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
namely the first ten propositions as they follow in order VVhich is vndoubtedly great pleasure to coÌ„sider also great increase furniture of knowledge VVhose PropositioÌ„s are set orderly after the propositioÌ„s of Euclide euery one of Barlaam correspoÌ„dent to the same of Euclide And doubtles it is wonderful to see how these two coÌ„trary kynds of quantity quantity discrete or number and quantity continual or magnitude which are the subiectes or matterâ— of Arithmitique and Geometry shoulde haue in them one and the same proprieties common to them both in very many points and affections although not in all For a line may in such sort be deuided that what proportion the whole hath to the greater parte the same shall the greater part haue to the lesse But that can not be in number For a number can not so be deuided that the whole number to the greater part thereof shall haue that proportion which the greater part hath to the lesse as Iordane very playnely proueth in his booke of Arithmetike which thynge Campane also as we shall afterward in the 9. booke after the 15. proposition see proueth And as touching these tenne firste propositions of the seconde booke of Euclide demonstrated by Barlaam in numbers they are also demoÌ„strated of Campane after the 15. proposition of the 9. booke whose demonstrations I mynde by Gods helpe to set forth when I shal come to the place They are also demoÌ„strated of Iordane that excellet learned authour in the first booke of his Arithmetike In the meane tyme I thought it not amisse here to set forth the demonstrations of Barlaam for that they geue great light to the seconde booke of Euclide besides the inestimable pleasure which they bring to the studious considerer And now to declare the first Proposition by numbers I haue put this example following Take two numbers the one vndeuided as 74. the other deuided into what partes and how many you list as 37. deuided into 20. 10. 5. and 2â— which altogether make the whole 37. Then if you multiply the number vndeuided namely 74 into all the partes of the number deuided as into 20. 10. 5. and 2. you shall produce 1480. 740. 370. 148. which added together make 2738 which self number is also produced if you multiplye the two numbers first geuen the one into the other As you see in the example on the other side set So by the aide of this Proposition is gotten a compendious way of multiplication by breaking of one of the numbers into his partes which oftentimes serueth to great vse in workingâ— chiâ—â—ly in the rule of proportions The demonstration of which proposition followeth in Barlaam But â—irst are put of the author these principles following ¶ Principles 1. A number is sâ—yd to multiply an other number when the number multiplied is so oftentymes added to it selfe as there be vnities in the number which multiplieth wherby is produced a certaine number which the number multiplied measureth by the vnities which are in the number which multipliâ—th 2. And the number produced of that a multiplication is called a plaine or superficiall number 3. A square number is that which is produced of the multiplicatian of any number into it selfe 4. Euery lesse number compared to a greater is sayd to be a part of the greater whether the lesse measure the greater or measure it not 5. Numbers whome one and the selfe same number measureth equally that is by one and the selfe same number are equall the one to the otheâ— 6. Numbers that are equemultiplâ—ces to one and the selfe same number that is which contayne one and the same number equally and alike are equall the one to the other The first Proposition Two numbers beyng geuen if the one of them be deuided into any numbers how many soeuer the playne or superficiall number which is produced of the multiplication of the two numbers first geuen the one into the other shall be equall to the superficiall nuÌ„bers which are produced of the multiplication of the number not deuided into euery part of the number deuided Suppose that there be two numbers AB and C. And deuide the number AB into certayne other numbers how many soeuer as into AD DE and EB Then I say that the superficiall number which is produced of the multiplication of the number C into the number AB is equall to the superficiall numbers which are produced of the multiplication of the number C into the nuÌ„ber AD and of C into DE and of C into EB For let F be the superficiall number produced of the multiplication of the number C into the number AB and let GH be the superficiall number produced of the multiplication of C into AD And let HI be produced of the multiplication of C into DE aâ—d finally of the multiplication of C into EB let there be produced the number IK Now forasmuch as AB multiplying the number C produced the number F therefore the number C measureth the number F by the vnities which are in the number AB And by the same reason may be proued that the number C doth also measure the number GH by the vnities which are in the number AD and that it doth measure the number HI by the vnities which are in the nuÌ„ber DF and finally that it measureth the number IK by the vnities which are in the number EB Wherefore the nuÌ„ber C measureth the whole number GK by the vnities which are in the number AB But it before measured the number F by the vnities which are in the number AB wherfore either of these numbers F and GK is equemultiplex to the number C. But numbers which are equemultiplices to one the selfe same numbers are equall the one to the other by the 6. definition Wherfore the number F is equall to the number GK But the number F is the superficiall number produced of the multiplication of the nuÌ„ber C into the number AB and the number GK is composed of the superficiall numbers produced of the multiplication of the nuÌ„ber C not deuided into euery one of the numbers AD DE and EB If therefore there be two numbers geuen and the one of them be deuided c. Which was required to be proued The 2. Theoreme The 2. Proposition If a right line be deuided by chaunce the rectangles figures which are comprehended vnder the whole and euery one of the partes are equall to the square whiche is made of the whole SVppose that the right line AB be by chaunse denided in the point C. Then I say that the rectangle figure comprehended vnder AB and BC together with the rectangle comprehended vnder AB and AC is equall vnto the square made of AB Describe by the 46. of the first vpon AB a square ADEB and by the 31 of the first by the point C draw a line parallel vnto either of these lines AD aâ—d BE and let the same
Poligonon figure of 24. sides Likewyse of the Hexagon AB and of the Pentagon AC shall be made a Poligonon figure of 30. sides one of whose sides shall subtend the arke BC. For the denomination of AB which is 6. excedeth the denomination of AC which is 5. onely by vnitie So also forasmuch as the denomination of AB which is 6. excedeth the denomination of AE which is 3. by 3. therefore the arke BE shall contayne 3. sides of a Poligonon figure of .18 sides And obseruing thys selfe same methode and order a man may finde out infinite sides of a Poligonon figure The end of the fourth booke of Euclides Elementes ¶ The fifth booke of Euclides Elementes THIS FIFTH BOOKE of Euclide is of very great commoditie and vse in all Geometry and much diligence ought to be bestowed therin It ought of all other to be throughly and most perfectly and readily knowne For nothyng in the bookes followyng can be vnderstand without it the knowledge of them all depende of it And not onely they and other writinges of Geometry but all other Sciences also and artes as Musike Astronomy Perspectiue Arithmetique the arte of accomptes and reckoning with other such like This booke therefore is as it were a chiefe treasure and a peculiar iuell much to be accompted of It entreateth of proportion and Analogie or proportionalitie which pertayneth not onely vnto lines figures and bodies in Geometry but also vnto soundes voyces of which Musike entreateth as witnesseth Boetius and others which write of Musike Also the whole arte of Astronomy teacheth to measure proportions of tymes and mouinges Archimides and Iordan with other writing of waightes affirme that there is proportion betwene waight and waight and also betwene place place Ye see therefore how large is the vse of this fift booke Wherfore the definitions also thereof are common although hereof Euclide they be accommodate and applied onely to Geometry The first author of this booke was as it is affirmed of many one Eudoxus who was Platos scholer but it was afterward framed and put in order by Euclide Definitions A parte is a lesse magnitude in respect of a greater magnitude when the lesse measureth the greater As in the other bookes before so in this the author first setteth orderly the definitions and declarations of such termes and wordes which are necessarily required to the entreatie of the subiect and matter therof which is proportion and comparison of proportions or proportionalitie And first he sheweth what a parte is Here is to be considered that all the definitions of this fifth booke be general to Geometry and Arithmetique and are true in both artes euen as proportion and proportionalitie are common to them both and chiefly appertayne to number neither can they aptly be applied to matter of Geometry but in respect of number and by number Yet in this booke and in these definitions here set Euclide semeth to speake of them onely Geometrically as they are applied to quantitie continuall as to lines superficieces and bodies for that he yet continueth in Geometry I wil notwithstanding for facilitie and farther helpe of the reader declare thē both by example in number and also in lynes For the clearer vnderstandyng of a parte it is to be noted that a part is taken in the Mathematicall Sciences two maner of wayes One way a part is a lesse quantitie in respect of a greater whether it measure the greater o◠no. The second way a part is onely that lesse quantitie in respect of the greater which measureth the greater A lesse quantitie is sayd to measure or number a greater quantitie when it beyng oftentymes taken maketh precisely the greater quantitie without more or lesse or beyng as oftentymes taken from the greater as it may there remayneth nothyng As suppose the line AB to contayne 3. and the lyne CD to contayne 9. thē doth the line AB measure the line CD for that if it be take certayne times namely 3. tymes it maketh precisely the lyne CD that is 9. without more or lesse Agayne if the sayd lesse lyne AB be taken from the greater CD as often as it may be namely 3. tymes there shall remayne nothing of the greater So the nūber 3. is sayde to measure 12. for that beyng taken certayne tymes namely foure tymes it maketh iust 12. the greater quantitie and also beyng taken from 12. as often as it may namely 4. tymes there shall remayne nothyng And in this meaning and signification doth Euclide vndoubtedly here in this define a part saying that it is a lesse magnitude in comparison of a greater when the lesse measureth the greater As the lyne AB before set contayning 3. is a lesse quantitie in comparison of the lyne CD which containeth 9. and also measureth it For it beyng certayne tymes taken namely 3. tymes precisely maketh it or taken from it as often as it may there remayneth nothyng Wherfore by this definition the lyne AB is a part of the lyne CD Likewise in numbers the number 5. is a part of the number 15. for it is a lesse number or quantitie compared to the greater and also it measureth the greater for beyng taken certayne tymes namely 3. tymes it maketh 15. And this kynde of part is called commonly pars metiens or mensurans that is a measuryng part some call it pars multiplicatina and of the barbarous it is called pars aliquota that is an aliquote part And this kynde of parte is commonly vsed in Arithmetique The other kinde of a part is any lesse quantitie in comparison of a greater whether it be in number or magnitude and whether it measure or no. As suppose the line AB to be 17. and let it be deuided into two partes in the poynt C namely into the line AC the line CB and let the lyne AC the greater part containe 12. and let the line BC the lesse part contayne 5. Now eyther of these lines by this definition is a part of the whole lyne AB For eyther of them is a lesse magnitude or quātity in cōparisō of the whole lyne AB but neither of thē measureth the whole line AB for the lesse lyne CB contayning 5. taken as oftē as ye list will neuer make precisely AB which contayneth 17. If ye take it 3. tymes it maketh only 15. so lacketh it 2. of 17. which is to litle If ye take it 4. times so maketh it 20. thē are there thre to much so it neuer maketh precisely 17. but either to much or to litle Likewise the other part AC measureth not the whole lyne AB for takē once it maketh but 12. which is lesse then 17. and taken twise it maketh 24. which are more then 17. by ◠So it neuer precisely maketh by takyng therof the whole AB but either more or lesse And this kynde of part they commonly call pars
Vnum quodque idea est quia vnum numero est that is euery thing therfore is that is therefore hath his being in nature and is that it is for that it is on in nomber According whereunto Iordane in that most excellent and absolute worke of Aâ—ithmeticke which he wrote defineth vnitie after this maner Vnitas est res per se discretio that is vnitie is properly and of it selfe the difference of any thing That is vnitie is that whereby euery thing doth properly and essentially differ and is an other thing from all others Certainely a very apt deâ—inition and it maketh playne the definition here set of Euclide 2 Number is a multitude composed of vnities As the number of three is a multitude composed and made of three vnities Likewise the number of fiue is nothing ells but the composition putting together of fiue vnities Although as was before sayde betwene a poynt in magnitude and vnitie in multitude there is great agreement and many thiâ—â—â—â— are comâ—on to them both for as a poynt is the beginning of magnitude so is vnitie the beginning of nomber And as a poynt in magnitude is indiuisible so is also vnitie in number indiuisible yet in this they differ and disagree There is no line or magnitude made of pointes as of his partes So that although a point be the beginning of a lyne yet is it no part therof But vnitie as it is the beginning of number so is it also a part therof which is somewhat more manifestly set of Boetius in an other dâ—ffinition of number which he geueth in his Arithmetike which is thus Numerus est quantitatâ— acernus ex vnitatibus profusus that is Number is a masse or heape of quantities produced of vnities which diffinition in substance is all one with the first wherin is said most plainly that the heape or masse that is the whole substance of the quantitie of number is produced made of vnities So that vnitie is as it were the very matter of number As foure vnities added together are the matter wherof the number 4. is made eche of these vnities is a part of the number foure namely a fourth part or a quarter Vnto this diffinition agreeth also the definition geuen of Iordane which is thus Number is a quantitie which gathereth together thinges seuered a sonder As fiue men beyng in themselues seuered and distincte are by the number fiue brought together as it were into one masse and so of others And although vnitie be no number yet it contayneth in it the vertue and power of all numbers and is set and taken for them In this place for the Farther elucidation of thinges partly before set and chiefly hereafter to be set because Euclide here doth make mention of diuers kyndes of numbers and also defineth the same is to be noted that number may be considered three maner of wayes First number may be considered absolutely without comparyng it to any other number or without applieng it to any other thing onely vewing â—nd paysing what it is in it selfe and in his owne nature onely and what partes it hath and what proprieties and passions As this number sixe may be considered absolutely in his owne nature that it is an euen number and that it is a perfect number and hath many mo conditions and proprieties And so conceiue ye of all other numbers whatsoeuer of 9. 12. and so forth An other way number may be coÌ„sidered by way of coÌ„parison and in respect of some other number either as equall to it selfe or as greater theÌ„ it selfe or as lesse theÌ„ it selfe As 12. may be coÌ„sidered as coÌ„pared to 12. which is equall vnto it or as to 24. which is greater then it for 12 is the halfe thereof of as to 6. which is lesse then it as beyng the double therof And of this consideration of numbers ariseth and springeth all kyndes and varieties of proportioÌ„ as hath before bene declared in the explanation of the principles of the fift booke so that of that matter it is needelesse any more to be sayd in this place Thus much of this for the declaration of the thinges following 3 A part is a lesse number in comparison to the greater when the lesse measureth the greater As the number 3 compared to the number 12. is a part For 3 is a lesse number then is 12. and moreouer it measureth 12 the greater number For 3 taken or added to it selfe certayne times namely 4 tymes maketh 12. For 3 foure tymes is 12. Likewise is â— a part of 8 2 is lesse then 8 and taken 4 tymes it maketh 8. For the better vnderstandyng of this diffinition and how this worde Parte is diuersly taken in Arithmetique and in Geometry read the declaration of the first diffinition of the 5. booke 4 Partes are a lesse number in respect of the greater when the lesse measureth not the greater As the number 3 compared to 5 is partes of 5 and not a part For the number 3 is lesse then the nuÌ„ber 5 and doth not measure 5. For taken once it maketh but 3. once 3 is 3 which is lesse then 5. and 3 taken twise maketh 6 which is more then 5. Wherfore it is no part of 5 but partes namely three fifth partes of 5. For in the number 3 there are 3 vnities and euery vnitie is the fifth part of 5. Wherfore 3 is three fifth partes of 5 and so of others 5 Multiplex is a greater number in comparison of the lesse when the lesse measureth the greater As 9 compared to 3 is multiplex the number 9 is greater then the number 3. And moreouer 3 the lesse number measureth 9 the greater number For 3 taken certaine tymes namely 3 tymes maketh 9. three tymes three is 9. For the more ample and full knowledge of this definition read what is sayd in the explanation of the second definition of the 5 booke where multiplex is sufficiently entreated of with all his kyndes 6 An euen number is that which may be deuided into two equal partes As the number 6 may be deuided into 3 and 3 which are his partes and they are equall the one not exceding the other This definition of Euclide is to be vnderstand of two such equall partes which ioyned together make the whole number as 3 and 3 the equall partes of 6 ioyned together make 6 for otherwise many numbers both euen and odde may be deuided into many equall partes as into 4. 5. 6â— or mo and therfore into 2. As 9 may be deuided into 3 and 3 which are his partes and are also equall for the one of them excedeth not the other yet is not therfore this number â— an euen number for that 3 and 3 these equall partes of 9 added together make not 9 but onely 6. Likewise taking the definition so generally euery number whatsoeuer should be an euen numberâ— for
the parts added together are not equal to the whole nor make the whole but make either more or lesse Wherefore of imperfect numbers there are two kindes the one is called abundanâ— or a building the other 〈◊〉 or wanting A number abunding is that whose partes being all added together make more then the wâ—â—ds number whose partes they are as 12. is an abundant number For all the parteâ— of 12. namely 6.4.3.2 and 1. added together make 16 which are more then 12. Likewise 18. is a number abunding all his partâ— nâ—mely 9.6.3.2 and 1. added together make 20. which are more then 18 and so of others A number diminute or wanting is that whose partes being all added together make lesse then the whole or number whose partes they are As 9. is a diminute or wanting number for all his partes namelyâ— 3. and 1. moe partes he hath not added together make onely 4 which are lesse then 9. Also 26. is a diminute nuÌ„ber all his partes namely 13.2.1 added togethâ—r make onely 16 which is a number much lesse then 26. And so of such like CAmpane and Flussates here adde certayne common sentences some of whichâ— for that they are in these three bookes following somtimes alledged I thought good here to annexe 1 The lesse part is that which hath the greater denomination and the greater part is that which hath the lesse denomination As the numbers 6. and 8. are either of them a part of the number 24 6. is a fourth part 4. times 6. is 24 and 8. is a third part 3. times 8. is 24. Now forasmuch as 4 which denominateth what part 6. is of 24 is greater then 3. which denominateth what part 8. is of 24. therefore is 6. a lesse part of 24â— then is 8. and so is 8. a greater part of 24. theÌ„ 6. is And so in others 2 Whatsoeuer numbers are equemultiplices to one the selfe same nuÌ„ber or to equall numbers are also equall the one to the other As if vnto the number 3 be taken two numbers containing the same number foure times that is being equemultiplices to the same number three the sayd two numbers shal be equall For 4. times 3. will euer be 12. So also will it be if vnto the two equal numbers 3. 3. be taken two numbers the one coÌ„taining the one number 3. foure times the other containing the other number 3. also foure times that is being equemultiplices to the equall numbers 3. and 3. 3 Those numbers to whome one and the selfe same number is equmultiplex or whose equemultiplices are equall are also equall the on to the other As if the number 18. be equemultiplex to any two numbers that is contayne any two numbers twise thrise fower times c As for example 3. times then are the sayd two numbers equall For 18. deuided by 3. will euer bring forth 6. So that that diuision made twise will bring forth 6. and 6. two equall numbers So also would it follow if the two numbers had equall equemultiplices namely if 18. and 18. which are equall numbers contayned any two numbers 3. times 4 If a number measure the whole and a part taken away it shall also measure the residue As if from 24. be taken away 9. there remaineth 15. And for as much as the number 3 measureth the whole number 24 also the number takeÌ„ away namely 9. it shall also measure the residue which is 15â— For 3. measureth 15 by fiue fiue times 3. is 15. And so of others 5 If a number measure any number it also measureth euery number that the sayd number measureth As the number 6. measuring the number 12. shall also measure all the numbers that 1â— measurethâ— as the numbers 24.36.48.60 and so forth which the number 12. doth measure by the numberâ— 2.3.4 and 5. And for as much as the number 12. doth measure the numbers 24.36.48 and 60. And the nuÌ„ber 6 doth measure the number 12. namely by 2. It followeth by this commoÌ„ sentence that the number 6. measureth eche of thâ—se numbers 24. 36.48 and 60. And so of others 6 If a number measure two numbers it shall also measure the number composed of them As the number 3 measureth these two numbers 6. and 9â— it measureth 6. by 2â— and 9. by 3. And therefore by this common sentence it measureth the number 15. which is composed of the numbers 6. and 9 namely it measureth it by 5. 7 If in numbers there be proportions how manysoeuer equall or the selfe same to one proportion they shall alâ—o be equall or the selfe same the one to the other As yf the proportion of the number 6. to the number 3. be as the proportion of the number 8. to the number 4 if also the proportion of the number 10. to the number 5. be as the proportion of the number 8. to the number 4 then shall the proportion of the number 6. to the number 3. be as the proportion of the number 10. is to the number 5 namely eche proportion is duple And so of others Euclide in his â— booke the 11. proposition demonstrated this also in continuall quantitie which although as touching that kinde of quantitie it might haue bene put also as a principle as in numbers he taketh it yet for that in all magnitudes theyr proportion can not be expressed as hath before bene noted shal be afterward in the tenth booke more at large made manifest therefore he demonstrateth it there in that place and proueth that it is true as touching all proportions generally whither they be rational or irrationall ¶ The first Proposition The first Theoreme If there be geuen two vnequall numbers and if in taking the lesse continually from the greater the number remayning do not measure the number going before vntill it shall come to vnitie then are those numbers which were at the beginning geuen prime the one to the other SVppose that there be two vnequal numbers AB the greater and CD the lesse and from AB the greater take away CD the lesse as oâ—ten as you can leauing FA and from CD take away FA as often as you can leauing the number GC And from FA take away GC as often as you can and so do continually till there remayne onely vnitie which let be HA. Then I say that no number measureth the numbers AB and CD For if it be possible let some number measure them and let the same be E. Now CD measuring AB leaueth a lesse number then it selfe which let be FA â— And FA measuring DC leaueth also a lesse then it selfe namely GC And GC measâ—ring FA leaueth vnitie HA. And forasmuch as the number E measureth DC and the number CD measureth the number BF therfore the number E also measureth BF and it measureth the whole number BA wherfore it also measureth that which remayneth namely the number FA 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
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
the whole line MG to the whole line EA by the 18. of the fifth Wherefore as MG the side of the cube is to EA the semidiameter so is the line FGHIM to the Octohedron ABKDLC inscribed in one the selfe same Sphere If therefore a cube and an Octohedron be contained in one and the selfe same Sphere they shall be in proportion the one to the other as the side of the cube is to the semidiameter of the Sphere which was required to be demonstrated A Corollary Distinctly to notefie the powers of the sides of the fiue solides by the power of the diameter of the sphere The sides of the tetrahedron and of the cube doo cut the power of the diameter of the sphere into two squares which are in proportion double the one to the other The octohedron cutteth the power of the diameter into two equall squares The Icosahedron into two squares whose proportion is duple to the proportion of a line diuided by an extreame and meane proportion whose lesse segmeÌ„t is the side of the Icosahedron And the dodecahedron into two squares whose proportion is quadruple to the proportion of a line diuided by an extreame and meane proportion whose lesse segment is the side of the dodecahedron For AD the diameter of the sphere contayneth in power AB the side of the tetrahedron and BD the side of the cube which BD is in power halfe of the side AB The diameter also of the sphere contayneth in power AC and CD two equall sides of the octohedron But the diameter contayneth in power the whole line AE and the greater segment thereof ED which is the side of the Icosahedron by the 15. of this booke Wheâ—fore their powers being in duple proportioÌ„ of that in which the sides are by the first corollary of the 20. of the sixth haue their proportion duple to the proportion of an extreame meane proportioÌ„ Farther the diameter coÌ„tayneth in power the whole line AF and his lesse segment FD which is the side of the dodecahedron by the same 15. of this booke Wherefore the whole hauing to the lesse â— double proportion of that which the extreame hath to the meane namely of the whole to the greater segment by the 10. diffinition of the fifth it followeth that the proportion of the power is double to the doubled proportion of the sides by the same first corollary of the 20. of the sixth that is is quadruple to the proportion of the extreame and of the meane by the diffinition of the sixth An aduertisment added by Flussas By this meanes therefore the diameter of a sphere being geuen there shall be geuen the side of euery one of the bodies inscribed And forasmuch as three of those bodies haue their sides commensurable in power onely and not in length vnto the diameter geuen for their powers are in the proportion of a square number to a number not square wherefore they haue not the proportion of a square number to a square number by the corollary of the 25. of the eight wherefore also their sides are incommensurabe in length by the 9. of the tenth therefore it is sufficient to compare the powers and not the lengths of those sides the one to the otherâ— which powers are contained in the power of the diameter namely from the power of the diameter let there ble taken away the power of the cube and there shall remayne the power of the Tetrahedron and taking away the power of the Tetrahedron there remayneth the power of the cube and taking away from the power of the diameter halfe the power thereof there shall be left the power of the side of the octohedron But forasmuch as the sides of the dodecahedron and of the Icosahedron are proued to be irrationall for the side of the Icosahedron is a lesse line by the 16. of the thirtenth and the side of the dedocahedron is a residuall line by the 17. of the same therfore those sides are vnto the diameter which is a rationall line set incommensurable both in length and in power Wherefore their comparison can not be diffined or described by any proportion expressed by numbers by the 8. of the tenth neither can they be compared the one to the other for irrational lines of diuers kindes are incoÌ„meÌ„surable the one to the other for if they should be commensurable they should be of one and the selfe same kinde by the 103. and 105. of the tenth which is impossible Wherefore we seking to compare them to the power of the diameter thought they could not be more aptly expressed then by such proportions which cutte that rationall power of the diameter according to their sides namely diuiding the power of the diameter by lines which haue that proportioÌ„ that the greater segment hath to the lesse to put the lesse segment to be the side of the Icosahedron deuiding the sayd power of the diameter by lines hauing the proportion of the whole to the lesse segment to expresse the side of the dodecahedron by the lesse segment which thing may well be done betwene magnitudes incommensurable The ende of the fourtenth Booke of Euclides Elementes after Flussas ¶ The fiftenth booke of Euclides Elementes THis finetenth and last booke of Euclide or rather the second boke of Appollonius or Hypsicles teacheth the inscription and circumscriptioÌ„ of the fiue regular bodies one within and about an other a thing vndoutedly plesant and delectable in minde to contemplate and also profitable and necessary in act to practise For without practise in act it is very hard to se and conceiue the constructions and demonstrations of the propositions of this booke vnles a man haue a very depe sharpe fine imagination Wherfore I would wish the diligent studeÌ„t in this booke to make the study thereof more pleasant vnto him to haue presently before his eyes the bodyes formed framed of pasted paper as I taught after the diffinitions of the eleuenth booke And then to drawe and describe the lines and diuisions and superficieces according to the constructions of the propositions In which descriptions if he be wary and diligent he shall finde all things in these solide matters as clere and as manifest vnto the eye as were things before taught only in plaine or superficial figures And although I haue before in the twelfth boke admonished the reader hereof yet bicause in this boke chiefly that thing is required I thought it should not be irkesome vnto him againe to be put in minde thereof Farther this is to be noted that in the Greke exemplars are found in this 15. booke only 5. propositions which 5. are also only touched and set forthe by Hypsicies vnto which Campane addeth 8. and so maketh vp the number of 13. Campane vndoubtedly although he were very well lerned and that generally in all kinds of learning yet assuredly being brought vp in a time of rudenes when all good letters were darkned barberousnes had
a right line coupling their centres being diuided by an extreame and meane proportion maketh the greater segment the right line which coupleth the centres of the next bases If by the centres of fiue bases set vppon one base be drawne a playne superficies and by the centres of the bases which are set vpon the opposite base be drawne also a playne superficies and then be drawne a right line coupling the centres of the opposite bases that right line is so cut that eche of his partes set without the playne superficies is the greater segment of that part which is contayned betwene the playnes The side of the dodecahedron is the greater segment of the line which subtendeth the angle of the pentagon A perpendicular line drawne from the centre of the dodecahedron to one of the bases is in power quintuple to half the line which is betwene the playnes And therfore the whole line which coupleth the centres of the opposite bases is in power quintuple to the whole line which is betwene the sayd playnes The line which subtâ—deth the angle of the base of the dodecahedroÌ„ together with the side of the base are in power quintuple to the line which is drawne from the ceÌ„tre of the circle which contayneth the base to the circumference A section of a sphere contayning three bases of the dodecahedron taketh a third part of the diameter of the sayd sphere The side of the dodecahedron and the line which subtendeth the angle of the pentagon are equall to the right line which coupleth the middle sections of the opposite sides of the dodecahedron ¶ The ende of the Elementes of Geometrie of the most auncient Philosopher 〈◊〉 of Megara The intent of this Preface Number Note the worde Vnit to expresse the Greke Monaâ— not Vnitie as we hauâ— all commonly till now vsed Magnitude A point A Line Magnitude Ano. 1488. ☞ Arithmetike Note * Anno. 1550. R. B. Note 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ☞ This noble Earle dyed Anno. 1554. skarse of 24. yeares of ageâ— hauing no issue by his wife Daughter to the Duke of Somerset Iustice. ☞ * Plato 7. de Rep. ☞ * Note The difference betwene Strataruhmetrie and Tacticie I.D. Fâ—endâ—â— you will finde it hard to performâ— my descriptiâ—n of â—his Fâ—ate But by Châ—râ—graphieâ— you may helpe your selfe some â—hat wherâ— thâ— Figures knowne in Sidâ—â—â—nd Angles are not Regular And whereâ— Resolution into Triangles can sâ—â—uâ— c. And yet you will finde it strange to deale thus generâ—lly with Arithmeticall figures and that for Battayle â—ay Their coâ—tentâ—â—â— differ so much from like Geometrâ—call Figurâ—s A marueilous Glasse ☞ S.W.P. ☞ Note 1. 2. 3. 5. 6. 7. 8. I.D. Read in Aristotle his 8. booke of Politikes the 5 6 and 7. chapters Where you shall haue some occasion farder to thinke of Musike than commonly is thought ☜ * Anno. 1548 and 1549. in Louayn Note I.D. The Cutting of a Sphare according to any proportion assigned may by this proposition be done Mechanically by tempering Lâ—quor to a certayne waight in respect of the waight of the Sphare 〈◊〉 Swyâ—â—â—ng A common errorâ— noted A paradox N. T. The wonderfull vse of these Propositions The practise Staticall to know the proportion betwene the Cube and the Sphare I. D. * For so haue you 256. partes of a Graine * The proportion of the Square to the Circle insâ—ribed * The Squaâ—ing of the Cirâ—le Mâ—â—haniâ—ally * To any Squirâ— gâ—uenâ— to 〈…〉 Note Squaring of the Circle without knowledge of the proportion betwene Circumference and Diameter To Dubble the Cube redily by Art Mechanicall depending vppon Demonstration Mathematicall I. D. The 4. sides of this Pyramiâ— must be 4. Isosceles Triangles â— likâ— and â—quall I. D. * In all workâ—ngeâ— with this Pyramis or Cone Let their Situationâ— be in all Pointeâ— and Conditions a like oâ— all one while you are about â—ne worke Els you will 〈◊〉 I. D. * Consider well whan you must put your wateâ—â— togyther and whan you must empty youâ— first waterâ— out of your Pyramiâ— or Cone Elâ— you will 〈◊〉 * Vitruuius Lib. 9. Cap. 3. ☞ God bâ— thanked â—or this Inuentionâ— the fruiâ—e â—nsuing * Note Note as concerning the Sphaericall Superâ—icies of the water ☞ * Note Note this Abridgeâ—ent of Dubbling 〈◊〉 Cube â—â— * Note * ☜ To giue Cubes one to the other in any proportion Rationall or Irrationall * Emptying the first The demonstrations of this Dubbling of the Cube and oâ— the rest I.D. * Here 〈…〉 of the water * By the 33. of the eleuenth books of Euclide I.D. * And your diligence in praâ—â—ise can â—o in waight of wateâ—â— peâ—forme it Therefore now you arâ— able to â—eue good reason of your whole doing * Note this Corollary * The great Commodities following of these new Inuentions * ☞ Such is the Fruite of the Mathematicall Sciences and Artes. MAN is the Lesse World. * ☜ Microcosmus * Lib. 3. Cap. 1. ☞ Saw Milles. * Atheneus Lib. 5. cap. 8. Proclus Pag. 18. To go to the bottom of the Sea without daunger Plutâ—â—â—bus in Marco Mâ—rcello Syâ—asius in Epistolis Polybius Plinius Quintâ—lianus T. Liuius * Athenaâ—s * Galeâ—us Anthemius Burning Glasses Gunnes 4. Reg. 20. A perpetuall Motion An obiection The Answer ☜ A Mathematicien Vitrunius VVho is an Architect * The Immaterialitie of perfect Architecture What Lineament is Note Anno. 1559. * Anno. 1551 De his quae Mundo mirabiliter eueniunt cap. 8. Tusc. â— * ☞ A Digression Apologeticall * A prouerb Fayre fisht and caught a Frog ☞ Psal. 140. Act. 7. C. Lib. 30. Cap. 1. ☜ R. B. ☞ Vniuersities ☜ ☞ The Ground platt of this Praeface in a Table The argumâ—â—â— of the first Booke Definition of a poynt Definition of a poynt after Pithagoras Definition of a liâ—â— An other definition of a line An other The endes of a line Difference of a point frâ—â—nity Vnitie is a part of number A poynt is no part of quantitie Definition of a right line DefinitioÌ„ therof after Plato An other definition An other An other An other An other VVhy Euclide here defineth not a crooked lyne Definition of a superficies A superficies may be deuided two wayes An other definition of a superficies The extremes of a superficies Another definition of a superficies Definition of a plaine superficies Another definition of a playne superficies NOTE Another definition of a playne superficies An other definition An other definition An other definition Definition of a playne angle Definition of a â—ecâ—ilined angle 〈◊〉 of angles VVhat a right angle VVhat also a perpendicular lyne iâ— VVhat an obtuse angle â—â— VVhat an acute angle is The limite of any thing No science of thinges infinite Definition of a figure Definition of a circle A circle the most perfect of all figures The centre of a circle Definition of a diameter Definition of a semicircle Definition of a section of a circle Definition of râ—â—â—â—lined figures
Construction Two cases in this Proposition First caseâ— Second case Demonstration Construction Demonstration Construction Demonstration An other way after Peliâ—arius Construction Demonstration Construction Demonstration Demonstration leading to an impossibilitie Three cases in this Propositiâ—n The third case Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration Construction Demonstration A Proposition added by Petarilius Note Construction Demonstration An other way also after Pelitarius Construction Demonstration An other way to do the samâ— after Pelitarius Demonstration Demonstrâ—tion Demonstration leading to an absurditie A Corollary Construction Demonstration Demonstration An â—ther way to do the same after OroÌ„tius An other way after Pelitarius Construction Demonstration An addition of Flussates * A Poligonon figure is a figure consisting of many sides The argument of this fift booke The first aâ—thor of this booke Eudoxus The first definition A part taken two maner of wayes The fiâ—st way The second way How a lesse quantity is sayd to measure a greater In what significatioÌ„ Euclide here taketh a part Parâ— metienâ— or mensuranâ— Pars multiplicatiâ—a Pars aliquota This kinde of part coÌ„monly vsed in Arithmetique The other kinde of part Pars constitâ—ens or componens Pars aliquanta The second definition Numbers very necessary for the vnderstanding of this booke and the other bookes following The tâ—ird definition Rational proportion deuided â—â—to two kindes Proportion of equality Proportion of inequality Proportioâ— of the greater to the lesse Multiplex Duple proportion Triple quadruple Quintuple Superperticular Sesquialtera Sesquitertia Sesquiquarta Superpartiens Superbipartiens Supertripartiens Superquadripartiens Superquintipartiens Multiplex superperticular Dupla Sesquialtera Dupla sesquitertia Tripla sesquialtera Multiplex superpartiens Dupla superbipartiens Dupla supertripartiens Tripla superbipartiens Tripla superquadâ—ipartiens How to knoâ— the denomination of any proportion Proportion of the lesse in the greater Submultiplex Subsuperparticular Subsuperpertient c. The fourth definition Example of this definition in magnitudes Example thereof in numbers Note The fifth definition An example of this â—efinition in magnitudes Why Euclide in defining of Proportion vsed multiplication The sixth definition An example of this definitiin in magnitudes An example in numbers An other example in numbers An other example in numbers Note this particle according to any multiplication An example where the equimultiplices of the first and third exceedâ— the equimultiplices of the second and fourth and yet the quantities geuen are not in one and the selfe same proportion A rule to produce equimultiplices of the first and third equall to the equimultiplices of the secondâ— and fâ—urth Example thereof The seuenth definition 9 12 3 4 Proportionality of two sortes contiâ—uall and discontinuall An example of continuall proportionalitie in numbers 16.8.4.2.1 In coutinnall proportionalitie the quantities cannot be of one kinde Discontinuall propâ—rtionalitie Example of discontinual proportionality in numbers In discoÌ„tinual proportionalitie the proportions may be of diuers kindes The eight definition An example of this definition in magnitudes An example in numbers Note The ninth definition An example of this definition in magnitudes Example â—n numbers The tenth definition A rule to adde proportions to proportions 8. 4. 2. 1. 2 2 2 1 1 1 The eleuenth definition Example of this definition in magnitudâ— Example in numbers The twelfâ—h definition Example of this deâ—inition in magnituds Example in numbers The thirtenth definition Example of this definition in magnituds Example in numbers The fourtenth definition Example of this definition in magnituds Example in numbers The fiâ—tâ—ne definition This is the coÌ„uerse of the former definition Example in magnitudes Example in numbers The sixtene definition An example of this definition in magnitudes An example in numbers The seueÌ„tenth definition An example of this definition in magnitudes An example in numbers Note The eighttenth definition An example of this definition in magnitudes Example in numbers The nintenth definition An example of this definition in magnitudes Example in numbers The 20. definition The 2â— defiâ—ition These two last definitions not found in the greeke examplers Construction Demonstration Demonstrationâ— Construction Demonstration Construction Demonstration ALemmae or an assumpt A Corollary Conuerse proportion Construction Demonstration Two cases in this Propotion The second The second part demonstrated The first part of this Proposition demonstrated The second part of the proposition demonstrated First differâ—câ— of the first part Demonstratiâ— of tâ—e same first differeÌ„ce Second diffeâ—ence Third diâ—â—erence The second part â—f this proposition The first parâ— of this Proposition demonstrated The second part proued The first part of this proposition proued The second part demonstrated Construction Demonstrationâ— Constrâ—ction Demonstrationâ— Construction Demonstration An addition of Campane Demonstration Construction Demonstration DemoÌ„stration of alternate proportion Construction Demonstration DemonstratioÌ„ of proportion by diuision Constrâ—ctionâ— Demonsâ—ration Demonstration of proportion by composition This proposition is the conuerse of the former DemonstratioÌ„â—eâ—aing to an â—mpossibility That which the fift of this booke proued only touching multiplices this proueth generally of all magnitudes ALemma A Corollary Conuersion of proportion This proposition pertaineth to Proportion of equality inordinate proportionality The second difference The third difference Thâ—r proposition pertaineth to Proportion of equality in perturbate proportionality The third difference Proportion of equality in ordinate proportionality Construction Demonstration When there are more then three magnitudes in either order Aâ—CDEâ—GH Proportion of equality in perturbate proprotionalitie Construction Demonstration Note That which the second propositioÌ„ of this booke proued only touching multiplices is here proued generally touching magnitudes An other demonstration of the same affirmatiuely An other demonstration of the same affirmatiuely An other demonstration of the same DemonstratioÌ„ leading to an impossibilitie An other demonstration of the same affirmatiuely Demonstration Demonstrationâ— Demonstration The argument of this sixth booke This booke necessary for the vse of instrumentes of Geometry The first definition The second deâ—inition Reciprocall figures called mutuall figures The third definition The fourth definition The fifth definition An other example Of substraction of proportion The sixth definition Demonstration of the first part Demonstâ—ation of the second part A Corollary added by Flussates The first part of this Theoreme Demonstration of the second part A Corollary added by Flussates Construction Demonstration of the first part Demonstratiâ—n of the second part which is the conuerse of the first Construction Demonstration This is the conuerse of the former proposition Construction Demonstration Constructioâ— The first part of this proposition Demonstration leading to an impossibilitie The second part of this proposition Construction Demonstration Construction Demonstration Construction Demonstration A Corollary out of Flussates By this and the former propoâ—ition may a right line be deuided into what partes soeuer you will. Construction Demonstration An other way after Pelitarius An otâ—eâ— way after Pelitarius Construction Demonstration An other way after Campane Construction Demonstratioâ— A proposition added by Pelitarius The
How often therfore these fiue sundry sortes of Operations do for the most part of their execution differre from the fiue operations of like generall property and name in our Whole numbers practisable So often for a more distinct doctrine we vulgarly account and name it an other kynde of Arithmetike And by this reason the Consideration doctrine and working in whole numbers onely where of an Vnit is no lesse part to be allowed is named as it were an Ariâ—hmetike by it selfe And so of the Arithmetike of Fractions In lyke sorte the necessary wonderfull and Secret doctrine of Proportion and proportionalytie hath purchased vnto it selfe a peculier maner of handlyng and workyng and so may seme an other forme of Arithmetike Moreouer the Astronomers for spede and more commodious calculation haue deuised a peculier maner of orderyng nuÌ„bers about theyr circular motions by Sexagenes and Sexagesmes By Signes Degrees and Minutes c. which commonly is called the Arithmetike of Astronomical or Phisicall Fractions That haue I briefly noted by the name of Arithmetike Circular By cause it is also vsed in circles not Astronomicall c. Practise hath led Numbers farder and hath framed them to take vpon them the shew of Magnitudes propertie Which is Incommensurabilitie and Irrationalitie For in puâ—e Arithmetike an Vnit is the common Measure of all Numbers And here NuÌ„bers are become as Lynes Playnes and Solides some tymes Rationall some tymes Irrationall â— And haue propre and peculier characters as √ ◠√ â— and so of other Which is to signifie Roâ—e Square Rote Cubik and so forth propre and peculier fashions in the fiue principall partes Wherfore the practiser estemeth this a diuerse Arithmetike from the other Practise bryngeth in here diuerse compoundyng of Numbers as some tyme two three foure or more Radicall nuÌ„bers diuersly knit by signes oâ— More Lesse as thus √ â— 12 + √ â— 15. Or â—hus √ â—â— 19 + √ â— 12 √ â— 2â— c. And some tyme with whole numbers or fractions of whole Number amoÌ„g them as 20 + √ â—â—4◠√ â— + 33 √ â— 10 √ â—â— 44 + 12 + √ â—9 And so infinitely may hap the varieâ—ie After this Both the one and the other hath fractions incident and so is this Arithmetike greately enlarged by diuerse exhibityng and vse of Compositions and mixtynges Consider howâ— I beyng desirous to deliuer the student from error and Cauillation do giue to this Practise the name of the Arithmetike of Radicall numbers Not of Irrationall or Surd Numbersâ— which other while are Rationall though they haue the Signe of a Rote before them which Arithmetike of whole Numbers most vsuall would say they had no such Roote and so account them Surd Numbers which generally spokeÌ„ is vntrue as Euclides tenth booke may teach you Therfore to call them generally Radicall Numbers by reason of the signe √ prefixed is a sure way and a sufficient generall distinction from all other ordryng and vsing of Numbers And yet beside all this Consider the infinite desire of knowledge and incredible power of mans Search and Capacitye how they ioyntly haue waded farder by mixtyng of speculation and practise and haue found out and atteyned to the very chief perfection almost of Numbers Practicall vse Which thing is well to be perceiued in that great Arithmeticall Arte of AEquation commonly called the Rule of Coss. or Algebra The Latines termed it Regulam Rei Census that is the Rule of the thyng and his value With an apt name comprehendyng the first and last pointes of the worke And the vulgar names both in Italian Frenche and Spanish depend in namyng it vpon the signification of the Latin word Res A thing vnleast they vse the name of Algebra And therin commonly is a dubble error The one of them which thinke it to be of Geber his inuentyng the other of such as call it Algebra For first though Geber for his great skill in Numbers Geometry Astronomy and other maruailous Artes mought haue semed hable to haue first deuised the sayd Rule and also the name carryeth with it a very nere likenes of Geber his name yet true it is that a Greke Philosopher and Mathematicien named Diophantus before Geber his tyme wrote 13. bookes therof of which six are yet extant and I had them to vse of the famous Mathematicien and my great frende Petrus Monâ—aureus And secondly the very name is Algiebar and not Algebra as by the Arabien Auicen may be proued who hath these precise wordes in Latine by Andreas Alpagus most perfect in the Arabik tung so translated Scientia faciendi Algiebar Almachabel i. Scientia inueniendi numerum ignotum per additionem Numeri diuisionem aequationem Which is to say The Science of workyng Algiebar and Almachabel that is the Science of findyng an vnknowen number by Addyng of a Number Diuision aequation Here haue you the name and also the principall partes of the Rule touched To name it The rule or Art of AEquation doth signiâ—ie the middle part and the State of the Rule This Rule hath his peculier Characters and the principal partes of Arithmetike to it appertayning do differe from the other Arithmeticall operations This Arithmetike hath NuÌ„bers Simple CoÌ„pound Mixt and Fractions accordingly This Rule and Arithmetike of Algiebar is so profound so generall and so in maner conteyneth the whole power of Numbers Application practicall that mans witt can deale with nothyngâ— more proffitable about numbers nor match with a thyng more mete for the diuine force of the Soule in humane Studies affaires or exercises to be tryed in Perchaunce you looked for long ere now to haue had some particular profe or euident testimony of the vse proffit and Commodity of Arithmetike vulgar in the Common lyfe and trade of men Therto then I will now frame my selfe But herein great care I haue least length of sundry profes might make you deme that either I did misdoute your zelous mynde to vertues schole or els mistrust your hable witts by some to gesse much more A profe then foure fiue or six such will I bryng as any reasonable man therwith may be persuaded to loue honor yea learne and exercise the excellent Science of Arithmetike And first who nerer at hand can be a better witnesse of the frute receiued by Arithmetike then all kynde of Marchants Though not all alike either nede it or vse it How could they forbeare the vse and helpe of the Rule called the Golden Rule Simple and Compoundeâ— both foâ—ward and backward How might they misse Arithmeticall helpe in the Rules of Felowshypâ— either without tyme or with tymeâ— and betwene the Marchant his â—actorâ— The Rulâ—â— of Baâ—tering in wares onelyâ— or part in wares and part in money would they gladly want Our Marchant venturers and Trauaylers ouer Sea how could they order their doynges iustly and without losse vnleast certaâ—ne and generall Rules for ExchauÌ„ge of money and Rechaunge were
