ—â—des therefore nedeth no inuention of myne to confirme the same and for that the sayd demonstration is ouer long here to be added I will refere you thether for the demonstration and here supply that which to Archimedes demonstration shall geue light and to your farther speculation and practise shal be a great ayde and direction Suppose K to be a sphere the greatest circle K in coÌ„teyned let be ABCE and his diameter BE ceÌ„ter D. Let the sphere K be cutte by a playne superficies perpâ—ndicularly erected vpon the sayd greatest circle ABCE let the section be the circle about AC And let the segmentes of the sphere be the one that wherein is ABC whose â—oppe is Bâ— and the other let be that wherein is AEC and his toppe let be E I say that a cone which hath his base the circle about AC held a line which to BF the heith of the segment whose toppe is B hath that proportion that a line compoâ—ed of DE the semidiameter of the sphere and EF the heith of the other remayning segment whose toppe is E hath to EF the heith of that other segment remayning is equall to the segment of the sphere K whose toppe is B. To make this cone take my easy order thus Frame your worke for the findâ—ng of the fourth proportionall lineâ— by making EF the first and a line composed of DE and EF the secondâ— and the third let be BF then by the 12. of the sixâ—h let the fourth proportionall line be found which let be FGâ— vpon F the center of the base of the segment whose toppe is B erect a line perpendicular equall to FG found and drawe the lines GA and GC and so make perfect the cone GAC I say that the cone GAC is equall to the segment of the sphere K whose toppe is B. In like maner for the other segmeÌ„t whose toppe is E to finde the heith due for a cone equal to it by the order of the Theoreme you must thus frame your lines let the first be BF the second DB and BF composed in one right line and the third must be EF where by the 12. of the sixth finding the fourth it shall be the heith to rere vpon the base the circle about AC to make an vpright cone equall to the segment whose toppe is E. ¶ Logistically ¶ The Logisticall finding hereof is most easy the diameter of the sphere being geuen and the portions of the diameter in the segmentes conteyned or axes of the segmentes being knowne Then order your numbers in the rule of proportion as I here haue made most playne in ordring of the lines for the â—ought heith will be the producte A Corollary 1. Hereby and other the premises it is euident that to any segment of a Sphere whose whole diameter is knowne and the Axe of the segment geuen An vpright cone may be made equall or in any proportion betwene two right lines assignedâ— and therefore also a cylinder may to the sayd segment of the Sphere be made equall â—r in any proportion geuen betwene two right lines A Corollary 2. Manifestly also of the former theoreme it may be inferred that a Sphere and his diameter being deuided by one and the same playne superficies to which the sayd diameter is perpendicularâ— the two segmentes of the Sphere are one to the other in that proportion in which a rectangle parallelipipedon hauing for his base the square of the greater part of the diameter and his heith a line composed of the lesse portion of the diameter and the semidiameter to the rectangle parallelipâ—pedon hauing for his base the square of the lesse portion of the diameter his heith a line composed of the semidiameter the greater part of the diameter A Theoreme 8. Euery Sphere to the cube made of his diameter is in maner as 11. to 21. As vpon the first and second propositioÌ„s of this booke I began my additions with the circle being the chiefe among playne figures and therein brought manifold considerations about circles as of the proportion betwene their circumferences and their diameters of the content or Area of circles of the proportion of circles to the squares described of their diameters of circles to be geuen in al proâ—portions to other circles with diuerse other most necessary problemes whose vse is partly there specified So haue I in the end of this booke added some such Problemes Theoremesâ— about the sphere being among solides the chiefe as of the same either in it selfe considered or to cone and cylinder compared by reason of