the Lengthes of dayes and nightes the Houres and times both night and day are knowne with very many other pleasant and necessary vses Wherof some are knowne but better remaine for such to know and vse who of a sparke of true fire can make a wonderfull bonfire by applying of due matter duely Of Astrologie here I make an Arâ—e seuerall from Astronomie â— not by new deuise but by good reason and authoritie for Astrologie is an Arte Mathematicall which reasonably demonstrateth the operations and effectes of the naturall beames of light and secrete influence of the Sterres and Planets in euery element and elementall body at all times in any Horizon assigned This Arte is furnished with many other great Artes and experiences As with perfecte Perspectiue Astronomie Cosmographie Naturall Philosophie of the 4. Elementes the Arte of Graduation and some good vnderstaÌ„ding in Musike and yet moreouer with an other great Arte hereafter following though I here set this before for some considerations me mouing Sufficient you see is the stuffe to make this rare and secrete Arte of and hard enough to frame to the Conclusion Syllogisticall Yet both the manifolde and continuall trauailes of the most auncient and wise Philosophers for the atteyning of this Arte and by examples of effectes to confirme the same hath left vnto vs sufficient proufe and witnesse and we also daily may perceaue That mans body and all other Elementall bodies are altered disposed ordred pleasured and displeasured by the Influentiall working of the Sunne Mone and the other Starres and Planets And therfore sayth Aristotle in the first of his Meteorologicall bookes in the second Chapter Est autem necessariò Mundus iste supernis lationibus ferè continuus Vt inde vis eius vniuersa regatur Ea siquidem Causa prima putanda omnibus est vnde motus principium existit That is This Elementall World is of necessitie almost next adioyning to the heauenly motions That from thence all his vertue or force may be gouerned For that is to be thought the first Cause vnto all from which the beginning of motion is And againe in the tenth Chapter Opâ—rtet igitur horum principia sumamus causas omnium similiter Principium igitur vy mouens praecipuumque omnium primum Circulus ille est in quo manifeste Solis latio c. And so forth His Meteorologicall bookes are full of argumentes and effectuall demonstrations of the vertue operation and power of the heauenly bodies in and vpon the fower Elementes and other bodies of them either perfectly or vnperfectly composed And in his second booke De Generatione Corruptione in the tenth Chapter Quocirca prima latiâ— Orâ—us Interitus causa non est Sed obliqui Circuli latio ea namque continua est duobus motibus fit In Englishe thus Wherefore the vppermost motion is not the cause of Generation and Corruption but the motion of the Zodiake for that both is continuall and is caused of two mouinges And in his second booke and second Chapter of hys Physikes Homo namque generat hominem atque Sol. For Man sayth he and the Sonne are cause of mans generation Authorities may be brought very many both of 1000. 2000. yea and 3000. yeares Antiquitie of great Philosophers Expert Wise and godly men for that Conclusion which daily and hourely we men may discerne and perceaue by sense and reason All beastes do feele and simply shew by their actions and passions outward and inward All Plants Herbes Trees Flowers and Fruites And finally the Elementes and all thinges of the Elementes composed do geue Testimonie as Aristotle sayd that theyr Whole Dispositions vertues and naturall motions depend of the Actiuitie of the heauenly motions and Influences Whereby beside the specificall order and forme due to euery seede and beside the Nature propre to the Indiuiduall Matrix of the thing produced What shall be the heauenly Impression the perfect and circumspecte Astrologien hath to Conclude Not onely by Apotelesmes 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 but by Naturall and Mathematicall demonstration 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 Whereunto what Sciences are requisâ—te without exception I partly haue here warned And in my Bropâ—deâ—â—â—â— besides other matter there disclosed I haue Mathematically furnished vp the whole Method To this our age not so carefully handled by any that euer I saw or heard of I was for â—1 yeares ago by certaine earnest disputations of the Learned Gâ—â—ardus Mâ—rcâ—tâ—â— and 〈◊〉 Gogaâ—a and other therto so prouoked and by my constant and inuincible zeale to the veritie in obseruations of Heauenly Influencies to the Minâ—te of time than so diligent And chiefly by the Supernaturall influence from the Starre of Iacob so directed That any Modest and Sober Student carefully and diligently selâ—ing for the Truth will both finde coÌ„fesse therin to be the Veritie of these my wordes And also become a Reasonable Reformer of three Sortes of people about these Influentiall Operations greatly erring from the truth Wherof the one is Light Beleuers the other Light Despisers and the third Light Practisers The first most coÌ„mon Sort thinke the Heauen and Sterres to be answerable to any their doutes or desires which is not so and in dede they to much ouer reache The Second sorte thinke no Influentiall vertue froÌ„ the heauenly bodies to beare any Sway in Generation and Corruption in this Elementall world And to the Sunne Mone and Sterres being so many so pure so bright so wonderfull bigge so farre in distance so manifold in their motions so constant in their periodes c. they assigne a sleight simple office or two and so allow vnto theÌ„ according to their capacities as much vertue and power Influentiall as to the Signe of the Sunne Mone and seuen Sterres hanged vp for Signes in London for distinction of houses such grosse helpes in our wordly affaires And they vnderstand not or will not vnderstand of the other workinges and vertues of the Heauenly Sunne Mone and Sterres not so much as the Mariner or Husband manâ— no not so much as the Elephant doth as the Cynocephalus as the Porâ—entine doâ—h nor will allow these perfect and incorruptible mighty bodies so much vertuall Radiation Force as they see in a litle peece of a Magnes stone which at great distance sheweth his operation And perchaunce they thinke the Sea Riuers as the Thames to be some quicke thing and sâ— to ebbe aâ—d slow run in and out of them selues at â—heiâ— owne fantasies God helpe God helpe Surely these men come to short and either are to dull or willfully blind or perhaps to malicious The third man is the common and vulgare Astrologien or Practiser who being not duely artificially and perfectly furnished yet either for vaine glory or gayne or like a simple dolt blinde Bayardâ— both in matter and maner erreth to the discredit of the Wary and modest Astrologien and to the robbing of those most noble corporall
Marke in your lines what numbers the water Cutteth Take the waight of the same Cube againeâ— in the same kinde of water which you had before put that also into the Pyramis or Cone where you did put the first Marke now againe in what number or place of the lines the water Cutteth them Two wayes you may conclude your purpose it is to wete either by numbers or lines By numbers as if you diuide the side of your Fundamentall Cube into so many aequall partes as it is capable of conueniently with your ease and precisenes of the diuision For as the number of your first and lesse line in your hollow Pyramis or Cone is to the second or greater both being counted from the vertex so shall the number of the side of your Fundamentall Cube be to the nuÌ„ber belonging to the Radicall side of the Cube dubble to your Fundamentall Cube Which being multiplied Cubik wise will sone shew it selfe whether it be dubble or no to the Cubik number of your Fundamentall Cube By lines thus As your lesse and first line in your hollow Pyramis or Cone is to the second or greater so let the Radical side of your FundameÌ„tall Cube be to a fourth proportionall line by the 12. proposition of the sixth boke of Euclide Which fourth line shall be the Rote Cubik or Radicall side of the Cube dubble to your Fundamentall Cube which is the thing we desired For this may I with ioy say EYPHKA EYPHKA EYPHKA thanking the holy and glorious Trinity hauing greater cause therto then Archimedes had for finding the fraude vsed in the Kinges Crowne of Gold as all men may easily Iudge by the diuersitie of the frute following of the one and the other Where I spake before of a hollow Cubik Coffen the like vse is of it and without waight Thus. Fill it with water precisely full and poure that water into your Pyramis or Cone And here note the lines cutting in your Pyramis or Cone Againe fill your coffen like as you did before Put that Water also to the firstâ— Marke the second cutting of your lines Now as you proceded before so must you here procede And if the Cube which you should Double be neuer so great you haue thus the proportion in small betwene your two litle Cubes And then the side of that great Cube to be doubled being the third will haue the fourth found to it proportionall by the 12. of the sixth of Euâ—lide Note that all this while I forget not my first Proposition Staticall here rehearsed that the Supersicies of the water is Sphaericall Wherein vse your discretion to the first line adding a small heare breadth more and to the second halfe a heare breadth more to his length For you will easily perceaue that the difference can be no greater in any Pyramis or Cone of you to be handled Which you shall thus trye For â—inding the swelling of the water aboue leuell Square the Semidiameter from the Centre of the earth to your first Waters Superficies Square then halfe the Subtendent of that watry Superficies which Subtendent must haue the equall partes of his measure all one with those of the Semidiameter of the earth to your watry Superficies Subtracte this square from the first Of the residue take the Rote Square That Roâ—e Subtracte from your first Semidiameter of the earth to your watry Superficies that which remaineth is the heith of the water in the middle aboue the leuell Which you will finde to be a thing insensible And though it were greatly sensible yet by helpe of my sixt Theoreme vpon the last Proposition of Euclides twelfth booke noted you may reduce all to a true Leuell But farther diligence of you is to be vsed against accidentall causes of the waters swelling as by hauing somwhat with a moyâ—t Sponge before made moyst your hollow Pyramis or Cone will preuent an accidentall cause of Swelling c. Experience will teach you abundantly with great ease pleasure and coÌ„moditie Thus may you Double the Cube Mechanically Treble it and so forth in any proportion Now will I Abridge your paine cost and Care herein Without all preparing of your Fundamentall Cubes you may alike worke this Conclusion For that was rather a kinde of Experimentall demoÌ„stration then the shortest way and all vpon one Mathematicall Demonstration depending Take water as much as conueniently will serue your turne as I warned before of your Fundamentall Cubes bignes Way it precisely Put that water into your Pyramis or Cone Of the same kinde of water then take againe the same waight you had before put that likewise into the Pyramis or Cone For in eche time your marking of the lines how the Water doth cut them shall geue you the proportion betwen the Radicall sides of any two Cubes wherof the one is Double to the other working as before I haue taught you sauing that for you Fundamentall Cube his Radicall side here you may take a right line at pleasure Yet farther proceding with our droppe of Naturall truth you may now geue Cubes one to the other in any proportioÌ„ geueÌ„ Rationall or Irrationall on this maner Make a hollow Parallelipipedon of Copper or Tinne with one Base waÌ„ting or open as in our Cubike Coffen FroÌ„ the bottome of that Parallelipipedon raise vp many perpendiculars in euery of his fower sides Now if any proportion be assigned you in right lines Cut one of your perpendiculars or a line equall to it or lesse then it likewise by the 10. of the sixth of Euclide And those two partes set in two sundry lines of those perpendiculars or you may set them both in one line making their beginninges to be at the base and so their lengthes to extend vpward Now set your hollow Parallelipipedon vpright perpendicularly steadie Poure in water handsomly to the heith of your shorter line Poure that water into the hollow Pyramis or Cone Marke the place of the rising Settle your hollow Parallelipipedon againe Poure water into it vnto the heith of the second line exactly Poure that water duely into the hollow Pyramis or Cone Marke now againe where the water cutteth the same line which you marked before For there as the first marked line is to the second So shall the two Radicall sides be one to the other of any two Cubes which in their Soliditie shall haue the same proportion which was at the first assigned were it Rationall or Irrationall Thus in sundry waies you may furnishe your selfe with such straunge and profitable matter which long hath bene wished for And though it be Naturally done and Mechanically yet hath it a good Demonstration Mathematicall Which is thisâ— Alwaies you haue two Like Pyramids or two Like Cones in the proportions assigned and like Pyramids or Cones are in proportion one to the other in the proportion of their Homologall sides or lines tripled Wherefore if to the first and second lines found
diameter is double to that square whose diameter it is The 34. Theoreme The 48. Proposition If the square which is made of one of the sides of a triangle be equall to the squares which are made of the two other sides of the same triangle the angle comprehended vnder those two other sides is a right angle SVppose that ABC be a triangle and let the square which is made of one of the sides there namely of the side BC be equall to the squares which are made of the sides BA and AC Then I say that the angle BAC is a right angle Rayse vp by the 11. propositioÌ„ from the point A vnto the right line AC a perpendicular line AD. And by the thirde proposition vnto the line AB put an equall line AD. And by the first peticion draw a right line from the point D to the poinâ— C. And forasmuch as the line DA is equall to the line AB the square which is made of the line DA is equall to the square whiche is made of the line AB Put the square of the line AC common to them both VVherefore the squares of the lines DA and AC are equal to the squares of the lines BA and AC But by the proposition going before the square of the line DC is equal to the squares of the lines AD and AC For the angle DAC is a right angle and the square of BC is by supposition equall to the squares of AB and AC VVherefore the square of DC is equall to the square of BC wherefore the side DC is equall to the side BC. And forasmuch as AB is equall to AD â—nd AC is common to them both therefore these two sides DA and AC are equall to these two sides BA and AC the one to the other and the base DC is equall to the base BCâ— wherfore by the 8. proposition the angle DAC is equall to the angle BAC But the angle DAC is a right angle wherefore also the angle BAC is a right angle If therefore the square which is made of one of the sides of a triangle be equall to the squares which are made of the two other sides of the same triangle the angle comprehended vnder those two other sides is a right angle which was required to be proued This proposition is the conuerse of the former and is of Pelitarius demonstrated by an argument leading to an impossibilitie after this maner The ende of the first booke of Euclides Elementes ¶ The second booke of Euclides Elementes IN this second booke Euclide sheweth what is a GnomoÌ„ and a right angled parallelogramme Also in this booke are set forth the powers of lines deuided euenly and vneuenly and of lines added one to an other The power of a line is the square of the same line that is a square euery side of which is equall to the line So that here are set forth the qualities and proprieties of the squares and right lined figures which are made of lines of their parts The Arithmetician also our of this booke gathereth many compendious rules of reckoning and many rules also of Algebra with the equatioÌ„s therein vsed The groundes also of those rules are for the most part by this second booke demonstrated This booke moreouer contayneth two wonderfull propositions one of an obtuse angled triangle and the other of an acute which with the ayde of the 47. proposition of the first booke of Euclide which is of a rectangle triangle of how great force and profite they are in matters of astronomy they knowe which haue trauayled in that arte VVherefore if this booke had none other profite be side onely for these 2. propositions sake it were diligently to be embraced and studied The definitions 1. Euery rectangled parallelogramme is sayde to be contayned vnder two right lines comprehending a right angle A parallelogramme is a figure of fower sides whose two opposite or contrary sides are equall the one to the other There are of parallelogrammes fower kyndes a square a figure of one side longer a Rombus or diamond and a Romboides or diamond like figure as before was sayde in the 33. definition of the first booke Of these fower sortes the square and the figure of one side longer are onely right angled Parallelogrammes for that all their angles are right angles And either of them is contayned according to this definition vnder two right lynes whiâ—h concurre together and cause the right angle and containe the same Of which two lines the one is the length of the figure the other the breadth The parallelogramme is imagined to be made by the draught or motion of one of the lines into the length of the other As if two numbers shoulde be multiplied the one into the other As the figure ABCD is a parallelograme and is sayde to be contayned vnder the two right lines AB and AC which contayne the right angle BAC or vnder the two right lines AC and CD for they likewise contayne the right angle ACD of which 2. lines the one namely AB is the length and the other namely AC is the breadth And if we imagine the line AC to be drawen or moued directly according to the leÌ„gth of the line AB or contrary wise the line AB to be moued directly according to the length of the line AC you shall produce the whole rectangle parallelogramme ABCD which is sayde to be contayned of them euen as one number multiplied by an other produceth a plaine and righte angled superficiall number as ye see in the figure here set where the number of sixe or sixe vnities is multiplied by the number of fiue or by fiue vnities of which multiplication are produced 30. which number being set downe and described by his vnities representeth a playne and a right angled number VVherefore euen as equall numbers multipled by equal numbers produce numbers equall the one to the other so rectangle parallelogrames which are comprehended vnder equal lines are equal the one to the other 2. In euery parallelogramme one of those parallelogrammes which soeuer it be which are about the diameter together with the two supplementes is called a Gnomon Those perticuler parallelogrames are sayde to be about the diameter of the parallelograme which haue the same diameter which the whole parallelograme hath And supplementes are such which are without the diameter of the whole parallelograme As of the parallelograme ABCD the partial or perticuler parallelogrames GKCF and EBKH are parallelogrames about the diameter for that ech of them hath for his diameter a part of the diameter of the whole parallelogramme As CK and KB the perticuler diameters are partes of the line CB which is the diameter of the whole parallelogramme And the two parallelogrammes AEGK and KHFD are supplementes because they are wythout the diameter of the whole parallelogramme namely CB. Now any one of those partiall parallelogrammes
about the diameter together with the two supplementes make a gnomon As the parallelograme EBKH with the two supplementes AEGK and KHFD make the gnomon FGEH Likewise the parallelogramme GKCF with the same two supplementes make the gnomon EHFG And this diffinition of a gnomon extendeth it selfe and is generall to all kyndes of parallelogrammes whether they be squares or figures of one side longer or Rhombus or Romboides To be shorte if you take away from the whole parallelogramme one of the partiall parallelogrammes which are about the diameter whether ye will the rest of the figure is a gnomon Campaâ—e after the last proposition of the first booke addeth this propositioÌ„ Two squares being geuen to adioyne to one of them a Gnomon equall to the other square which for that as then it was not taught what a Gnomon is I there omitted thinking that it might more aptly be placed here The doing and demonstration whereof is thus Suppose that there be two squares AB and CD vnto one of which namely vnto AB it is required to adde a Gnomon equall to the other square namely to CD Produce the side BF of the square AB directly to the point E. and put the line FE equall to the side of the square CD And draw a line from E to A. Now then forasmuch as EFA is a rectangle triangle therefore by the 47. of the first the square of the line EA is equall to the squares of the lines EF FA. But the square of the line EF is equall to the square CD the square of the side FA is the square AB Wherefore the square of the line AE is equall to the two squares CD and AB But the sides EF and FA are by the 21. of the first longer then the side AE and the side FA is equall to the side FB Wherfore the sides EF and FB are longer theÌ„ the side AE Wherefore the whole line BE is longer then the line AE From the line BE cut of a line equall to the line AE which let be BC. And by the 46. proposition vpon the line BC describe a square which let be BCGH which shal be equal to the square of the line AE but the square of the line AE is equal to the two squares AB and DC Wherefore the square BCGH is equal to the same squares Wherfore forasmuch as the square BCGH is composed of the square AB and of the gnomon FGAH the sayde gnomon shal be equall vnto the square CD which was required to be done An other more redy way after Pelitarius Suppose that there be two squares whose sides let be AB and BC. It is required vnto the square of the line AB to adde a gnomon equall to the square of the line BC. Set the lines AB and BC in such sort that they make a right angle ABC And draw a line froÌ„ A to C. And vpoÌ„ the line AB describe a square which let be ABDE And produce the line BA to the point F and put the line BF equall to the line AC And vpon the line BF describe a square which let be BFGH which shal be equal to the square of the line AC wheÌ„ as the lines BF and AC are equal and therefore it is equal to the squares of the two lines AB and BC. Now forasmuch as the square BFGH is made complete by the square ABDE and by the gnomon FEGD the gnomon FEGD shal be equal to the square of the line BC which was required to be done The 1. Theoreme The 1. Proposition If there be two right lines and if the one of them be deuided into partes howe many soeuer the rectangle figure comprehended vnder the two right lines is equall to the rectangle figures whiche are comprehended vnder the line vndeuided and vnder euery one of the partes of the other line SVppose that there be two right lynes A and BC and let one of them namely BC be deuided at all aduentures in the pointes D and E. Then I say that the rectangle figure comprehended vnder the lines A and BC is equall vnto the rectangle figure comprehended vnder the lines A and BD vnto the rectangle figure which is coÌ„prehended vnder the lines A and DE and also vnto the rectangle figure which is comprehended vnder the lines A and EC For from the pointe Brayse vp by the 11. of the first vnto the right line BC a perpendiculer line BF vnto the line A by the third of the first put the line BG equall and by the point G by the 31. of the first draw a parallel line vnto the right line BC and let the same be GM and by the selfe same by the points D E and C draw vnto the line BG these parallel lines DK EL and CH. Now then the parallelograme BH is equall to these parallelogrammes BK DL and EH But the parallelograme BH is equall vnto that which is contayned vnder the lines A and BC. For it is compreheÌ„ded vnder the lines GB BC and the line GB is equall vnto the line A And the parallelograme BK is equall to that which is contayned vnder the lines A and BD for it is comprehended vnder the line GB and BD and BG is equall vnto A And the parallelograme DL is equall to that which is contayned vnder the lines A and DE for the line DK that is BG is equal vnto A And moreouer likewise the parallelograme EH is equall to that which is contained vnder the lines A EC VVherfore that which is compreheÌ„ded vnder the lines A BC is equall to that which is comprehended vnder the lines A BD vnto that which is compreheÌ„ded vnder the lines A and DE and moreouer vnto that which is comprehended vnder the lines A and EC If therfore there be two right lines and if the one of them be deuided into partes how many soeuer the rectangle figure comprehended vnder the two right lines is equall to the rectangle figures which are comprehended vnder the line vndeuided and vnder euery one of the partes of the other line which was required to be demonstrated Because that all the Propositions of this second booke for the most part are true both in lines and in numbers and may be declared by both therefore haue I haue added to euery Proposition conuenient numbers for the manifestation of the same And to the end the studious and diligent reader may the more fully perceaue and vnderstand the agrement of this art of Geometry with the science of Arithmetique and how nere deare sisters they are together so that the one cannot without great blemish be without the other I haue here also ioyned a little booke of Arithmetique written by one Barlaam a Greeke authour a man of greate knowledge In whiche booke are by the authour demonstrated many of the selfe same proprieties and passions in number which Euclide in this his second boke hath demonstrated in magnitude
definition of the first the line MB is equall vnto the line MK put the line MD common to the both wherfore these two lines MK and MD are equall to these two lines BM and MD the one to the other and the angle KMD is by the 23. of the first equall to the angle BMD VVhereâ—ore by the 4. of the first the base DK is equall to the base DB. Or it may thus be demonstrated Draw by the first petition a line from M to N. And for asmuch as by the 15. definition of the first the line KM is equall vnto the line MN and the line MD is common to them both And the base KD is equall to the base DN by supposition therefore by the 8. of the first the angle KMD is equall to the angle DMN But the angle KMD is equall to the angle BMD Wherfore the angle BMD is equall to the angle NMD the lesse vnto the greater which is impossible Wherefore from the poynt D can not be drawen vnto the circumference ABC on eche side of DG the lest more then two equall right lines If therefore without a circle be taken any poynt and from that poynt be drawen into the circle vnto the circumference certaine right lines of which let one be drawen by the centre and let the rest be drawen at all adventures the greatest of those right lines which fall in the concauitie or hollownes of the circumference of the circle is that which passeth by the centre And of all the other lines that line which is nigher to the line which passeth by the centre is greater then that which is more distant But of those right lines which end in the conuexe part of the circumference that line is the lest which is drawen from the poynt to the dimetient and of the other lines that which is nigher to the least is alwayes lesse then that which is more distant And from that poynt can be drawen vnto the circumference on ech side of the lest only two equall right lines which was required to be proued Thys Proposition is called commonly in old bookes amongest the barbarous Caâ—dâ— Panonis that is the Peacockes taile ¶ A Corollary Hereby it is manifest that the right lines which being drawen from the poynt geuen without the circle and fall within the circle are equally distant from the least or from the greatest which is drawen by the centre are equall the one to the other but contrarywyse if they be vnequally distant whether they light vpon the concaue or conuexe circumference of the circle they are vnequall The 8. Theoreme The 9. Proposition If within a circle be taken a poynt and from that poynt be drawen vnto the circumference moe then two equall right lines the poynt taken is the centre of the circle SVppose that the circle be ABC and within it let there be taken the poynt D. And from D let there be drawen vnto the circumference ABC moe then two equall right lines that is DA DB and DC Then I say that the poynt D is the centre of the circle ABC Draw by the first petition these right lines AB and BC and by the 10. of the first deuide theÌ„ into two equall partes in the poyntes E and F namely the line AB in the poynt E and the line BC in the poynt F. And draw the lines ED and FD and by the second petition extend the lines ED and FD on eche side to the poyntes K G and H L. And for asmuch as the line AE is equall vnto the line EB and the line ED is common to them both therefore these two sides AE and ED are equall vnto these two sides BE and ED and by supposition the base DA is equall to the base DB. Wherfore by the 8. of the first the angle AED is equall to the angle BED Wherfore eyther of these angles AED and BED is a right angle Wherefore the line GK deuideth the line AB into two equall partes and maketh right angles And for asmuch as if in a circle a right line deuide an other right line into two equall partes in such sort that it maketh also right angles in the line that deuideth is the centre of the circle by the Correllary of the first of the third Therfore by the same Correllary in the line GK is the centre of the circle ABC And by the same reason may we proue that in the line HL is the centre of the circle ABC and the right lines GK and HL haue no other poynt common to them both besides the poynt D. Wherefore the poynt D is the centre of the circle ABC If therefore within a circle be taken a poynt and from that point be drawen vnto the circumference more then two equall right lines the poynt taken is the centre of the circle which was required to be proued ¶ An other demonstration Let there be taken within the circle ABC the poynt D. And from the poynt D let there be drawen vnto the circumference more then two equall right lines namely DA DB and DC Then I say that the poynt D is the centre of the circle For if not then if it be possible let the point E be the centre and draw a line from D to E and extend DE to the poyntes F and G. Wherefore the line FG is the diameter of the circle ABC And for asmuch as in FG the diameter of the circle ABC is taken a poynt namely D which is not the centre of that circle therefore by the 7. of the third the line DG is the greatest and the line DC is greater then the line DB and the line DB is greateâ— then the line DA. But the lines DC DB DA are also equall by supposition which is impossible Wherefore the poynt E is not the centre of the circle ABC And in like sort may we proue that no other poynt besides D. Wherefore the poynt D is the centre of the circle ABC which was required to be proued The 9. Theoreme The 10. Proposition A circle cutteth not a circle in moe pointes then two FOr if it be possible let the circle ABC cut the circle DEF in mo pointes then two that is in B G H F. And drawe lines froÌ„ B to G and from B to H. And by the 10. of the first deuide either of the lines BG BH into two equall partes in the pointes K and L. And by the 11. of the first from the poynt K raise vp vnto the line BH a perpendicular line KC and likewise from the poynt L raise vp vnto the line BG a perpendicular line LM and extend the line CK to the poynt A and LNM to the poyntes X and E. And for asmuch as in the circle ABC the right line AC deuideth the right line BH into two equall partes and maketh right angles therfore by the 3. of the third in the line AC is the centre of
the circle ABC Agayne for asmuch as in the selfe same circle ABC the right line NX that is the line ME deuideth the right line BG into two equall partes and maketh right angles therefore by the third of the third in the line NX is the centre of the circle ABC And it is proued that it is also in the line AC And these two right lines AC and NX meete together in no other poynt besides O. Wherefore the poynt O is the centre of the circle ABC And in like sort may we proue that the poynt O is the centre of the circle DEF Wherefore the two circles ABC and DEF deuiding the one the other haue one and the same centre which by the 5. of the third is impossible A circle therfore cutteth not a circle in moe poyntes then two which was required to be proued An other demonstration to proue the same Suppose that the circle ABC do cut the circle DGF in mo poyntes then two that is in B G F and H. And by the first of the third take the centre of the circle ABC and let the same be the poynt K. And draw these right lines KB KG and KF Now for asmuch as within the circle DEF is taken a certaine poynt K and from that poynt are drawen vnto the circumference moe then two equall right lines namely KB KG and KF therefore by the 9. of the third K is the centre of the circle DEF And the poynt K is the centre of the circle ABC Wherefore two circles cutting the one the other haue one and the same centre which by the 5. of the third is impossible A circle therfore cutteth not a circle in moe pointes then two which was required to be demonstrated The 10. Theoreme The 11. Proposition If two circles touch the one the other inwardly their centres being geuen a right line ioyning together their centres and produced will fall vpon the touch of the circles SVppose that these two circles ABC and ADE do touch the one the other in the poynt A. And by the first of the third take the centre of the circle ABC and let the same be F and likewise the centre of the circle ADE and let the same be G. Then I say that a right line drawen from F to G and being produced will fall vpon the poynt A. For if not then if it be possible let it fall as the line FGDH doth And draw these right lines AF AG. Now for asmuch as the lines AG and GF are by the 20. of the first greater then the line FA that is then the line FH take away the line GF which is common to them both Wherefore the residue AG is greater then the residue GH But the line DG is equall vnto the line GA by the 15. definition of the first Wherefore the line GD is greater then the line GH the lesse then the greater which is impossible Wherfore a right line drawen from the poynt F to the poynt G and produced falleth not besides the poynt A which is the point of the touch Wherefore it falletâ— vpon the touch If therefore two circles touch the one the other inwardly their centres being geuen a right line ioyning together their centres and produced will fall vpon the touch of the circles which was required to be proued An other demonstration to proue the same But now let it fall as GFC falleth and extend the line GFC to the poynt H and drawe these right lines AG and AF. And for asmuch as the lines AG and GF are by the 20. of the first greater then the line AF. But the line AF is equall vnto the line CF that is vnto the line FH Take away the line FG common to them both Wherfore the residue AG is greater then the residue GH that is the line GD is greater then the line GH the lesse greater then the greater which is impossible Which thing may also be proued by the 7. Proposition of this booke For for asmuch as the line HC is the diameter of the circle ABC in it is taken a poynt which is not the centre namely the poynt G therefore the line GA is greater then the line GH by the sayd 7. Proposition But the line GD is equall to the line GA by the definition of a circle Wherefore the line GD is greater then the line GH namely the part grâ—ater then the whole which is impossible The 11. Theoreme The 12. Proposition If two circles touch the one the other outwardly a right line drawen by their centres shall passe by the touch SVppose that these two circles ABC and ADE do touch the one the other outwardly in the poynt A. And by the third of the third take the centre of the circle ABC and let the same be the poynt F and likewise the centre of the circle ADE and let the same be the poynt G. Then I say that a right line drawen from the poynt F to the poynt G shall passe by the poynt of the touch namely by the poynt A. For if not then if it be possible let it passe as the right line FCDG doth And draw these right lines AF AG. And for asmuch as the poynt F is the centre of the circle ABC therfore the line FA is equall vnto the line FC Againe for asmuch as the poynt G is the centre of the circle ADE therefore the line GA is equall to the line GD And And it is proued that the line FA is equall to the line FC Wherefore the lines FA and AG are equall vnto the lines FC and GD Wherefore the whole line FG is greater then the lines FA and AG. But it is also lesse by the 20. of the first which is impossible Wherfore a right line drawen from the poynt F to the poynt G shall passe by the poynt of the touch namely by the poynt A. If therefore two circles touch the one the other outwardly a right line drawen by their centres shall passe by the touch which was required to be demonstrated ¶ An other demonstration after Pelitarius Suppose that the two circles ABC and DEF do touch the one the other outwardly in the poynt A And let G be the centre of the circle ABC From which poynt produce by the touch of the circles the line GA to the poynt F of the circumference DEF Which for asmuch as it passeth not by the centre of the circle DEF as the aduersary affirmeth draw from the same centre G an other right line GK which if it be possible let passe by the centre of the circle DEF namely by the poynt H cutting the circumference ABC in the poynt B the circuÌ„ference DEF in the poynt D let the opposite poynt therof be in the point K. And for asmuch as froÌ„ the poynt G taken without the circle DEF is drawen the line GK passing by the centre H and
when perpendicular lines drawen frō the centre to those lines are equall by the 4. definition of the third Wherfore the lines AB and CD are equally distant from the centre But now suppose that the right lines AB and CD be equally distant from the centre that is let the perpendicular line EF be equall to the perpendicular line EG Then I say that the line AB is equall to the line CD For the same order of construction remayning we may in like sort proue that the line AB is double to the line AF and that the line CD is double to the line CG And for asmuch as the line AE is equall to the line CE for they are drawen from the centre to the circumference therfore the square of the line AE is equall to the square of the line CE. But by the 47. of the first to the square of the line AE are equall the squares of the lines EF and FA. And by the selfe same to the square of the line CE are equall the squares of the lines EG and GC Wherfore the squares of the lines EF and FA are equall to the squares of the lines EG and GC Of which the square of the line EG is equall to the square of the line EF for the line EF is equall to the line EG Wherefore by the third common sentence the square remayning namely the square of the line AF is equall to the square of the line CG Wherefore the line AC is equall vnto the line CG But the line AB is double to the line AF and the line CD is double to the line CG Wherefore the line AB is equall to the line CD Wherefore in a circle equall right lines are equally distant from the centre And lines equally distant from the centre are equall the one to the other which was required to be proued ¶ An other demonstration for the first part after Campane Suppose that there be a circle ABDC whose centre let be the poynt E. And draw in it two equall lines AB and CD Then I say that they are equally distant from the centre Draw from the centre vnto the lines AB and CD these perpendicular lines EF and EG And by the 2. part of the 3. of this booke the line AB shall be equally deuided in the poynt F. and the line CD shall be equally deuided in the poynt G. And draw these right lines EA EB EC and ED. And for asmuch as in the triangle AEB the two sides AB and AE are equall to the two sides CD and CE of the triangle CED the base EB is equall to the base ED. therefore by the 8. of the first the angle at the point A shall be equall to the angle at the point C. And for asmuch as in the triangle AEF the two sides AE and AF are equall to the two sides CE and CG of the triangle CEG and the angle EAF is equall to the angle CEG therefore by the 4. of the first the base EF i◠equall to the base EG which for asmuch as they are perpendicular lines therefore the lines AB CD are equally distant frō the centre by the 4. definition of this booke The 14. Theoreme The 15. Proposition In a circle the greatest line is the diameter and of all other lines that line which is nigher to the centre is alwayes greater then that line which is more distant SVppose that there be a circle ABCD and let the diameter thereof be the line AD and let the centre thereof be the poynt E. And vnto the diameter AD let the line BC be nigher then the line FG. Then I say that the line AD is the greatest and the line BC is greater then the line FG. Draw by the 12. of the first from the centre E to the lines BC and FG perpendicular lines EH and EK And for asmuch as the line BC is nigher vnto the centre then the line FG therfore by the 4. definition of the third the line EK is greater then the line EH And by the third of the first put vnto the line EH an equall line EL. And by the 11. of the first from the point L raise vp vnto the line EK a perpendicular line LM and extend the line LM to the poynt N. And by the first petition draw these right lines EM EN EF and EG And for asmuch as the line EH is equall to the line EL therefore by the 14. of the third and by the 4. definition of the same the line BC is equall to the line MN Againe for asmuch as the line AE is equall to the line EM and the line ED to the line EN therefore the line AD is equall to the lines ME and EN But the lines ME and EN are by the 20. of the first greater then the line MN Wherefore the line AD is greater then the line MN And for asmuch as these two lines ME and EN are equall to these two lines FE and EG by the 15. definition of the first for they are drawen from the centre to the circumference and the angle MEN is greater then the angle FEG therefore by the 24. of the first the base MN is greater then the base FG. But it is proued that the line MN is equall to the line BC Wherefore the line BC also is greater then the line FG. Wherefore the diameter AD is the greatest and the line BC is greater then the line FG. Wherefore in a circle the greatest line is the diameter and of all the other lines that line which is nigher to the centre is alwaies greater then that line which is more distant which was required to be proued ¶ An other demonstration after Campane In the circle ABCD whose centre let be the poynt E draw these lines AB AC AD FG and HK of which let the line AD be the diameter of the circle Then I say that the line AD is the greatest of all the lines And the other lines eche of the one is so much greater then ech of the other how much nigher it is vnto the centre Ioyne together the endes of all these lines with the centre by drawing these right lines EB EC EG EK EH and EF. And by the 20. of the first the two sides EF and EG of the triangle EFG shall be greater then the third side FG. And for asmuch as the sayd sides EF EG are equall to the line AD by the definition of a circle therefore the line AD is greater then the line FG. And by the same reason it is greater then euery one of the rest of the lines if they be put to be bases of triangles for that euery two sides drawen frō the centre are equall to the line AD. Which is the first part of the Proposition Agayne for asmuch as the two sides EF and EG of the triangle EFG are equall to the
say that the lyne GFH which by the correllary of the 16. of this booke toucheth the circle is a parallel vnto the line AB For forasmuch as the right line CF fallyng vpon either of these lines AB GH maketh all the angles at the point â— right angles by the 3. of this boke and the two angles at the point Fare supposed to be right angles therfore by the 29. of the first the lines AB and GH are parallels which was required to be done And this Probleme is very commodious for the inscribing or circumscribing of figures in or about circles The 16. Theoreme The 18. Proposition If a right lyne touch a circle and from the centre to the touch be drawen a right line that right line so drawen shal be a perpendicular lyne to the touche lyne SVppose that the right line DE do touch the circle ABC in the point C. And take the centre of the circle ABC and let the same be F. And by the first petition from the poynt F to the poynt C drawe a right line FC Then I say that CF is a perpendicular line to DE. For if not draw by the 12. of the first from the poynt F to the line DE a perpendicular line FG. And for asmuch as the angle FGC is a right angle therefore the angle GCF is an acute angle Wherefore the angle FGC is greater then the angle FCG but vnto the greater angle is subtended the greater side by the 19. of the first Wherefore the line FC is greater then the line FG. But the line FC is equall to the line FB for they are drawen from the centre to the circumference Wherfore the line FB also is greater then the line FG namely the lesse then the greater which is impossible Wherefore the line FG is not a perpendicular line vnto the line DE. And in like sort may we proue that no other line is a perpendicular line vnto the line DE besides the line FC Wherfore the line FC is a perpendicular line to DE. If therefore a right line touch a circle from y centre to the touch be drawen a right line that right line so drawen shall be a perpendicular line to the touch line which was required to be proued ¶ An other demonstration after Orontius Suppose that the circle geuen be ABC which let the right lyne DE touch in the point C. And let the centre of the circle be the point F. And draw a right line from F to C. Then I say that the line FC is perpendicular vnto the line DE. For if the line FC be not a perpeÌ„diculer vnto the line DE then by the conuerse of the x. definition of the first boke the angles DCF FCE shal be vnequall therfore the one is greater then a right angle and the other is lesse then a right angle For the angles DCF and FCE are by the 13. of the first equall to two right angles Let the angle FCE if it be possible be greater then a right angle that is let it be an obtuse angle Wherfore the angle DCF â—hal be an acute angle And forasmuch as by suppositioÌ„ the right line DE toucheâ—h the circle ABC therefore it cutteth not the circle Wherefore the circumference BC falleth betwene the right lines DC CF therfore the acute and rectiline angle DCF shall be greater then the angle of the semicircle BCF which is contayned vnder the circumfereÌ„ce BC the right line CF. And so shall there be geueÌ„ a rectiline acute angle greater then the angle of a semicircle which is contrary to the 16. proposition of this booke Wherfore the angle DCF is not lesse then a right angle In like sort also may we proue that it is not greater then a right angle Wherfore it is a right angle and therfore also the angle FCE is a right angle Wherefore the right line FC is a perpendicular vnto the right line DE by the 10. definition of the firstâ— which was required to be proued The 17. Theoreme The 19. Proposition If a right lyne doo touche a circle and from the point of the touch be raysed vp vnto the touch lyne a perpendicular lyne in that lyne so raysed vp is the centre of the circle SVppose that the right line DE do touch the circle ABC in the point C. And from C raise vp by the 11. of the first vnto the line DE a perpendicular line CA. Then I say that in the line CA is the centre of the circle For if not then if it be possible let the centre be without the line CA as in the poynt F. And by the first petition draw a right line from C to F. And for asmuch as a certaine right line DE toucheth the circle ABC and from the centre to the touch is drawen a right line CF therefore by the 18. of the third FC is a perpendicular line to DE. Wherefore the angle FCE is a right angle But the angle ACE is also a right angle Wherefore the angle FCE is equall to the angle ACE namely the lesse vnto the greater which is impossibleâ— Wherefore the poynt F is not the centre of the circle ABC And in like sort may we proue that it is no other where but in the line AC If therefore a right line do touch a circle and from the point of the touch be raised vp vnto the touch line a perpendicular line in that line so raised vp is the centre of the circle which was required to be proued The 18. Theoreme The 20. Proposition In a circle an angle set at the centre is double to an angle set at the circumference so that both the angles haue to their base one and the same circumference SVppose that there be a circle ABC and at the centre thereof namely the poynt E let the angle BEC be set at the circumference let there be set the angle BAC and let them both haue one and the same base namely the circumference BC. Then I say that the angle BEC is double to the angle BAC Draw the right line AE and by the second petition extend it to the poynt F. Now for asmuch as the line AE is equall to the line EB for they are drawen from the centre vnto the circumference the angle EAB is equall to the angle EBA by the 5. of the first Wherefore the angles EAB and EBA are double to the angle EAB But by the 32. of the same the angle BEF is equall to the angles EAB and EBA Wherefore the angle BEF is double to the angle EAB And by the same reason the angle FEC is double to the angle EAC Wherefore the whole angle BEC is double to the whole angle BAC Againe suppose that there be set an other angle at the circumference and let the same be BDC And by the â—irst petition draw a line from D to E. And by the second petition extend
and it is in the segment ABC which is greater then the semicircle And forasmuch as in the circle there is a figure of foure sides namely ABCD. But if within a circle be described a figure of foure sides the angles therof which are opposite the one to the other are equall to two right angles by the 22. of the third Wherfore by the same the angles ABC and ADC are equall to two right angles But the angle ABC is lesse then a right angle Wherfore the angle remayning ADC is greater then a right angle and it is in a segment which is lesse then the semicircle Againe forasmuch as the angle comprehended vnder the right lines AC and AF is a right angle therfore the angle comprehended vnder the right line CA and the circumference ADC is lesse then a right angle Wherfore in a circle an angle made in the semicircle is a right angle but an angle made in the segment greater then the semicircle is lesse then a right angle and an angle made in the segment lesse then the semicircle is greater then a right angle And moreouer the angle of the greater segment is greater then a right angle the angle of the lesse segment is lesse then a right angle which was required to be demonstrated An other demonstration to proue that the angle BAC is a right angle Forasmuch as the angle AEC is double to the angle BAE by the 32. of the first for it is equall to the two inward angles which are opposite But the inwarde angles are by the 5. of the first equall the one to the other and the angle AEB is double to the angle EAC Wherfore the angles AEB and AEC are double to the angle BAC But the angles AEB and AEC are equall to two right angles Whâ—rfore the angle BAC is a right angle Which was required to be demonstrated Correlary Hereby it is manifest that if in a triangle one angle be equall to the two other angles remayning the same angle is a right angle for that the side angle to that one angle namely the angle which is made of the side produced without the triangle is equall to the same angles but when the side angles are equall the one to the other they are also right angles ¶ An addition of Pelitarius If in a circle be inscribed a rectangle triangle the side opposite vnto the right angle shall be the diameter of the circle Suppose that in the circle ABC be inscribed a rectangle triangle ABC whose angle at the point B let be a right angle Then I say that the side AC is the diameter of the circle For if not then shall the centre be without the line AC as in the point E. And draw a line from the poynt A to the point E produce it to the circumference to the point D and let AED be the diameter and draw a line from the point B to the point D. Now by this 31. PropositioÌ„ the angle ABD shall be a right angle and therefore shall be equall to the right angle ABC namely the part to the whole which is absurde Euen so may we proue that the centre is in no other where but in the line AC Wherfore AC is the diameter of the circle which was required to be proued ¶ An addition of Campane By thys 31. Proposition and by the 16. Proposition of thys booke it is manifest that although in mixt angles which are contayned vnder a right line and the circumference of a circle there may be geuen an angle lesse greater then a right angle yet can there neuer be geueÌ„ an angle equall to a right angle For euery section of a circle is eyther a semicircle or greater then a semicircle or lesse but the angle of a semicircle is by the 16. of thys booke lesse then a right angle and so also is the angle of a lesse section by thys 31. Proposition Likewise the angle of a greater section is greater then a right angle as it hath in thys Proposition bene proued The 28. Theoreme The 32. Proposition If a right line touch a circle and from the touch be drawen a right line cutting the circle the angles which that line and the touch line make are equall to the angles which consist in the alternate segmentes of the circle SVppose that the right line EF do touch the circle ABCD in the point B and from the point B let there be drawen into the circle ABCD a right line cutting the circle and let the same be BD. Then I say that the angles which the line BD together with the touch line EF do make are equall to the angles which are in the alternate segmentes of the circle that is the angle FBD is equall to the angle which consisteth in the segment BAD and the angle EBD is equall to the angle which consisteth in the segment BCD Raise vp by the 11. of the first from the point B vnto the right line EF a perpendicular line BA And in the circumference BD take a point at all aduentures and let the same be C. And draw these right lines AD DC and CB. And for asmuch as a certaine right line EF toucheth the circle ABC in the point B and from the point B where the touch is is raysed vp vnto the touch line a perpendicular BA Therfore by the 19. of the third in the line BA is the centre of the circle ABCD. Wherfore the angle ADB being in the semicircle is by the 31. of the third a right angle Wherefore the angles remayning BAD and ABD are equall to one right angle But the angle ABF is a right angle Wherefore the angle ABF is equall to the angles BAD and ABD Take away the angle ABD which is common to them both Wherefore the angle remayning DBF is equall to the angle remayning BAD which is in the alternate segment of the circle And for asmuch as in the circle is a figure of fower sides namely ABCD therfore by the 22. of the third the angles which are opposite the one to the other are equall to two right angles Wherfore the angles BAD and BCD are equall to two right angles But the angles DBF and DBE are also equall to two right angles Wherefore the angles DBF and DBE are equall to the angles BAD and BCD Of which we haue proued that the angle BAD is equall to the angle DBF Wherefore the angle remayning DBE is equall to the angle remayning DCB which is in the alternate segment of the circle namely in the segment DCB If therfore a right line touch a circle and from the touch be drawen a right line cutting the circle the angles which that line and the touch line make are equall to the angles which consist in the alternate segmentes of the circle which was required to be proued In thys Proposition may be two cases For the line drawen from
constituens or componens Because that it with some other part or partes maketh the whole As the lyne CB together with the line AC maketh the whole lyne AB Of the barbarous it is called pars aliquanta In this signification it is taken in Bâ—rlaâ—â— in the beginnyng of his booke in the definition of a part when he saith Euery lesse number compared to a greater is sayd to be a part of the greater whether the lesse measure the greater or measure it not Multiplex is a greater magnitude in respect of the lesse when the lesse measureth the greater As the line CD before set in the first example is multiplex to the lyne AB For that CD a lyne contayning 9. is the greater magnitude and is compared to the lesse namely to the lyne AB contayning 3. and also the lesse lyne AB measureth the greater line CD for taken 3. tymes it maketh it as was aboue sayde So in numbers 12. is multiplex to 3 for 12 is the greater number and is compared to the lesse namely to 3. which 3. also measureth it for 3 taken 4 tymes maketh 12. By this worde multiplex which is a terme proper to Arithmetike and number it is easy to consider that there can be no exact knowledge of proportion and proportionalitie and so of this fifth booke wyth all the other bookes followyng without the ayde and knowledge of numbers Proportion is a certaine respecte of two magnitudes of one kinde according to quantitie Proportion rationall is deuided into two kindes into proportion of equalitie and into proportion of inequalitie Proportion of equalitie is when one quantitie is referred to an other equall vnto it selfe as if ye compare 5 to 5 or 7 to 7 so of other And this proportion hath great vse in the rule of Cosse For in it all the rules of equations tende to none other ende but to finde out and bring forth a nuÌ„ber equall to the number supposed which is to put the proportion of equalitie Proportion of inequalitie is when one vnequall quantity is compared to an other as the greater to the lesse as 8. to 4 or 9. to 3 or the lesse to the greater as 4. to 8 or 3. to 9. Proportion of the greater to the lesse hath fiue kindes namely Multiplex Superparticular Superpartiens Multiplex superperticular and Multiplex superpartiens Multiplex is when the antecedent containeth in it selfe the consequent certayne times without more or lesse as twice thrice foure tymes and so farther And this proportion hath vnder it infinite kindes For if the antecedent contayne the consequent iustly twise it is called dupla proportion as 4 to 2. If thrice tripla as 9. to 3. If 4. tymes quadrupla as 12. to 3. If 5. tymes quintupla as 15. to 3. And so infinitely after the same maner Superperticular is wheÌ„ the antecedeÌ„t containeth the consequent only once moreouer some one part therof as an halfe a third or fourth c. This kinde also hath vnder it infinite kindes For if the antecedent containe the consequent once and an halfe therof it is called Sesquialtera as 6. to 4 if once and a third part Sesquitertia as 4. to 3 if once and a fourth part Sesquiquarta as 5. to 4. And so in like maner infinitely Superpartiens is wheÌ„ the antecedent coÌ„taineth the consequent onely once moreouer more partes then one of the same as two thirdes three fourthes foure fifthes and so forth This also hath infinite kindes vnder it For if the antecedent containe aboue the consequent two partes it is called Superbipartiens as 7. to 5. If 3. partes Supertripartiens as 7. to 4. If 4. partes Superquadripartiens as 9. to 5. If 5. partes Superquintipartiens as 11. to 6. And so forth infinitely Multiplex Superperticular is when the antecedent containeth the consequent more then once and moreouer onely one parte of the same This kinde likewise hath infinite kindes vnder it For if the antecedent containe the consequent twise and halfe therof it is called dupla Sesquialtera as 5. to 2. If twise and a third Dupla Sesquitertia as 7. to 3. If thrice and an halfe Tripla sesquialtera as 7. to 2. If foure times and an halfe Quadrâ—pla Sesquialtera as 9. to 2. And so goyng on infinitely Multiplex Superpartient is when the antecedent contayneth the consequent more then once and also more partes then one of the consequent And this kinde also hath infinite kindes vnder it For if the antecedent containe the consequent twise and two partes ouer it is called dupla Superbipartiens as 8. to 3. If twice and three partes dupla Supertripartiens as 11. to 4. If thrice and two partes it is named Tripla Superbipartiens as 11. to 3. If three tymes and foure partes Treble Superquadripartiens as 31. to 9. And so forth infinitely Here is to be noted that the denomination of the proportion bâ—twene any two numbers is had by deuiding of the greater by the lesse For the quotient oâ— number produced of that diuision is euen the denomination of the proportion Which in the first kinde of proportion namely multiplex is euer a whole number and in all other kindes of proportion it is a broken number As if ye will know the denomination of the proportion betwene 9 and 3. Deuide 9. by 3. so shall ye haue in the quotient 3. which is a whole number and is the denomination of the proportion and sheweth that the proportion betwene 9. 3. is Tripla So the proportion betwene 12. and 3. is quadrupla for that 12. beyng deuided by 3. the quotient is 4. and so of others in the kinde of multiplex And although in this kinde the quotient be euer a whole number yet properly it is referred to vnitie and so is represented in maner of a broken number as 5 â— and 4 â— for vnitie is the denomination to a whole number Likewise the denomination of the proportion betwene 4 and 3 is 1. 1 â— for that 4 deuided by 3. hath in the quotient 1 1 â— one and a third part of which third part it is called sesquitercia so the proportion betwene 7 and 6. is 1 â…™ one and a sixt of which fixt part it is called sesquisexta and so of other of that kinde Also betwene 7 and 5 the denomination of the proportion is 1 â— â— one and two fifthes which denomination coÌ„sisteth of two parts namely of the munerator and denominator of the quotient of 2. and 5 of which two fifthes it is called superbipartiens quintas for 2 the numerator sheweth the denomination of the number of the partes and 5. the denominator sheweth the denominatioÌ„ what parts they are so of others Also the denomination betwene
of the one also to the residue of the other shal be equemultiplex as the whole is to the whole which was required to be proued The 6. Theoreme The 6. Proposition If two magnitudes be â—quemultiplices to two magnitudes any parâ—es taken away of them also be aequemultiplices to the same magnitudes the residues also of them shal vnto the same magnitudes be either equall or equemultiplices SVppose that there be two magnitudes AB and CD equemultiplices to two magnitudes E and F and let the partes takeÌ„ away of the magnitudes AB and CD namely AG and CH be equemultiplices to the same magnitudes E and F. Then I say that the residues GB and HD are vnto the selfe same magnitudes E and F either equall or els equemultiplices And in like sort may we proue that if GB be multiplex to E HD also shal be so multiplex vnto F. If therfore there be two magnitudes equemultiplices to two magnitudes and any parts taken away of them be also equemultiplices to the same magnitudes the residues also of them shall vnto the same magnitudes be either equall or equemultiplices which was required to be proued The 7. Theoreme The 7. Proposition Equall magnitudes haue to one the selfe same magnitude one and the same proportion And one and the same magnitude hath to equall magnitudes one and the selfe same proportion SVppose that A and B be equall magnitudes and take any other magnitude namely C. Then I say that either of these A and B haue vnto C one and the same proportion and that C also hath to either of these A and B one and the same proportion I say moreouer that C hath to either of these A and B one and the same proportion For the same order of constructioÌ„ remaining we may in like sort proue that D is equal vnto E there is taken an other multiplex to C namely F. Wherefore if F exceede D it also excedeth E and if it be equall it is equall and if it be lesse it is lesse But F is multiplex to C and D E are other equemultiplices to A and B. Wherfore as C is to A so is C to B. Wherfore equall magnitudes haue to one and the same magnitude one and the same proportion and one and the same magnitude hath to equall magnitudes one and the selfe same proportion which was required to be demonstrated The 8. Theoreme The 8. Proposition Vnequall magnitudes beyng taken the greater hath to one and the same magnitude a greater proportion then hath the lesse And that one and the same magnitude hath to the lesse a greater proportion then it hath to the greater SVppose that AB and C be vnequall magnitudes of which let AB be the greater and C the lesse And let there be an other magnitude whatsoeuer namely D. Then I say that AB hath vnto D a greater proportion then hath C to D and also that D hath to C a greater proportion then it hath to AB For forasmuch as AB is greater then C let there be taken a magnitude equall vnto C namely BE. Now then the lesse of these two magnitudes AE and EB being multiplied will at the length be greater then D. I say moreouer that D hath to C a greater proportion then D hath to AB For the same order of construction still remayning we may in like sort proue that N is greater then K and that it is not greater then FH And N is multiplex to D and FH and K are certayne other equemultiplices to AB and C. Wherfore D hath to C a greater proportion then D hath to AB ¶ For that Orontius seemeth to demonstrate this more plainly therefore I thought it not amisse here to set it Suppose that there be two vnequall magnitudes of which leâ— Aâ— bâ— the greâ—ter and C the lesse and let there be a certaine other magnitude namely D. Then I say first that AB hath to D a greater proportion then hath C to D. For forasmuch as by supposition AB is greater then the magnitude C therefore the magnitude AB contayâ—â—th the same magnitude C and an other magnitude besides Let Eâ— be equall vnto C and let AE be the part remayning of the same magnitude Now AE and EB are eyther vnequall or equall the one to the other First let them be vnequallâ— and leâ— AE be lesse then EB And vnto AE the lesse take any multiplex whatsoâ—uâ—r so that it be greater theÌ„ the magnitude D and let the same be FG. And how multiplex FG is to AE so multiplex let GH be to Eâ— â— and K to C. Agayne take the duple of D â— which let be L and then the triple and leâ— the same be M. And so forward alwayâ—s adding one vntill there be produced suâ—h a multiplex to D which shall be nâ—xt greater then GH that is which amongest the 〈◊〉 oâ— D â— by the conâ—inâ—all addition of one doth first beginne to exceâ—de GH and let thâ—â—â—me be N â— which let â—e quadruple to D. Now then the multiplex GH is the next multiplex lesse then N â— and theâ—fore â—s not lesse then M that is is either equall vnto it or greater then it And forasmuch â—s FG is equemultiplex to AE as GH is to Eâ— therefore how multiplex Fâ— is to AE so multiplex is FH to AB by the first of the fift But how multiplex FG is to AE so multiplex is K to C therefore how multiplex FH is to AB so multiplex is K to C. Moreouer forasmuch as GH and K are equemultiplices vnto EB and C â— and EB is by construction equall vnto C therfore by the common sentence GH is equâ—ll vnto K. But GH is not lesse then M as hath before bene shewed and FG â— was put to be greater then D. Wherefore the whole FH is greater then these two D and M. But D and M are equall vnto N. For N is quadruple to D. And M being triple to D doth together with D make quadruple vnto D. Wherefore FH is greater then N. Farther K is proued to be equall to GH Wherefore K is lesse then N. But FH and K are equemultiplices vnto AB and C vnto the first magnitude I say and the third and N is a certaine other multiplex vnto D which representeth the second the fourth magnitude And the multiplex of the first excedeth the multiplex of the second but the multiplex of the third excedeth not the multiplex of the fourth Wherefore AB the first hath vnto D the second a greater proportion then hath C the third to D the fourth by the 8. definition of thys booke But if AE be greater then EB let EB the lesse be multiplied vntill there be produced a multiplex greater then the magnitude D which let be GH And how multiplex GH is to EB so multiplex let FG be to AE and K also to C. Then take vnto D
such a multiplex as is next greater then FG and againe let the same be N which let be quadruple to D. And in like sort as before may we proue that the whole FH is vnto AB equemultiplex as GH is to EB and also that FH K are equemultiplices vnto AB â— C and finally that GH is equall vnto K. And forasmuch as the multiplex N is next greater then FG therefore FG is not lesse then M. But GH is greater then D by construction Wherefore the whole FH is greater then D and M and so consequently is greater then N. But K excedeth not N for K is equall to GH for how multiplex K is to EB the lesse so multiplex is FG to Aâ— the greater Bâ—t those magnitudes which are equemultipliceâ— vnto vnequall magnitudâ—s are according to the same proportion vnequall Wherefore K is lesse then FG and therefore iâ— much lesse then N. Wherefore againe the multiplex of the first exceedeth the multiplex of the second but the multiplex of the third excedeth not thâ— multiplex of the fourth Wherefore by the 8. definition of the fift Aâ— the first hath to D the second a greater proportion then hath C the third to D the fourth But now if AE be equall vnto EB eyther of them shall be equall vnto C. Wherfore vnto either of thosâ— three magnitudes take equemultiplices greater then D. So that let FG be multiplex to AE and GH vnto EB and K agayne to C which by the 6. coÌ„mon sentence shall be equall the one to the other Let N also be multiplex to D and be next greater then euery one of them namely let it be qâ—adruplâ— to D. This coÌ„struction finished we may again proue that FH and K are equemultiplices to AB and C and that FH the multiplex of thâ— first magnitude exceedeth N the multiplex of the second magnitudeâ—â—nd thaâ— K tâ—â—â—ultiplex of the third excedeth not the multiplex of the fourth Wherfore we may conclude that AB hath vnto D a greater proportion then hath C to D. Now also I say that the self same magnitude D hath vnto the lesse magnitude C a greater proportion theÌ„ it hath to the greater AB And this may plainly be gathered by the foresayd discourse without chaunging the order of the magnitudes of the equemultiplices For seing that euery way it is before proued that FH excedeth N and K is exceeded of the selfe same N therefore conuersedly N excedeth K but doth not excede FH But N is multiplex to D that is to the first and third magnitude and K is multiplex to the second namely to C â— and FH is multiplex to the fourth namely to AB â— Wherefore the multiplex of the first excedeth the multiplex of the second but the multiplex of the third excedeth not the multiplex of the fourth Wherefore by the 8. definition of this fift booke D the first hath vnto C the second a greater proportion then hath D the third to AB the fourth which was required to be proued The 9. Theoreme The 9. Proposition Magnitudes which haue to one and the same magnitude one and the same proportion are equall the one to the other And those magnitudes vnto whome one and the same magnitude hath one and the same proportion are also equall SVppose that either of these two magnitudes A and B haue to C one and the same proportion Then I say that A is equall vnto B. For if it be not then either of these A and B should not haue to C one the same proportioÌ„ by the 8. of the fifth but by supposition they haue wherefore A is equall vnto B. Againe suppose that the magnitude C haue to either of these magnitudes A and B one and the same proportion Then I say that A is equall vnto B. For if it be not C should not haue to either of these A and B one and the same proportion by the former proposition but by supposition it hath wherfore A is equall vnto B. Wherfore magnitudes which haue to one and the same magnitude one and the same proportion are equall the one to the other And thosâ— magnitudes vnto whome one and the same magnitude hath one and the same proportion are also equall which was required to be proued The 10. Theoreme The 10. Proposition Of magnitudes compared to one and the same magnitude that which hath the greater proportion is the greater And that magnitude wherunto one and the same magnitude hath the greater proportion is the lesse SVppose that A haue to C a greater proportion then B hath to C. Then I say that A is greater then B. For if it be not then either A is equall vnto B or lesse then it But A cannot be equal vnto B for then either of these A and B should haue vnto C one and the same proportion by the 7 of the fifth but by supposition they haue not wherfore A is not equall vnto B. Neither also is A lesse then B for theÌ„ should A haue to C a lesse proportion then hath B to C by the 8. of the fifth but by supposition it hath not Wherfore A is not lesse then B. And it is also proued that it is not equall wherfore A is greater then B. Agayne suppose that C haue to B a greater proportion then C hath to A. Then I say that B is lesse then A. For if it be not then is it either equall vnto it or els greater but B cannot be equall vnto A for then should C haue to either of these A and B one and the same proportion by the 7. of the fifth but by supposition it hath not wherfore B is not equall vnto A. Neither also is B greater then A for then should C haue to B a lesse proportion then it hath to A by the 8. of the fifth but by supposition it hath not wherefore B is not greater then A. And it was proued that it is not equall vnto A wherfore B is lesse then A. Wherfore of magnitudes compared to one and the same magnitude that which hath y greater proportion is the greater And that magnitude wherunto one and the same magnitude hath the greater proportion is the lesse Which was required to be proued The 11. Theoreme The 11. Proposition Proportions which are one and the selfe same to any one proportion are also the selfe same the one to the other SVpppose that as A is to B so is C to D and as C is to D so is E to F. Then I say that as A is to B so is E to F. Take equemultiplices to A C and E which let be G H K. And likewise to B D and F take any other equemultiplices which let be L M and N. And because as A is to B so is C to D and to A and G are taken equemultiplices G H to B and D are take certaine other equemultiplices L
multiplex is MP composed of the third and sixt to FD. And for that as AB is to BE so is CD to DF and to AB CD are taken equemultiplices GK and LN and likewise to EB and FD are taken certaine other equemultiplices that is HO and MP If therefore GK exceede HO then LN also exceedeth MP and if it be equall it is equall and if it be lesse it is lesse by the conuersion of the sixt definition of the fift Let GK exceede HO Wherefore KH common to them both being taken away the residue GH shall exceede the residue KO But if GK exceede HO then doth LN exceede MP Wherefore let LN excede MP and MN which is coÌ„mon to the both being takeÌ„ away y residue LM shall exceede the residue NP. Wherefore if GH exceede KO then shall LM exceede NP. And in like sort may we proue that if GH be equall vnto KO then LM shall be equall vnto NP and if it be lesse it shall be lesse but GH and LM are equemultiplices to AE and GF and likewise KO and NP are certayne other equemultiplices to EB and FD. Wherfore as AE is to EB so â—CF to FD by the sixâ— definition of the fift If composed magnitudes therefore be proportionall then also deuided they shall be proportionall which was required to be demonstrated The 18. Theoreme The 18. Proposition If magnitudes deuided be proportionall then also composed they shall be proportionall SVppose that the magnitudes deuided being proportionall be AE EB CF FD so that as AE is to EB so let CF be to FD. Then I say that composed also they shall be proportionall that is as AB is to BE so is CD to DF. For if AB be not vnto BE as CD is to FD then shall AB be vnto BE as CD is either vnto a magnitude lesse then FD on vnto a magnitude greater Let it first be vnto a lesse namely to DG And forasmuch as as AB is to BE so is CD to DG the composed magnitudes therefore are proportionall wherefore deuided also they shall be proportionall by the 17. of the first Wherefore as AE is to EB so is CG to GD But by supposition as AE is to EB so is CF to FD. Wherefore by the 11. of the fift as CG is to GD so is CF to FD. Now then there are foure magnitudes CG GD CF and FD of which the first CG is greater then the third CF. Wherefore by the 14. of the fift the second GD is greater then the fourth FD. But it is also put to be lesse then it which is impossible Wherfore it can not be that as AB is to BE so is CD to a magnitude lesse then FD. In like sort may we proue that it can not be so to a magnitude greater then FD. For by the same order of demonstration it would follow that FD is greater then the sayd greater magnitude which is impossible Wherfore it must be to the selfe same If therefore magnitudes deuided be proportionall then also composed they shall be proportionall which was required to be proued The 19. Theoreme The 19. Proposition If the whole be to the whole as the part taken away is to the part taken away then shall the residue be vnto the residue as the whole is to the whole SVppose that as the whole AB is to the whole CD so is the part taken away AE to the part taken away CF. Then I say that the residue EB shall be vnto the residue FD as the whole AB is to the whole CD For for that as the whole AB is to the whole CD so is AE to CE therfore alternately also by the 16. of the fift as AB is to AE so is CD to CF. Aâ—d for that when magnitudes composed are proportionall the same deuided also are proportionall by the 17. of the fift â— therefore as BE is to EA so is DF to FC Wherfore alternately also by the 16. of the fift as BE is to DF so is EA to FC But as AE is to CF so by supposition is the whole AB to the whole CD Wherefore the residue EB shall be vnto the residue FD as the whole AB is to the whole CD If therfore the whole be to the whole as the part taken away is to the part taken away then shall the residue be vnto the residue as the whole is to the whole which was required to be proued ¶ ALemma or Assumpt And forasmuch as by supposition as AB is to CD so is AE to CF and alternately as AB is to AE so is CD to CF. And now it is proued that as AB is to CD so is EB to FD. Wherefore againe alternately as AB is to â—B so is CD to FD. Wherefore it followeth that as AB is to AE so is CD to CF and againe as the same AB is to EB so is the same CD to DF. Corollary And hereby it is manifest that if magnitudes composed be proportionall then also by conuersion of Proportion which of some is called Proportion by Eversion and which is as before it was defined wheÌ„ the antecedent is compared to the excesse wherein the antecedent exceedeth the consequent they shall be proportionall The 20. Theoreme The 20. Proposition If there be three magnitudes in one order and as many other magnitudes in an other order which being taken two and two in eche order are in one and the same proportion and if of equalitie in the first order the first be greater then the third then in the second order the first also shall be greater then the third and if it be equall it shall be equall and if it be lesse it shall be lesse SVppose that there be three magnitudes in one order namely A B C let there be as many magnitudes in an other order which let be D E F which being taken two and two in ech order let be in one and the same proportion that is as A is to B so let D be to E and as B is to C so let E be to F. But now if A be equall vnto C D also shall be equall vnto F. For then A and C haue vnto B one and the same proportion by the first part of the seuenth of this booke And for that as A is to B so is D to E and as C is to B so is F to E therefore D and F haue vnto E one and the same proportion Wherefore by the first part of the 9 of this booke D is equall vnto F. But now suppose that A be lesse then C. Then also shall D be lesse then F. For by the 8. of this booke C shall haue vnto B a greater proportion then hath A to B. But as A is to B so is D to E by supposition and as C is to B so haue we proued is F to E. Wherefore F hath vnto E a greater
proportion then hath D to E. Wherefore by the first part of the 10. of this booke F is greater then D. If therefore there be three magnitudes in one order and as many other magnâ—tudes in an other order which being taken two and two in ech order are in one and the same proportion and if of equalitie in the first order the first be greater then the third then in the second order also the first shall be greater then the third and if it be equall it shall be equall and if it be lesse it shall be lesse which was required to be proued The 21. Theoreme The 21. Proposition If there be three magnitudes in one order and as many other magnitudes in an other order which being taken two and two in eche order are in one and the same proportion and their proportion is perturbate if of equalitie in the first order the first be greater then the third theÌ„ in the second order the first also shall be greater then the third and if it be equall it shall be equall and if it be lesse it shall be lesse SVppose that there be three magnitudes in one order namely A B C and let there be as many other magnitudes in an other which let be D E F which being taken two two in ech order let be in one and the same proportion and let their proportion be perturbate So that as A is to B so let E be to F as B is to C so let D be to E and of equalitie let A be greater then G. Then I say that D also is greater thâ—n E and if it be equall it is equall and if it be lesse it is lesse Likewise if A be lesse then C D also is lesse then F. For then C shall haue vnto B a greater proportion then hath A to B by the 8. of the fift Wherefore E also â—ath vnto D a greater proportion then it â—ath to F. Whereâ—ore by the second part of the 10. of this booke D is lesse then F. If therefore there be three magnitudes in one order as many other magnitudes in an other order which being taken two and two in ech order are in one the same proportion their proportion is perturbate and if of equalitie in the first order the first be greater then the third then in the second order the first also shall be greater then the thirdâ— and if it be equall it shall be equall and if it be lesse it shall be lesse which was required to be proued The 22. Theoreme The 22. Proposition If there be a number of magnitudes how many soeuer in one order and as many other magnitudes in an other order which being taken two and two in ech order are in one and the same proportion they shall also of equalitie be in one and the same proportion SVppose that there be a certaine number of magnitudes in one order As for example A B C and let there be as many other magnitudes in an other order which let be D E F which being taken two and two let be in one and the same proportion So that as A is to B so let D be to E and as B is to C so let E be to F. Then I say that of equalitie they shall be in the same proportion that is as A is to Câ— so is D to F. Take vnto A and D equemultiplices G H and likewise to B E take any other equemultiplices whatsoeuer namely K and Lâ— and moreouer vnto C and F take any other equemultiplices also what soeuer namely M and N. And forasmuch as as A is to B so is D to E and vnto A and D are taken equemultiplices G and H and likewise vnto B and E are taken certaine other equemâ—ltiplices K and L therfore by the 4. of the fift as G is to Kâ— so is H to L. And by the same reason as K is to M so is L to N. Seing therefore that there be in order three magnitudes G K M as many other magnitudes in an other order namely H L N which being compared two to two are in one and the same proportion therefore of equalitie by the 20. of the fift if N exceede M then shall H exceede G and if it be equall it shall be equall and if it be lesse it shall be lesse But G and H are equemultiplices vnto A and D and M and N are certaine other equemultiplices vnto C and F therfore by the 6. definition of the fift as A is to C so is D to F. So also if there be more magnitudes then three in â—ither order the first of the one order shall be to the last as the first of the other order is to the last As if there were foure in one order namely ABCD and other foure in the other order namely EFGH we may with three magnitudes A B C and E F G proue that as A is to C so is E to G And then leauing out in either order the second and taking the fourth as leauing out B and F and taking D and H we may proue by these three and three A C D and E G H that as A is to D so is E to H. And obseruing this order thys demonstration will serue how many soeuer the magnitudes be in either order If therefore there be a number of magnitudes how many soeuer in one order and as many other magnitudes in an other order which being taken two and two in eche order are in one and the same proportion they shall also of equalitie be in one and the same proportion which was required to be demonstrated The 23. Theoreme The 23. Proposition If there be three magnitudes in one order and as many other magnitudes in an other order which beyng taken two two in eche order are in one and the same proportion and if also their proportion be perturbate then of equalitie they shall be in one and the same proportion SVppose that there be in one order three magnitudes namely A B C let be takeÌ„ in an other order as many other magnituds which let be D E F which being taken two and two in eche order let be in one and the same proportion and suppose that their proportioÌ„ be perturbate So that as A is to B so let E be F and as B is to C so let D be to E. TheÌ„ I say that as A is to C so is D to F. Take vnto A B D equemultiplices and let the same be GHK and likewise vnto C E F take any other equemultiplices whatsoeuer and let the same be LMN And forasmuch as G and H are equemultiplices vnto A and B but the partes of equemultiplices are in the same proportion that their equemultiplices are by the 15. of the fift wherfore as A is to B so is G to H. And by the
therefore alternately as AB is to E so is CD to F by the 16. of the fift But AB is greater then E Wherfore also CD is greater then F Which thing may also be proued by the 14. of the same Now for that as AB is to CD so is E to F but E is equall vnto AG and F is equall vnto CH therefore as AB is to CD so is AG to CH and forasmuch as the whole AB is to the whole CD so is the part taken away AG to the part taken away CH therefore the residue GB by the 1â— of the fift is vnto the residue HD as the whole AB is to the whole CD But AB the first is greater then CD the third Wherfore GB the second is greater then HD the fourth by the 14. of the fift And forasmuch as AG is equall vnto E CH is equall vnto F therefore AG and F are equall vnto CH and E. And forasmuch as if vnto thinges vnequall be added thinges equall all shall be vnequall by the fourth common sentence therefore seing that GB and DH are vnequall and GB is the greater if vnto GB be added AG and F and likewise if vnto HD be added CH E there shall be produced AB and F greater then CD E. If therefore there be foure magnitudes proportionall the greatest and the least of them shall be greater theÌ„ the other remayning which was required to be demonstrated Here follow certayne propositions added by Campane which are not to be contemned and are cited euen of the best learned namely of Iohannes Regio montanus in the Epitome which he writeth vpon Ptolome ¶ The first Proposition If there be foure quantities and if the proportion of the first to the second be greater then the proportion of the third to the fourth then contrariwise by conuersion the proportion of the second to the first shall be lesse then the proportion of the fourth to the third It may also be demonstrated directly For let E be vnto B as C is to D. Then coÌ„uersedly B is to E as D is to C. And forasmuch as A is greater then E by the first part of the tenth of this booke therfore by the second part of the 8 of the same B hath vnto A a lesse proportion then hath B to E. Wherfore by the 13. of the same B hath vnto A â— lesse proportion then hath D to C which was required to be proued ¶ The second Proposition If there be foure quantities and if the proportion of the first to the second be greater then the proportion of the third to the fourth then alternately the proportion of the first to the third shall be greater then the proportion of the second to the fourth This may also be demonstrated affirmatiuely let E be vnto B as C is to D. Now theÌ„ by the first part of the tenth of this booke E is lesse then A wherfore by the first parte of the 8. of the same the proportion of A to C is greater then the proportion of E to C. But alternately E is to C as B is to D. Wherfore by the 13. of the same A hath to C a 〈…〉 ¶ The third Proposition If there be foure quantities and if the proportion of the first tâ— the second be greater then the proportion of the third to the fourth then by composition also the proportioÌ„ of the fâ—â—th and second to the second shall be greater then the proportioâ— of the third and fourthâ— to the fourth This may also be demonstrated aâ—firmatâ—â—ely Forasmuch as the proportion of A to B is greater then the proportion of C to D let E be vnto B as C is to D. And so by the first part of the 10. of this booke E shall be lesse then A. And therfore by the common sentence EB shall be lesse then AB Wherfore by the first part of the 8. of the same AB hath vnto B a greater proportion then hath EB to B. But by composition EB is to B as CD is to D. For by supposition E is vnto B as is to D. Wherfore by the 12. of this booke AB hath to B a greater proportion then hath CD to D which was required to be proued ¶ The fourth Proposition If there be foure quantities and if the proportion of the first and the second to the second be greater then the proportion of the third and fourth to the fourth then by diuision also the proportion of the first to the second shall be greater thâ—n the proportion of the thirde to the fourth Suppose that the proportion of AB to B be greater then the proportion of CD to D. Then I say that by diuision also the proportion of A to B is greater then the proportion of C to D. For it cannot be the same For then by composition AB should be to B as CD is to D. Neither also can it be lesse for if the proportion of C to D be greater then the proportion of A to B then by the former proposition the proportion of CD to D should be greater then the proportion of AB to B which is contrary also to the suppositioÌ„ Wherfore the proportion of A to B is neither one and the same with the proportion of C to D 〈…〉 it Wherefore it is greater then it which was required to be proued The same may also be proued affirmatiuely Suppose that EB be vnto B as CD is to D. Now then by the first part of the 10. of the fifth EB shall be lesse then AB and therefore by the common sentence E is lesse then A wherfore by the first part of the 8. of this booke the proportion of E to B is lesse then the proportion of A to B but as E is to B so is C to D wherfore the proportion of C to D is lesse then the proportion of A to B. Wherfore the proportion of A to B is greater then the proportion of C to D which was required to be proued ¶ The fifth Proposition If there be foure quantities and if the proportion of the first and the second to the second be greater then the proportion of the third and the fourth to the fourth then by euersion the proportion of the first and second to the first shall be lesse then the proportion of the third and fourth to the third Suppose that the proportion of AB to B be greater then the proportion of CD to D. Then I say that by euersion the proportion of AB to A is lesse then the proportion of CD to C. For by diuision by the former proposition the proportion of A to B is greater then the proportion of C to D. Wherefore by the first of these propositions conuersedly B hath vnto A a lesse proportioÌ„ theÌ„ hath D to C. Wherfore by the 3. of the same by composition the proportion of AB to A is lesse theÌ„ the
proportion of CD to C which was required to be proued ¶ The sixt Proposition If there be taken three quantities in one order and as many in an other order and if the proportion of the first to the second in the first order be greater then the proportion of the first to the second in the latter order then also the proportion of the first to the third in the first order shall be greater then the proportion of the first to the third in the latter order Suppose that there be three quaÌ„tities in one order A B C as many other quaÌ„tities in an other order D E F. And let the proportion of A to B in the first order be greater then the proportion of D to E in the second order and let also the proportion of B to C in the first order be greater then the proportion of E to F in the second order Then I say that the proportion of A to C in the first order is greater theÌ„ the proportion of D to F in the second order For let G be vnto C as E is to F. Now then by the first part of the 10 of this booke G shall be lesse then B. And therefore by the second parte of the 8. of the â—â—me the proporâ—â—on of A to G iâ— greater thâ—n the proportâ—on of â— to â— Whâ—rfore the proportion of A to G is muche greater thâ—n the proportion of D to E. Now then let â— be 〈…〉 D is to E. Wherfore by the first part of the 10â— of the same A is greatâ—r theÌ„ H. And therfore by the first part of the 8. of the same the proportion of A to C is greater then the proportion of H to C. But by proportion of equality H is to C as D is to F for H is to G as D iâ— to E and G is to C as E is to F. Wherfore by the 12. of the same A hath to C a greater proportion then hath D to F which was required to be proued ¶ The seuenth Proposition If there be taken three quantities in one order and as many other in an other order and if the proportion of the second to the third in the first order be greater then the proportion of the first to the second in the latter order if also the proportion of the first to the second in the first order be greater then the proportion of the second to the third in the latter order then shall the proportion of the first to the third in the first order be greater then the proportion of the first to the third in the latter order Suppose that there be three quaÌ„tities in one order A B C and as many other in an other order D E F. And let the proportion of B to C in the first order be greater then the proportion of D to E in the second order and let also the proportion of A to B in the first order be greater then the proportion of E to F in the second order Then I say that A hath to C a greater proportion then hath D to F. This pertaineth to proportion of equalitie For let G be vnto C as D is to E. And by the first part of the 10. of this bokeâ— G shal be lesse theÌ„ â— And therfore by the second part of the 8. of the same the proportioÌ„ of A to G is greater then the proportion of A to B. Wherfore A hath vnto G a much greater proportioÌ„ then hath â— to F. Now then let H be vnto G as E is to F. And by the first part of the 10. of the same A shal be greater then H. And by the first part of the 8. of the same the proportion of A to C is greater then the proportion of H to C. But by the 23. of the same the proportion of H to C is as the proportion of D to F for G is to C as D is to E and H is to G as E is to F. Wherfore by the 12. of the same the proportion of A to C is greater then the proportion of D to F which was required to be proued ¶ The eight Proposition If the proportion of the whole to the whole be greater thâ—n the proportion of a part taken away to a part taken away theÌ„ shall the proportion of the residue vnto the residue be greater then the proportion of the whole to the whole Suppose that there be two quantities AB C D from which let there be cutte of these magnitudes AE and CF and let the residue be EB and FD. And let the proportioÌ„ of AB to CD be greater then the proportion of AE to CF. Then I say that the proportion of EB to FD is greater then the proportion of AB to CD For by the second of these propositions now added alternately the proportion of Aâ— to A◠〈◊〉 greater then the proportion of CD to CF. And therfore by euersion of proportion by the 5. of the same the proportion of AB to Eâ— is lesse then the proportion of CD to FD. Wherfore agayne alternately the proportion of AB to CD is lesse then the proportioÌ„ of EB to FD which was required to be proued ¶ The ninth Proposition If quantities how many soeuer in one order be compared to as many other in an other order and if there be a greater proportion of euery one that goeth before to that wherunto it is referred then of any that followeth to that wherunto it is referred the proportion of them all taken together vnto all the other taken together shall be greater then the proportion of any that followeth to that wherunto it is compared and also then the proportion of all them taken together to all the other taken together but shall be lesse then the proportion of the first to the first Suppose that there be three quantities in one order A B C as many other in an other order D E F. And let the proportioÌ„ of A to D be greater theÌ„ the proportioÌ„ of B to E let also the proportioÌ„ of B to E be greater then the proportioÌ„ of C to F. TheÌ„ I say that the proportioÌ„ of ABC takeÌ„ al together to DEF takeÌ„ altogether is greater theÌ„ the proportion of B to E and also then the proportion of C to F more ouer theÌ„ the proportion of B C takeÌ„ together to EF takeÌ„ together but is lesse then the proportioÌ„ of A to D For forasmuch as A hath to D a greater proportioÌ„ theÌ„ hath B to E therfore alternately A hath to B a greater proportion then hath D to E wherfore by coÌ„position AB hath to B a greater proportioÌ„ theÌ„ hath DE to E. And againe alternately AB hath to DE a greater proportion then hath B to E. Wherefore by the former proposition A hath to â— a greater proportion then hath AB to DE. And by the same reason may it be proued that hath to E a greater
triangle vnto the section deuideth the angle of the triangle into two equall partes This construction is the halfe part of that Gnomical figure described in the 43. proposition of the first booke which Gnomical figure is of great vse in a maner in all Geometrical demonstrations The 4. Theoreme The 4. Proposition In equiangle triangles the sides which coÌ„taine the equall angles are proportionall and the sides which are subtended vnder the equall angles are of like proportion SVppose that there be two equiangle triangles ABC and DCE and let the angle ABC of the one triangle be equall vnto the angle DCE of the other triangle and the angle BAC equall vnto the angle CDE and moreouer the angle ACB equall vnto the angle DEC Then I say that those sides of the triangles ABC DCE which include the equall angles are proportionall and the side which are subtended vnder the equall angles are of like proportion For let two sides of the sayd triangles namely two of those sides which are subtended vnder equall angles as for example the sides BC and CE be so set that they both make one right line And because the angles ABC ACB are lesse then two right angles by the 17. of the first but the angle ACB is equall vnto the angle DEC therfore the angles ABC DEC are lesse theÌ„ two right angles Wherefore the lines BA ED being produced will at the length meete together Let them meete and ioyne together in the poynt F. And because by supposition the angle DCE is equall vnto the angle ABC therfore the line BF is by the 28. of the first a parallell vnto tâ—e line CD And forasmuch as by supposition the angle ACB is equall vnto the angle DEC therefore againe by the 28. of the first the line AC is a parallell vnto the line FE Wherefore FADC is a parallelogramme Wherfore the side FA is equall vnto the side DC and the side AC vnto the side FD by the 34. of the first And because vnto one of the sides of the triangle BFE namely to FE is drawen a parallell line AC therefore as BA is to AF so is BC to CE by the 2. of the sixt But AF is equall vnto CD Wherfore by the 11. of the fift as BA is to CD so is BC to CE which are sides subtended vnder equall angles Wherefore alternately by the 16. of the fift as AB is to BC so is DC to CE. Againe forasmuch as CD is a parallell vnto BF therefore againe by the 2. of the sixt as BC is to CE so is FD to DE. But FD is equall vnto AC Wherefore as BC is to CE so is AC to DE which are also sides subtended vnder equall angles Wherfore alternately by the 16. of the fift â—s BC is to CA so is CE to EDâ— Wherfore forasmuch as it hath bene demonstrated that as AB is vnto BCâ— so is DC vnto CEâ— but as DC is vnto CA so is CE vnto EDâ— it followeth of equalitie by the 22. of the fift that â—s BA is vnto AC so is CD vnto DEâ— Wherfore in eqâ—iangle triangleâ— the sides which include the equall angles are proportionall and the sides which are subtâ—nded vnder the equall angles are of like proportion â—hich was required to be demonstrated The 5. Theoreme The 5. Proposition If two triangles haue their sides proportionall the triangâ—â—s are equiangle and those angles in theÌ„ are equall vnder which are subtended sides of like proportion SVppose that there be two triangles ABC DEF hauing their sides proportionall as AB is to BC so let DE be to EF as BC is to AC so let EF be to DF and moreouer as BA is to AC so let ED be to DF. Then I say that the triangle ABC is equiangle vnto the triangle DEF and those angles in them are equall vnder which are subtended sides of like proportion that is the angle ABC is equall vnto the angle DEF and the angle BCA vnto the angle EFD and moreouer the angle BAC to the angle EDF Vpon the right line EF and vnto the pointes in it E F describe by the 23. of the first angles equall vnto the angles ABC ACB which let be FEG and EFG namely let the angle FEG be equall vnto the angle ABC and let the angle EFG be equall to the angle ACB And forasmuch as the angles ABC and ACB are lesse then two right angles by the 17. of the first therefore also the angles FEG and EFG are lesse then two right angles Wherefore by the 5. petition of the first the right lines EG FG shall at the length concurre Let theÌ„ concurre in the poynt G. Wherefore EFG is a triangle Wherefore the angle remayning BAC is equall vnto the angle remayning EGF by the first Corollary of the 32. of the first Wherfore the triangle ABC is equiangle vnto the triangle GEF Wherefore in the triangles ABC and EGF the sides which include the equall angles by the 4. of the sixt are proportionall and the sides which are subtended vnder the equall angles are of like proportion Wherefore as AB is to BC so is GE to EF. But as AB is to BC so by supposition is DE to EF. Wherefore as DE is to EF so is GE to EF by the 11. of the fift Wherefore either of these DE and EG haue to EF one and the same proportion Wherefore by the 9. of the fift DE is equall vnto EG And by the same reason also DF is equall vnto FG. Now forasmuch as DE is equall to EG and EF is common vnto them both therefore these two sides DE EF are equall vnto these two sides GE and EF and the base DF is equall vnto the base FG. Wherefore the angle DEF by the 8. of the first is equall vnto the angle GEF and the triangle DEF by the 4. of the first is equall vnto the triangle GEF and the rest of the angles of the one triangle are equall vnto the rest of the angles of the other triangle the one to the other vnder which are subtended equall sides Wherefore the angle DFE is equall vnto the angle GFE and the angle EDF vnto the angle EGF And becauseâ— the angle FED is equall vnto the angle GEF but the angle GEF is equall vnto the angle ABC therefore the angle ABC is also equall vnto the angle FED And by the same reason the angle ACB is equall vnto the angle DFEâ— and moreouer the angle BAC vnto the angle EDF Wherefore the triangle ABC is equiangle vnto the triangle DEF If two triangles therefore haue their sides proportionall the triangles shall be equiangle those angles in them shall be equall vnder which are subtended sides of like proportion which was required to be demonstrated The 6. Theoreme The 6. Proposition If there be two triangles wherof the one hath one angle equall to one angle of
was required to be demonstrated After this Proposition Campane demonstrateth in numbers these foure kindes of proportionalitie namely proportion conuerse composed deuided and euerse which were in continual quantitie demonstrated in the 4. 17. 18. and 19. propositions of the fift booke And first he demonstrateth conuerse proportion in this maner But if A be greater then B C also is greater then D and what part or partes B is of A the selfe same part or partes is D of C. Wherefore by the same definition as B is to A so iâ— D to C which was required to be proued Proportionalitie deuided is thus demonstrated Suppose that the number AB be to the number B as the number CD is to the number D. Then I say that deuided also as A is to B so is C to D. For for that as AB is to B so is CD to D therâ—fore alternately by the 14. of this booke as AB is to CD so is B to D. Wherefore by the 11. of this booke as AB is to CD so is A to C. Wherefore as B is to D so is A to C and for that as A is to C so is B to D theâ—efore alternately as A is to B so is C to D. Proportionalitie composed is thus demonstrated If A be vnto B as C is to D then shall AB be to B as CD is to D. For alternately A is to C as B is to D. Wherefore by the 13. of this booke as AB namely all the antecedentes are to CD namely to all the consequentes so is B to D namely one of the antecedentes to one of the consequentes Wherfore alteâ—nately as AB is to B so is CD to D. Euerse proportionalitie is thus proued Supposâ— that AB be to B as CD is to D then shall AB be to A as CD is to C. For alternately AB is to CD aâ— B is to D. Wherefâ—râ— by the 13. of this booâ—â— Aâ— is â— CD as A is to C. Wherefore alternately AB iâ— to A aâ— CD iâ— to C whiâ—h was required to be proued ¶ A proportion here added by Campane If the proportion of the first to the second be as the proportion of the third to the foâ—rth and if the proportiâ—n of â—he fift to the second be as the propâ—rtion of the sixt to the fourth then the proportion of the first and the fifth taken together shall be to the second as the proportion â—f the third and the sixt taken together to the fourth And after the same maner may you proue the conuerse of this Proposition If B be to A as D is to Câ— and if also B be vnto E as D is to F Then shall B be to AE as D is to CF. For by conuerse proportionalitie A is to B as C is to D. Wherefore of equalitie A is to E as C is to F. Wherefore by composition A and E are to E as C and F are to F. Wherefore conuersedly E is to A and E as F is to C and F. But by supposition B is to E as D is to F. Wherefore agayne by Proportion of equalitie B is to A and E as D is to C and F which was required to be proued A Corollary By this also it is manifest that if the proportion of numbers how many soeuer vnto the first be as the proportion of as many other numbers vnto the second then shall the proportion of the numbers composed of all the numbers that were antecedentes to the first be to the first as the number composed of all the numbers that were antecedentes to the second is to the second And also conuersedly if the proportion of the first to nuÌ„bers how many soeuer be as the proportion of the second to as many other numbers then shall the proportion of the first to the number composed of all the numbers that were consequentes to it selfe be as the proportion of the second to the number composed of all the numbers that were consequenâ—es to it selfe ¶ The 13. Theoreme The 15. Proposition If vnitie measure any number and an other number do so many times measure an other number vnitie also shall alternately so many times measure the third number as the second doth the fourth SVppose that vnitie A do measure the number BC and let an other nuÌ„ber D so many times measure some other nuÌ„ber namely EF. Then I say that alternately vnitie A shall so many times measure the number D as the number BC doth measure the number EF. For forasmuch as vnitie A doth so many times measure BC as D doth EF therefore how many vnities there are in BC so many numbers are there in EF equall vnto D. Deuide I say BC into the vnities which are in it that is into BG GH and HC And deuide likewise EF into the numbers equall vnto D that is into EK KL and LF Now then the multitude of these BG GH and HC is equall vnto the multitude of these EK KL LF And forasmuch as these vnities BG GH and HC are equall the one to the other and these numbers EK KL LF are also equall the one to the other and the multitude of the vnities BG GH and HC are equall vnto the multitude of the numbers EK KL LF therefore as vnitie BG is to the number EK so is vnitie GH to the number KL and also vnitie HC to the number LF Wherfore by the 12â— of the seueâ—th as one of the antecedeâ—tâ—â—â—s to one of the consequentes so are all the antecedenâ—es to all the consequentes Wherfore as vnitie BG is to the number EK so is the number BC to the number EF. But vnitie BG is equall vnto vnitie A and the number EK to the number D. VVherefore by the 7. common sentence as vnitie A is to the number D so is the number BC to the number EF. VVherefore vnitie A measureth the nuÌ„ber D so many times as BC measureth EF by the 21 definition of this booke which was required to be proued ¶ The 14. Theoreme The 16. Proposition If two numbers multiplying them selues the one into the other produce any numbers the numbers produced shall be equall the one into the other SVppose that there be two numbers A and B and let A multiplying B produce C and let B multiplying A produce D. Thâ—n I say that the number Câ— equall vnto the nâ—mber D. Take any vnitie namelyâ— E. And forasmuch as A multiplying B produced C therefore B measureth C by the vnities which are in A. And vnitie E measureth the number A by those vnities which are in the number A. VVhereâ—ore vnitie E so many times measureth A as B measureth C. VVherefore alternately by the 15. of the seuenth vnitie E measureth the number B so many times as A measureth C. Againe for that B multiplying A produced D therefore A measureth D by thâ— vnities which are in B. And vnitie E
denomination of the number B. For how often B measureth A so many vnities let there be in C. And let D be vnitie And forasmuch as B measureth A by those vnities which are in C and vnitie D measureth C by those vnities which are in C therefore vnitie D so many times measureth the number C as B doth measure A. Wherefore alternately by the 15. of the seuenth vnitie D so many times measureth B as C doth measure A. Wherfore what part vnitie D is of the number B the same part is C of A. But vnitie D is a part of B hauing his denomination of B. VVherfore C also is a part of A hauing his denomination of B. VVherfore A hath C as a part taking his denomination of B which was required to be proued The meaning of this Proposition is that if three measure any number that number hath a third part and if foure measure any number the sayd number hath a fourth part And so forth ¶ The 35. Theoreme The 40. Proposition If a number haue any part the number wherof the part taketh his denomination shall measure it SVppose that the number A haue a part namely B and let the part B haue his denomination of the number C. Then I say that C measureth A. Let D be vnitie And forasmuch as B is a part of A hauing his denomination of C and D being vnitie is also a part of the number C hauing his denominatioÌ„ of C therefore what part vnitie D is of the number C the same part is also B of A wherefore vnitie D so many â—â—â—es measureth the number C as B measurâ—â—â— A. Whâ—refâ—re â—â—â—ernâ—â—ely by â—â—e 15. â—f the sâ—â—â—nth vnitie D so many â—â—mes mâ—â—â—â—reth the nuâ—â—beâ— B 〈◊〉 C measâ—â—eth A. Wherefâ—â—â— C measureth A which was requiâ—â—d to bâ— proued This Proposition is the conuerse of the former and the meaning therof is that euery number hauing a third part is measured of three and hauing a fourth part is measured of foure And so forth ¶ The 6. Probleme Th 41. Proposition To finde out the least number that containeth the partes geuen SVppose that the partes geuen be A B C namely let A be an halfe part B a third part C a fourth part Now it is required to finde out the least nuÌ„ber which coÌ„taineth the partes A B C. Let the said partes A B C haue their denominations of the numbers D E F. And take by the 38. of the seuenth â—he least number which the numbers D E F measure and let the same be G. And forasmuch as the numbers D E F measure the number G therfore the number G hath paâ—tes denominated of the nuÌ„bers D E F by the 39. of the seueÌ„th But the parts A B C haue their denominatioÌ„ of the numbers D E F. Wherfore G hath those partes A B C. I say also that it is the least number which hath these partes For if G be not the least number which containeth those partes A B C then let there be some number lesse then G which containeth the saide partes A B C. And suppose the same to be the number H. And forasmuch as H hath the said partes A B C therfore the numbers that the partes A B C take their denominations of shall measure H by the 40. of the seuenth But the numbers wheroâ— the partes A B C take their denominations of are D E F. Wherfore the numbers D E F mâ—asuâ—e thâ— number H which is lesse then G which is impossible For G is supposed to be tâ—â— lâ—â—sâ— number that the numbers D E F do measure Wherfore there is no number lesse then G which containeth these partes A B C which was required to be done Corrolary Hereby it is manifest that if there be taken the least number that numbers how many soeuer do measure the sayd number shall be the least which hath the partes denominated of the sayd numbers how many soeuer Campane after he hath taught to finde out the first least number that conâ—ayneth the partes geuen teacheth also to finde out the second least number that is which except the least of all is lesse then all other and also the third least and the fourth c. The second is found out by doubling the number G. For the numbers which measure the nuÌ„ber G â—hall also measure the double therof by the 5. commoâ— sentence of the seuenth But there cannot be geuen a number greater then the number G lesse then the double therof whom the partes geuen shall â—â—asureâ— For forasmuch as the partes geuen do â—easurâ— the whole namely which is lesse then the double and they also measure the part taken away namely the number G they should also measure the residue namely a number lesse then G which is proued to be the lest number that they do measure which is impossibleâ— wherefore the second number which the said partes geuen do measureâ— must exceeding G needes reache to the double of G and the third to the treble and the fourth to the quadruple and so inâ—initely for those partes can neuer measure any number lesse then the number G. By this Proposition also it is easie to find out the least number containing the partes geuen of partes As if we would finde out the least number which contayneth one third 〈◊〉 â—â— an halâ—e part and one fourth part of a third part reduce the said dâ—â—ers fractiâ—â— into simple fraction by the common 〈◊〉 of reducing of frâ—â—â—ions namely the 〈◊〉 of an halâ—e into a 〈◊〉 part of an wholeâ— and â—he fourth of thiâ—d into a twelfth part of anâ— whole And then by this Probleme search out the least number which contayneth a sixâ— part and a twelfth part and so haue you done The end of the seuenth booke of Euclides Elementes ¶ The eighthe booke of Euclides Elementes AFter that Euclide hath in the seuenth booke entreated of the proprieties of numbers in generall and of certayne kindes thereof more specially and of prime and composed numbers with others now in this eight booke he prosecuteth farther and findeth out and demonstrateth the properties and passions of certayne other kindes of numbers as of the least numbers in proportion and how such may be found out infinitely in whatsoeuer proportion which thing is both delectable and to great vse Also here is entreated of playne numbers and solide and of theyr sides and proportion of them Likewise of the passions of numbers square and cube and of the natures and conditions of their sides and of the meane proportionall numbers of playne solide square and cube numbers with many other thinges very requisite and necessary to be knowne ¶ The first Theoreme The first Proposition If there be numbers in continuall proportion howmanysoeuer and if their extremes be prime the one to the other they are the least of all numbers that haue one and the same proportion with them SVppose that the numbers
number is made of vnities and therfore cannot a point be a common part of all lines and measure them as vnitie is a common part of all numbers and measureth them Vnitie taken certayne tymes maketh any number For there are not in any number infinite vnities but a point taken certayne tymes yea as often as ye list neuer maketh any line for that in euery line there are infinite pointes Wherfore lines figures and bodies in Geometry are oftentymes incommensurable and irrationall Now which are rationall and which irrationall which commensurable and which incommensurable how many and how sundry sortes and kindes there are of them what are their natures passions and properties doth Euclide most manifestly shew in this booke and demonstrate them most exactly This tenth booke hath euer hitherto of all men and is yet thought accompted to be the hardest booke to vnderstand of all the bookes of Euclide Which coÌ„mon receiued opinion hath caused many to shrinke and hath as it were deterred them from the handeling and treatie thereof There haue bene in deede in times past and are presently in these our dayes many which haue delt and haue taken great and good diligence in commenting amending and restoryng of the sixe first bookes of Euclide and there haue stayed themselues and gone no farther beyng deterred and made afrayde as it seemeth by the opinion of the hardnes of this booke to passe forth to the bookes following Truth it is that this booke hath in it somewhat an other strauÌ„ger maner of matter entreated of theÌ„ the other bokes before had and the demonstrations also thereof the order seeme likewise at the first somewhat straunge and vnaccustomed which thinges may seeme also to cause the obscuritie therof and to feare away many from the reading and diligent study of the same so much that many of the well learned haue much complayned of the darkenes and difficultie thereof and haue thought it a very hard thing and in maner impossible to attayne to the right and full vnderstanding of this booke without the ayde and helpe of some other knowledge and learnyng and chiefly without the knowledge of that more secret and subtill part of Arithmetike commonly called Algebra which vndoubtedly first well had and knowne would geue great light therunto yet certainly may this booke very well be entred into and fully vnderstand without any straunge helpe or succour onely by diligent obseruation of the order and course of Euclides writinges So that he which diligently hath perused and fully vnderstandeth the 9. bookes goyng before and marketh also earnestly the principles and definitions of this â—enth booke he shal well perceiue that Euclide is of himselfe a sufficient teacher and instructer and needeth not the helpe of any other and shall soone see that this tenth booke is not of such hardnes and obscuritie as it hath bene hetherto thought Yea I doubt not but that by the trauell and industry taken in this translation and by addicions and emendations gotten of others there shall appeare in it no hardnes at all but shall be as easie as the rest of his bookes are Definitions 1 Magnitudes commensurable are suchwhich one and the selfe same measure doth measure First he sheweth what magnitudes are commensurable one to an other To the better and more cleare vnderstanding of this definition note that that measure whereby any magnitude is measured is lesse then the magnitude which it measureth or at least equall vnto it For the greater can by no meanes measure the lesse Farther it behoueth that that measure if it be equall to that which is measured taken once make the magnitude which is measured if it be lesse then oftentimes taken and repeted it must precisely render and make the magnitude which it measureth Which thing in numbers is easely sene for that as was before said all numbers are commensurable one to an other And although Euclide in this definition comprehendeth purposedly onely magnitudes which are continuall quantities as are lines superficieces and bodies yet vndoubtedly the explication of this and such like places is aptly to be sought of numbers as well rationall as irrationall For that all quantities commensurable haue that proportion the one to the other which number hath to numbers In numbers therfore 9 and 12 are commensurable because there is one common measure which measureth them both namely the number 3. First it measureth 12 for it is lesse then 12. and being taken certaine times namely 4 times it maketh exactly 12 3 times 4 is 12 it also measureth 9 for it is lesse then 9. and also taken certaine times namely 3 times it maketh precisely 9 3 times 3 is 9. Likewise is it in magnitudes if one magnitude measure two other magnitudes those two magnitudes so measured are said to be commensurable As for example if the line C being doubled do make the line B and the same lyne C tripled do make the line A then are the two lines A and B lines or magnitudes commensurable For that one measure namely the line C measureth theÌ„ both First the line C is lesse theÌ„ the line A and alsolesse theÌ„ the line B also the line C taken or repeted certaine times namely 3 times maketh precisely the line A and the same line C taken also certain times namely two times maketh precisely the line B. So that the line C is a common measure to them both and doth measure them both And therfore are the two lines A and B lines commensurable And so imagine ye of magnitudes of other kyndes as of superficiall figures and also of bodies 2 Incommensurable magnitudes are such which no one common measure doth measure This diffinition neadeth no explanation at all it is easely vnderstanded by the diffinition going before of lines commensurable For contraries are made manifest by comparing of the one to the other as if the line C or any other line oftentimes iterated doo not render precisely the line A nor the line B theÌ„ are the lines A and B incommensurable Also if the line C or any other line certayne times repeted doo exactly render the line A and doo not measure the line B or if it measure the line B and measureth not also the line A the lines A and B are yet lines incoÌ„mensurable so of other magnitudes as of superficieces and bodyes 3 Right lines commensurable in power are such whose squares one and the selfe same superficies area or plat doth measure To the declaration of this diffinition we must first call to minde what is vnderstanded ment by the power of a line which as we haue before in the former bookes noted is nothing ells but the square thereof or any other plaine figure equall to the square therof And so great power habilitie â—s a line said to haue as is the quantitie of the square which it is able to describe or a figure superficial equal to the square
therof 〈◊〉 This iâ— also to be noted that of lines some are commensurable in length the one to the other and some are commensurable the one to the other in power Of lines commensurable in length the one to the other was geuen an example in the declaration of the first diffinitioÌ„ namely the lines A and B which were commensurable in length one and the selfe measure namely the line C measured the length of either of them Of the other kinde is geuen this diffinition here set for the opening of which take this example Let there be a certaine line namely the line BC and let the square of that line be the square BCDE Suppose also an other line namely the line FH let the square thereof be the square FHIK and let a certayne superficies namely the superficies A measure the square BCDE taken 16. times which is the number of the litle areas squares plats or superficieces coÌ„tained and described within the sayd squares ech of which is equall to the superficie A. Agayne let the same superficies A measure the square FHIK 9. times taken according to the number of the fieldâ—s or superficieces contayned and described in the same Ye see theÌ„ that one and the selfe same superficies namely the superficies A is a common measure to both these squares and by certayne repeticions thereof measureth them both Wherefore the two lines BC and FH which are the sides or lines producing these squares and whose powers these squares are are by this diffinition lines commensurable in power 4 Lines incommensurable are such whose squares no one plat or superficies doth measure This diffinition is easy to be vnderstanded by that which was sayd in the diffinition last set before this and neadeth no farther declaration And thereof take this example If neither the superficies A nor any other superficies doo measure the two squares B CDE and FHIK or if it measure the one â—â—rely BCDE and not the other FHIK or if it measure the square FHIK and not the square BCDE the two lines BC and FH are in power incommensurable and therfore also incommeÌ„surable in length For whatsoeuer lines are incommeÌ„surable in power the same are also incommensurable in length as shall afterward in the 9. proposition of this booke be proued And therfore such lines are here defined to be absolutely incommensurable These thinges thus standing it may easely appeare that if a line be assigned and layd before vs there may be innumerable other lines commensurable vnto it and other incommensurable vnto it of commensurable lines some are commensurable in length and power and some in power onely 5 And that right line so set forth is called a rationall line Thus may ye see how to the supposed line first set may be compared infinite lines some commensurable both in length power and some commensurable in power onely and incommensurable in length and some incommensurable both in power in length And this first line so set whereunto and to whose squares the other lines and their squares are compared is called a rationall line commonly of the most part of writers But some there are which mislike that it should be called a rationall line that not without iust cause In the Greeke copy it is called 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 rete which signifieth a thing that may be spokeÌ„ expressed by word a thing certayne graunted and appoynted Wherefore Flussates a man which bestowed great trauell and diligence in restoring of these elementes of Euclide leauing this word rationall calleth this line supposed and first set a line certaine because the partes thereof into which it is deuided are certaine and known and may be expressed by voyce and also be coumpted by number other lines being to this line incommensurable whose parts are not distinctly known but are vncertayne nor can be expressed by name nor assignd by number which are of other men called irrationall he calleth vncertaine and surd lines Petrus Montaureus although he doth not very wel like of the name yet he altereth it not but vseth it in al his booke Likewise wil we doo here for that the word hath bene and is so vniuersally receiued And therefore will we vse the same name and call it a rationall line For it is not so great a matter what names we geue to thinges so that we fully vnderstand the thinges which the names signifie This rationall line thus here defined is the ground and foundation of all the propositions almost of this whole tenth booke And chiefly from the tenth proposition forwardes So that vnlesse ye first place this rationall line and haue a speciall and continuall regard vnto it before ye begin any demonstration ye shall not easely vnderstand it For it is as it were the touch and triall of all other lines by which it is known whether any of them be rationall or not And this may be called the first rationall line the line rationall of purpose or a rationall line set in the first place and so made distinct and seuered from other rationall lines of which shall be spoken afterwarde And this must ye well commit to memory 6 Lines which are commensurable to this line whether in length and power or in power onely are also called rationall This definition needeth no declaration at all but is easily perceiued if the first definition be remembred which â—heweth what magnitudes are commensurable and the third which â—heweth what lines are commensurable in power Here notâ— how aptly naturally Euclide in this place vseth these wordes commensurable either in length and power or in power onely Because that all lines which are commensurable in length are also commensurable in powerâ— when he speaketh of lines commensurable in leÌ„gth he euer addeth and in power but when he speaketh of lines commensurable in power he addeth this worde Onely and addeth not this worde in length as he in the other added this worde in power For not all lines which are commensurable in power are straight way commensurable also in length Of this definition take this example Let the first line rationall of purpose which is supposed and laide forth whose partes are certaine known and may be expressed named and nuÌ„bred be AB the quadrate wherof let be ABCD then suppose againe an other lyne namely the line EF which let be commensurable both in length and in power to the first rationall line that is as before was taught let one line measure the length of eche line and also lâ—t one superâ—icies measure the two squares of the said two lines as here in the example is supposed and also appeareth to the eie then is the line E F also a rationall line Moreouer if the lyne EF be commensurable in power onely to the rationall line AB first set and supposed so that no one line do measure the two lines AB and EF As in example yâ— see to be for
that the line EF is made equall to the line AD which is the diameter of the square ABCD of which square the line AB is a side it is certayne that the â—ide of a square is incoÌ„meÌ„surable in leÌ„gth to the diameter of the same square if there be yet founde any one superficies which measureth the two squares ABCD and EFGH as here doth the triangle ABD or the triangle ACD noted in the square ABCD or any of the foure triangles noted in the square EFGH as appeareth somwhat more manifestly in the second example in the declaration of the last definition going before the line EF is also a rational line Note that these lines which here are called rationall lines are not rational lines of purpose or by supposition as was the first rationall line but are rationall onely by reason of relation and comparison which they haue vnto it because they are commensurable vnto it either in length and power or in power onely Farther here is to be noted that these wordes length and power and power onely are ioyned onely with these wordeâ— commensurable or incommensurable and are neuer ioyned with these woordes rationall or irrationall So that no lines can be called rational in length or in power nor like wise can they be called irrationall in length or in power Wherin vndoubtedly Campanus was deceiued who vsing those wordes speaches indifferently caused brought in great obscuritie to the propositions and demonstrations of this boke which he shall easily see which marketh with diligence the demonstrations of Campanus in this booke 7 Lines which are incommensurable to the rationall line are called irrationall By lines incommensurable to the rationall line supposed in this place he vnderstandeth such as be incommensurable vnto it both in length and in power For there are no lines incommensurable in power onely for it cannot be that any lines should so be incommenâ—urable in power onely that they be not also incommensurable in length What so euer lines be incommeâ—surable in power the same be also incommensurable in length Neither can Euclide here in this place meane lines incommensurable in length onely for in the diffinition before he called them rationall lines nâ—ither may they be placed amongst irrationall lines Wherfore it remayneth that in this diffintion he speaketh onely of those lines which are incommensurable to the rationall line first geuen and supposed both in length and in power Which by all meanes are incommensurable to the rationall line therfore most aptly are they called irrationall lines This diffinition is easy to be vnderstanded by that which hath bene sayd before Yet for the more plainenes see this example Let the â—â—rst rationall line supposed be the line AB whose square or quadrate let be ABCD. And let there be geuen an other line EF which lâ—t be to the rationall line incommensurable in length and power so that let no one line measure the length of the two lines AB and EF and let the square of the line EF be EFGH Now if also there be no one superficies which measureth the two squares ABCD and EFGH as is supposed to be in this example theÌ„ is the line EF an irrationall line which word irrational As before did this word rational misliketh many learned in this knowledge of Geometry Flussates as he left the word rationall and in steade thereof vsed this word certaine so here he leaueth the word irrationall and vseth in place thereof this word vncertaine and euer nameth these lines vncertaine lines Petrus Montaureus also misliking the word irrationall would rather haue them to be called surd lines yet because this word irrationall hath euer by custome and long vse so generally bene receiuedâ— he vseth continually the same In Greeke such lines are called 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 alogoi which signifieth nameles vnspeakeable vncertayne in determinate and with out proportion not that these irrationall lines haue no proportion at all either to the first rationall line or betwene them selues but are so named for that theyr proportions to the rationall line cannot be expressed in number That is vndoubtedly very vntrue which many write that their proportions are vnknowne both to vs and to nature Is it not thinke you a thing very absurd to say that there is any thing in nature and produced by nature to be hidde from nature and not to be knowne of nature it can not be sayd that their proportions are vtterly hidde and vnknowne to vs much lesse vnto nature although we cannot geue them their names and distinctly expresse them by numbers otherwise should Euclide haue taken all this trauell and wonderfull diligence bestowed in this bookeâ— in vaine and to no vseâ— in which he doth nothing ellâ— but teach the proprieties and passions of these irrationall linesâ— and sheweth the proportions which they haue the one to the other Here is also to be noted which thing also Tartalea hath before diligently notedâ— that Campanus and many other writers of Geometryâ— ouer much â—â—â—ed and were deceiued in that they wrote and taught that all these lines whose squares were not sâ—gnified and mought be expressed by a square number although they mighâ— by any other number as by 11. 12. 14. and such others not square numbers are irrationall lines Which is manifestly repugnant to the groundes and principles of Euclide who wil that all lines which are commensurable to the rationall line whether it be in length and power or in power onely should be rationall Vndoubtedly this hath bene one of the chiefest and greatest causes of the wonderfull confusion and darkenes of this booke which so hath tossed and tormoyled the wittes of all both writers and readers masters and scholers and so ouerwhelmed them that they could not with out infinite trauell and sweate attayne to the truth and perfect vnderstanding thereof 8 The square which is described of the rationall right line supposed is rationall Vntill this diffinition hath Euclide set forth the nature and proprietie of the first kinde of magnitude namely of lines how they are rationall or irrationall now he bâ—ginneth to â—hew how the second kinde of magnitudes namely superficies are one to the other rationall or irrationall This diffinition is very playne Suppose the line AB to be the rationall line hauing his parts and diuisions certaynely knowne the square of which line let be the square ABCD. Now because it is the square of the rationall line AB it is also called rationall and as the line AB is the first rationall line vnto which other lines compared are coumpted rationall or irrationall so is the quadrat or square thereof the â—irst rationall superficies vnto which all other squares or figures compared are coumpted and named rationall or irrationall 9 Such which are commensurable vnto it are rationall In this diâ—â—inition where it is sayd such as are commensurable to the square of the rationall line are not vnderstand onely other squares or
quadrates but all other kindes of rectiline figures playne plats superficieses What so euer so that if any such figure be coÌ„mensurable vnto that rationall squareâ— it is also rationall As suppose that the square of the rationall line which is also rationall be ABCD suppose 〈◊〉 so some other square as the square EFGH to be commensurable to the same theÌ„ is the square EFGH also rational So also if the rectiline figure KLMN which is a figure on the one side longer be commensurable vnto the sayd square as is supposed in this exampleâ— it is also a rational superficies and so of all other superficieses 10 Such which are incommensurable vnto it are irrationall Where it is sayd in this diffinition such which are incommensurable it is generally to be taken as was this word coÌ„mensurable in the diffinitioÌ„ before For al superficieses whether they be squares or figures on the one side longer or otherwise what maner of right lined figure so euer it be if they be incommensurable vnto the rationall square supposed theÌ„ are they irrationall As let thâ— square ABCD be the square of the supposed rationall line which square therefore is also rationall suppose also also an other square namely the square E suppose also any other figure as for example sake a figure of one side longer which let be F Now if the square E and the figure F be both incommensurable to the rationall square ABCD then is 〈◊〉 of these figures E F irrationall And so of other 11 And these lines whose poweres they are are irrationall If they be squares then are their sides irrationall If they be not squares but some other rectiline figures then shall the lines whose squares are equall to these rectiline figures be irrationall Suppose that the rationall square be ABCD. Suppose also an other square namely the square E which let be incoÌ„meÌ„surable to the rationall square therefore is it irrationall and let the side or line which produceth this square be the line FG then shall the line FG by this diffinition be an irrationall line because it is the side of an irrationall square Let also the figure H being a figure on the one side longer which may be any other rectiline figure rectangled or not rectangled triangle pentagone trapezite or what so euer ells be incommensurable to the rationall square ABCD then because the figure H is not a square it hath no side or roote to produce it yet may there be a square made equall vnto it for that all such figures may be reduced into triangles and so into squares by the 14. of the second Suppose that the square Q be equall to the irrationall figure H. The side of which figure Q let be the line KL then shall the line KL be also an irrational line because the power or square thereof is equal to the irrationall figure H and thus conceiue of others the like These irrationall lines and figures are the chiefest matter and subiect which is entreated of in all this tenth booke the knowledge of which is deepe and secret and pertaineth to the highest and most worthy part of Geometrie wherein standeth the pith and mary of the hole science the knowlede hereof bringeth light to all the bookes following with out which they are hard and cannot be at all vnderstoode And for the more plainenes ye shall note that of irrationall lines there be diâ—ers sortes and kindes But they whose names are set in a table here following and are in number 13. are the chiefe and in this teÌ„th boke sufficiently for Euclides principall purpose discoursed on A mediall line A binomiall line A first bimediall line A second bimediall line A greater line A line containing in power a rationall superficies and a mediall superficies A line containing in power two mediall superficieces A residuall line A first mediall residuall line A second mediall residuall line A lesse line A line making with a rationall superficies the whole superficies mediall A line making with a mediall superficies the whole superficies mediall Of all which kindes the diffinitions together with there declarations shal be set here after in their due places ¶ The 1. Theoreme The 1. Proposition Two vnequall magnitudes being geuen if from the greater be taken away more then the halfe and from the residue be againe taken away more then the halfe and so be done still continually there shall at length be left a certaine magnitude lesser then the lesse of the magnitudes first geuen SVppose that there be two vnequall magnitudes AB and C of which let AB be the greater Then I say that if from AB be taken away more then the halfe and from the residue be taken againe more then the halfe and so still continually there shall at the length be left a certaine magnitude lesser then the lesse magnitude geueÌ„ namely then C. For forasmuch as C is the lesse magnitude therefore C may be so multiplyed that at the length it will be greater then the magnitude AB by the 5. definition of the fift booke Let it be so multiplyed and let the multiplex of C greater then AB be DE. And deuide DE into the partes equall vnto C which let be DF FG and GE. And from the magnitudes AB take away more then the halfe which let be BH and againe from AH take away more then the halfe which let be HK And so do continually vntill the diuisions which are in the magnitude AB be equall in multitude vnto the diuisions which are in the magnitude DE. So that let the diuisions AK KH and HB be equall in multitude vnto the diuisions DF FG and GE. And forasmuch as the magnitude DE is greater then the magnitude AB and from DE is taken away lesse then the halfe that is EG which detraction or taking away is vnderstand to be done by the former diuision of the magnitude DE into the partes equall vnto C for as a magnitude is by multiplication increased so is it by diuision diminished and from AB is taken away more then the halfe that is BH therefore the residue GD is greater then the residue HA which thing is most true and most easie to conceaue if we remeÌ„ber this principle that the residue of a greater magnitude after the taking away of the halfe or lesse then the halfe is euer greater then the residue of a lesse magnitude after the taking away of more then the halfe And forasmuch as the magnitude GD is greater then the magnitude HA and from GD is taken away the halfe that is GF and from AH is taken away more then the halfe that is HK therefore the residue DF is greater then the residue AK by the foresayd principle But the magnitude DF is equall vnto the magnitude C by supposition Wherefore also the magnitude C is greater then the magnitude AK Wherefore the magnitude AK is lesse then the magnitude C. Wherefore of the magnitude
magnitude A is commensurable vnto the magnitude C therefore by the 5. of the tenth A hath vnto C such proportion as number hath to number Let A haue vnto C that proportion that the number D hath to the number E. Againe forasmuch as B is commensuâ—able vnto C therefore by the selfe same C hath vnto B that proportion that number hath to number Let C haue vnto B that proportion that the number F hath vnto the number G. Now then take the least numbers in continuall proportion and in these proportions geuen namely that the number D hath to the number E and that the number F hath to the number G by the 4. of the eight which let be the numbers â— K L. So that as the number D is to the number E so let the number H be to the number K and as the nuÌ„ber F is to the nuÌ„ber G so let the nuÌ„ber K be to the nuÌ„ber L. Now for that as A is to C so is D to E but as D is to E so is H to K therfore as A is to C so is H to K. Againe for that as C is to B so is F to G but as F is to G so is K to L therefore as C is to B so is K to L. But it is now proued that as A is to C so is H to K. Wherefore of equalitie by the 22. of the fift as A is to B so is the number H to the number L. Wherefore A hath vnto B such proportion as number hath to number Wherefore by the sixt of the tenth the magnitude A is commensurable vnto the magnitude B. Magnitudes therefore commensurable to one and the selfe same magnitude are also commensurable the one to the other which was required to be proued ¶ An Assumpt If there be two magnitudes compared to one and the selfe same magnitude and if the one of them be commensurable vnto it and the other incommensurable those magnitudes are incommensurable the one to the other SVppose that there be two magnitudes namely A and B and let C be a certayne other magnitude And let A â—e commensurable vnto C and let B be commeÌ„surable vnto the selfe same C. Then I say that the magnitude A is incommensurable vnto B. For if A be commensurable vnto B forasmuch as A is also commeÌ„surable vnto C therefore by the 12. of the tenth B is also commeÌ„surable vnto C which is contrary to the supposition ¶ The 10. Theoreme The 13. Proposition If there be two magnitudes commensurable and if the one of them be incommensurable to any other magnitude the other also shall be incommensurable vnto the same SVppose that these two magnitudes A B be commensurable the one to the other and let the one of them namely A be incommensurable vnto an other magnitude namely vnto C. Then I say that the other magnitude also namely B is incommensurable vnto C. For if B be commensurable vnto C then forasmuch as A is commensurable vnto B therefore by the 12. of the tenth the magnitude A also is commensurable vnto the magnitude C. But it is supposed to be incommensurable vnto it which is impossible Wherefore the magnitudes B and C are not commensurable Wherefore they are incommensurable If therefore there be two magnitudes commensurable and if the one of them be incommensurable to any other magnitude the other also shal be incommensurable vnto the same which was required to be proued ¶ A Corollary added by Montaureus Magnitudes commensurable to magnitudes incommeÌ„surable are also incommensurable the one to the other Suppose that the magnitudes A and B be incommensurable the one to the other and let the magnitudâ— C be coÌ„mensurable to A and let the magnitude D be coÌ„mensurable vnto B. Then I say that the magnituâ—â—s C and D are incommensurable the one to the other For A and C are commensurable of which the magnitude A is incommensurable vnto B wherefore by this 13. proposition the magnitudes C and B are also incommensurable but the magnitudeâ— B and D are coÌ„mensurable wherefore by the same or by the former assumpt the magnitudes C and D are incommensurable the one to the other This corollary Theon vseth often times as in the 22. 26. and 36 propositions of this booke and in other propositions also ¶ An Assumpt Two vnequall right lines being geuen to fiâ—de out how much the greater is in power more then the lesse And like in sorte two right lines being geuen by this meanes may be founde out a right lyne which contayneth them both in power Suppose that the two right lines geuen be AD and DB. It is required to â—inde out a right lyne that contayneth them both in power Let the lines AB and DB be so put that they comprehend a right angle ADB and draw a right line from A to B. Now agayne it is manifest by the 47. of the â—irst that the line AB contayneth in power the lines AD and DB. ¶ The 11. Theoreme The 14. Proposition If there be sower right lines proportionall and if the first be in power more then the second by the square of a right line commensurable in length vnto the first the third also shal be in power more then the fourth by the square of a right line commensurable vnto the third And if the first be in power more then the second by the square of a right line incommensurable in length vnto the first the third also shall be in power more then the fourth by the square of a right line incommensurable in length to the third SVppose that these foure right lines A B C D be proportionall As A is to B so let C be to D. And let A be in power more then B by the square of the line E. And likewise let C be in power more then D by the square of the line F. Then I say that if A be commensurable in length vnto the line E C also shall be commensurable in length vnto the line F. And if A be incommensurable in length to the line E C also shall be incommensurable in length to the line F. For for that as A is to B so is C to D therefore as the square of the line A is to the square of the line B so is the square of the line C to the square of the line D by the 22. of the sixt But by supposition vnto the square of the line A are equall the squares oâ— the lines E and B and vnto the square of the line C are equall the squares of the of the lines D and F Wherefore as the squares of the lines E and B which are equall to the square of the line A are to the square of the line B so are the squares of the lines D and F which are equall to the square of the line C to the square of the line D by the seuenth of the fift Wherfore
Now forasmuch as D measureth AB and BC it also measureth the whole magnitude AC And it measureth AB Wherefore D measureth these magnitudes CA and AB Wherefore CA AB are commensurable And they are supposed to be incoÌ„mensurableâ— which is impossible Wherfore no magnitude measureth AB and BC. Wherefore the magnitudes AB and BC are incommensurable And in like sort may they be proued to be incommensurable if the magnitude AC be supposed to be incommensurable vnto BC. If therefore there be two magnitudes incommensurable composed the whole also shall be incommensurable vnto either of the two partes component and if the whole be incommensurable to one of the partes component those first magnitudes shall be incommensurable which was required to be proued ¶ A Corollary added by Montaureus If an whole magnitude bee incommensurable to one of the two magnitudes which make the whole magnitude it shall also be incommensurable to the other of the two magnitudes For if the whole magnitude AC be incoÌ„mensurable vnto the magnitude BC then by the 2 part of this 16. Theorâ—me the magnitudes AB and BC shall be incommensurable Wherefore by the first part of the same Theoreme the magnitude AC shall be incommensurable to either of these magnitudes AB and BC. This Corollary 〈◊〉 vseth in the demonstration of the â—3 Theoreme also of other Propositions ¶ An Assumpt If vpon a right line be applied a parallelogramme wanting in figure by a square the parallelogramme so applied is equall to that parallelogramme which is contayned vnder the segmentes of the right line which segmentes are made by reason of that application This Assumpt I before added as a Corollary out of Flussates after the 28. Proposition of the sixt booke ¶ The 14. Theoreme The 17. Proposition If there be two right lines vnequall and if vpon the greater be applied a parallelogramme equall vnto the fourth part of the square of the lesse line and wanting in figure by a square if also the parallelogramme thus applied deuide the line where vpon it is applied into partes commensurable in length then shall the greater line be in power more then the lesse by the square of a line commensurable in length vnto the greater And if the greater be in power more then the lesse by the square of a right line commensurable in length vnto the greater and if also vpon the greater be applied a parallelograÌ„me equall vnto the fourth part of the square of the lesse line and wanting in figure by a square then shall it deuide the greater line into partes commensurable But now suppose that the line BC be in power more then the line A by the square of a line commensurable in length vnto the line BC. And vpon the line BC let there be applied a rectangle parallelograme equall vnto the fourth part of the square of the line A and wanting in figure by a square and let the sayd parallelograme be that which is contained vnder the lines BD and DC Then must we proue that the line BD is vnto the line DC commensurable in length The same constructions and suppositions that were before remayning we may in like sort proue that the line BC is in power more then the line A by the square of the line FD. But by suppositioÌ„ the line BC is in power more theÌ„ the line A by the square of a line coÌ„mensurable vnto it in length Wherfore the line BC is vnto the line FD coÌ„mensurable in length Wherefore the line composed of the two lines BF and DC is coÌ„mensurable in length vnto the line FD by the second part of the 15. of the tenth Wherefore by the 12. of the tenth or by the first part of the 15. of the tenth the line BC is commensurable in length to the line composed of BF and DC But the whole line conposed BF and DC is commensurable in length vnto DC For BF as before hath bene proued is equall to DC Wherefore the line BC is commensurable in length vnto the line DC by the 12. of the tenth Whâ—â—â—fore also the line BD is commensurable in length vnto the line DC by the second part of thâ— 15. of the teâ—th If therfore there be two right lines vnequall and if vpon the greater be appliâ—d a parallelograme equall vnto the fourth part of the square of the lesse and wanting in figure by a square if also the parallelograme thus applied deuide the line whereupon it is applied into partes commensurable in length then shall the greater line be in power more then the lesse by the square of a line commensurable in length vnto the greater And if the greater be in power more then the lesse by the square of a line commeÌ„surable in length vnto the greater and if also vpon the greater be applied a parallelograme equall vnto the fourth part of the square made of the lesse and wanting in figure by a square then shall it deuide the greater line into partes commensurable in length which was required to be proued Campanâ— after this proposition reacheth how we may redily apply vpon the line BC a parallelograme equall to the fourth part of the square of halfe of the line A and wanting in figure by a square after this maner Deuide the line BC into two lines in such sort that halfe of the line A shal be the meane proportionall betwene those two lines which is possible when as the line BC is supposed to be greater then the line A and may thus be done Deuide the line BC into two equal partes in the point E and describe vpon the line BC a semicircle BHC And vnto the line BC and from the point C erect a perpâ—dicular line CK and put the line CK equall to halfe of the line Aâ— And by the point K draw vnto the line EC a parallel line KH cutting the semicircle in the point H which it must needes cut foâ—asmuch as the line BC is greater then the line A And froÌ„ the point H draw vnto the line BC a perpendicular liâ—e HD which line HDâ— forasmuch as by the 34 of the first it is equall vnto the line KC shall also be equall to halfe of the line A draw the lines BH and HC Now then by the â—â— of the third the angle BHC is a right aâ—gle Wherefore by the corollary of the eight of the sixt booke the line HD is the meane proportionall betwene the lines BD and DC Wherefore the halfe of the line A which is equall vnto the line HD is the meane proportionall betwene the lines BD and DC Wherefore that which is contained vnder the lines BD and DC is equall to the fourth part of the square of the line A. And so if vpon the line BD be described a rectangle parallelograme hauing his other side equall to the line DC there shal be applied vpon the line BC a rectangle parallelograme equall vnto the square of halfe of the line A and wanting in figure by
28. Proposition To finde out mediall right lynes commensurable in power onely contayning a mediall parallelogramme LEt there be put three rationall right lines commensurable in power only namely A B and C and by the 13. of the sixt take the meane proportional betwene the lines A and B let thâ— same be D. And as the line B is to the line C so by the 12. of the sixt let the line D be to the line E. And forasmuch as the lines A and B are rationall commensurable in power onely therefore by the 21. of the tenth that which is contained vnder the lines A and B that is the square of the line D is mediall Wherfore D is a mediall line And forasmuch as the lines B and C are commensurable in power onely and as the line B is to the line C so is the line D to the line E wherfore the lines D and E are commensurable in power onely by the corollary of the tenth of this booke but D is a mediall line Wherefore E also is a mediall line by the 23. of this booke Wherfore D E are mediall lines commensurable in power onely I say also that they containe a mediall parallelograme For for that as the line B is to the line C so is the line D to the line E therfore alternately by the 16 of the fift as the line B is to the line D so is the line C to the line E. But as the lyne B is to the line D so is the line D to the line Aâ— by conuerse proportion which is proued by the corollary of the fourth of the fifth Wherfore as the line D is to the line A so is the line C to the line E. Wherfore that which is contained vnder the lines A C is by the 16. of the sixâ— equall to that which is contayned vnder the lines D E. But that which is contained vnder the lines A and C is medial by the 21. of the tenth Wherfore that which is coÌ„tained vnder the lines D and E is mediall Wherfore there are found out mediall lines commensurable in power onely containing a mediall superficies which was required to be done An Assumpt To finde out two square numbers which added together make a square number Let there be put two like superficiall numbers AB and BC which how to finde out hath bene taught after the 9. proposition of this booke And let them both be either euen numbers or odde And let the greater number be AB And forasmuch as if from any euen number be taken away an euen number or froÌ„ an odde number be taken away an odde number the residue shall be euen by the 24. and 26 of the ninth If therfore from AB being an euen number be taken away BC an euen number or from AB being an odde number be taken away BC being also odde the residue AC shall be euen Deuide the number AC into two equall partes in D wherefore the number which is produced of AB into BC together with the square number of CD is by the sixt of the second as Barlaam demonstrateth it in numbers equall to the square number of BD. But that which is produced of AB into BC is a square nuÌ„ber For it was proued by the first of the ninth that if two like plaine numbers multiplieng the one the other produce any nuÌ„ber the number produced shal be a square number Wherfore there are found out two square numbers the one being the square number which is produced of AB into BC and the other the square number produced of CD which added together make a square number namely the square number produced of BD multiplied into himselfe forasmuch as they were demoÌ„strated equall to it A Corollary And hereby it is manifest that there are found out two square numbers namely the 〈◊〉 the square number of BD and the other the square number of CD so that that numbâ—r wherin th one excedeth the other the number I say which is produced of AB into BC is also a square number namely when Aâ— BC are like playne numbers But when they are not like playne numbers then are there found out two square numbers the square number of BD and the square number of DC whose excesse that is the number wherby the greater excedeth the lesse namely that which is produced of AB into BC is not a square number ¶ An Assumpt To finde out two square numbers which added together make not a square number Let AB and BC be like playne numbers so that by the first of the ninth that which is produced of AB into BC is a square number and let AC be an euen number And deuide Câ— into two equall parâ—es in D. Now by that which hath before bene sayd in the former assumpt it is manifest that the square number produced of AB into BC together with the square number of CD is equall to the square number of BD. Take away from CD vnitie DE. Wherfore that which is produced of AB into BC together with the square of CE is lesse then the square number of BD. Now then I say that the square numâ—er produced of AB into BC added to the square number of CE make not a square number For if they do make a square number then that square number which they make is either greater theÌ„ the square number of BE or equall vnto it or lesse then it First greater it cannot be for it is already proued that the square number produced of AB into BC together with the square number of CE is lesse then the square number of BD. But betwene the square number of BD and the square number of BE there is no meane square number For the number BD excedeth the number BE onely by vnitie which vnitie can by no meanes be deuided into numbers Or if the number produced of AB into BC together with the square of the nuÌ„ber CE should be greater then the square of the number BE then should the selfe same number produced of AB into BC together with the square of the number CE be equall to the square of the number BD the contrary wherof is already proued Wherfore if it be possible let that which is produced of AB into BC together with the square number of the number CE be equall to the square number of BE. And let GA be double to vnitie DE that is let it be the number two Now forasmuch as the whole number AC is by supposition double to the whole number CD of which the number AG is double to vnitie DE therfore by the 7. of the seuenth the residue namely the number GC is double to the residue namely to the number EC Wherfore the number GC is deuided into two equall partes in E. Wherefore that which is produced of GB into BC together with the square number of CE is equall to the square nuÌ„ber
contayned vnder the lines BD and DC is equall to the square of the line DA. As touching the fourth that the parallelogramme contained vnder the lines BC and AD is equall to the parallelogramme contained vnder the lines BA and AC is thus proued For forasmuch as as we haue already declared the triangle ABC is like and therefore equiangle to the triangle ABD therefore as the line BC is to the line AC so is the line BA to the line AD by the 4. of the sixt But if there be foure right lines proportionall that which is contained vnder the first and the last is equall to that which is contained vnder the two meanes by the 16. of the sixt Wherefore that which is contained vnder the lines BC and AD is equall to that which is contayned vnder the lines BA and AC I say moreouer that if there be made a parallelogramme complete contained vnder the lines BC and AD which let be EC and if likewise be made complete the parallelogramme contained vnder the lines BA and AC which let be AF it may by an other way be proued that the parallelogramme EC is equall to the parallelogramme AF. For forasmuch as either of them is double to the triangle ACB by the 41. of the first and thinges which are double to one and the selfe same thing are equall the one to the other Wherefore that which is contained vnder the lines BC and AD is equall to that which is contained vnder the lines BA and AC 2. ¶ An Assumpt If a right line be deuided into two vnequall partes as the greater part is to the lesse so is the parallelogramme contayned vnder the whole line and the greater part to the parallelogramme contayned vnder the whole line and the lesse part This Assumpt differeth litle from the first Proposition of the sixt booke 3. ¶ An Assumpt If there be two vnequall right lines and if the lesse be deuided into two equall partes the parallelogramme contained vnder the two vnequall lines is double to the parallelogramme contained vnder the greater line halfe of the lesse line Suppose that there be two vnequall right lines AB and BC of which leâ— AB be the greater and deuide the line BC into two equall partes in the point D. Thâ—n I say that the parallelogramme contained vnder the lines AB BC is double to the parallelogramme contained vnder the lines AB and BD. From the point B raise vp vpon the right line BC a perpendicular line BE and let BE be equall to the line BA And drawing from the point C and D the lines CG and DF parallels and equall to BE and then drawing the right line GFE the figure is complete Nâ—â— for that aâ—â—he line DB is to the line DC so is the parallelogramme BF to the parallelogramme DG by the 1. of the sixt therâ—ore by composition of proportion as the whole line BC is to the line DC so is the parallelogramme BG to the parallelogramme DG by the 18. of the fift But the line BC is double to the line DC Wherefore the parallelogramme BG is double to the parallelogramme DG But the parallâ—logramme BG is contained vnder the lines AB and BC for the line AB is equall to the line BE and the parallelogramme DG is contayned vnder the lines AB and BD for the line BD is equall to the line DC and the line AB to the line DF which was required to be demonstrated ¶ The 10. Probleme The 33. Proposition To â—inde out two right lines incommensurable in power whose squares added together make a rationall superficies and the parallelogramme contained vnder them make a mediall superficies TAke by the 30. of the tenth two rationall right lines commensurable in power onely namely AB and BC so that let the line AB being the greater be in power more then the line BC being the lesse by the square of a line incommensurable in length vnto the line AB And by the 10. of the first deuide the line BC into two equall partes in the point D. And vpon the line AB apply a parallelogramme equall to the square either of the line BD or of the line DC and wanting in figure by a square by the 28. of the sixth and let that parallelogramme be that which is contained vnder the lines AE and EB And vpon the line AB describe a semicircle AFB And by the 11. of the first from the point E raise vp vnto the line AB a perpendiculer line EF cutting the circumference in the point F. And draw lines from A to F and from F to B. And forasmuch as there are two vnequall right lines AB and BC and the line AB is in power more then the line BC by the square of a line incommensurable in lâ—ngth vnto AB and vpon the line AB is applied a parallelograme equall to the fourth part oâ— the square of the line BC that is to the square of the halfe of the line BC and wanting in â—igure by a square and the said parallelogramme is that which is contained vnder the lines AE and EB wherfore by the 2. part of the 18. of the tenth the line AE is incommeÌ„surable in length vnto the line EB But as the line AE is to the line EB so is the parallelogramme contained vnder the lines BA and AE to the parallelogramme contayned vnder the lines AB and BE by the second assumpt before put And that which is contained vnder the line BA and AE is equall to the square of the line AF by the second part of the first assumpt before put And that which is contained vnder the lines AB and BE is by the first part of the same assumpt equall to the square of the line BF Wherfore the square of the line AF is incommâ—nsurable to the square of the line BF Wherfore the lines AF and BF are incommensurable in power And forasmuch as AB is a rationall line by supposition therfore by the 7 definition of the tenth the square of the line AB is rationall Wherefore also the squares of the lines AF and FB added together make a rationall superficies For by the 47. of the first they are equal to the square of the line AB Again forasmuch as by the third part of the first assumpt going before that which is contained vnder the lines AE and EB is equall to the square of the line EF. But by supposition that which is contained vnder the lines AE and EB is equall to the square of the line BD. Wherfore the line FE is equall to the line BD. Wherfore the linâ— BG is double to the line â— E. Wherfore by the third assumpt going before that which is contained vnder the lines AB and BC is double to that which is contained vnder the lines AB and EF. But that which is contained vnder the lines AB and BC is by supposition mediall
length and so likewise should the lines AD and DB be For euery line measureth it selfe and any other line equall to it selfe Moreouer the line DB is either one and the same with the line ACâ— that is is equall to the line AC oâ— els it is greater-then the line AC either els it is lesse then it If DB be equall to the line AC then putting the line DB vpon the line AC eche endes of the one shall agree with eche endes of the other Wherfore putting the point B vpon the point A the point D also shall fall vpon the point C and the line AD which is the rest of the line AC shall also be equall to the line CB which is the rest of the line DB. Wherfore the line AB is deuided into his names in the point C. And so also shal the line AB being deuided in the point D be deuided in the self â—ame point that the self same line AB was before deuided in the point C which is coÌ„trary to the suppositioÌ„ For by suppositioÌ„ it was deuided in sundry pointes namely in C D. But if the line DB be greaterâ— the the line AC let the line AB be deâ—ided into two equal partes in the point E. Wherfore the points C D shal not equally be distant froÌ„ the point E Now by the first assupt going before this propositioÌ„ that which is coÌ„posed of the squares of the lines AD DB is greater theÌ„ that which is composed of the squares of the lines AC CBâ— But that which is composed of the squares of the lines AD DB together with that which is coÌ„tained vnder the lines AD DB twise is equall to that which is composed of the squares of the lines AC CB together with that which is contained vnder the lines AC and CB twise for either of them is equall to the square of the whole line AB by the 4. of the second wherefore how much that which is coÌ„posed of the squares of the lines AD and DB added together is greater then that which is composed of the squares of the lines AC and CB added together so much is that which is contained vnder the lines AC and CB twise greater then that which is contained vnder the lines AD and DB twise But that which is composed of the squares of the lines AD and DB excedeth that which is composed of the squares of the lines AC and CB by a rationall superficies by the 2. assumpt going before this propositionâ— For that which is composed of the squares of the lines AD and DB is rationall and so also is that which is composed of the squares of the lines AC and CB for the lines AD and DB are put to be rationall commensurable in power onely and so likewise are the lines AC and CB. Wherfore also that which is contained vnder the lines AC and CB twise exceedeth that which is contained vnder the lines AD DB twise by a rational superficies wheÌ„ yet notwithstaÌ„ding they are both medial superficieces by the 21. of the tenth which by the 26. of the same is impossible And if the line DB be lesse then the line AC we may by the like demonstration proue the selfe same impossibilitie Wherfore a binomiall line is in one point onely deuided into his names Which was required to be demonstrated 〈…〉 ollary added by Flussates Two ration 〈…〉 surable in power onely being added together cannot be equall to two other rationall line 〈…〉 in power onely added together For either of them should make a binomia 〈…〉 so should a binomiall line be deuided into his names in moe poyntes then onâ—â—â—ch by this proposition is proued to be impossible The like shall follow in the fiue 〈◊〉 irrationall lines as touching their two names ¶ The 31. Probleme The 43. Proposition A first bimediall line is in one poynt onely deuided into his names SVppose that AB be a first bimediall line and let it be deuided into his partes in the point C so that let the lines AC and CB be mediall coÌ„mensurable in power onely and containing a rationall superficies Then I say that the line AB can not be deuided into his names in any other poynt then in C. For if it be possible let it be deuided into his names in the poynt D so that let AD DB be mediall lines commensurable in power onely comprehending a rationall superficies Now forasmuch as how much that which is contayned vnder the lines AD and DB twise diâ—ferreth from that which is contayned vnder the lines AC and CB twise so much differreth that which is composed of the squares of the lines AD and DB from that which is composed of the squares of the lines AC and CB but that which is contayned vnder the lines AD and DB twise differreth from that which is contayned vnder the lines AC and CB twise by a rationall superficies by the second assumpt going before the 41. of the tenth For either of those superficieces is rationall Wherefore that which is composed of the squares of the lines AC and CB differeth from that which is composed of the squares of the lines AD and DB by a rationall superficies when yet they are both mediall superficieces which is impossible Wherefore a first bimediall line is in one poynt onely deuided into his names which was required to be proued ¶ The 32. Theoreme The 44. Proposition A second bimediall line is in one poynt onely deuided into his names SVppose that the line AB being a second bimediall line be deuided into hys names in the poynt C so that let the lines AC and CB be mediall lines commensurable in power onely comprehending a mediall superficies It is manifest that the poynt C deuideth not the whole line AB into two equall partes For the lines AC and CB are not commensurable in length the one to the other Now I say that the line AB cannot be deuided into his names in any other poynt but onely in C. For if it be possible let it be deuided into his names in the poynt D so that let not the line AC be one and the same that is let it not be equall with the line DB. But let it be greater then it Now it is manifest by the first assumpt going before the 42. proposition of this booke that the squares of the lines AC and CB are greater then the squares of the lines AD and DB. And also that the lines AD and DB are mediall lines commensurable in power onely comprehending a mediall supersicies Take a rationall line EF. And by the 44. of the first vpon the line EF apply a rectangle parallelograme EK equall to the square of the line AB From which parallelograme take away the parallelograme EG equall to the squares of the lines AC and CB Wherefore the residue namely the parallelograme HK
the definition of a first binomiall line seâ— before the 48. proposition of this booke the line DG is a first binomiall line which was required to be proued This proposition and the fiue following are the conuerses of the sixe former propositions ¶ The 43. Theoreme The 61. Proposition The square of a first bimediall line applied to a rationall line maketh the breadth or other side a second binomiall line SVppose that the line AB be a first bimediall line and let it be supposed to be deuided into his partes in the point C of which let AC be the greater part Take also a rationall line DE and by the 44. of the first apply to the line DE the parallelograÌ„me DF equall to the square of the line AB making in breadth the line DG Then I say that the line DG is a second binomiall line Let the same constructions be in this that were in the Proposition going before And forasmuch as the line AB is a first bimediall line and is deuided into his partes in the point C therefore by the 37. of the tenth the lines AC and CB are mediall commensurable in power onely coÌ„prehending a rationall superficies Wherfore also the squares of the lines AC and CB are mediall Wherefore the parallelogramme DL is mediall by the Corollary of the 23. of the tenth and it is applied vppon the rationall line DE. Wherefore by the 22. of the tenth the line MD is rationall and incommensurable in length to the line DE. Againe forasmuch as that which is coÌ„tayned vnder the lines AC and CB twise is rationall therefore also the parallelogramme MF is rationall and it is applied vnto the rationall line ML Wherefore the line MG is rationall and commensurable in length to the line ML that is to the line DE by the 20. of the tenth Wherefore the line DM is incommensurable in length to the line MG and they are both rationall Wherefore the lines DM and MG are rationall commensurable in power onely Wherefore the whole line DG is a binomiall line Now resteth to proue that it is a second binomiall line Forasmuch as the squares of the lines AC and CB are greater then that which is contayned vnder the lines AC and CB twise by the Assumpt before the 60. of this booke therefore the parallelogramme DL is greater then the parallelogrrmme MF Wherefore also by the first of the sixt the line DM is greater then the line MG And forasmuch as the square of the line AC is commensurable to the square of the line CB therefore the parallelogramme DH is commensurable to the parallelogramme KL Wherefore also the line DK is commensurable in length to the line KM And that which is contayned vnder the lines DK and KM is equall to the square of the line MN that is to the fourth part of the square of the line MG Wherefore by the 17. of the tenth the line DM is in power more then the line MG by the square of a line commensurable in length vnto the line DM and the line MG is commensurable in length to the rationall line put namely to DE. Wherefore the line DG is a second binomiall line which was required to be proued ¶ The 44. Theoreme The 62. Proposition The square of a second bimediall line applied vnto a rationall line maketh the breadth or other side therof a third binomiall lyne SVppose that AB be a second bimediall line and let AB be supposed to be deuided into his partes in the point C so that let AC be the greater part And take a rationall line DE. And by the 44. of the first vnto the line DE apply the parallelogramme DF equall to the square of the line AB and making in breadth the line DG Then I say that the line DG is a third binomiall line Let the selfe same constructions be in this that were in the propositions next going before And forasmuch as the line AB is a second bimediall line and is deuided into his partes in the point C therfore by the 38. of the tenth the lines AC and CB are medials commensurable in power only compreheÌ„ding a mediall superficies Wherfore that which is made of the squares of the lines AC and CB added together is mediall and it is equall to the parallelogramme DL by construction Wherefore the parallelogramme DL is mediall and is applied vnto the rationall line DE wherfore by the 22. of the tenth the line MD is rationall and incommensurable in length to the line DE. And by the lyke reason also the line MG is rationall and incommensurable in length to the line ML that is to the line DE. Wherfore either of these lines DM and MG is rational and incommensurable in length to the line DE. And forasmuch as the line AC is incommensurable in length to the line CB but as the line AC is to the line CB so by the assumpt going before the 22. of the tenth is the square of the line AC to that which is contained vnder the lines AC and CB. Wherfore the square of the line AC is incâ—mmmensurable to that which is contayned vnder the lines AC and CB. Wherfore that that which is made of the squares of the lines AC and CB added together is incommensurable to that which is contained vnder the lines AC and CB twise that is the parallelogramme DL to the parallelogramme MF Wherfore by the first of the sixt and 10. of the tenth the line DM is incommensurable in length to the line MG And they are proued both rationall wherfore the whole line DG is a binomiall line by the definition in the 36. of the tenth Now resteth to proue that it is a third binomiall line As in the former propositions so also in this may we conclude that the line DM is greater then the line MG and that the line DK is commensurable in length to the line KM And that that which is contained vnder the lines DK and KM is equall to the square of the line MN Wherfore the line DM is in power more then the line MG by the square of a line commensurable in length vnto the line DM and neither of the lines DM nor MG is commensurable in length to the rational line DE. Wherfore by the definition of a third binomiâ—ll line the line DG is a third binomiall line which was required to be proued ¶ Here follow certaine annotations by M. Dee made vpon three places in the demonstration which were not very euident to yong beginners †The squares of the lines AC and Câ— are medials 〈◊〉 iâ— taught after the 21â— of this tenth and therâ—ore forasmuch as they are by supposition commeÌ„surable th' one to the other by the 15. of the teÌ„th the compound of them both is commensurable to ech part But the partes are medials therfore by the coâ—ollary of the 23. of the tenth the compound shall be
in like sort be deuided as the line AB is by that which hath bene demonstrated in the 66. Proposition of this bookeâ— let it be so deuided in the poynt E. Wherefore it can not be so deuided in any other poynt by the 42â— of this booke And for that the line AB â—â— to the line DZ as the line AG is to the line DE but the lines AG DE namely the greater names are commensurable in length the one to the other by the 10. of this booke for that they are commensurable in length to 〈◊〉 and the selfe same rationall line by the first definition of binomiall lines Wherefore the lines AB and DZ are commensurable in length by the 13. of this booke But by supposition they are commensurable in power onely which is impossible The selfe same demonstration also will serue if we suppose the line AB to be a second binomial line for the lesse names GB and EZ being commensurable in length to one and the selfe same rationall line shall also be commensurable in length the one to the other And therefore the lines AB and DZ which are in the selfe same proportion with them shall also be commensurable in length the one to the other which is contrary to the supposition Farther if the squares of the lines AB and DZ be applyed vnto the rationall line CF namely the parallelogrammes CT and HL they shall make the breadthes CH and HK first binomiall lines of what order soeuer the lines AB DZ whose squares were applyed vnto the rational line are by the 60. of this booke Wherefore it is manifest that vnder a rationall line and a first binomiall line are confusedly contayned all the powers of binomiall lines by the 54. of this booke Wherfore the onely commensuration of the powers doth not of necessitie bryng forth one and the selfe same order of binomiall lines The selfe same thyng also may be proued if the lines AB and DZ be supposed to be a fourth or fifth binomiall line whose powers onely are conmmensurable namely that they shall as the first bring forth binomiall lines of diuers orders Now forasmuch as the powers of the lines AG and GB and DE and EZ are commensurable proportionall it is manifest that if the line AG be in power more then the line GB by the square of a line commensurable in length vnto AG the line DE also shall be in power more then the line EZ by the square of a line commensurable in length vnto the line DE by the 16. of this booke And so shall the two lines AB and DZ be eche of the three first binomiall lines But if the line AG be in power more then the line GB by the square of a line incommensurable in length vnto the line AG the line DE shall also be in powâ—r 〈◊〉 then the line EZ by the square of a line incomensurable in length vnto the line DE by the selfâ— same Pâ—oposition And so shall eche of the lines AB and DZ be of the three last binomiall lines But why it is not so in the third and sixt binomiall lines the reason is For that in them neither of the nameâ— is commensurable in length to the rationall line put FC ¶ The 50. Theoreme The 68. Proposition A line commensurable to a greater line is also a greater line SVppose that the line AB be a greater line And vnto the line AB let the line CD be commensurable Then I say that the line CD also is a greater line Deuide the line AB into his partes in the point E. Wherfore by the 39. of the tenth the lines AE and EB are incommensurable in power hauing that which is made of the squares of them added together rationall and that which is contained vnder theÌ„ mediall And let the rest of the construction be in this as it was in the former And for that as the line AB is to the line CD so is the line AE to the line CF thâ— line EB to the line FD but the line AB is commensurable to the line CD by suppositioÌ„ Wherfore the line AE is commensurable to the line CF and the line EB to the line FD. And for that as the line AE is to the line CF so is the line EB to the line FD. Therfore alternately by the 16. of the fift as the line AE is to the line EB so is the line CF to the line FD. Wherfore by composition also by the 18. of the fift as the line AB is to the line EB so is the line CD to the line FD. Wherefore by the 22. of the sixt as the square of the line AB is to the square of the line EB so is the square of the line CD to the square of the line FD. And in like sort may we proue that as the square of the line AB is to the square of the line AE so is the square of the line CD to the square of the line CF. Wherfore by the 11. of the fift as the square of the lyne AB is to the squares of the lines AE and EB so is the square of the line CD to the squares of the lines CF and FD. Wherfore alternately by the 16. of the fift as the square of the line AB is to the square of the line CD so are the squares of the lines AE and EB to the squares of the lines CF and FD. But the square of the line AB is commensurable to the square of the line CD for the line AB is commensurable to the line CD by suppositioÌ„ Wherfore also the squares of the lines AE and EB are commensurable to the squares of the lines CF and FD. But the squares of the lines AE and EB are incommensurable and being added together are rationall Wherfore the squares of the lines CF and FD are incommensurable being added together are also rationall And in like sort may we proue that that which is contained vnder the lines AE and EB twise is commensurable to that which is contained vnder the lines CF and FD twise But that which is contained vnder the lines AE and EB twise is mediall wherfore also that which is contained vnder the lines CF and FD twise is medial Wherfore the lines CF and FD are incommensurable in power hauing that which is made of the squares of them added together rationall and that which is contained vnder theÌ„ mediall Wherfore by the 39. of the tenth the whole line CD is irrationall is called a greater line A line therfore commensurable to a greater line is also a greater line An other demonstration of Peter Montaureus to proue the same Suppose that the line AB be a greater line and vnto it let the line CD be commensurable any way that is either both in length and in power or els in power onely Then I say that the line CD also is a greater
line Deuide the line AB into his partes in the point E. and let the rest of the construction be in this as it was in the former And for that as the line AB is to the line CD so is the line AE to the lyne CF and the line EB to the line FD therfore as the line AE is to the lyne CF so is the line EB to the line FD but the line AB is commensurable to the line CD Wherfore also the lyne AE is commensurable to the lyne CF and likewise the line EB to the line FD. And for thâ—â— as the line AE is to the line CF so is the line EB to the line FD therfore alternately as the lyne AE is to the line EB so is the line CF to the lyne FD. Wherfore by the 22. of the sixt as the square of the lyne AE is to the square of the line EB so is the square of the line CF to the square of the line FD. Wherfore by composition by the 18. of the fift as that which is made of the squares of the lynes Aâ— and Eâ— added together is to the square of the lyne EB so is that which is made of the squareâ— of the lyneâ— Câ— and FD added together to the square of the lyne FD. Wherefore by contrary proportion as the square of the line EB is to that which is made of the squares of the lines AE and Eâ— added together so is the square of the lyne FD to that which is made of the squares of the lynes CF and FD added together Wherfore alternately as the square of the line EB is to the square of the lyne FD so is that which is made of the squares 〈◊〉 the lâ—nes AE and EB added together to that whiche is made of the squares of the lynes CF and FD added together But the square of the lyne EB is coÌ„mensurable to the square of the lyne FD for it hath already bene proued that the lines EB and FD are coÌ„meÌ„surable Wherfore that which is made of the squares of the lines AE EB added together is commeÌ„surable to that which is made of the squares of Câ— FD added together But that which is made of the squares of the lines AE and EB added together is rationall by suppositioÌ„ Wherfore that which is made of the squares of the lynes CF and FD added together is also rationall And as the lyne AE is to the lyne EB so is the line CF to the lyne FD But as the lyne AE is to the lyne EB so is the square of the line A 〈…〉 contayned vnder the lynes AE and EB therfore at the lyne CF is to the lyne FD so is the square of the lyne AE to the parallelogramme contayned vnder the lines AE and EB as the lyne CF is to the lyne FD so is the square of the lyne CF to the parallelograÌ„me contayned vnder the lynes â—F FD. Wherfore as the square of the lyne AE is to the parallelograÌ„me conâ—â—â—â—ed vnder the lines AE and EB so is the square of the lyne CF to the parallelogramme coÌ„tayned vnder the lynes CF and FD. Wherâ—or◠〈◊〉 â—s the square of the line AE is to the square of the lyne CF so is the parallelogramme contained vnder the lynes AE and EB to the parallelogramme 〈◊〉 vndeâ— the lines â—â— and â—â— But the square of the lyne AE is commensurable to the square of the lyne CF for it is already prâ—â—â—d that the lynes AE and CF are commeÌ„surable Wherefore the parallelogramme contayned vnder the lynes AE and EB is commensurable to the parallelogramme contayned vnder the lynes CF and FD. But the parallelogramme contayned vnder the lines AE and EB is mediall by suppoâ—ition Wherfore the parallelogramme contayned vnder the lynes CF and â—D also is mediall And as it hath already bene proued as the line AE is to the lyne EB so is the lyne CF to the lyne FD. But the lyne AE was by supposition incommensurable in power to the line EB Wherfore by the 10. of the tenth the lyne CF is incommensurable in power to the lyne FD. Wherfore the lynes CF and FD are incommensurable in power hauing that which is made of the squares of them added together rationall and that which is contayned vnder them mediall Wherfore the whole lyne CD is by the 39. of the tenth a greater lyne Wherfore a lyne commensurable to a greater lyne is also a greater lyne which was required to be demonstrated An other more briefe demonstration of the same after Campane Suppose that A be a greater line vnto which let the line B be commeÌ„surable either in length and power or in power onely And take a rational line CD And vpon it apply the superficies Câ— equall to the square of the line A and also vpoÌ„ the line FE which is equall to the rationall line CD apply the parallelogramme FG equall to the square of the line B. And forasmuch as the squares of the two lines A and â— are commensurable by supposition the superficies Câ— shal be commensurable vnto the superficies FG and therefore by the first of the sixt and tenth of this booke the line DE is commensurable in length to the line GB And forasmuch as by the â—3 of this booke the line DE is a fourth binomiall line therefore by the â—6 of this booke the line GE is also a fourth binomiall line wherefore by the 57. of this booke the line B which contayneth in power the superficies FG is a greater line ¶ The 51. Theoreme The 69. Proposition A line commensurable to a line contayning in power a rationall and a mediall is also a line contayning in power a rationall and a mediall SVppose that AB be a line contayning in power a rationall and a mediall And vnto the line AB let the line CD be commensurable whether in length and power or in power onely TheÌ„ I say that the line CD is a line coÌ„tayning in power a rationall a mediall Duide the line AB into his parts in the poynt E. Wherfore by the 40. of the tenth the lines AE and EB are incommensurable in power hauing that which is made of the squares of them added together medial and that which is contayned vnder theÌ„ nationall Let the same construction be in this that was in the former And in like sort we may proue that the lines CF and FD are incommensurable in power and that that which is made of the square of the lines AE and EB is commensurable to that which is made of the squares of the lines CF and FD and that that also which is contayned vnder the lines AE and EB is commeÌ„surable to that which is contayned vnder the lines CF and FD. Wherefore that which is made of the squares of the lines CF and FD is mediall and that which is contayned vnder the lines CF and FD is rationall Wherefore the whole line CD is a line contayning in
to the same and so the line BD is a sixt residuall line and the line KH is a sixt binomiall line Wherfore KH is a binomiall line whose names KF and FH are commensurable to the names of the residuall line BD namely to BC and CD and in the selfe same proportion and the binomiall line KH is in the selfe same order of binomiall lines that the residuall BD is of residuall lines Wherefore the square of a rationall line applied vnto a residuall line maketh the breadth or other side a binomiall line whose names are commensurable to the names of the residuall line and in the selfe same proportion and moreouer the binomiall line is in the selfe same order of binomiall lines that the residuall line is of residuall lines which was required to be demonstrated The Assumpt confirmed Now let vs declare how as the line KH is to the line EH so to make the line HF to the line FE Adde vnto the line KH directly a line equall to HE and let the whole line be KL and by the tenth of the sixt let the line HE be deuided as the whole line KL is deuided in the point H let the line HE be so deuided in the point F. Wherfore as the line KH is to the line HL that is to the line HE so is the line HF to the line FE An other demonstration after Flussas Suppose that A be a rationall line and let BD be a residuall line And vpon the line BD apply the parallelogramme DT equall to the square of the line A by the 45. of the first making in breadth the line BT Then I say that BT is a binominall line such a one as is required in the proposition Forasmuch as BD is a residuall line let the line coÌ„ueniently ioyned vnto it be GD Wherfore the lines BG and GD are rationall commensurable in power onely Vpon the rationall line BG apply the parallelogramme BI equall to the square of the line A and making in breadth the line BE. Wherefore the line BE is rationall and commensurable in length to the line BG by the 20. of the tenth Now forasmuch as the parallelogrammes BI and TD are equall for that they are eche equall to the square of the line A therfore reciprokally by the 14. of the sixth as the line BT is to the line BE so is the line BG to the line BD. Wherefore by conuersion of proportion by the corrollary of the 19. of the fifth as the line BT is to the line TE so is the line BG to the line GD As the line BG is to the line GD so let the line TZ be to the line ZE by the corrollary of the 10. of the sixth Wherefore by the 11. of the fifth the line BT is to the line TE as the line TZ is to the line ZE. For either of them are as the line BG is to the line GD Wherefore the residue BZ is to the residue ZT as the whole BT is to the whole TE by the 19. of the fifth Wherefore by the 11. of the fifth the line BZ is to the line ZT as the line ZT is to the line ZE. Wherfore the line TZ is the meane proportionall betwene the lines BZ and ZE. Wherefore the square of the first namely of the line BZ is to the square of the second namely of the line ZT as the first namely the line BZ is to the third namely to the line ZE by the corollary of the 20. of the sixth And for that as the line BG is to the line GD so is the line TZ to the line ZE but as the line TZ is to the line ZE so is the line BZ to the line ZT Wherefore as the line BG is to the line GD so is the line BZ to the line ZT by the 11. of the fifth Wherfore the lines BZ and ZT are commensurable in power onely as also are the lines BG and GD which are the names of the residuall line BD by the 10. of this booke Wherfore the right lines BZ and ZE are coÌ„mensurable in length for we haue proued that they are in the same proportion that the squares of the lines BZ and ZT are And therefore by the corollary of the 15. of this booke the residue BE which is a rationall line is commensurable in length vnto the same line BZ Wherefore also the line BG which is commensurable in length vnto the line BE shall also be commensurable in length vnto the same line EZ by the 12. of the tenth And it is proued that the line RZ is to the line ZT commensurable in power onely Wherefore the right lines BZ and ZT are rationall commensurable in power onely Wherefore the whole line BT is a binomiall line by the 36. of this booke And for that as the line BG is to the line GD so is the line BZ to the line ZT therefore alternately by the 16. of the fifth the line BG is to the line BZ as the line GD is to the line ZT But the line BG is commensurable in length vnto the line BZ Wherefore by the 10. of this booke the line GD is commensurable in length vnto the line ZT Wherefore the names BG and GD of the residuall line BD are commensurable in length vnto the names BZ and ZT of the binomial line BT and the line BZ is to the line ZT in the same proportion that the line BG is to the line GD as before it was more manifest And that they are of one and the selfe same order is thus proued If the greater or lesse name of the residuall line namely the right lines BG or GD be coÌ„mensurable in length to any rationall line put the greater name also or lesse namely BZ or ZT shal be commensurable in length to the same rationall line put by the 12. of this booke And if neither of the names of the residuall line be commensurable in length vnto the rationall line put neither of the names of the binomiall line shal be commensurable in length vnto the same rationall line put by the 13. of the tenth And if the greater name BG be in power more then the lesse name by the square of a line commensurable in length vnto the line BG the greater name also BZ shal be in power more then the lesse by the square of a line commensurable in length vnto the line BZ And if the one be in power more by the square of a line incommensurable in length the other also shal be in power more by the square of a line incommensurable in length by the 14. of this booke The square therefore of a rationall line c. which was required to be proued ¶ The 90. Theoreme The 114. Proposition If a parallelograÌ„me be coÌ„tained vnder a residuall line a binomiall lyne whose names are commensurable to the names of the residuall line and in the selâ—e same proportion the lyne which contayneth in power
angles BAC CAD DAB be equall the one to the other then is it manifest that two of them which two so euer be taken are greater then the third But if not let the angle BAC be the greater of the three angles And vnto the right line AB and from the poynt A make in the playne superficies BAC vnto the angle DAB an equall angle BAE And by the 2. of the first make the line AE equall to the line AD. Now a right line BEC drawne by the poynt E shall cut the right lines AB and AC in the poyntes B and C draw a right line from D to B and an other from D to C. And forasmuch as the line DA is equall to the line AE and the line AB is common to theÌ„ both therefore these two lines DA and AB are equall to these two lines AB and AE and the angle DAB is equall to the angle BAE Wherefore by the 4. of the first the base DB is equall to the base BE. And forasmuch as these two lines DB and DC are greater then the line BC of which the line DB is proued to be equall to the line BE. Wherefore the residue namely the line DC is greater then the residue namely then the line EC And forasmuch as the line DA is equall to the line AE and the line AC is common to them both and the base DC is greater then the base EC therefore the angle DAC is greater then the angle EAC And it is proued that the angle DAB is equall to the angle BAE wherfore the angles DAB and DAC are greater then the angle BAC If therefore a solide angle be contayned vnder three playne superficiall angles euery two of those three angles which two so euer be taken are greater then the third which was required to be proued In this figure ye may playnely behold the former demonstration if ye eleuate the three triangles ABD Aâ—C and ACD in such â—orâ—that they may all meete together in the poynt A. The 19. Theoreme The 21. Proposition Euery solide angle is comprehended vnder playne angles lesse then fower right angles SVppose that A be a solide angle contayned vnder these superficiall angles BAC DAC and DAB Then I say that the angles BAC DAC and DAB are lesse then fower right angles Take in euery one of these right lines ACAB and AD a poynt at all aduentures and let the same be B C D. And draw these right lines BC CD and DB. And forasmuch as the angle B is a solide angle for it is contayned vnder three superficiall angles that is vnder CBA ABD and CBD therefore by the 20. of the eleuenth two of them which two so euer be taken are greater then the third Wherefore the angles CBA and ABD are greater then the angle CBD and by the same reason the angles BCA and ACD are greater then the angle BCDâ— and moreouer the angles CDA and ADB are greater then the angle CDB Wherefore these sixe angles CBA ABD BCA ACD CDA and ADB are greater theÌ„ these thre angles namely CBD BCD CDB But the three angles CBD BDC and BCD are equall to two right angles Wherefore the sixe angles CBA ABD BCA ACD CDA and ADB are greater theÌ„ two right angles And forasmuch as in euery one of these triangles ABC and ABD and ACD three angles are equall two right angles by the 32. of the first Wherefore the nine angles of the thre triangles that is the angles CBA ACB BAC ACD DAC CDA ADB DBA and BAD are equall to sixe right angles Of which angles the sixe angles ABC BCA ACD CDA ADB and DBA are greater then two right angles Wherefore the angles remayning namely the angles BAC CAD and DAB which contayne the solide angle are lesse then sower right angles Wherefore euery solide angle is comprehended vnder playne angles lesse then fower right angles which was required to be proued If ye will more fully see this demonstration compare it with the figure which I put for the better sight of the demonstration of the proposition next going before Onely here is not required the draught of the line AE Although this demonstration of Euclide be here put for solide angles contayned vnder three superficiall angles yet after the like maner may you proceede if the solide angle be contayned vnder superficiall angles how many so euer As for example if it be contayned vnder fower superficiall angles if ye follow the former construction the base will be a quadrangled figure whose fower angles are equall to fower right angles but the 8. angles at the bases of the 4. triangles set vpon this quadrangled figure may by the 20. proposition of this booke be proued to be greater then those 4. angles of the quadrangled figure As we sawe by the discourse of the former demonstration Wherefore those 8. angles are greater then fower right angles but the 12. angles of those fower triangles are equall to 8. right angles Wherefore the fower angles remayning at the toppe which make the solide angle are lesse then fower right angles And obseruing this course ye may proceede infinitely ¶ The 20. Theoreme The 22. Proposition If there be three superficiall plaine angles of which two how soeuer they be taken be greater then the third and if the right lines also which contayne those angles be equall then of the lines coupling those equall right lines together it is possible to make a triangle SVppose that there be thre superficial angles ABC DEF and GHK of which let two which two soeuer be taken be greater then the third that is let the angles ABC and DEF be greater then the angle GHK and let the angles DEF and GHK be greater then the angle ABC and moreouer let the angles GHK and ABC be greater then the angle DEF And let the right lines AB BC DE EF GH and HK be equall the one to the other and draw a right line from the point A to the point C and an other from the point D to the point F and moreouer an other from the point G to the point K. Then I say that it is possible of three right lines equall to the lines AC DF and GK to make a triangle that is that two of the right lynes AC DF and GK which two soeuer be taken are greater then the third Now if the angles ABC DEF and GHK be equall the one to the other it is manifest that these right lines AC DF and GK being also by the 4. of the first equall the one to the other it is possible of three right lines equall to the lines AC DF and GK to make a triangle But if they be not equall let them be vnequall And by the 23. of the first vnto the right line HK and at the point in it H make vnto the angle ABC an equall angle KHL. And by the â— of the first to one of the lines
fourth The second and third are to be found which may betwene A B be two meanes in continuall propoâ—tion as now suppose such two lines found and let them be C and D. Wherefore by Euclides Corollary as A is to B if A were taken as first so shall the Parallelipipedon described of A be to the like Parallelipipedon and in like sort described of C being the second of the fower lines in continuall proportion it is to weâ—e A C D and B. Or if B shall be taken as first and that thus they are orderly in continuall proportion B D C A then by the sayd Corollary as B is to A so shall the Parallelipipedon described of B be vnto the like Parallelipipedon and in like sort described oâ— D. And vnto a Parallelipipedon of A or B at pleasure described may an other of C or D be made like and in like sort situated or described by the 27. of this eleuenth booke Wherefore any two right lines being geueÌ„ c which was required to be done Thus haue I most briefly brought to your vnderstanding if first B were double to A then what Parallelipipedon soeuer were described of A the like Parallelipipedon and in like sort described of C shall be double to the Parallelipipedon described of A. And so likewise secondly if A were double to B the Parallelipipedon of D shoulde be double to the like of B described both being like situated Wherefore if of A or B were Cubes made the Cubes of C and D are proued double to them as that of C to the Cube of A and the Cube of D to the Cube of B in the second case And so of any proportion els betwene A and B. Now also do you most clerely perceaue the Mathematicall occasion whereby first of all men Hippocrates to double any Cube geuen was led to the former Lemma Betwene any two right lines geuen to finde two other right lines which shall be with the two first lines in continuall proportion After whose time many yeares diuine Plato Heron Philo Appollonius Diâ—â—lâ—â— Pappus Sporus Menechâ—us Archytas Tarentinus who made the wodden doue to slye Eratoâ—â—hene Nicomedes with many other to their immortall fame and renowme published diuers their witty deuises methods and engines which yet are extant whereby to execute thys Problematicall Lemma But not withstanding all the trauailes of the â—oresayd Philosophers and Mathematiciens yea and all others doinges and contriuinges vnto this day about the sayd Lemma yet there remaineth sufficient matter Mathematically so to demonstrate the same that most exactly readily it may also be Mechanically practisâ—d that who soeuer shall achieue that feate shall not be counted a second Archimedes but rather a perâ—les Mathematicien and Mathematicorum Princeps I will sundry wayes in my briefe additions and annotations vpon Euclide excite you thereto yea and bring before your eyes sundry new wayes by meinuented and in this booke so placed as matter thereof to my inuentions appertayning may geue occasion Leauing the farther full absolute my concluding of the Lemma to an other place and time which will now more coÌ„pendiously be done so great a part therof being before hand in thys booke published ¶ A Corollary added by Flussas Parallelipipedons consisting vpon equall bases are in proportion the one to the other as their altitudes are For if those altitudes be cut by a plaine superficies parallel to the bases the sections shall be in proportion the one to the other as the sections of the bases cut by the 25. of this booke Which sections of the bases are the one to the other in that proportion that their sides or the altitudes of the solides are by the â—irst of the sixt Wherefore the solides are the one to the other as their altitudes are But if the bases be vnlike the selfe same thing may be proued by the Corollary of the 25. of this booke which by the 25. Proposition was proued in like bases ¶ The 29. Theoreme The 34. Proposition In equall Parallelipipedons the bases are reciprokall to their altitudes And Parallelipipedons whose bases are reciprokall to their altitudes are equall the one to the other But now againe suppose that the bases of the Parallelipipedons AB and CD be reciprokall to their altitudes that is as the base EH is to the base NP so let the altitude of the solide CD be to the altitude of the solide AB Then I say that the solide AB is equall to the solide CD For againe let the standing lines be erected perpendicularly to their bases And now if the base EH be equall to the base NP but as the base EH is to the base NP so is the altitude of the solide CD to the altitude of the solide AB Wherefore the altitude oâ— the solide CD is equall to the altitude of the solide AB But Parallelipipedons consisting vpon equall bases and vnder one and the selfe same altitude are by the 31. of the eleuenth equall the one to the other Wherefore the solide AB is equall to the solide CD But now suppose that the base EH be not equall to the base NP but let the base EH be the greater Wherefore also the altitude of the solide CD that is the line CM is greater then the altitude of the solide AB that is then the line AG. Put againe by the 3. of the first the line CT equall to the line AG and make perfecte the solide CZ Now for that as the base EH is to the base NP so is the line MC to the line AG. But the line AG is equall to the line CT Wherefore as the base EH is to the base NP so is the line CM to the line CT But as the base EH is to the base NP so by the 32. of the eleuenth is the solide AB to the solide CZ â—or the solides AB and CZ are vnder equall altitudes And as the line CM is to the line CT so by the 1. of the sixt is the base MP to the base Pâ—T and by the 32. of the eleuenth the solide CD to the solide CZ Wherefore also by the 11. and 9. of the fift as the solide AB is to the solide CZ so is the solide CD to the solide CZ Wherfore either of these solides AB and CD haue to the solide CZ one and the same proportion Wherefore by the 7. of the fift the solide AB is equall to the solide CD which was required to be demonstrated Againe suppose that the bases of the Parallelipipedons AB and CD be reciprokall to their altitudes that is as the base EH is to the base NP so let the altitude of the solide CD be to the altitude of the solide AB Then I say that the solide AB is equall to the solide CD For the same order of construction remayning for that as the base EH is to the base NP so is the altitude of the solide CD to the altitude
and so of such like Who can not readily fall into Archimedes reckoning and account by his method To finde the proportion of the circumference of any circle to his diameter to be almost triple and one seuenth of the diameter but to be more then triple and ten one seuentithes that is to be lesse then 3 1 â— and more then 3 10 71. And where Archimedes vsed a Poligonon figure of 96. sides he that for exercise sake or for earnest desire of a more nerenes will vse Polygonon figures of 384. sides or more may well trauaile therein till either wearines cause him stay or els he finde his labour fruitles In deede Archimedes concluded proportion of the circumference to the diameter hath hitherto serued the vulgare andâ— Mechanicall worâ— men wherewith who so is not concented let his owne Methodicall trauaile satisfie his desire or let him procure other therto For narrower termes of greater and lesse found and appointed to the circumference will also winne to the Area of the circle a nearer quantitie seing it is well demâ—â—â—rated of Archimedâ—s that a triangle rectangle of whose two sides contayning the right angle one is equall to the semidiameter of the circle and the other to the circumference of the same is equall to the Area of that circle Vpon which two Theoremes it followeth that the square made of the diameter is in that proportion to the circle very neare in which 14 is to 11â— Wherefore euery circle is eleuen fowertenthes well neare of the square about him described The one side then of that square deuide into 14. equall partes and from that point which endeth the eleuenth part drawe to the opposite side a line parallel to the other sidâ—s and so make peâ—fecte the parallelogramme Then by the last Proposition of the second booke vnto that parallelogramme whose one side hath those 11. equall partes make a square equall Then is it euident that square to be equall to the circle about which the first square is described As ye may here beholde in these figures Gentle frend the great desire which I haue that both with pleasure and also profite thou mayest spend thy time in these excellent studies doth cause me here to furnishe thee somewhat extraordinarily about the circle not onely by pointing vnto thee the welspring of Archimedes his so much wondreâ— at and iustly commended trauaile in the former 3. Theoremes here repeated but also to make thee more apt to vnderstand and practise this and other bookes following where vse of the circle may be had in any consideration as in Cones Cylinders and Spheres c. ¶ A Corollary 1. By Archimedes second Theoreme as I haue here alleaged them it is manifest that a parallelogramme contained either vnder the semidiameter and halfe the circumference or vnder the halfe semidiameter and whole circumference of any circle is equall to the circle by the 41. of the first and first of the six◠¶ A Corollary 2. Likewise it is euident that the parallelogramme contayned vnder the semidiameter and halfe of any portion of the circumference of a circle geuen is equall to that sector of the same circle to which the whole portion of the circumference geuen doth belong Or you may vse the halfe semidiameter and the whole portion of the circumference as sides of the said parallelogramme The farther winning and inferring I commiâ—â—â— to your skill care and â—â—udyâ— But in an other sort will I geue you newe ayde and instruction here ¶ A Theoreme Of all circles the circumferences to their owne diameters haue one and the same proportion in what one circle soeuer they are assigned That is as Archimedes hath demonstrated almost as 22. to 7 or nearer if nearer be fouâ—d vntill the very precise proportion be demonstrated Which what soeuer it be in all circumferences to their proper diameters will be demonstrated one and the same A Corollary 1. Wherefore if two circles be propounded which suppose to be A and B as the circumference of A is to the circumference of B so is the diameter of A to the diameter of B. For by the former Theoreme as the circumference of A is to his own diameter so is the circumference of B to his own diameter Wherfore alternately as the circumference of A is to the circumference of B so is the diameter of A to the diameter of B Which was required to be demonstrated A Corollary 2. It is now then euident that we can geue two circles whose circumferences one to the other shall haue any proportion geuen in two right lines The great Mechanicall vse besides Mathematicall considerations which these two Corollaryes may haue in Wheeles of Milles Clockes Cranes and other engines for water workes and for warres and many other purposes the earnest and wittie Mechanicien will soone boult out gladly practise â— Iohn Dee ¶ The 2. Theoreme The 2. Proposition Circles are in that proportion the one to the other that the squares of their diameters are SVppose that there be two circles ABCD and EFGH and let their diameters be BD and FH Then I say that as the square of the line DB is to the square of the line FH so is the circle ABCD to the circle EFGH For if the circle ABCD be not vnto the circle EFGH as the square of the line BD is to the square of the line FH then the square of the line BD shall be to the square of the line FH as the circle ABCD is to a superficies either lesse then the circle EFGH or greater First let the square of the line BD be to the square of the line FH as the circle ABCD is to a superficies lesse then the circle EFGH namely to the superficies S. Describe by the 6. of the fourth in the circle EFGH a square EFGH Now this square thus described is greater then the halfe of the circle EFGH For if by the pointes E F G H we drawe right lines touching the circle the square EFGH is the halfe of the square described about the circle but the square described about the circle is greater then the circle Wherefore the square EFGH which is inscribed in the circle is greater then the halfe of the circle EFGH Deuide the circumferences EF FG GH and HE into two equall partes in the pointes K L M N. And drawe these right lines EK KF FL LG GM MH HN and NE. Wherefore euery one of these triangles EKF FLG GMH and HNE is greater then the halfe of the segmeÌ„t of the circle which is described about it For if by the pointes K L M N be drawen lines touching the circle and then be made perfecte the parallelograÌ„mes made of the right lines EF FG GH HE euery one of the triangles EKF FLG GMH HNE is the halfe of the parallelogragraÌ„me which is described about it by the 41. of the first but the segmeÌ„t described about it is lesse then the parallelogramme Wherefore euery one
of these triangles EKF FLG GMH and HNE is greater then the halfe of the segment of the circle which is described about it Now then deuiding the circumferences remaining into two equall partes and drawing right lines from the pointes where those diuisions are made so continually doing this we shall at the length by the 1. of the tenth leaue certaine segmentes of the circle which shall be lesse then the excesse wherby the circle EFGH excedeth the superficies S. For it hath bene proued in the first Proposition of the tenth booke that two vnequall magnitudes being geuen if from the greater be taken away more then the halfe and likewise againe from the residue more then the halâ—e and so continually there shall at the length be left a certaine magnitude which shall be lesse then the lesse magnitude geuen Let there be such segmentes left let the segmentes of the circle EFGH namely which are made by the lines EK KF FL LG GM MH HN and NE be lesse then the excesse whereby the circle EFGH excedeth the superficies S. Wherefore the residue namely the Poligonon figure EKFLGMHN is greater then the superficies S. Inscribe in the circle ABCD a Poligonon figure like to the Poligonon figure EKFLGMHN and let the same be AXBOCPDR Wherefore by the Proposition next going before as the square of the line BD is to the square of the line FH so is the Poligonon figure AXBOCPDR to the Poligonon figure EKFLGMHN But as the square of the line BD is to the square of the line FG so is the circle ABCD supposed to be to the superficies S. Wherefore by the 11. of the fift as the circle ABCD is to the superficies S so is the Poligonon figure AXBOCPDR to the Poligonon figure EKFLGMHN Wherefore alternately by the 16. of the fift as the circle ABCD is to the Poligonon figure described in it so is the superficies S to the Poligonon figure EKFLGMHN But the circle ABCD is greater then the Poligonon figure described in it Wherefore also the superficies S is greater then the Poligonon figure EKFLGHMN but it is also lesse which is impossible Wherefore as the square of the line BD is to the square of the line FH so is not the circle ABCD to any superficies lesse then the circles EFGH In like sort also may wproue that as the square of the line FH is to the square of the line BD so is not the circle EFGH to any superficies lesse then the circle ABCD. I say namely that as the square of the line BD is to the square of the line FH so is not the circle ABCD to any superficies greater theÌ„ then the circle EFGH For if it be possible let it be to a greater namely to the superficies S. Wherfore by conuersion as the square of the line FH is to the square of the line BD so is the superficies S to the circle ABCD. But as the sâ—perficies S is to the circle ABCD so is the circle EFGH to some supeâ—ficies lâ—sse theÌ„ the circle ABCD. Wherefore by the 11. of the fift as the square of the line FH is to the square of the line BD so is the circle EFGH to some superficies lesse then the circle ABCD which is in the first case proued to be impossible Wherefore as the square of the line BD is to the square of the line FH so is not the circle ABCD to any superficies greater then the circle EFGH And it is also proued that it is not to any lesse Wherefore as the square of the lâ—ne BD is to the square of the line FH so is the circle ABCD to the circle EFGH Wherefore circles are in that proportion the one to the other that the squares of their diameters are which was required to be proued ¶ An Assumpt I say now that the superficies S being greater then the circle EFGH as the superficies S is to the circle ABCD so is the circle EFGH to some superficies lesse then the circle ABCD. For as the superficies S is to the circle ABCD so let the circle EFGH be to the superficies T. Now I say that the superficies T is lesse then the circle ABCD. For for that as the superficies S is to the circle ABCD so is the circle EFGH to the superficies T therefore alternately by the 16. of the fift as the superficies S is to the circle EFGH so is the circle ABCD to the superficies T. But the superficies S is greater then the circle EFGH by supposition Wherefore also the circle ABCD is greater then the superficies T by the 14. of the fift Wherefore as the superficies S is to the circle ABCD so is the circle EFGH to some superficies lesse then the circle ABCD which was required to be demonstrated ¶ A Corollary added by Flussas Circles haue the one to the other that proportion that like Poligonon figures and in like sort described in them haue For it was by the first Proposition proued that the Poligonon figures haue that proportioÌ„ the one to the other that the squares of the diameters haue which proportion likewise by this Propositionâ— the circles haue ¶ Very needefull Problemes and Corollaryes by Master Ihon Dee inuented whose wonderfull vse also be partely declareth A Probleme 1. Two circles being geueÌ„ to finde two right lines which haue the same proportion one to the other that the geuen circles haue oâ—e to the otherâ— Suppose A and B to be the diameters of two circles geuen I say that two right lines are to be fouÌ„de hauing that proportioÌ„ that the circle of A hath to the circle of B. Let to A B by the 11 of the sixth a third proportionall line be found which suppose to be C. I say now that A hath to C that proportion which the circle of A hath to the circle of B. For forasmuch as A B and C are by construction three proportionall lines the square of A is to the square of B as A is to C by the Corollary of the 20. of the sixth â— but as the square of the line A is to the square of the line B so is the circle whose diameter is the line A to the circle whose diameter is the line B by this second of the eleueÌ„th Wherfore the circles of the lineâ— A and B are in the proportion of the right lines A and C. Therefore two circles beâ—ng geuen we haue found two right lines hauing the same proportion betwene theÌ„ that the circles geuen haue one to the other which ought to be done A Probleme 2. Two circles being geuen and a right line to finde an other right line to which the line geueÌ„ shall haue that proportion which the one circle hath to the other Suppose two circles geueÌ„ which let be A B a right line geueÌ„ which let be C I say that an other right line is to be â—ounde to which the line C shall haue that proportion that
last proposition in the second booke and also of the 31. in the sixth booke ¶ A Probleme 5. Two vnequall circles being geuen to finde a circle equall to the excesse of the greater to the lesse Suppose the two vnequal circles geueÌ„ to be ABC DEF let ABC be the greater whose diameter suppose to be AC the diameter of DEF suppose to be DF. I say a circle must be found equal to that excesse in magnitude by which ABC is greater thâ— DEF By the first of the fourth in the circle ABC Apply a right line equall to DF whose one end let be at C and the other let be at B. FroÌ„ B to A draw a right line By the 30. of the third it may appeare that ABC is a right angle and thereby ABC the triangle is rectangled wherfore by the first of the two corollaries here before the circle ABC is equall to the circle DEF For BC by construction is equall to DF and more ouer to the circle whose diameter is AB That circle therefore whose diameter is AB is the circle conteyning the magnitude by which ABC is greater then DEF Wherefore two vnequal circles being geuen we haue found a circle equall to the excesse of the greater to the lesse which ought to be doone A Probleme 6. A Circle being geuen to finde two Circles equall to the same which found Circles shall haue the one to the other any proportion geuen in two right lines Suppose ABC a circle geuen and the proportion geuen let it be that which is betwene the two right lines D and E. I say that two circles are to be found equall to ABC and with al one to the other in the proportioÌ„ of D to E. Let the diameter of ABC be AC As D is to E so let AC be deuided by the 10. of the sixth in the poynt F. At F to the line AC let a perpeÌ„dicular be drawne FB and let it mete the circuÌ„fereÌ„ce at the poynt B. From the poynt B to the points A and C let right lines be drawne BA and BC. I say that the circles whose diameteâ— are the lines BA and BC are equall to the circle ABC and that those circles hauing to their diameters the lines BA and BC are one to the other in the proportion of the line D to the line E. For first that they are equal it is euident by reason that ABC is a triangle rectangle wherfore by the 47. of the first the squares of BA and BC are equall to the square of AC And so by this second it is maniâ—est the two circles to be equall to the circle ABC Secondly as D is to â— so is AF to FC by construction And as the line AF is to the line FC so is the square of the line â—A to the square of the line BC. Which thing we will briefely pâ—oue thus The parallelogramme contayned vnder AC and AF is equall to the square of BA by the Lemma after the 32. of the tenth booke and by the same Lemma or Assumpt the parallelogramme contayned vnder AC and â—C is equall to the square of the line BC. Wherfore as the first parallelogramme hath it selfe to the secondâ— so hath the square of BA equall to the first parallelogramme it selfe to the square of BC equall to the second parallelogramme But both the parallelograÌ„mes haue one heigth namely the line AC and bases the lines AF and FC wherefore as AF is to FC so is the parallelogâ—amme contayned vnder AC AF to the parallelogramme contayned vnder AC FC by the fiâ—st of the sixth And therefore as AF is to FC so is the square of BA to the square of BC. And as the square of BA is to the square of BC so is the circle whose diameter is BA to the circle whose diameter is BC by this second of the twelfth Wherefore the circle whose diameter is BA is to the circle whose diameter is BC as D is to E. And before we proued them equall to the circle ABC Wherâ—fore a circle being geuen we haue found two circles equall to the same which haue the one to the other any proportion geuen in two right lines Which ought to be done Note Heâ—e may you perâ—eiue an other way how to execute my first probleme for if you make a right angle conteyned of the diameters geueÌ„ as in this figure suppose them BA and BC and then subtend the right angle with the line AC and from the right angle let fall a line perpendicular to the base AC that perpâ—ndicular at the point of his fall deuideth AC into AF and FC of the proportion required A Corollary It followeth of thinges manifestly proued in the demonstration of this probleme that in a triangle rectangle if from the right angle to the base a perpendicular be let fall the same perpendicular cutteth the base into two partes in that proportion one to the other that the squares of the righâ— lines conteyning the right angle are in one to the other those on the one side the perpendicular being compared to those on the other both square and segment A Probleme 7. Betwene two circles geuen to finde a circle middell proportionall Let the two circles geuen be ACD and BEF I say that a circle is to be fouÌ„d which betwene ACD and BEF is middell proportionall Let the diameter of ACD be AD and of BEF let Bâ— be the diameter betwene AD and BF finde a line middell proportionall by the 13. of the sixth which let be HK I say that a circle whose diameter is HK is middell proportionall betwene ACD and BEF To AD HK and BF three right lines in continuall proportion by construction let a fourth line be found to which BF shal haue that proportion that AD hath to HK by the 12. of the sixth let that line be â— It is manifest that the â—ower lines AD HK BF and L are in continuall proportion For by coÌ„struction as AD is to HK so is Bâ— to L. And by construction on as AD is to HK so is HK to BF wherefore HK is to BF as BF is to L by the 11. of the fifth wherfore the 4. lines are in continuall proportion Wherefore as the first is to the third that is AD to BF so is the square of the first to the square of the second that is the square of AD to the square of HK by the corollary of the 20. of the sixth And by the same corollary as HK is to L so is the square of HK to the square of BF But by alternate proportion the line AD is to BF as HK is to L wherefore the square of AD is to the square of HK as the square of HK is to the square of BF Wherefore the square of HK is middell proportionall betwene the square of AD and the square of BF But as the squares are
with the altitudes of the sayd pyramids A and B shall be equall by the 6. of this booke Wherefore by the first part of this proposition the bases of the pyramids C to D are reciprokall with the altitudes of D to C. But in what proportion are the bases C to D in the same are the bases A to B forasmuch as they are equall And in what proportion are the altitudes of D to C in the same are the altitudes of B to A which altitudes are likewise equall Wherefore by the 11. of the fifth in what proportion the bases A to B are in the same reciprokally are the altitudes of the pyramids B to A. In like sort by the second part of this proposition may be proued the conuerse of this corollary The same thing followeth also in a Prisme and in a sided columne as hath before at large bene declared in the corollary of the 40. proposition of the 11. booke For those solides are in proportioÌ„ the one to the other as the pyramids or parallelipipedons for they are either partes of equemultiplices or equemultiplices to partes The 10. Theoreme The 10. Proposition Euery cone is the third part of a cilinder hauing one and the selfe same base and one and the selfe same altitude with it SVppose that there be a cone hauing to his base the circle ABCD and let there be a cilinder hauing the selfe same base and also the same altitude that the cone hath Then I say that the cone is the third part of the cilinder that is that the cilinder is in treble proportion to the cone For if the cilinder be not in treble proportion to the cone then the cilinder is either in greater proportions then triple to the cone or els in lesse First let it be in greater then triple And describe by the 6. of the fourth in the circle ABCD a square ABCD. Now the square ABCD is greater then the halfe of the circle ABCD For if about the circle ABCD we describe a square the square described in the circle ABCD is the halfe of the square described about the circle And let there be Parallelipipedon prismes described vpon those squares equall in altitude with the cilinder But prismes are in that proportion the one to the other that their bases are by the 32. of the eleuenth and 5. Corollary of the 7. of this booke Wherefore the prisme described vpon the square ABCD is the halfe of the prisme described vpon the square that is described about the circle Now the clinder is lesse then the prisme which is made of the square described abouâ— the circle ABCD being equal in altitude with it for it contayneth it Wherfore the prisme described vpon the square ABCD and being equall in altitude with the cylinder is greater then half the cylinder Deuide by the 30. of the third the circumferences AB BC CD and DA into two equall parts in the points E F G H And draw these right lines AE EB BF FC CG GD DH HA. Wherfore euery one of these triangles AEB BFC CGD and DHA is greater then halfe of that segment of the circle ABCD which is described about it as we haue before in the 2. proposition declared Describe vpon euery one of these triangles AEB BFC CGD and DHA a prisme of equall altitude with the cylinder Wherefore euery one of these prismes so described is greater then the halfe part of the segment of the cylinder that is set vpon the sayd segments of the circle For if by the pointes E F G H be drawen parallell lines to the lines AB BC CD and DA and then be made perfect the parallelogrammes made by those parallell lines and moreouer vpon those parallelograÌ„mes be erected parallelipipedons equall in altitude with the cylinder the prismes which are described vpon eche of the triangles AEB BFC CGD and DHA are the halfes of euery one of those parallelipipedons And the segments of the cylinder are lesse then those parallelipipedons so described Wherefore also euery one of the prismes which are described vpon the triangles AEB BFC CGD and DHA is greater then the halfe of the segment of the cylinder set vpon the sayd segment Now therefore deuiding euery one of the circumferences remaining into two equall partes and drawing right lines and raysing vp vpon euery one of these triangles prismes equall in altitude with the cylinder and doing this continually we shall at the length by the first of the tenth leaue certaine segments of the cylinder which shal be lesse then the excesse whereby the cylinder excedeth the cone more then thrise Let those segments be AE EB BF FC CG GD DH and HA. Wherfore the prisme remayning whose base is the poligonon â—igure AEBFCGDH and altitude the selfe same that the cylinder hath is greater then the cone taken three tymes But the prisme whose base is the poligonon figure AEBFCGDH and altitude the selfe same that the cylinder hath is treble to the pyramis whose base is the poligonon figure AEBFCGDA and altitude the selfe same that the cone hath by the corollary of the 3. of this booke Wherfore also the pyramis whose base is the poligonon figure AEBFCGDH and toppe the self same that the cone hath is greater then the cone which hath to his base the circle ABCD. But it is also lesse for it is contayned of it which is impossible Wherefore the cylinder is not in greater proportion then triple to the cone I say moreouer that the cylinder is not in lesse proportion then triple to the coneâ— For if it be possible let the cylinder be in lesse proportion then triple to the cone Wherefore by conuersion the cone is greater then the third part of the cylinder Describe now by the sixth of the fourth in the circle ABCD a square ABCD. Wherefore the square ABCD is greater then the halfe of the circle ABCD vpon the square ABCD describe a pyramis hauing one the selfe same altitude with the cone Wherfore the pyramis so described is greater theÌ„ halfe of the cone For if as we haue before declared we describe a square about the circle the square ABCD is the halfe of the square described about the circle and if vppon the squares be described parallelipipedons equall in altitude with the cone which solides are also called prismes the prisme or parallelipipedon described vpoÌ„ the square ABCD is the halfe of the prisme which is described vpoÌ„ the square described about the circle for they are the one to the other in that proportioÌ„ that their bases are by the 32. of the eleueÌ„th 5. corollary of the 7. of this booke Wherfore also their third parts are in the self same proportion by the 15. of the fift Wherfore the pyramis whose base is the square ABCD is the halfe of the pyramis set vpon the square described about the circle But the pyramis set vpon the square described about the circle is greater then the cone whome
FD. Wherfore cones cylinders consisting vpon equal bases are in proportion the one to the other as their altitudes which was required to be demonstrated ¶ The 15. Theoreme The 15. Proposition In equall Cones and Cylinders the bases are reciprokall to their altitudes And cones and Cylinders whose bases are reciprokall to their altitudes are equall the one to the other SVppose that these cones ACL EGN or these cylinders AX EO whose bases are the circles ABCD EFGH and axes KL and MN which axes are also the altitudes of the cones cylinders be equall the one to the other TheÌ„ I say that the bases of the cylinders XA EO are reciprokal to their altitudes that is that as the base ABCD is to the base EFGH so the altitude MN to the altitude KL For the altitude KL is either equall to the altitude MN or not First let it be equall But the cylinder AX is equal to the cylinder EQ But cones and cylinders consisting vnder one and the selfe same altitude are in proportion the one to the other as their bases are by the 11. of the twelueth Wherfore the base ABCD is equall to the base EFGH Wherefore also they are reciprokal as the base ABCD is to the base EFGH so is the altitude MN to the altitude KL But now suppose that the altitude LK be not equall to the altitude M. N but let the altitude MN be greater And by the 3. of the first from the altitude MN take away PM equall to the altitude KL so that let the line PM be put equal to the line KL And by the point P let there be extended a playne superâ—icies TVS which let cut the cylinder EO and be a parallell to the two opposite playne superâ—icieces that is to the circles EFGH and RO. And making the base the circle EFGH the altitude MP imagine a cylinder ES. And for that the cylinder AX is equall to the cylinder EO and there is an other cylinder ES therfore by the 7. of the fift as the cylinder AX is to the cylinder ES so is the cylinder EO to the cylinder ES. But as the cylinder AX is to the cylinder ES so is the base ABCD to the base EFGH For the cylinders AX and ES are vnder one and the selfe same altitude And as the cylinder EO is to the cylinder ES so is the altitude MN to the altitude MP For cylinders coÌ„sisting vpoÌ„ equall bases are in proportion the one to the other as their altitudes Wherfore as the base ABCD is â—o the base EFGH so is the altitude MN to the altitude MP But the altitude PM is equall to the altitude KL Wherefore as the base ABCD is to the base EFGH so is the altitude MN to the altitude KL Wherefore in the equall cylinders AX and EO the bases are reciprokall to their altitudes But now suppose that the bases of the cylinders AX and EO be reciprokal to their altitudes that is as the base ABCD is to the base EFGH so is the altitude MN to the altitude KL Then I say that the cylinder AX is equall to the cylinder EO For the selfe same order of constructioÌ„ remayning for that as the base ABCD is to the base EFGH so is the altitude MN to the altitude KL but the altitude KL is equall to the altitude PM Wherefore as the base ABCD is to the base EFGH so is the altitude MN to the altitude PM But as the base ABCD is to the base EFGH so is the cylinder AX to the cylinder ES for they are vnder equall altitudes and as the altitude MN is to the altitude PM so is the cylinder EO to the cylinder ES by the 14. of the twelueth Wherefore also as the cylinder AX is to the cylinder ES so is the cylinder EO to the cylinder ES. Wherefore the cylinder AX is equall to the cylinder EO by the 9. of the fift And so also is it in the cones which haâ—â— the selfe same bases and altitudes with the cylinders Wherefore in equall cones and cylinders the bases are reciprokall to their altitudes c. which was required to be demonstrated A Corrollary added by Campane and Flussas Hitherto hath bene shewed the passions and proprieties of cones and cylinders whose altitudes fall perpendicularly vpon the bases Now will we declare that cones and cilinders whose altitudes fall obliquely vpon their bases haue also the selfe same passions and proprieties which the foresayd cones and cilinders haue Forasmuch as in the tenth of this booke it was sayd that euery Cone is the third part of a cilinder hauing one and the selfe same base one the selfe same altitude with it which thing was demoÌ„strated by a cilinder geuen whose base is cut by a square inscribed in it and vpon the sides of the square are described Isosceles triangles making a poligonon figure and againe vpon the sides of this poligonon figure are infinitely after the same maner described other Isosceles triangles taking away more theÌ„ the halfe as hath ofteÌ„times bene declared therfore it is manifest that the solides set vpon these bases being vnder the same altitude that the cilinder inclined is and being also included in the same cilinder do take away more then the halfe of the cilinder and also more theÌ„ the halfe of the residue as it hath bene proued in erected cylinders For these inclined solides being vnder equall altitudes and vpon equall bases with the erected solides are equall to the erected solides by the corollary of the â—0 of the eleuenth Wherfore they also in like sort as the erected take away more then the halfe If therfore we coÌ„pare the inclined cilinder to a cone set vpon the selfe same base and hauing his altitude erected and reason by an argument leading to an impossibilitie by the demonstration of the tenth of this booke we may proue that the sided solide included in the inclined cylinder is greater then the triple of his pyramis and it is also equall to the same which is impossible And this is the first case wherein it was proued that the cilinder not being equall to the triple of the cone is not greater then the triple of the same And as touching the second case we may after the same maner conclude that that â—ided solide contayned in the cylinâ—er is greater then the cylinder which is very absurdâ— Wherefore if the cylinder be neither greater then the triple of the cone nor lesse it must nedes be equall to the same The demonstration of these inclined cylinders most playnely followeth the demonstration of the erected cylinders for it hath already bene proued that pyramids and sided solides which are also called generally Prismes being set vpon equall bases and vnder one and the selfe same altitude whether the altitude be erected or inclined are equall the one to the other namely are in proportion as their bases are by
line which subtendeth the angle ZOB to the third line which subtendeth the angle ZKB But by construction BO is equall to BK therefore OZ is equall to KZ And the third alâ—o is equall to the third Wherefore the point Z in respecte of the two triangles rectangles OZB and KZB determineth one and the same magnitude iâ— the line BZ Which can not be if any other point in the line BZ were assigned nearer or farther of from the point B. One onely poynt therefore is that at which the two perpendiculars KZ and OZ fall But by construction OZ falleth at Z the point and therefore at the same Z doth the perpendicular drawen from K fall likewyse Which was required to be demonstrated Although a briefe monition mought herein haue serued for the pregnant or the humble learner yet for them that are well pleased to haue thinges made plaine with many wordes and for the stiffenecked busie body it was necessary with my controlment of other to annexe the cause reason therof both inuincible and also euident A Corollary 1. Hereby it is manifest that two equall circles cutting one the other by the whole diameter if from one and the same end of their common diameter equall portions of their circumferences be taken and from the pointes ending those equall portions two perpendiculars be let downe to their common diameter those perpendiculars shall fall vpon one and the same point of their common diameter 2. Secondly it followeth that those perpendiculars are equall ¶ Note From circles in our first supposition eche to other perpendicularly erected we procede and inferre now these Corollaries whether they be perpendicularly erected or no by reasou the demonstration hath a like force vpon our suppositions here vsed ¶ The 16. Theoreme The 18. Proposition Spheres are in treble proportion the one to the other of that in which their diameters are SVppose that there be two spheres ABC and DEF and let their diameters be BC and EF. Then I say that the sphere ABC is to the sphere DEF in treble proportion of that in which the diameter BC is to the diameter EF. For if not then the sphere ABC is in treble proportion of that in which BC is to EF either to some sphere lesse then the sphere DEF or to some sphere greater First let it be vnto a lesse namely to GHK And imagine that the spheres DEF and GHK be both about one and the selfe same centre And by the proposition next going before describe in the greater sphere DEF a polihedron or a solide of many sides not touching the superficies of the lesse sphere GHK And suppose also that in the sphere ABC be inscribed a polihedron like to the polihedron which is in the sphere DEF Wherefore by the corollary of the same the polihedron which is in the sphere ABC is to the polihedron which is in the sphere DEF in treble proportion of that in which the diameter BC is to the diameter EF. But by supposition the sphere ABC is to the sphere GHK in treble proportion of that in which the diameter BC is to the diameter EF. Wherefore as the sphere ABC is to the sphere GHK so is the polihedroÌ„ which is described in the sphere ABC to the polihedroÌ„ which is described in the sphere DEF by the 11. of the fift Wherfore alternately by the 16. of the fift as the sphere ABC is to the polihedron which is described in it so is the sphere GHK to the polihedron which is in the sphere DEF But the sphere ABC is greater then the polihedroÌ„ which is described in it Wherfore also the sphere GHK is greater then the polihedroÌ„ which is in the sphere DEF by the 14. of the fift But it is also lesse for it is contayned in it which impossible Wherefore the sphere ABC is not in treble proportioÌ„ of that in which the diameter BC is to the diameter EF to any sphere lesse then the sphere DEF In like sort also may we proue that the sphere DEF is not in treble proportion of that in which the diameter EF is to the diameter BC to any sphere lesse then the sphere ABC Now I say that the sphere ABC is not in treble proportioÌ„ of that in which the diameter BC is to the diameter EF to any sphere greater theÌ„ the sphere DEF For if it be possible let it be to a greater namely to LMN Wherfore by conuersion the sphere LMN is to the sphere ABC in treble proportion of that in which the diameter EF is to the diameter BC. But as the sphere LMN is to the sphere ABC so is the sphere DEF to some sphere lesse theÌ„ the sphere ABC as it hath before bene proued for the sphere LMN is greater then the sphere DEF Wherfore the sphere DEF is in treble proportioÌ„ of that in which the diameter EF is to the diameter BC to some sphere lesse theÌ„ the sphere ABC which is proued to be impossible Wherefore the sphere ABC is not in treble proportion of that in which BE is to EF to any sphere greater theÌ„ the sphere DEF And it is also proued that it is not to any lesse Wherefore the sphere ABC is to the sphere DEF in treble proportion of that in which the diameter BC is to the diameter EF which was required to be demonstrated A Corrollary added by Flussas Hereby it is manifest that spheres are the one to the other as like Polihedrons and in like sort described in them are namely eche are in triple proportion of that in which the diameters A Corollary added by Mâ— Dee It is then euident how to geue two right lines hauing that proportion betwene them which any two spheres geuen haue the one to the other For if to their diameters as to the first and second lines of fower in continuall proportion you adioyne a third and a fourth line in continuâ—ll proportion as I haue taught before The first and fourth lines shall aunswere the Pâ—obleme How generall this rule is in any two like solides with their correspondent or Omologall lines I neede not with more wordes declare ¶ Certaine Theoremes and Problemes whose vse is manifolde in Spheres Cones Cylinders and other solides added by Ioh. Dee A Theoreme 1. The whole superficies of any Sphere is quadrupla to the greatest circle in the same sphere contayned It is needeles to bring Archimedes demonstration hereof into this place seing his boke of the Sphere and Cylinder with other his woâ—kes are euery where to be had and the demoÌ„stration therof easie A Theoreme 2. Euery sphere is quadruplâ— to that Cone whose base is the greatest circle height the semidiameter of the same sphere This is the 32. Proposition of Archimedes fiâ—st booke of the Sphere and Cylinder A Probleme 1. A Sphere being geuen to make an vpright Cone equall to the same or in any other proportioâ— geuen betwene two right lines And as concerning the other part of
also deuided into two equall parts being cylinders which two equall cylinders let be IG and FK the axe of IG suppose to be HN and of FK the axe to be NM And for that FG is an vpright cylinder and at the poynt N cut by a playne Superficies parallell to his opposite bases the common section of that playne superficies and the cylindâ—r FG must be a circle equall to his base FLB and haue his center the point N. Which circle let be IOK And seing that FLB is by supposition equall to the greatest circle in A IOK also shall be equall to the greatest circle in A contained Also by reason MH is by supposition equal to the diameter of A and NH by constructioÌ„ half of MH it is manifest that NH is equall to the semidiameter of A. If therefore you suppose a cone to haue the circle IOK to hiâ— base and NH to his heith the sphere A shall be to that Cone quadrupla by the 2. Theoreme Let that cone be HIOK Wherefore A is quadrupla to HIOK And the Cylinder IG hauing the same base with HIOK the circle IOK and the same heith the right line NH is triple to the cone HIOK by the 10. of this twelfth booke But to IG the whole cylinder FG is double as is proued Wherefore FG is triple and triple to the cone HIOK that is sextuple And A is proued quadrupla to the same HIOK Wherefore FG is to HIOK as 6. to 1 and A is to HIOK as 4. to 1 Therfore FG is to A as 6 to 4 which in the least termes is as 3 to 2. but 3 to 2 is the termes of sesquialtera proportion Wherefore the cylinder FG is to A sesquialtera in proportion Secondly forasmuch as the superficies of a cylinder his two opposite bases excepted is equall to that circle whose semidiameter is middell proportionall betwene the side of the cylinder and the diameter of his base as vnto the 10. of this booke I haue added But of FG the side BG being parallell and equall to the axe MH must also be equall to the diameter of A. And the base FLB being by supposition equall to the greatest circle in A contained must haue his diameter FB equal to the sayd diameter of A. The middle proportional therfore betwene BG and FB being equall eche to other shalâ— be a line equall to either of them As i◠〈◊〉 set BG and FB together as one line and vpon that line composed as a diameter make a semicirclâ— and from the center to the circumference draw a linâ— perpendicular to the sayd diameter by the 〈◊〉 of the sixth that perpendicular is middel proportional betwene FB and BG the semidiametersâ— and he him selfe also a semidiameter and therfore by the definition of a circle equall to FB and likewise to BG And a circle hauing his semidiameter equall to the diameter FB is quadruple to the circle FLB For the square of euery whole line is quadruple to the squâ—re of his halfe line as may be proued by the 4. of the second and by the second of this twelfth circles are one to the other as the squares of their diameters are Wherfore the superficies cylindricall of FG alone is quadrupla to his base FLB But if a certayne quantity be dupla to one thing and an other quadrupla to the same one thing those two quantities together are sextupla to the same one thing Therefore seing the base opposite to FLB being equall to to FLB added to FLB maketh that coÌ„pound double to FLB that double added to the cylindricall superâ—icies of FG doth make a superficies sextupla to FLB And the superficies of A is quadrupla to the same FLB by the first Theoreme Therefore the cylindricall superficies of FG with the superficieces of his two bases is to the superficies FLB as 6 to 1 and the superficies of A to FLB is as 4 to 1. Wherfore the cylindricall superficies of FG his two bases together are to the superficies of A as 6 to 4 that is in the smallest termes as 3 to 2. Which is proper to sesâ—uialtera proportion Thirdly it is already made euident that the superficies cylindrical of FG onely by it self is quadrupla to FLB And also it is proued that the superficies of the sphere A is quadrupla to the same FLB Wherefore by the 7. of the fifth the cylindricall superficies of FG is equall to the superficies of A. Therfore euery cylinder which hath his base the greatest circle in a sphere and heith equal to the diameter of that sphere is sesquialtera to that spere Also the superficies of that cylinder with his two bases is sesquialtera to the superficies of the sphere and without his two bases is equall to the superficies of the sphere which was to be demonstrated The Lemma If A be to C as 6 to 1 and B to C as 4 to 1 A is to B as 6 to 4. For seing B is to C as 4 to 1 by supposition therefore backward by the 4. of the fifth C is to B as 1 to 4. Imagine now two orders of qnantities the first A C and B the second 6 1 and 4. Forasmuch as A is to C as 6 to 1 by supposition and C is to B as 1 to 4 as we haue proued wherfore A is to B as 6 to 4 by the 22 of the fift Therfore if A be to C as 6 to 1 and B to C as 4 to 1 A is to B as 6 to 4. which was to be proued Note Sleight things some times lacking euideÌ„t proufe brede doubt or ignorance And I nede not warnâ— you how genâ—rall this demonstration is for if you put in the place of 6 and 4 any other numbers the like manner of conclusion will follow So likewise in place of 1. any other one number may be as if A be to C as â— to 5 and B vnto C be as 7 to 5 A shall be to B as 6 to 7. c. A Probleme 4. To a Sphere geuen to make a cylinder equall or in any proportion geuen betwene two right lines Suppose the geuen Sphere to be A and the proportion geuen to be that betwene X and Y. I say that a cylinder is to be made equall to Aâ— or els in the same proportion to A that is betwene X to Y. Let a cylinder be made such one as the Theoreme next before supposed that shall haue his base equall to the greatest circle in A and height equall to the diameter of A Let that cylinder bâ— the vpright cylinder BC. Leâ— the one side of BC be the right line QC Deuide QC into three equal partâ— of which let QE containe two and let the third part be CE. By the point E suppose a plaine parallel to the bases of BC to passe through the cylinder BC cutting the same by the circle DE. I say that the cylinder BE is equall to the Sphere A. For seing BC being an
conteyned in A the sphere be the circle BCD And by the probleme of my additions vpon the second proposition of this booke as X is to Y so let the circle BCD be to an other circle found let that other circle be EFG and his diameter EG I say that the sphericall superficies of the sphere A hath to the sphericall superficies of the sphere whose greatest circle is EFG or his equall that proportion which X hath to Y. For by construction BCD is to EFG as X is to Y and by the theoreme next beforeâ— as BCD is to â—FG so is the spherical superficies of A whose greatest circle is BCD by supposition to the sphericall superficies of the sphere whose greatest circle is EFG wherefore by the 11. of the fifth as X is to Y So is the sphericall superficies of A to the sphericall superficies of the sphere whose greatest circle is EFG wherefore the sphere whose diameter is EG the diameter also of EFG is the sphere to whose sphericall superficies the sphericall superficies of the sphere A hath that proportion which X hath to Y. A sphere being geuen therefore we haue geuen an other sphere to whose sphericall superficies the superficies sphericall of the sphere geuâ— hath any proportion geuen betwene two right lines which ought to be done A Probleme 10. A sphere being geuen and a Circle lesse then the greatest Circle in the same Sphere conteyned to coapt in the Sphere geuen a Circle equall to the Circle geuen Suppose A to be the sphere geuen and the circle geuen lesse then the greatest circle in A conteyned to be FKG I say that in the Sphere A a circle equall to the circle FKG is to be coapted First vnderstand what we meane here by coapting of a circle in a Sphere We say that circle to be coapted in a Sphere whose whole circumference is in the superficies of the same Sphere Let the greatest circle in the Sphere A conteyned be the circle BCD Whose diameter suppose to be BD and of the circle FKG let FG be the diameter By the 1. of the fourth let a line equall to FG be coapted in the circle BCD Which line coapted let be BE. And by the line BE suppose a playne to passe cutting the Sphere A and to be perpendicularly erected to the superficies of BCD Seing that the portion of the playne remayning in the sphere is called their common section the sayd section shall be a circle as before is proued And the common section of the sayd playne and the greatest circle BCD which is BE by supposition shall be the diameter of the same circle as we will proue For let that circle be BLEM Let the center of the sphere A be the point H which H is also the ceÌ„ter of the circle BCD because BCD is the greatest circle in A conteyned From H the center of the sphere A let a line perpendicularly be let fall to the circle BLEM Let that line be HO and it is euident that HO shall fall vpon the common section BE by the 38. of the eleuenth And it deuideth BE into two equall parts by the second part of the third proposition of the third booke by which poynt O all other lines drawne in the circle BLEM are at the same pointe O deuided into two equall parts As if from the poynt M by the point O a right line be drawne one the other side comming to the circumference at the poynt N it is manifest that NOM is deuided into two equall partes at the poynt O by reason if from the center H to the poyntes N and M right lines be drawne HN and HM the squares of HM and HN are equall for that all the semidiameters of the sphere are equal and therefore their squares are equall one to the other and the square of the perpendicular HO is common wherefore the square of the third line MO is equall to the square of the third line NO and therefore the line MO to the line NO So therefore is NM equally deuided at the poynt O. And so may be proued of all other right lines drawne in the circle BLEM passing by the poynt O to the circumference one both sides Wherefore O is the center of the circle BLEM and therefore BE passing by the poynt O is the diameter of the circle BLEM Which circle I say is equal to FKG for by construction BE is equall to FG and BE is proued the diameter of BLEM and FG is by supposition the diameter of the circle FKG wherefore BLEM is equall to FKG the circle geuen and BLEM is in A the sphere geueÌ„ Wherfore we haue in a sphere geuen coapted a circle equall to a circle geuen which was to be done A Corollary Besides our principall purpose in this Probleme euidently demonstrated this is also made manifest that if the greatest circle in a Sphere be cut by an other circle erected vpon him at right angles that the other circle is cut by the center and that their common section is the diameter of that other circle and therefore that other circle deuided is into two equall partes A Probleme 11. A Sphere being geuen and a circle lesse then double the greatest circle in the same Sphere contained to cut of a segment of the same Sphere whose Sphericall superficies shall be equall to the circle geuen Suppose K to be a Sphere geuen whose greatest circle let be ABC and the circle geuen suppose to be DEF I say that a segment of the Sphere K is to be cut of so great that his Sphericall superficies shall be equall to the circle DEF Let the diameter of the circle ABC be the line AB At the point A in the circle ABC coapt a right line equall to the semidiameter of the circle DEF by the first of the fourth Which line suppose to be AH From the point H to the diameter AB let a perpendicular line be drawen which suppose to be HI Produce HI to the other side of the circumference and let it come to the circumference at the point L. By the right line HIL perpendicular to AB suppose a plaine superficies to passe perpendicularly erected vpon the circle ABC and by this plaine superficies the Sphere to be cut into two segmentes one lesse then the halfe Sphere namely HALI and the other greater then the halfe Sphere namely HBLI I say that the Sphericall superficies of the segment of the Sphere K in which the segment of the greatest circle HALI is contayned whose base is the circle passing by HIL and toppe the point A is equall to the circle DEF For the circle whose semidiameter is equall to the line AH is equall to the Sphericall superficies of the segment HAL by the 4. Theoreme here added And by construction AH is equall to the semidiameter of the circle DEF therefore the Sphericall superficies of the segment of the Sphere K cut of by the
lines added together shal be quintuple to the square made of the halfe of the whole lyne SVppose that the right line AB be deuided by an extreme and meane proportioÌ„ in the point C. And let the greater segment therof be AC And vnto AC adde directly a ryght line AD and let AD be equall to the halfe of the line AB Then I say that the square of the line CD is quintuple to the square of the line DA. Describe by the 46. of the first vpon the lines AB and DC squares namely AE DF. And in the square DF describe and make complete the figure And extend the line FC to the point G. And forasmuch as the line AB is deuided by an extreme and meane proportion in the point C therefore that which is contayned vnder the lines AB and BC is equall to the square of the line AC But that which is contayned vnder the lines AB and BC is the parallelogramme CE and the square of the line AC is the square HF. Wherefore the parallelogramme CE is equall to the square HF. And forasmuch as the line BA is double to the line AD by constructâ—on 〈◊〉 the lyne BA is equall to the line KA and the line AD to the lyne AH therefore also the lyne KA is double to the line AH But as the lyne KA is to the line AH so is the parallelogramme CK to the parallelogramme CH Wherefore the parallelogramme CK is double to the parallelogramme CH. And the parallelogrammes LH and CH are double to the parallelogramme CH for supplementes of parallelogrammes are bâ— the 4â— of the first equall the one to the other Wherefore the parallelogramme CK is equall to the parallelogrammes LH CH. And it is proued that the parallelogramme CE is equall to the square FH Wherefore the whole square AE is equall to the gnâ—mon MXN And forasmuch as the line BA iâ— double to the line AD therefore the square of the line BA is by the 20. of the sixth quadruple to the square of the line DA that is the square AE to the square DH But the square AE is equall to the gnomoÌ„ MXN wherefore the gnomoÌ„ MXN is also quadruple to the square DH Wherefore the whole square DF is quintuple to the square DH But the square DF iâ— the square of the line CD and the square DH is the square of the line DA. Wherefore the square of the line CD is quintuple to the square of the line DA. If therefore a right line be deuided by an extreame and meane proportion and to the greater segment be added the halfe of the whole line the square made of those two lines added together shal be quintuple to the square made of the halfe of the whole line Which was required to be demonstrated Thys proposition is an other way demonstrated after the fiueth proposition of this booke The 2. Theoreme The â— Proposition If a right line be in power quintuple to a segment of the same line the double of the sayd segment is deuided by an extreame and meane proportion and the greater segment thereof is the other part of the line geuen at the beginning Now that the double of the line AD that is AB is greater then the line AC may thus be proued For if not then if if it be possible let the line AC be double to the line AD wherefore the square of the line AC is quadruple to the square of the line AD. Wherefore the squares of the lines AC and AD are quintuple to the squares of the line AD. And it is supposed that the square of the line DC is quintuple to the square of the line AD wherefore the square of the line DC is equall to the square of the lines AC and AD which is impossible by the 4. of the second Wherefore the line AC is not double to the line AD. In like sorte also may we proue that the double of the line AD is not lesse then the line AC for this is much more absurd wherefore the double of the line AD is greater theÌ„ the line ACâ— which was required to be proued This proposition also is an other way demonstrated after the fiueth proposition of this booke Two Theoremes in Euclides Method necessary added by M. Dee A Theoreme 1. A right line can be deuided by an extreame and meane proportion but in one onely poynt Suppose a line diuided by extreame and meane proportion to be AB And let the greater segment be AC I say that AB can not be deuided by the sayd proportion in any other point then in the point C. If an aduersary woulde contend that it may in like sort be deuided in an other point let his other point be supposed to be D making AD the greater segment of his imagined diuision Which AD also let be lesse then our AC for the first discourse Now forasmuch as by our aduersaries opinion AD is the greater segment of his diuided lineâ— the parallelogramme conteyned vnder AB and DB is equall to the square of AD by the third definition and 17. proposition of the sixth Booke And by the same definition and proposition the parallelogramme vnder AB and CB conteyned is equall to the square of our greater segment AC Wherefore as the parallelogramme vnder AB and Dâ— is to the square of AD so i◠〈◊〉 parallelogramme vnder AB and CB to the square of AC For proportion of equality is concluded in them both But forasmuch as Dâ—â— iâ— byâ— supposition greater theÌ„ CB the parallelograÌ„me vnder AB and DB is greater then the parallelogramme vnder AC and CB by the first of the sixth for AB is their equall heith Wherefore the square of AD shal be greater then the square of AC by the 14. of the fifth But the line AD is lesse then the line AC by supposition wherefore the square of AD is lesse then the square of AC And it is concluded also to be greater then the square of AC Wherefore the square of AD is both greater then the square of ACâ— and also lesse Which is a thing impossible The square therefore of AD is not equall to the parallelogramme vnder AB and DB. And therefore by the third definition of the sixth AB is not deuided by an extreame and meane proportion in the point D as our aduersary imagined And Secondly in like sort will the inconueniency fall out if we assigne AD our aduersaries greater segment to be greater then our AC Therefore seing neither on the one side of our point C neither on the other side of the same point C any point can be had at which the line AB can be deuided by an extreame and meane proportion it followeth of necâ—ssitie that AB can be deuided by an extreame and meane proportion in the point C onely Therefore a right line can be deuided by an extreame
third proposition we will vse the same suppositions and constructions there specified so farre as they shall serue our purpose Beginning therefore at the conclusion we must infer the part of the proposition before graunted It was concluded that the square of the line DB is quintuple to the square of the line DC his owne segment Therefore DN the square of DB is quintuple to GF the square of DC But the squaâ—e of AC the double of DC which is RS is quadruple to GF by the second Corollary of the 20. of the sixth and therefore RS with GF are quintuple to GF and so it is euident that the square DN is equall to the square RS together with the square GF Wherefore from those two equalles taking the square GF common to them both remayneth the square RS equall to the Gnomon XOP But to the Gnomon XOP the parallelogramme CE is equall Wherefore the square of the line AC which is RS is equâ—ll to the parallelograÌ„me Câ— Which parallelogamme is coÌ„tained vnder BE equall to AB and CB the part remayning of the first line gâ—uen which was DB. And the line AB is made of the double of the segment DC and of CBâ— the other part of the line DB first gouen Wherefore the double of the segment DC with CB the part remayning which altogether is the whole line AB is to AC the double of the segment DC as that same AC is to CB by the second part of the 16. of the sixth Therfore by the 3. definitioÌ„ of the sixth booke the whole line AB is deuided by an extreme and meane proportion AC the double of the segmeÌ„t DC being middell proportionall is the greater part therof Wheâ—efore if a right line be quintuple in power c. as in the proposition which was to be demonstrated Or thus it may be demonstrated Forasmuch as the square DN is quinâ—uple to the square GF I meane the square of DB the line geueÌ„ to the square oâ— DC the segmeÌ„t And the same square DN is equall to the parallelograÌ„me vnder AB CB with the square made of the line DC by the sixth of the second for vnto the line AC equally deuided the line CB is as it were adioyned Wherefore the parallelogramme vnder AB CB together with the square of DC which is GF is quintuple to the square GF made oâ— thâ— line DC Taking then that square GF â—rom the parallelogramme vnder AB CB that parallelogramme vnder AB CB remayning alone is but quadruple to the sayd square of the line DC But by the 4. of the second or the second Corollary of the 20. of the sixth RS â—he square of the line AC is quadrupla to the same square GFâ— Wherfore by the 7. of the fifth the square of the line AC is equall to the parallelogramme vnder AB CB and so by the second part of the 16. of the sixth AB AC and CB are three lines in continuall proportion And seing AB is greater theÌ„ AC the same AC the double of the line DC shall be greater then the part BC remayning Wherfore by the 3. definition of the sixth AB composed or made of the double of DC and the other part of DB remaining is deuided by an extreme and middel proportion and also his greater segment is AC the double of the segment DC Wherfore If a right line be quintuple in power c. as in the propositionâ— which was to be demonstrated A Theoreme 2. If a right line deuided by an extreme and meane proportion be geuen and to the great segment â—herof he directly adioyned a line equal to the whole line geuen that adioyned line and the said greater segment do make a line diuided by extreme and meane proportion whose greater segment is the line â—dioyned Suppose the line geuen deuided by extreame and meane proportion to be AB deuided in the point C and his greater segment let be AC vnto AC directly adioyne a line equall to AB let that be AD I say that AD together with AC that is DC is a deuided by extreme and middel proportion whose greater segment is AD the line adioyned Deuide AD equally in the point E. Now forasmuch as AE is the halfe of AD by construction it is also the halfe of AB equall to AD by construction Wherfore by the 1. of the thirtenth the square of the line composed of AC and AE which â—ne is EC is quintuple to the square of the line AE Wherefore the double of AE and the line AC composed as in one right line is a line deuided by extreme and meane proportion by the conuerse of this third by me demonstrated and the double of AE is the greater segment But DC is the line composed of the double of AE the line AC and with all AD is the double of AE Wherfore DC is a line deuided by extreme and meane proportion and AD iâ— hiâ— greater segment If a right line therefore deuided by extreme and meane proportion be geuen and to the greater segment thereof be directly adioyned a line equall to the whole line geuen that adioyned line and the sayd greater segment do make a line diuided by extreame and meane proportion whose greater segment is the line adioyned Which was required to be demonstrated Two other briefe demonstrations of the same Forasmuch as AD is to AC as AB is to AC because AD is equall to AB by construction but as AB is to AC so is AC to CB by supposition Therefore by the 11. of the fifth as AC is to CB so is AD to AC Wherefore as AC and CB which is AB is to CB so is AD and AC which is DC to AC Therefore euersedly as AB is to AC so is DC to AD. And it is proued AD to be to AC as AC is to CB. Wherefore as AB is to AC and AC to CB so is DC to AD and AD to AC But AB AC and CB are in continuall proportion by supposition Wherfore DC AD and AC are in continuall proportion Wherefore by the 3. definition of the sixth booke DC is deuided by extreme and middell proportion and his greatest segment is AD. Which was to be demonstrated Note from the marke how this hath two demonstrations One I haue set in the margent by ¶ A Corollary 1. Vpon Euclides third proposition demonstrated it is made euident that of a line deuided by extreame and meane proportion if you produce the lesse segment equally to the length of the greater the line therby adioyned together with the sayd lesse segment make a new line deuided by extreame and middle proportion Whose lesse segment is the line adioyned For if AB be deuided by extreme and middell proportion in the point C AC being the greater segment and CB be produced from the poynt B making a line with CB equall to AC which let be CQ and the
line thereby adioyned let be BQ I say that CQ is a line also deuided by an extreame and meane proportion in the point B and that BQ the line adioyned is the lesse segment For by the thirde it is proued that halfe AC which let be CD with CB as one line composed hath his powre or square quintuple to the powre of the segment CD Wherfore by the second of this booke the double of C D is deuided by extreme and middell proportionâ— and the greater segment thereof shal be CB. But by construction CQ is the double of CD for it is equall to AC Wherefore CQ is deuided by extreme and middle proportion in the point B and the greater segment thereof shal be CB. Wherefore BQ is the lesse segment which is the line adioyned Therefore a line being deuided by extreme and middell proportion if the lesse segment be produced equally to the length of the greater segment the line thereby adioyned together with the sayd lesse segment make a new line deuided by extreme meane proportion whoâ—e lesse segment is the line adioyned Which was to be demonstrated ¶ A Corollary 2. Ifâ— from the greater segment of a line diuided by extreme and middle proportion a line equall to the lesse segment be cut of the greater segment thereby is also deuided by extreme and meane proportion whose greater segmentâ— shall be 〈◊〉 that part of it which is cut of For taking from AC a line equall to CB let AR remayne I say that AC is deuided by an extreme and meane proportion in the point R and that CR the line cut of is the greater segment For it is proued in the former Corollary that CQ is deuided by extreme and meane proportion in the point B. But AC is equall to CQ by construction and CR is equall to CB by construction Wherefore the reâ—idue AR is equall to BQ the residue Seing therfore the whole AC is equall to the whole CQ and the greater part of AC which is CR is equal to CB the greater part of CQ and the lesse segmeÌ„t also equall to the lesse and withall seing CQ is proued to be diuided by extreme meane proportion in the point B it foloweth of necessity that AC is diuided by extreme and meane proportion in the point R. And seing CB is the greater segment of CQ CR shall be the greater segment of AC Which was to be demonstrated A Corollary 3. It is euident thereby a line being diuided by extreme and meane proportion that the line wheâ—â—by the greater segment excedeth the lesse together with the lesse segment do make a line diuided by extreme and meane proportion whose lesse segment is the sayd line of exceesse or difference betwene the segments Iohn Dee ¶ Two new wayes to deuide any right line geuen by an extreme and meane proportion demonstrated and added by M. Dee A Probleme To deuide by an extreme and meane proportion any right line geuen in length and position Suppose a line geuen in length and positionâ— to be AB I say that AB is to be deuided by an extreme and meane proportion Deuide AB into two equall parts as in the point C. Produce AB directly from the point B to the point D making BD equal to BC. To the line AD and at the point D let a line be drawen perpendicular by the 11. of the first which let be DF of what length you will From DF and at the point D cut of the sixth parte of DF by the 9. of the sixth And let that sixth part be the line DG Vppon DF as a diameter describe a semicircle which let be DHF From the point G rere a line perpendicular to DF which suppose to be GH and let it come to the circumference of DHF in the point H. Draw right lines HD and HF. Produce DH from the point H so long till a line adioyned with DH be equall to HF which let be DI equall to HF. From the point H to the point B the one ende of our line geuen let a right line be drawen as HB From the point I let a line be drawen to the line AB so that it be also parallel to the line HB Which parallel line suppose to be IK cutting the line AB at the point K. I say that AB is deuided by an extreme meane proportion in the point K. For the triangle DKI hauing HB parallel to IK hath his sides DK and DI cut proportionally by the 2. of the sixth Wherefore as IH is to HD so is KB to BD. And therfore compoundingly by the 18. of the fiueth as DI is to DH so is DK to DB. But by construction DI is equall to HF wherefore by the 7. of the fifth DI is to DH as HF is to DH Wherefore by the 11. of the fifth DK is to DB as HF is to DH Wherefore the square of DK is to the square of DB as the square of HF is to the square of DH by the 22. of the sixth But the square of HF is to the square of DH as the line GF is to the line GDâ— by my corrollary vpon the 5 probleme of my additions to the second proposition of the twelfth Wherefore by the 11. of the fifth the square of DK is to the square of DB as the line GF is to the line GD But by construction GF is quintuple to GD Wherefore the square of DK is quintuple to the square of DB and therefore the double of DB is deuided by an extreme and meane proportioÌ„ and BK is the greater segment therof by the 2. of this thirtenth Wherefore seing AB is the double of DB by construction the line AB is deuided by an extreme and meane proportion and his greater segment is the line BK Wherefore AB is deuided by an extreme and meane proportion in the point K. We haue therefore deuided by extreme and meane proportion any line geuen in length and position Which was requisite to be done The second way to execute this probleme Suppose the line geuen to be AB Deuide Aâ— into two equall parts as suppose it to be done in the point C. Produce AB from the point B adioyning a line equall to BC which let be BD. To the right line AD and at the point D erect a perpendicular line equall to BD let that be DE. Produce ED froÌ„ the point D to the point F making DF to contayne fiue such equall partes as DE is one Now vpon EF as a diameter describe a semicircle which let 〈◊〉 EKF and let the point where the circumference of EKF doth cut the line AB be the point K. I say that AB is deuided in the point K by an extreme and meane proportion For by the 13. of the sixth ED DK DF are three lines in continuall proportion DK being the middle proportionall â— Wherefore by the corollary of the 20. of the sixth as ED is to DF so is
the square of ED to the square of DK but by construction ED is subquintuple to DF. Wherefore the square of ED is subquintuple to the square of DK And therefore the square of DK is quintuple to the square of ED. And ED is equall to ED by construction therefore the square of DK is quintuple to the square of E D. Wherefore the double of BD is deuided by an extreme and meane proportion whose greater segment is BK â— by the second of this thirteâ—th But by construction AB is the double of â—D â— Wherefore AB is diuided by extreme and meane proportion and his greater segment is BK and thereby K â— the point of the diuision We haue therefore deuided by extreme and meane proportion any right line geuen in length and position Which was to be done Noteâ— Ech of these wayes may well be executed But in the first you haue this auantage that the diameter is taken at pleasure Which â—n the second way is euer iust thrise so long as the line geuen to be deuided Iohn Dee ¶ The 4. Theoreme The 4. Proposition If a right line be deuided by an extreame and meane proportion the squares made of the whole line and of the lesse segmeÌ„t are treble to the square made of the greater segment SVppose that the right line AB be deuided by an extreame meane proportioÌ„ in the point C. And let the greater segment thereof be AC Then I say that the squares made of the lines AB and BC are treble to the square of the line AC Describe by the 46. of the first vpon the line AB a square ADEB And make perfect the figure Now forasmuch as the line AB is deuided by an extreame and meane proportion in the point C and the greater segmeÌ„t thereof is the line AC therefore that which is contayned vnder the lines AB and BC is equall to the square of the line AC But that which is contayned vnder the lines AB and CB is the parallelogramme AK and the square of the line AC is the square FD. Wherefore the parallelogramme AK is equall to the square FD. And the parallelogramme AF is equall to the parallelogramme FE put the square CK common to them both wherfore the whole parallelograÌ„me AK is equall to the whole parallelogramme CE Wherefore the parallelogrammes CE and AK are double to the parallelogramme AK Bâ—t the parallelogrammes AK and CE are the gnomon LMN and the square CK Wherefore the gnomon LMN and the square CK are double to the parallelogramme AK But it is proued that the parallelogramme AK is equal to the square DF. Wherefore the gnomon LMN and the square CK are double to the square DF. Wherefore the gnomon LMN and the squares CK and DF are treble to the square DF. But the gnomon LMN and the squares CK and DF are the whole square AE together with the square CK which are the squares of the lines AB and BC. And DF is the square of the line AC Wherefore the squares of the lines AB and BC are treble to the square of the line AC If therefore a right line be deuided by an extreame and meane proportion the squares made of the whole line and of the lesse segment are treble to the square made of the greater segment which was required to be proued Looke for an other demonstration of this proposition after the fifth proposition of this booke ¶ The 5. Theoreme The 5. Proposition If a right line be deuided by an extreame and meane proportion and vnto it be added a right â—ine equall to the greater segment the whole right line is deuided by an extreame and meane proportion and the greater segment thereof is the right line geuen at the beginning SVppose that the right line AB be deuided by an extreame and meane proportion in the point C and let the greater segment thereof be AC And vnto the line AB adde the line AD equall to the line AC Then I say that the line Dâ— is deuided by an extreame and meane proportion in the point A and the greater segment thereof is the right line pâ—t at the beginning namely AB Describe by the 46. of the first vpon on the line AB a square AE and make perfect the figure And forasmuch as the line AB is deuided by an extreame and meane proportion in the point C therefore that which is contayned vnder the lines AB and BC is equall to the square of the line AC But that which is contayned vnder the lines AB and BC is the parallelogramme CEâ— and the square of the â—â—ne AC is the square CH. Wherefore the parallelogramme CE is equall to the square CH. But vnto the square CH is equall the square DH by the first of the sixth and vnto the parallelogramme CE is equall the parallelogramme HE. Wherefore the parallelogramme DH is equall to the parallelogramme HE. Adde the parallelogramme HB common to them both Wherefore the whole parallelogramme DK is equall to the whole square AE And the parallelogramme DK is that which is contayned vnder the lines BD and DA for the line AD is equall to the line DL the square AE is the square of the line AB Wherfore that which is contayned vnder the lines AD and DB is equall to the square of the line AB Wherefore as the line DB is to the line BA so is the line BA to the line AD by the 17. of the sixth But the line DB is greater then the line BA Wherefore the line BA is greater then the line AD. Wherefore the line BD is deuided by an extreame and meane proportion in the point A and his greater segment is the line AB If therefore a right line be deuided by an extreame and meane proportion and vnto â—t be added a right line equall to the greater segment the whole right line is deuided by an extreame and meane proportion and the greater segment therof is the right line geuen at the beginning which was required to be demonstrated This proposition is agayne afterward demonstrated A Corollary added by Campane Hereby it is 〈…〉 from the greaâ—â—◠〈◊〉 of a line deuided by an extreame meane proportion be 〈◊〉 away 〈◊〉 segment the sayd great a segment shall be deuided by an extreame and meane proportion and the greater segment thereof shall be the line taken away As let the line â—â— be deuided by an extreame and meanâ— proportion in the point C. And leâ— the 〈…〉 line 〈…〉 A. D. I say that AC is also deuided by an extreame and meanâ— proportion in the point D and that his greater portion is DC For by the definitino of a line so deuided AB is to AC as AC is to CB. But as AC is to CB so is AC to DC by the 7. of the 〈…〉 is equall to CB wherefore by the 11. of the fifth as AB is to AC so is AC to CD
the line AB is diuided by an extreme and meane proportion in the poynt C and the greater segment thereof is the line ACâ— therfore that which is contayned vnder the lines AB and BC is equall to the square of the line AC Wherfore that which is coÌ„tayned vnder the lines AB and BC twise is double to the square of AC Wherfore that which is contayned vnder the lines AB and BC twise together with the square of the line AC is treble to the square of the line AC But that which is contayned vnder the lines AB and BC twise together with the square of the line AC is the squares of the lines AB and BC by the 7. of the second Wherefore the squares of the lines AB and BC are treble to the square of the line AC which was required to be demonstrated Resolution of the 5. Theoreme Suppose that a certaine right line AB be diuided by an extreme and meane proportion in the point C. And let the greater segment therof be the line AC And vnto the line AB adde a line equall to the line AC and let the same be AD. Theâ— I say that the line DB is diuided by an extreme and meane proportion in the point A. And the greater segment therof is the line AB For forasmuch as the line DB is diuided by an extreme meane proportion in the point A and the greater segment thereof is the line AB therfore as the line DB is to the line BA so is the line BA to the line AD but the line AD is equall to the line AC wherefore as the line DB is to the line BA so is the line BA to the line AC Wherfore by conuersioÌ„ as the line BD is to the line DA so is the line AB to the line BC by the corollary of the 19. of the fifth wherfore by diuision by the 17. of the fifth as the line BA is to the line AD â—o is the line AC to the line CB. But the line AD is equall to the line AC Wherfore as the line BA is to the line AC so is the line AC to the line CB. And so it is indeede for the line AB is by supposition diuided by an extreme and meane proportion in the point C. Composition of the 5. Theoreme Now forasmuch as the line AB is diuided by an extreme and meane proportion in the point C therefore as the line BA is to the line AC so is the line AC to the line CB but the line AC is equall to the line AD. Wherefore as the line BA is to the line AD so is the line AC to the line CB. Wherfore by composition by the 18. of the fifth as the line BD is to the line DA so is the line AB to the line BC. Wherefore by conuersion by the corollary of the 19. of the fiueth as the line DB is to the line BA so is the line BA to the line AC but the line AC is equall to the line AD. Wherefore as the line DB is to the line BA so is the line BA to the line AC Wherfore the line DB is deuided by an extreme and meane proportion in the point A and his greater segment is the line AB which was required to be demonstrated An Aduise by Iohn Dee added SEing it is doubteles that this parcel of Resolution and Composition is not of Euclides doyng it can not â—ustly be imputed to Euclide that he hath therby eyther superfluitie or any part disproportioned in his whole Composition Elementall And though for one thing one good demonstration well suffiseth for stablishing of the veritie yet oâ— one thing diuersly demonstrated to the diligent examiner of the diuerse meanes by which that varietie ariseth doth grow good occasions of inuenting demonstrations where matter is more straunge harde and barren Also though resolution were not in all Euclide before vsed yet thankes are to be geuen to the Greke Scholic writter who did leaue both the definition and also so short and easy examples of a Method so auncient and so profitable The antiquity of it is aboue 2000. yeares it is to weâ—e euer since Plato his time and the profite therof so great that thus I finde in the Greeke recorded 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 Proclus hauing spoken of some by nature excellent in inuenting demonstrations pithy and breif sayeth Yet are there Methods geuen for that purpose And in dede that the best which by Resolution reduceth the thing inquired of to an vndoubted principle Which Method Plato taught Leodamas as iâ— reported And he is registred thereby to haue bene the inuenter of many things in Geometry And verely in Problemes it is the chief ayde for winning and ordring a demonstration first by Supposition of the thing inquired of to be done by due and orderly Resolution to bring it to a stay at an vndoubted veritie In which point of Art great abundance of examples are to be seen in that excellent and mighty Mathematiciâ—n Archimedes in his expositor Eutocius in Menaechmus likewise and in Diocles booke de Pytiâ—s and in many other And now for as much as our Euclide in the last six Propositions of this thirtenth booke propoundeth and concludeth those Problemes which were the ende Scope and principall purpose to which all the premisses of the 12. bookes and the rest of this thirtenth are directed and ordered It shall be artificially done and to a great commodity by Resolution backward from these 6. Problemes to returne to the first definition of the first booke I meane to the definition of a point Which is nothing hard to do And I do counsaile all such as desire to atteinâ— to the profound knowledge of Geometrie Arithmeticke or any braunche of the sciences Mathematicall so by Resolution discreatly and aduisedly to resolue vnlose vnioynt and disseauer euery part of any worke Mathematicall that therbyâ— aswell the due placing of euery verity and his proofe as also what is either superfluous or wanting may euidently appeare For so to inuent there with to order their writings was the custome of them who in the old time were most excellent And I for my part in writing any Mathematicall conclusion which requireth great discourse at length haue found by experience the commoditie of it such that to do other wayes were to me a confusion and an vnmethodicall heaping of matter together besides the difficulty of inuenting the matter to be disposed and ordred I haue occasion thus to geue you friendely aduise for your beâ—ofeâ— because some of late haue inueyed against Euclide or Theon in this place otherwise than I would wish they had The 6. Theoreme The 6. Propositionâ— If a rationall right line be diuided by an extreme and meane proportion eyther of the segments is an irrationall line of that kinde which is called a residuall line SVppose that AB beyng a rationall line be deuidedly 〈◊〉 extreme
geuen wherin were contained the former solides and to proue that the side of the Icosahedron is an irrationall line of that kinde which is called a lesse line Now forasmuch as the lines QP QV QT QS and QR do eche subtend right angles contayned vnder the sides of an equilater hexagon of an equilater decagon inscribed in the circle PRSTV or in the circle EFGHK which two circles are equall therfore the sayd lines are eche equal to the side of the pentagon inscribed in the foresayd circle by the 10. of this booke and are equall the one to the other by the 4. of the first for all the angles at the poynt W which they subtend are right angles Wherefore the fiue triangles QPV QPR QRS QST and QTV which are contayned vnder the sayd lines QV QP QR QS QT and vnder the sides of the pentagon VPRST are equilater and equal to the ten former triangles And by the same reason the fiue triangles opposite vnto them namely the triangles YML YMN YNX YXO and YOL are equilater and equal to the said ten triangles For the lines YL YM YN YX and YO do subtend right angles coÌ„tayned vnder the sides of an equilater hexagon and of an equilater decagoÌ„ inscribed in the circle EFGHK which is equall to the circle PRSTV Wherefore there is described a solide contayned vnder 20. equilater triangles Wherefore by the last diffinition of the eleuenth there is described an Icosahedron Now it is required to comprehend it in the sphere geuen and to proue that the side of the Icosahedron is an irrationall line of that kinde which is called a lesse line Forasmuch as the line ZW is the side of an hexagon the line WQ is the side of a decagon therfore the line ZQ is diuided by an extreme and meane proportion in the point W and his greater segmeÌ„t is ZW by the 9. of the thirteÌ„th Wherfore as the line QZ is to the line ZW so is the line ZW to the line WQ But the ZW is equall to the line ZL by construction and the line WQ to the line ZY by construction also Wherefore as the line QZ is to the line ZL so is the line ZL to the line ZY and the angles QZLâ— and LZY are right angles by the 2. diffinition of the eleuenth If therfore we draw a right line from the poynt L to the poynt Q the angle YLQ shal be a right angle by reasoÌ„ of the likenes of the triangles YLQ and ZLQ by the 8. of the sixth Wherfore a semicircle described vpoÌ„ the line QY shal passe also by the point L by the assumpts added by Campane after the 13. of this booke And by the same reasoÌ„ also for that as the line QZ is the line ZW so is the line ZW to the line WQ but the line ZQ is equall to the line YW and the line ZW to the line PW wherefore as the line YW is to the line WP so is the line PW to the line WQ And therefore agayne if we draw a right line from the poynt P to the point Y the angle YPQ shal be a right angle Wherfore a semicircle described vpon the line QY shal passe also by the point P by the former assumpts if the diameter QY abiding fixed the semicircle be turned round about vntil it come to the selfe same place from whence it began first to be moued it shall passe both by the point P and also by the rest of the pointes of the angles of the Icosahedron and the Icosahedron shal be comprehended in a sphere I say also that it is contayned in the sphere geuen Diuide by the 10. of the first the line ZW into two equall parts in the point a. And forasmuch as the right line ZQ is diuided by an extreme and meane proportion in the point W and his lesse segment is QW therefore the segment QW hauing added vnto it the halfe of the greater segment namely the line Wa is by the 3. of this booke in power quintuple to the square made of the halfe of the greater segment wherefore the square of the line Qa is quintuple to the square of the line â—W But vnto the square of the Qa the square of the line QY is quadruple by the corollary of the 20. of the sixth for the line QY is double to the line Qa and by the same reason vnto the square of the WA the square of the line ZW is quadruple Wherefore the square of the line QY is quintuple to the square of the line ZW by the 15. of the fiueth And forasmuch as the line AC is quadruple to the line CB therefore the line AB is quintuple to the line CB. But as the line AB is to the line BC so is the square of the line AB to the square of the line BD by the 8 of the sixth and corollary of the 20. of the same Wherfore the square of the line AB is quintuple to the square of the line BD And it is is proued that the square of the line QY is quintuple to the square of the line ZW And the line BD is equall to the line ZW for either of them is by position equall to the line which is drawen from the centre of the circle EFGHK to the circumference Wherefore the line AB is equall to the YQ But the line AB is the diameter of the sphere geuen Wherefore the line YQ which is proued to be the diameter of the sphere contayning the Icosahedron is equall to the diameter of the sphere geuen Wherefore the Icosahedron is contayned in the sphere geueÌ„ Now I say that the side of the Icosahedron is an irrationall line of that kinde which is called a lesse line For forasmuch as the diameter of the sphere is rational and is in power quintuple to the square of the line drawen froÌ„ the centre of the circle OLMNX wherefore also the line which is drawen from the centre of the circle OLMNX is rationall wherefore the diameter also being coÌ„mensurable to the same line by the 6. of the tenth is rationall But if in a circle hauing a rationall line to his diameter be described an equilater pentagon the side of the pentagon is by the 11. of this booke an irrationall line of that kinde which is called a lesse line But the side of the pentagon OLMNX is also the side of the Icosahedron described as hath before ben proued Wherfore the side of the IcosahedroÌ„ is an irrationall line of that kinde which is called a lesse line Wherefore there is described an Icosahedron and it is contayned in the sphere geuen and it is proued that the side of the' Icosahedron is an irrationall line of that kind which is called a lesse line Which was required to be done and to be proued A Corollary Hereby it is manifest that the diameter of the sphere is in power quintuple to the line which is drawen from the centre of the circle to
double to the side of the Octohedron the side is in power sequitertia to the perpeÌ„diclar line by the 12. of this booke wherfore the diameter thereof is in power duple superbipartiens tertias to the perpendicular line Wherfore also the diameter and the perpeÌ„dicular line are rationall and commensuâ—able by the 6. of the tenth As touching an Icosahedron it was proued in the 16. of this booke that the side thereof is a lesse line when the diameter of the sphere is rationall And forasmuch as the angle of the inclination of the bases thereof is contayned of the perpendicular lines of the triangles and subtended of the right line which subtendeth the angle of the Pentagon which contayneth fiue sides of the Icosahedron and vnto the perpendicular lines the side is commensurable namely is in power sesquitertia vnto them by the Corollary of the 12. of this booke therefore the perpendicular lines which contayne the angles are irrationall lines namely lesse lines by the 105. of the tenth booke And forasmuch as the diameter contayneth in power both the side of the Icosahedron and the line which subtendeth the foresayd angle if from the power of the diameter which is rationall be taken away the power of the side of the Icosahedron which is irrationall it is manifest that the residue which is the power of the subtending line shal be irrationall For if it shoulde be rationall the number which measureth the whole power of the diameter and the part taken away of the subtending line should also by the 4. common sentence of the seuenth measure the residue namely the power of the side which is irrationall for that it is a lesse line which were absurd Wherefore it is manifest that the right lines which compose the angle of the inclination of the bases of the Icosahedron are Irrationall lines For the subtending line hath to the line contayninge a greater proportion then the whole hath to the greater segment The angle of the inclination of the bases of a dodecahedron is contayned vnder two perpendiculars of the bases of the dodecahedron and is subtended of that right line whose greater segment is the side of a Cube inscribed in the dodecahedron which right line is equall to the line which coupleth the sections into two equal parts of the opposite sides of the dodecahedron And this coupling line we say is an irrationall line for that the diameter of the sphere contayneth in power both the coupling line and the side of the dodecahedron but the side of the dodecahedron is an irrationall line namely a residuall line by the 17. of this booke Wherefore the residue namely the coupling line is an irrationall line as it is â—asy to proue by the 4. coÌ„mon sentence of the seueÌ„th And that the perpeÌ„dicular lines which contayne the angle of the inclination are irrationall is thus proued By the proportion of the subtending line of the foresayd angles of inclination to the lines which containe the angle is found out the obliquitie of the angle For if the subtending line be in power double to the line which contayneth the angle then is the angle a right angle by the 48. of the first But if it be in power lesse then the double it is an acute angle by the 23. of the second But if it be in power more then the double or haue a greater proportion then the whole hath to the greater segmeÌ„tâ— the angle shal be an obtuse angle by the 12. of the second and 4. of the thirtenth By which may be proued that the square of the whole is greater then the double of the square of the greater segment This is to be noted that that which Flussas hath here taught touching the inclinations of the bases of the â—iue regular bodies Hypsicles teacheth after the 5 proposition of the 15. booke Where he confesseth that he receiued it of one Isidorus and seking to make the mater more cleare he endeuored himselfe to declare that the angles of the inclination of the solides are geuen and that they are either acute or obtuse according to the nature of the solide although â—uclidâ— in all his 15. bookes hath not yet shewed what a thing geuen is Wherefore Flussas framing his demoÌ„stration vpon an other ground procedeth after an other maner which semeth more playne and more aptly hereto be placed then there Albeit the reader in that place shal not be frustrate of his also The ende of the thirtenth Booke of Euclides Elementes ¶ The fourtenth booke of Euclides Elementes IN this booke which is commonly accompted the 14. booke of Euclide is more at large intreated of our principal purpose namely of the comparison and proportion of the fiue regular bodies customably called the 5. figures or formes of Pythagoras the one to the other and also of their sides together eche to other which thinges are of most secret vse and inestimable pleasure and commoditie to such as diligently search for them and attayne vnto them Which thinges also vndoubtedly for the woorthines and hardnes thereof for thinges of most price are most hardest were first searched and found out of Philosophers not of the inferior or meane sort but of the depest and most grounded Philosophers and best exercised in Geometry And albeit this booke with the booke following namely the 15. booke hath bene hetherto of all men for the most part and is also at this day numbred and accompted amoÌ„gst Euclides bookes and supposed to be two of his namely the 14. and 15. in order as all exemplars not onely new and lately set abroade but also old monumentes written by hand doo manifestly witnes yet it is thought by the best learned in these dayes that these two bookes are none of Euclides but of some other author no lesse worthy nor of lesse estimation and authoritie notwithstanding then Euclide Apollonius a man of deepe knowledge a great Philosopher and in Geometrie maruelous whose woÌ„derful bookes writteÌ„ of the sections of cones which exercise occupy thewittes of the wisest and best learned are yet remayning is thought and that not without iust cause to be the author of them or as some thinke Hypsicles him selfe For what can be more playnely then that which he him selfe witnesseth in the preface of this booke Basilides of Tire sayth Hypsicles and my father together scanning and peysing a writing or books of Apollonius which was of the comparison of a dodecahedron to an Icosahedron inscribed in one and the selfe same sphere and what proportion these figures had the one to the other found that Apollonius had fayled in this matter But afterward sayth he I found an other copy or booke of Apollonius wherein the demonstration of that matter was full and perfect and shewed it vnto them whereat they much reioysed By which wordes it semeth to be manifest that Apollonius was the first author of this booke which was afterward set forth by Hypsicles For so his owne wordes after in
one and the selfe same sphere LEt the diameter of the sphere geuen be AB and let the bases of the Icosahedron and Dodecahedron described in it be the triangle MNR and the pentagon FKH and about them let there be described circles by the 5. and 14. of the fourth And let the lines drawne from the centres of those circles to the circumferences be LN and OK Then I say that the lines LN and OK are equal and therfore one and the selfe same circle containeth both those figures Let the right line AB be in power quintuple to some one right line as to the line CG by the Corollary of the 6. of the tenth And making the ceÌ„tre the poynt C the space CG describe a circle DZG And let the side of a pentagon inscribed in that circle by the 11. of the fourth be the line ZG And let EG subtending halfe of the arke ZG be the side of a Decagon inscribed in that circle And by the 30. of the sixt diuide the line CG by an extreme meane proportion in the poynt I. Now forasmuche as in the 16. of the thirtenth it was proued that this line CG vnto whome the diameter AB of the sphere is in power quintuple is the line which is drawne from the centre of the circle which containeth fiue angles of the Icosahedron and the side of the pentagon described in that circle DZG namely the line ZG is side of the Icosahedron described in the Sphere whose diameter is the line AB therefore the right line ZG is equal to the line MN which was put to be the side of the IcosahedroÌ„ or of his triaÌ„gular base Moreouer by the 17. of the thirtenth it was manifest that the right line â—H which subtendeth the angle of the pentagon of the Dodecahedron inscribed in the foresayde sphere is the side of the Cube inscribed in the self same sphere For vpon the angles of the cube were made the angles of the Dodecahedron Wherefore the diameter AB is in power triple to FH the side of the Cube by the 15. of the thirtenth But the same line AB is by supposition in power quintuple to the line CG Wherefore fiue squares of the line CG are equal to thre squares of the line FH for eche is equal to one and the self same square of the line AB And forasmuche as EG the side of the Decagon cutteth the right line CG by an extreme and meane proportion by the corollary of the 9. of the thirtenth Likewise the line HK cutteth the line FH the side of the Cube by an extreeme and meane proportion by the Corollary of the 17. of the thirtenth therfore the lines CG and FH are deuided into the self same proportions by the second of this booke and the right lines CI and EG which are the greater segmentes of one and the selfe same line CG are equal And forasmuche as fiue squares of the line CG are equal to thre squares of the lines FH therefore fiue squares of the line GE are equal to thre squares of the line HK for the lines GE and HK are the greater segmeÌ„ts of the lines CG and FH Wherefore fiue squâ—reâ— of the lineâ— CG GE are equal to the squares of the 〈◊〉 â—H HK by the 1â— of the â—ift But vnto the squares of the lines CG and GEâ— is â—qual the squâ—re of thâ—â—ine ZG by the 10. of the thirteÌ„th and vnto the line ZG the line MN was equal wherfore fiue squares of the line MN are equall to three squares of the lines FH HK But the squares of the lines â—â— and HK 〈◊〉 quintuple to the square of the line OK which is drawne from the centre by the third of this booke Wherfore thre squares of the lines FH and HK make 15. squares of the line OK And forasmuch as the square of the line MN is triple to the square of the line LN which is drawne from the centre by the 12. of the thirtenth therfore fiue squares of the line MN are equal to 15. squares of the line LN But fiue squares of the line MN are equal vnto thre squares of the lines FH and HK Wherefore one square of the line LN is equall to one square of the line OK being eche the fiuetenth part of equal magnitudes by the 15. of the fifâ—â— Wherfore the lines LN and OK which are drawne from the centers are equal Wherefore also the circles NRM and FKH which are described of those lines are equal And those circles contayne by supposition the bâ—ses of the Dodecahedron and of the Icosahedron described in one and the selfe same sphere Wherfore one and the selfe same circle c. aâ— in thâ— proâ—â—sition which was required to be proued The 5. Proposition If in a circle be inscribed the pentagon of a Dodecahedron and the triangle of an Icosahedron and from the centre to one of theyr sides be drawne a perpendicular line That which is contained 30. times vnder the side the perpendicular line falling vpon it is equal to the superficies of that solide vpon whose side the perpendicular line falleth SVppose that in the circle AGE be described the pentagon of a Dodecahedron which let be ABGDE and the triangle of an Icosahedron described in the same sphere which let be AFH And let the centre be the poynt C. â—â—on which draw perpendicularly the line CI to the side of the Pentagon and the line CL to the side of the triangle Then I say that the rectangle figure contained vnder the lines CI and GD 30. times is equal to the superficies of the Dodecahedron and that that which is coÌ„tained vnder the lines CL AF 30. times is equal to the superâ—icies of the IcosahedroÌ„ described in the same sphere Draw these right lines CA CF CG and CD Now forasmuch as that which is coÌ„tained vnder the base GD the altitude IC is double to the triangle GCD by the 41. of the first And fiue triangles like and equal to the triangle GCD do make the pentagon ABGDE of the Dodecahedron wherfore that which is contained vnder the lines GD and IC fiue times is equal to two pentagoÌ„s Wherfore that which is contained vnder the lines GD and IC â—0 times is equal to the 12. pentagons which containe the superficies of the Dodecahedron Againe that which is contained vnder the lynes CL and AF is double to the triangle ACF wherefore that which is contained vnder the lines CL and AF three times is equal to two suche triangles as AFH is which is one of the bases of the Icosahedron for the triangle ACF is the third part of the triangle AFH as it is easie to proue by the 8. 4. of the first Wherfore that which is coÌ„tained vnder the lines CL and AF. 30 times times is equall to 10. such triangles as AFH iâ— which containe the superficies of the Icosahedron And forasmuch as one and the selfe same
spherâ— containeth the Dodecahedron of this pentagon and the Icosahedron of this triangle by the 4. of this booke â— and the line CL falleth perpendiculaâ—ly vpon the side of the Icosahedron and the line CI vpon the side of the Dodecahedron that which is 30. times contained vnder the side and the perpendicular line falling vpon it is equal to the â—uperficies of that solide vpon whose side the perpendiculâ—r falleth If therefore in a circle c. as in the proposition which was required to be demonstrated A Corollary The superficieces of a Dodecahedron and of an Icosahedron described in one and the selfe same sphere are the one to the other as that which is contained vnder the side of the one and the perpendicular line drawne vnto it from the centre of his base to that which is contained vnder the side of the other and the perpendicular line drawne to it from the centre of his base For aâ— thirtyâ— timâ—s is to thirty times so is once to once by the 15. of thâ— fifth The 6. Proposition The superficies of a Dodecahedron is to the superficies of an Icosahedron described in one and the selfe same sphere in that proportion that the side of the Cube is to the side of the Icosahedron contained in the self same sphere SVppose that there be a circle ABG in it by the 4. of this boke let there be inscribed the sideâ— of a Dodecahedron and of an Icosahedron contained in onâ— and the selfe same sphere And let the side oâ— the Dodecahedron be AG and the side of the Icosahedron be DG And let the centre be the poynt E from which draw vnto those sâ—des perpendicular lines EI and EZ And produce the line EI to the poynt B and draw the linâ— BG And let the side of the cube contained in the self same sphere be GC Then I say that the superficies of the Dodecahedron iâ— to the superficies of the Icosahedron as the line â—G iâ— to the liâ—â— GD For forasmuche as the line EI beinâ— diuided by an extreme and meane proportion the greater segment thâ—rof shall be the linâ— EZ by the corollary of the first of this booke and the line CG being diuided by an extreme and meane proportion his greater segment is the line AG by the corollary of the 17. of the thirtenth Wherefore the right lines EI and CG â—râ— cut proportionally by the second of this bâ—oke Whâ—râ—fore as the line CG is to the line AG so is the line EI to the line EZ Wherâ—fore that which it contained vnder the extreames CG and EZ is â—quall to that which iâ— contaynâ—d vnder the meanes AG and EI. by the 16. of the sixth But as that which iâ— contained vnder the linâ—â— CG and â—Z is to that which is contained vnder the lines DG and EZ so by the first of the sixth iâ— the linâ— CG to the line DG for both those parallelogrames haue oâ—â— and the selfe same altitude namely the line EZ Wherfore as that which is contained vnder the lines EI and AG which iâ— proued equal to that which is contained vnder the lineâ— CG and EZ is to that which is contained vnder the lines DG and EZ so is the line CG to the liâ—â— DG But as that which is contained vnder the lines EI and AG is to that which is contained vnder the lines DG and EZ so by the corollary of the former proposition is the superficies of the Dodecahedron to the superficies of the Icosahedron Wherfore as the superficies â—â— the Dodecahedron is to the superficies of the Icosahedron so is CG the side of the cube to GD the side of the Icosahedron The superficies therefore of a Dodecahedron is to the superficiesâ— c. as in the proposition which was required to be proued An Assumpt The Pentagon of a Dodecahedron is equall to that which is contained vnder the perpendicular line which falleth vpon the base of the triangle of the Icosahedron and fiue sixth partes of the side of the cube the sayd three solides being described in one and the selfe same sphere Suppose that in the circle ABEG the pentagon of a Dodecahedron be Aâ—CIG and let two sides thereof AB and AG be subtended of the right line BG And let the triangle of the Icosahedron inscribed in the selfe same sphere by the 4. of this booke be AFH And let the centre of the circle be the poynt D and let the diameter be ADE cutting FH the side of the triangle in the poynt Z and cutting the line BG in the poynt K. And draw the right line BD. And from the right line KG cut of a third part TG by the 9. of the sixth Now then the line BG subtending two sides of the Dodecahedron shal be the side of the cube inscribed in the same sphere by the 17. of the thirtenth and the triangle of the Icosahedron of the same sphere shal be Aâ—H by the 4. of this booke And the line AZ which passeth by the centre D shall fall perpendicularly vpon the side of the triangle For forasmuch as the angles GAE BAE are equall by the 27. of the thirdâ— for they are see vpon equall circumferences therefore the â—ases BK and KG are by the â— of the first equall Wherefore the line BT contayneth 5. sixth partes of the line BG Then I say that that which is contayned vnder the lines AZ and BT is equall to the pentagon Aâ—Câ—G For forasmuch as the line â—Z is sesqâ—ialter to the line AD for the line Dâ— is diuided into two equall partes in the poynt Z by the corollary of the â—2â— of the thirtenth Likewise by construction the line KG is sesquialter to the line KT therefore as the line AZ is to the line AD so is the line KG to the 〈◊〉 â—T Wherefore that which is contayned vndeâ— the 〈◊〉 AZ and KT is equall to that which is contayned vnder the meanes AD and KG by the 16. of the sixth But vnto the line KG is the line â—K â—roued equall Wherefore that which is contayned vnder the lines AZ and KT is equall to that which is contayned vnder the lines AD and BK But that which is contayned vnder the lines AD and BK is by the 41. of the first double to the triangle ABD Wherefore that which is contayned vnder the lines AZ and KT is double to the same triangle ABD And forasmuch as the pentagon ABCIG contayneth◠〈…〉 equall â—o the triangle ABD and that which is contayned vnder the lines AZ and KT contayneth two such triangles therefore the pentagon ABCIG is duple sesquialter to the rectangle parallelogramme contayned vnder the lines AZ and KT And 〈…〉 1. of the sixth that which is coÌ„teyned vnder the lines AZ and BT is to that which is contayned vnder the lines AZ and KT as the base BT is to the base â—â—Tâ— therefore that which is contayned vnder the lines AZ
and BT is duple sesquialter to that which is contayned vnâ—â—r the line AZ KT But vnto that which is contayned vnder the lines AZ and KT the pentagon ABCIG is proued duple sesquialter Wherfore the pentagon ABCIG of the Dodecahedron is equall to that which is contayned vnder the perpendicular line AZ and vnder the line BT which is fiue sixe partes of the line BG ¶ The 7. Proposition A right line diuided by an extreame and meane proportion what proportion the line contayning in power the whole line and the greater segment hath to the line contayning in power the whole and the lesse segment the same hath the side of the cube to the side of the Icosahedron contayned in one and the same sphere TAke a circle ABE and in it by the 11. of the fourth inscribe an equilater pentagon BZECH and by the second of the same an equilater triangle ABI And let the centre thereof be the poynt G. And drawe a line from G to B. And diuide the line GB by an extreame and meane proportion in the poynt D by the 30. of the sixth And let the line ML contayne in power both the whole line GB and his lesse segment BD by the corollary of the 13. of the tenth And draw the right line Bâ— subâ—eÌ„diâ—g the angle of the pentagon which shall be the side of the cube by the corollary of the 17. of the thirtenth â— and the line BI shall be the side of the Icosahedron and the line â—Z the side of the Dodecahedron by the 4. of this booke Then I say that BE the side of the cube is to BI the side of the Icosahedron as the line contayning in power the lines BG GD is to the line contayning in power the lines GB and BD. For forasmuch as by the 12. of the thirtenth the line BI is in power triple to the line BG and by the 4. of the same the squares of the line GB BD are triple to the square of the line GD Wherefore by the 15. of the fifth the square of the line BI is to the squares of the lines GB BD namely triple to triple as the square of the line Bâ— is to the square of the line GD namely as one is to one But as the square of the line BG is to the square of the GD so is the square of the line BE to the square of the line BZ For the lines BG GD and BE BZ are in one and the same proportion by the second of this booke For BZ is the greater segment of the line BE by the corollary of the 17. of the thirtenth Wherefore the square of the line BE is to the square of the line BZ as the square of the line BI is to the squares of the lines BG and BD. Wherefore alternately the square of the line BE is to the square of the line BI as the square of the line BZ is to the squares of the lines GB and BD. But the square of the line BZ is equall to the squares of the lines BG and GD by the 10. of the thirtenth For the line BG is equall to the side of the hexagon and the line GD to the side of the decagon by the corollary of the 9. of the same Wherefore the squares of the lines BG and GD are to the squares of the lines Gâ— and BD as the square of the line BE is to the square of the line BI But the line ZB contayneth in power the lines BG and GD and the line ML contayneth in power the lines GB and BD by construction Wherefore as the line ZB which contayneth in power the whole line BG and the greater segment GD is to the line ML which contayneth in in power the whole line GB and the lesse segment BD so is BE the side of the cube to BI the side of the Icosahedron by the 22. of the sixth Wherefore a right line diuided by an extreame and meane proportion what proportion the line contayning in power the whole line and the greater segment hath to the line contayning in power the whole line and the lesse segment the same hath the side of the cube to the side of the Icosahedron coÌ„tayned in one and the same sphere which was required to be proued ¶ The 8. Proposition The solide of a Dodecahedron is to the solide of an Icosahedron as the side of a Cube is to the side of an Icosahedron all those solides being described in one and the selfe same Sphere FOrasmuch as in the 4. of this booke it hath bene proued that one and the self same circle containeth both the triangle of an Icosahedron and the pentagon of a Dodecahedron described in one and the selfe same Sphere Wherefore the circles which coÌ„taine those bases being equall the perpendiculars also which are drawen from the centre of the Sphere to those circles shall be equall by the Corollary of the Assumpt of the 16 of the twelfth And therefore the Pyramids set vpon the bases of those solides haue one and the selfe same altitude For the altitudes of those Pyramids concurrâ— in the centre Wherefore they are in proportion as their bases are by the 5. and 6. of the twelfth And therefore the pyramids which compose the Dodecahedron arâ— to the pyramids which compose the Icosahedron as the bases are which bases are the superficieces of those solides Wherefore their solides are the one to the other as their superficieces are But the superficies of the Dodecahedron is to the superficies of the Icosahedron as the side of the cube is to the side of the Icosahedron by the 6. of this booke Wherfore by the 11. of the fifth as the solide of the Dodecahedron is to the solide of the Icosahedron so is the side of the cube to the side of the Icosahedron all the said solides being inscribed in one and the selfe same Sphere Wherefore the solide of a Dodecahâ—dron is to the solide of an Icosahedron as the side of a cube is to the side of an Icosahedron all those solides being described in one and the self same Sphere which was required to be proued A Corollary The solide of a Dodecahedron is to the solide of an Icosahedron as the superficieces of the one are to the superficieces of the other being described in one and the selfe same Sphere Namely as the side of the cube is to the side of the Icosahedron as was before manifest for they are resolued into pyramids of one and the selfe same altitude ¶ The 9. Proposition If the side of an equilater triangle be rationall the superficies shall be irrationall of that kinde which is called Mediall SVppose that ABG be an equilater triangle and from the point A draw vnto the side BG a perpendicular line AD and let the line AB be rationall Then I say that the superficies ABG is mediall Forasmuch as the line AB is in power
DEF whose side let be DE and let the right line subtending the angle of the pentagon made of the sides of the Icosahedron be the line EF. Then I say that the side ED is in power double to the line H the lesse of those segmentes Forasmuch as by that which was demonstrated in the 15. of this booke it was manifest that ED the side of the Icosahedron is the greatâ—r segment of the line EFâ— and that the diameter DF containeth in power the two lines ED and EF namely the whole and the greater segment but by suppoâ—ition the side AB coÌ„taineth in power the two lines C H ioined together in the self same proportioÌ„ Wherefore the line EF is to the line ED as the line C is to the line H by the â— oâ— this bokeâ— And altâ—rnaâ—â—y by the 16. of the fiueth the line EF is to the line C as the line ED is to the line H. And forasmuche as the line DF containeth in power the two lines ED and EF and the line AB containeth in power the two lines C and H therefore the squares of the lines EF and ED are to the square of the line DF as the squares of the lines C and H to the square AB And alternately the squares of the lines EF and â—D are to the squares of the lines C and H as the square of the line DF is to the square of the line ABâ— But DF the diameter is by the 14. of the thirtenâ—h iâ— power double to AB the side of the octohedron inscribed by supposition in the same sphere Wherefore the squares of the lines EF and ED are double to the squares of the lines C and H. And therfore one square of the line ED is double to one square of the line H by the 12. of the fifth Wherfore ED the side of the Icosahedron is in power duple to the line H which is the lesse segment If therfore the poweâ— of the side of an octohedron be expressed by two right lines ioyned together by an extreme and meane proportion the side of the Icosahedron contained in the same sphere shal be duple to the lesse segment The 17. Proposition If the side of a dodecahedron and the right line of whome the said side is the lesse segment be so set that they make a right angle the right line which containeth in power halfe the line subtending the angle is the side of an Octohedron contained in the selfe same sphere SVppose that AB be the side of a Dodecahedron and let the right line of which that side is the lesse segment be AG namely which coupleth the opposite sides of the Dodecahedron by the 4. corollary of the 17. of the thirtenth and let those lines be so set that they make a right angle at the point A. And draw the right line BG And let the line D containe in power halfe the line BG by the first proposition added by Flussas after the laste of the sixth Then I say that the line D is the side of an Octohedron contayned in the same sphere Forasmuche as the line AG maketh the greater segment GC the side of the cube contained in the same sphere by the same 4. corollary of the 17. of the thirtenth and the squares of the whole line AG. and of the lesse segment AB are triple to the square of the greater segment GC by the 4. of the thirtenth Moreouer the diameter of the sphere is in power triple to the same line GC the side of the cube by the 15. of the thirtenth Wherfore the line BG is equal to the 〈◊〉 For it conâ—â—ineth in power the two lines AB and AG by the 47. of the first and therefore it containeth in power the triple of the line GC But the side of the Octohedron contained in the same sphere is in power triple to halfe the diameter of the sphere by the 14. of the thirtenth And by suppoâ—â—tion the line D contaiâ—â—â—â— in powâ—â— the halfe of the line BG Wherefore the line D containing in power the halfe of the same diameter is the side of an octohedron If therfore the side of a Dodecahâ—dron and the right line of whome the said side is the lesse segment be so set that they make a right angle the right line which containeth in power halfe the line subtending the angle is the side of an Ocâ—â—â—edron contained in the selfe same sphere Which was required to be proued A Corollary Vnto what right line the side of the Octoâ—edron is in power sesquialter vnto the same line the side of the Dodecahedron inscribed in the same sphere is the greater segment For the side of the Dodecahedron is the greater segment of the segment CG vnto which D the side of the Octohedron is in power sesquiâ—lter that is is halfe of the power of the line BG which was triple vnto the line CG ¶ The 18. Proposition If the side of a Tetrahedron containe in power two right lines ioyned together by an extreme and meane proportion the side of an Icosahedron described in the selfe same Sphere is in power sesquialter to the lesse right line SVppose that ABC be a Tetrahedron and let his side be AB whose power let be diuided into the lines AG and GB ioyned together by an extreme and meane proportion namely let it be diuided into AG the whole line and GB the greater seâ—ment by the Corollary of the first Proposition added by Flussas after the last of the sixth And let ED be the side of the Icosahedron EDF contained in the selfe same Sphere And let the line which subtendeth the angle of the Pentagon described of the sides of the Icosahedron be EF. Then I say that ED the side of the Icosahedron is in power sesquialter to the lesse line GB Forasmuch as by that which was demonstrated in the 15. of this booke the side ED is the greâ—ter segment of the line EF which subtendeth the angle of the Pentagon But as the whole line EF is to the greater segment ED so is the same grâ—â—ter segment to the lesse by the 30. of the sixth and by supposition AG was the whole line and Gâ— the greater segment Wherefore as EF is to ED so is AG to Gâ— by the second of the fouretenth And alternately the line EF is to the line AG as the line ED is to the line GB And forasmuch as by supposition the line AB containeth in power the two lines AG and GB therefore by the 4â— of the first the angle AGB is a right angle But the angle DEF is a right angle by that which was demonstrated in the 15. of this booke Wherefore the triangles AGâ— and FED are equiangle by the â— of the sixth Wherefore their sides are proportionall namely as the line ED is to the line GB so is the line FD to the line AB by the 4. of the sixth But by that which hath before
right linâ—s Now then multiply the 20. triangles into the sides of one of the triangles and so shall there be produced 6â— â—he halfe of which is 30. And so many sides hath an Icosahedron And in like sort in a dodecahedron forasmuch as 12. pentagons make a dodecahedron and euery pentagon contayneth â— right linesâ— multiply â—â— into 12. and there shall be produced 60. the halfe of which is 30. And so many are the sides of a dodecahedron And the reason why we take the halfe iâ— for that euery side whether it be of a triangle or of a pentagon or of a square as in a cube â—s taken twise And by the same reason may you finde out how many sides are in a cube and in a pyramis and in an octohedron But now agayne if ye will finde out the number of the angles of euery one of the solide figures when ye haue done the same multiplication that ye did before diâ—idâ— the same sides by the number of the plaine superficieces which comprehend one of the angles of the solides As for example forasmuch as 5. triangles contayne the solide angle of an Icosahedron diuide 60. by 5. and there will come forth 12. and so many solide angles hath an Icosahedâ—on In a dodecahedron forasmuch as three pentagons comprehend an angle diuide 60. by 3. and there will come forth 20 and so many are the angles of a dodecahedron And by the same reason may you finde out how many angles are in eche of the rest of the solide figures If it be required to be knowne how one of the plaines of any of the fiue solides being geuen there may be found out the inclination of the sayd plaines the one to the other which contayne eche of the solides This as sayth Isidorus our greate master is foâ—â—d out after this maner It is manifest that in a cube the plaines which contayne iâ— do◠〈◊〉 the one the other by a right angle But in a Tetrahedron one of the triangles being geuen let the endes of one of the sides of the sayd triangle be the centers and let the space be the perpendicular line drawne from the toppe of the triangle to the base and describe circumferâ—nces of a circle which shall cutte the one the other and from the intersection to the centers draw right lines which shall containe the inclination of the plaines coÌ„tayning the Tetrahedron In an Octoâ—edron take one of the sides of the triangle therâ—of and vpon it describe a square and draw the diagonall line and making the centres the endes of the diagonall line and the space likewise the perpendicular line drawne from the toppe of the triangle to the base describe circumferences and agayne from the common section to the centres draw right lines and they shall contayne the inclination sought for In an Icosahedron vpon the side of one of the triangles thereof describe a pentagon and draw the line which subtendeth one of the angles of the sayd pentagon and making the centres the endes of that line and the space the perpendicular line of the triangle describe circumferences and draw from the common intersectioâ— of the circumferences vnto the centres right lines and they shall contayne likewise the inclination of the plaines of the icosahedron In a dodecahedron take one of the pentagons and draw likewise the line which subtendeth one of the angles of the pentagon and making the centres the endes of that line and the space the perpendicular line drawne from the section into two equall partes of that line to the side of the pentagon which is parallel vnto it describe circumferences and from the point of the intersection of the circumferences draw vnto the centres right lines and they shall also containe the inclination of the plaines of the dodecahedron Thus did this most singular learned man reason thinking the deâ—onstration in euery one of them to be plaine and cleare But to make the demonstration of them manifest I think it good to declare and make open his wordesâ— and first in a Tâ—trahedronâ— The ende of the fiuetenth Booke of Euclides Elementes after 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ¶ The 6. Proposition The 6. Probleme In an Octohedron geuen to inscribe a trilater equilater Pyramis SVppose thaâ— the Octohedron whereâ—â— the Tetrahedron is required to be insâ—riâ—â—â— be ABGDEI Take 〈…〉 bases of the Octoâ—â—dron that is 〈…〉 close in the loweâ—â— triangle BGD namely AEâ— HED IGD and let the fourth be AIB which is opposite to the lowest triangle before put namely to EGD And take the centres of those fower bases which let be the pointes H C N â— And vpon the triangle HCN erecte a pyramis HCNL Now â—orasmuch as these two bases of the Octohedron namely AGE and ABI are set vpon the right lines EG and BI which are opposite the one to the otherâ— in the square GEBI of the Octohedron from the poinâ— A draâ—e by the centres of the bases namely by the centres H L perpendicular lines AHF ALK cutting the lines EG and BI 〈◊〉 two equall partes in the pointes F K by the Corollary of the 1â— of the thirtenth Wherfore a right line drawen froÌ„ the point F to the point K shall be a parallel and equall to the sides of the Octohedron namely to â—â— and GI by the 33. of the first And the right line HL which cutteth the 〈…〉 AF AK proportionally for AH and AL are drawen from the centres of equall circles to the circumferences is a parallel to the right line FK by the 2. of the sixth and also to the sides of the Octohedron namely to Eâ— and IG by the 9. of the eleuenth Wherefore as the line AF is to the line AH so is the line FK to the line HL by the 4. of the sixth For the triangles AFK and AHL are like by thâ— Corollary of the 2. of the sixth But the line AF is in sesquialter proportion to the line AH for the side EG maketh HF the halfe of the right line AH by the Corollary of the 12. of the thirtenth Wherfore FK or GI the side of the Octohedron is sesquialter to the righâ—line HL. And by the same reason may we proue that the sides of the Octohedron are sesquialter to the rest of the right lines which make the pyramis HNCI namely to the right lineâ—â— N NC CI LN and CH wherefore those right lines are equall and therefore the triangleâ— which are described of them namely the triangles HCN HNL NCL and CHL. which make the pyramis HNCL are equall and equilater And forasmuch as the angles of the same pyramis namely the angles H C N L do end in the centres of the bases of the Octohedron therefore it is inscribed â—o the same Octohedron by the first definition of this booke Wherefore in an Octohedron â—euen is inscribed a trilâ—ter equilaâ—â—â—â—â—â—amisâ— which was required to â—e donâ— A Corollary The bases of a Pyramis inscribed in an Octohedron are parallels
side GD the angles M N vnder the side AB the angles T S vnder the side BG the angles P O and vnder the side AG the angles R Q so there rest 4. angles whose true place we will now appoynt Forasmuch as a cube contayned in one and the selfe same sphere with a dodecahedron is inscribed in the same dodecahedron as it was manifest by the 17. of the thirtenth and 8. of this booke it followeth that a cube and a dodecahedron circumscribed about it are contayned in one and the selfe same bodies for that their angles concurre in one and the selfe same poyntes And it was proued in the 18. of this booke that 4. angles of the cube inscribed in the pyramis are set in the middle sections of the perpendicularâ— which are drawne from the solide angles of the pyramis to the opposite bases wherefore the other 4. angles of the dodecahedron are also as the angles of the cube set in those middle sections of the perpendiculars Namely the angle V is set in the middest of the perpendicular AHâ— the angle Y in the middest of the perpendicular BF the angle X in the middest of the perpendicular GE and lastly the angle D in the middest of the perpendicular D which is drawne from the toppe of the pyramis to the opposite base Wherefore those 4. angles of the dodecahedron may be sayd to be directly vnder the solide angles of the pyramis or they may be sayd to be set at the perpendiculars Wherefore the dodecahedron after this maner set is inscribed in the pyramis geuen by the first diffinition of this booke for that vpoÌ„ euery one of the bases of the pyramis are set an angle of the dodecahedroÌ„ inscribed Wherefore in a trilater equilater pyramis is inscribed a dodecahedron The 21. Probleme The 21. Proposition In euery one of the regular solides to inscribe a Sphere IN the 13. of thâ— thirtenth and thâ— other 4. propositioâ—â— following iâ— was declared that â—he â—â— regular solidesâ—â—re so contaâ—â—ed in a sphere that â—ight linâ—â— drawne from the cenâ—â—â— oâ— the 〈…〉 of 〈◊〉 solide inscribed are equall Which right lines therefore make pyramids whose â—oppes are the centre of the sphere or of the solide and the basâ—â—â—â—e cuâ—â—â— one of the bases of those solides And 〈…〉 solide â—quall and like the one to the other and described in equall circles those cirâ—les shall cutte the sphere for the angles which touch the circumference of the circle touch also the superficies of the sphere Wherefore perpeÌ„diculars drawne from the centre of the sphere to the bases or to the playne superficieces of the equall circles are equall by the corollary of the assumpt of the 1â— of the twelfth Wherefore making the centre the 〈◊〉 of the sphere which 〈◊〉 the solide and thâ— space some one of the equall perpendicularâ— dâ—scribâ— a sphere and it shall touch euery one of the bases of 〈◊〉 solide 〈…〉 perficies of the sphere passe beyond those bases when as those pâ—â—peâ—diculars 〈…〉 are drawne from the centre to the bases by the 3. corollary of the saâ—â—â—â—â—umpt Wherâ—fore â—e haue iâ— euery one of the regular bodies inscribed a sphere which regular boâ—â—â— are in number one i◠〈◊〉 by the corollary of the 1â— of the 〈◊〉 A Corollary The regular figures inscribed in spheres and also the spheres circumscribed about them or contayning them haue one and the selfe same centre Namely their pyramids the â—ngles of whose bâ—ses touch the superâ—â—â—â—â—â— of thâ—â—â—here doo from those angles cause equall right lines to be drawâ—â— to one and â—he selfe 〈◊〉 poynâ— making the topâ—â—â— of the pyramidâ— in the same poynt and therefore they 〈…〉 thâ— câ—â—tres of the spheres in the selfe same toppes when 〈◊〉 the right lines drawne from those angles to the croâ—â—ed superficies whâ—rein are 〈◊〉 the angles of the bases of the pyramidâ— are equallâ— An adueâ—â—â—sment of Flussas â— Of these solides onely the Octohedron receaueth the other solides inscribed one with 〈…〉 other For the Octohedron contayneth the Icosahedron inscribed in it and the same Icosahedron contayneth the Dodecahedron inscribed in the same Icosahedron and the same dodecahedron contayneth the cube inscribed in the same Octohedron and 〈…〉 â—â—râ—â—mscribeth the Pyramis inscribed in the sayd Octohedron But this happâ—neth not in the other solides The ende of the fiuetenth Booke of Euclides Elemenâ—â—â— after Caâ—paâ—â— and 〈◊〉 ¶ The sixtenth booke of the Elementes of Geometrie added by Flussas IN the former fiuetenth booke hath bene taught how to inscribe the fiue regular solides one with in an other Now semeth to rest to coÌ„pare those solidâ— so inscribed one to an other and to set forth their passionâ— and proprieties which thing Flussas considering in this sixteÌ„th booke added by him hath excellently well and most conningly performed For which vndoubtedly he hath of all them which haue a loue to the Mathematicals deserued much prayse and commendacion both for the great traâ—ailes and paynâ—s which it is most likely he hath taâ—â—n in iâ—uenting such straunge and wonderfull propositions with their demonstrations in this booke contayned as also for participating and communicating abrode the same to others Which booke also that the reader should want nothing conducing to the perfection of Euclides Elements I haue with some trauaile translated for the worthines â—hereof haue added it aâ— a sixtenth booke to the 15. bookes of Euclide Vouchsafe therefore gentle reader diligently to read and peyse it for in it shall you finde noâ— onely matter strange and delectable but also occasion of inuention of greater things pertayning to the natures of the fiue regular solidâ—s◠¶ The 1. Proposition A Dodecahedron and a cube inscribed in it and a Pyramis inscribed in the same cube are contained in one and the selfe same sphere FOr the angles of the pyramiâ— are seâ— in the angâ—es of the cube wherein it is inscribed by the first of the fiuetenthâ— and all the angles of the cube are set in the angles of the dodecahedâ—â—â— circumscribed 〈…〉 〈◊〉 the 8. of the fiuetenth And all the angles of the Dodecahedron are set in the superficies of the sphere by the 17. of the thirtenth Wherefore those three solides inscribed one within an other are contained in one and the selfe same sphere by the first diffinition of the fiuetenth A dodecahedron therfore and a cube inscribed in it and a pyramis inscribed in the same cube are contained 〈…〉 â—â—lfe same sphere 〈…〉 These three solides li 〈…〉 elfe same Icosahedron or Octohedron or Pyramis 〈…〉 me Icosahedron by the 5.11 12. of the fiuetenth and they ar 〈…〉 ctohedron by the 4. 6. and 16. of the same lastly they are inscribed in 〈…〉 the first 18. and 19. of the same For the angles of all these solide 〈…〉 the circumscribed Icosahedron or octohedron or pyramis ¶ The 〈…〉 The proportion of a Dodecahedron circumscribed about a cube to a DodecahedroÌ„ inscribed in the same cube is
Cube FOr forasmuch as the side of the pyramis inscribed in the cube subteÌ„deth two sides of the cube which containe a right angle by the 1. of the fiuetenth it is manifest by the 47. of the first that the side of the pyramis subteÌ„ding the said sides is in power duple to the side of the cube Wherefore also the square of the side of the cube is the halfe of the square of the side of the pyramis The side therefore of a cube containeth in power halfe the side of an equilater triangular pyramis inscribed in the said cube ¶ The 7. Proposition The side of a Pyramis is duple to the side of an Octohedron inscribed in it FOrasmuch as by the 2. of the fiuetenth it was proued that the side of the Octohedron inscribed in a pyramis coupleth the midle sections of the sides of the pyramis Wherefore the sides of the pyramis and of the Octohedron are parallels by the Corollary of the 39. of the first and therefore by the Corollary of the 2. of the sixth they subtend like triangles Wherfore by the 4. of the sixth the side of the pyramis is double to the side of the Octohedron namely in the proportion of the sides The side therefore of a pyramis is duple to the side of an Octohedron inscribed in it ¶ The 8. Proposition The side of a Cube is in power duple to the side of an Octohedron inscribed in it IT was proued in the 3. of the fiuetenth that the diameter of the Octohedron inscribed in the cube coupleth the centres of the opposite bases of the cube Wherefore the said diameter is equall to the side of the cube But the same is also the diameter of the square made of the sides of the Octohedron namely is the diameter of the Sphere which containeth it by the 14. of the thirtenth Wherefore that diameter being equall to the side of the cube is in power double to the side of that square or to the side of the Octohedron inscribed in it by the 47. of the first The side therefore of a Cube is in power duple to the side of an Octohedron inscribed in it which was required to be proued ¶ The 9. Proposition The side of a Dodecahedron is the greater segment of the line which containeth in power halfe the side of the Pyramis inscribed in the sayd Dodecahedron SVppose that of the Dodecahedron ABGD the side be AB and let the base of the cube inscribed in the Dodecahedron be ECFH by the â—â— of the fiuetenth And let the side of the pyramis inscribed in the cube be CH by the 1. of the fiuetenth Wherefore the same pyramis is inscribed in the Dodecahedron by the 10. of the fiuetenth Then I say that AB the side of the Dodecahedron is the greater segment of the line which containeth in power halfe the line CH which is the side of the pyramis inscribed in the Dodecahedron For forasmuch as EC the side of the cube being diuided by an extreme and meane proportion maketh the greater segment the line AB the side of the Dodecahedron by the â—â—rst Corollary of the 17. of the thirtenth For they are contâ—ined in one and the selfe same Sphere by the first of this booke and the line EC the side of the cube contayneth in power the halfe of the side CH by the 6. of this booke Wherefore AB the side of the Dodecahedron is the greater segment of the line EC which containeth in power the halfe of the line CH which is the side of the Dodecahedron inscribed in the pyramis The side therefore of a Dodecahedron is the greater segment of the line which containeth in power halfe the side of the Pyramis inscribed in the said Dodecahedron ¶ The 10. Proposition The side of an Icosahedron is the meane proportionall betwene the side of the Cube circumscribed about the Icosahedron and the side of the Dodecahedron inscribed in the same Cube SVppose that there be a cube ABFD in which let there be inscribed an icosahedron CLIGOR by the 14. of the fiuetenth Let also the Dodecahedron inscribed in the same be EDMNPS by the 13. of the same Now forasmuch as CL the side of the Icosahedron is the greater segmeÌ„t of AB the side of the cube circumscribed about it by the 3. Corollary of the 14. of the fiuetenth and the side ED of the DodecahedroÌ„ inscribed in thesame cube is the lesse segmeÌ„t of the same side AB of the cube by the 2. Corollary of the 13. of the fiuetenth it followeth that AB the side of the cube being diuided by an extreme and meane proportion maketh the greater segment CL the side of the Icosahedron inscribed in it and the lesse segment ED the side of the Dodecahedron likewise inscribâ—d in it Wherefore as the whole line AB the side of the cube is to the greater segment CL the side of the Icosahedron so is the greater segment CL the side of the Icosahedron to the lesse segment EDâ— the side of the Dodecahedron by the third definition of the sixth Wherefore the side of an Icosahedron is the meane proportionall betwene the side of the cube circumscribed about the Icosahedron and the side of the Dodecahedron inscribed in the same cube ¶ The 11. Proposition The side of a Pyramis is in power Octodecuple to the side of the cube inscribed in it FOr by that which was demonstrated in the 18. of the fiuetenth the side of the pyramis is triple to the diameter of the base of the cube inscribed in it and therefore it is in power nonecuple to the same diameter by the 20. of the sixth But the diamer is in power double to the side of the cube by the 47. of the first And the double of nonecuple maketh Octodecuple Wherefore the side of the pyramis is in power Octodecuple to the side of the cube inscribed in it ¶ The 12. Proposition The side of a Pyramis is in power Octodecuple to that right line whose greater segment is the side of the Dodecahedron inscribed in the Pyramis FOrasmuch as the Dodecahedron and the cube inscribed in it are set in one and the sâ—lfâ— same pyramis by the Corollary of the first of this booke and the side of the pyramis circumscribed about the cube is in power octodecuple to the side of the cube inscribed by the former Proposition but the greater segment of the selfe same side of the cube is the side of the Dodecahedron which containeth the cube by the Corollary of the 17. of the thirtenth Wherfore the side of the pyramis is in power octodecuple to that right line namely to the side of the cube whose greater segment is the side of the Dodecahedron inscribed in the pyramis ¶ The 13. Proposition The side of an Icosahedron inscribed in an Octohedron is in power duple to the lesse segment of the side of the same
Octohedron FOrasmuch as in the 17. of the fiuetenth it was proued that the side of an Icosahedron inscribed in a pyramis coupleth together the two sections which are produced by an extreme and meane proportion of the side of the Octohedron which make a right angle and that right angle is contained vnder the lesse segmentes of the sides of the Octohedron and is subtended of the side of the Icosahedron inscribed it followeth therefore that the side of the Icosahedron which subtendeth the right angle being in power equall to the two lines which containe the said angle by the 47. of the first is in power duple to euery one of the lesse segmeÌ„tes of the side of the Octohedron which containe a right angle Wherefore the side of an Icosahedron inscribed in an Octohedron is in power duple to the lesse segment of the â—ide of the same Octohedron ¶ The 14. Proposition The sides of the Octohedron and of the Cube inscribed in it are in power the one to the other in quadrupla sesquialter proportion SVppose that ABGDE be an Octohedron and let the cube inscribed in it be FCHI Then I say that AB the side of the Octohedron is in power quadruple sesquialter to FI the â—ide of the cube Let there be drawen to BE the base of the triangle ABE a perpendicular AN and againe let there be drawen to the same base in the triangle Gâ—E the perpendicular GN which AN GN shall passe by the centres F and I and the line AF is duple to the line FN by the Corollary of the 12. of the thirtenth Wherfore the line AO is duple to the line OE by the 2. of the sixth For the lines FO and NE are parallels And therefore the diameter AG is triple to the line FI. Wherfore the power of AG is noncuple to the power of FI. But the line AG is in power duple to the side AB by the 14. of the thirtenth Wherefore the square of the line AB being ing the halfe of the square of the line AG which is noncuple to the square of the line FI iâ— quadruple sesquialter to the square of the line FI. The sides therefore of the Octohedâ—â—â—â—nd of the cube inscribed in itâ— are in power the one to the other in quadruple sesquialter proportion ¶ The 1â— Proposition The side of the Octohedron is in power quadruple sesquialter to that right line whose greater segment is the side of the Dodecahedron inscribed in the same Octohedron FOrasmuch as in the 14. of this booke it was proued that the side of the Octohedron is in power quadruple sesquialter to the side of the cube inscribed in it but the side of the cube being cut by an extreme and meane proportion maketh the greater segment the side of the Dodecahedron circumscribed about it by the 3. Corollary of the 13. of the fiuetenth therefore the side of the Octohedron is in power quadruple sesquialter to that right line namely to the side of the cube whose greater segment is the side of the Dodecahedron inscribed in the cube But the Dodecahedron and the cube inscribed one within an other arâ— inscribed in one and the selfe same Octohedron by the Corollary of the first of this booke The side therefore of the Octohedron is in power quadruple sesquialter to that right line whose greater segment is the side of the Dodecahedron inscribed in the same Octohedron ¶ The 16. Proposition The side of an Icosahedron is the greater segment of that right line which is in power duple to the side of the Octohedron inscribed in the same Icosahedron SVppose that there be an Icosahedron ABGDFHEC whose side let be BG or â—Câ— and let the Octohedron insâ—â—ibâ—d in it be AKDâ— and let the side therof be AL. Then I say that the side â—C is the greater segment of that right line which is in power duple to the side AL. For forasmuch as figures inscribed and circumscribed haue oâ—e the selfsame centre by the Corollary of the â—1 of the fiuetenth let the same be the point I. Now right lineâ— drawen by thâ—◠〈◊〉 to the midle sections of the opposite sides namely the lines AID and KIL do in the point I â—ut 〈…〉 the other inâ—â— two â—quall 〈◊〉 and perpendicularly by the Corollary of the 14. of the fiuetenth and forasmuch as they couple the midle sections of the opposite lines BG and HF therfore they cut them perpendiularly wherefore also the lines BG 〈…〉 are parallels by the 4. Corollary of the 14. of the 〈…〉 Now then draw a line from B to H and the sayd â—â—ne BH shall be equall and parallel to the line KL by the 33. of the first But the line BH subtendeth â—wâ— sides of the pentagon which is composed of the sides of the Icosahedron namely the sides BA and AH Wherfore the line BH being cut by an extreme and meane proportion maketh the greater segment the side of the pentagon by the 8. of the thirtenth which side is also the side of the Icosahedron namely EC And vnto the line BH the line KLâ— is equall and the line KL is in power duple to AL the side of the Octohedron by the 47. of the first for in the square AKDL the angle KAL is a right angle Wherefore EC the side of the Icosahedron is the greater segment of the line BH or KL which is in power duple to AL â—he side of the Octohedron inscribed in the Icosahedron Wherefore the side of an Icosahedron is the greater segment of that right line which is in power duple to the side of the Octohedron inscribed in the same Icosahedron ¶ The 17. Proposition The side of a Cube is to the side of a Dodecahedron inscribed in it in duple proportion of an extreame and meane proportion FOr it was manifesâ— by the â— corollary of the 13. of the fiuetenth that the side of a cube diuided by an extreame and meane prâ—portion maketh the lesse segment the side of the dodecahedron inscribed in it but the whole is to the lesse segment in duple proportion of that in which it is to the greater by the 10. diffinitioÌ„ of the fifth For the whole the greater segmeÌ„t and the lesse are lines in continuall proportion by the 3. diffinition of the sixth Wherefore the whole namely the side of the cube is to the side of the dodecahedron inscribed in it namely to his lesse segment in duple propoâ—tion of an extreame and meane proportion ' namely of that which the whole hath â—o the greater segmenâ— by the 2. of the fourtenth ¶ The 18. Proposition The side of a Dodecahedron is to the side of a Cube inscribed in it in conuerse proportion of an extreame and meane proportion IT was proued in the 3. corollary of the 13. of the fiuetenth that the side of a Dodecahedâ—on circumscribed about a Cube is the greater segment of the side of the same Cube Wherefore the whole
Demonstration * The circles so made or so considered in the sphere are called the greatest circles All other not hauing the center of the sphere to be their center alsoâ— are called lesse circles Note these descriptions * An other Corollary * An other Corollary Construction * This is also proued in the Asâ—umpt before added out oâ— Flussas Note what a greater or greatest circle in a Spere is First part of the Construction Noteâ— * You know full well that in the superficies of the sphere â—â—â—ly the circumferences of the circles are but by thâ—se circumferences the limitatioÌ„ and assigning of circles is vsed and so the circumference of a circle vsually called a circle which in this place can not offend This figure is restored by M. Dee his diligence For in the greeke and Latine Euclides the line GL the line AG and the line KZ in which three lynes the chiefe pinch of both the demonstrations doth stand are vntruely drawen as by comparing the studious may perceaue Note You must imagine 〈◊〉 right line AX to be perpeÌ„dicular vpon the diameters BD and CE though here AC the semidiater seme to be part of AX. And so in other pointes in this figure and many other strengthen your imagination according to the tenor of constructions though in the delineatioÌ„ in plaine sense be not satisfied Note BO equall to BK in respect of M. Dee his demonstration following †Note â—his point Z that you may the better vnderstand M. Dee his demoÌ„stration Second part of the construction Second part of the demonstration ✚ Which of necessity shall fall vpon Z as M. Dee proueth it and his profe is set after at this marke ✚ following I. Dee * But AZ is greater theÌ„ AG as in the former propositioÌ„ KM was euident to be greater then KG so may it also be made manifest that KZ doth neyther touch nor cut the circle FGâ—H An other proue that the line AY is greater theÌ„ the line AG. * This as an assumpt is presently proued Two cases in this proposition The first case Demonstration leading to an impossibilitie Second case * As it is â—asiâ— to gather by the â—â—â—umpt put after the secoâ—â— of this booâ—â— Note a generall rule The second part of the Probleme two wayes executed An vpright Cone The second part of the Probleme The second â—aâ—â— oâ— the â—robleme ☜ * This may easely be demonstrated as in thâ— 17. proposition the section of a sphere was proued to be a circle * For taking away all doubt this aâ— a Lemma afterward is demâ—strated A Lemma as it were presently demonstrated Construction Demonstration The second part of the Probleme * Construction Demonstration An other way of executing this probleme The conuerse of the assuÌ„pt A great error commonly maintained Betwene straight and croked all maner of proportioÌ„ may be geuen Construction Demonstration The diffiniâ—ioÌ„ of a circâ—e â—â—apâ—â—d in a spâ—erâ— Construction Demonstration This is manifest if you consider the two triangles rectangles HOM and HON and then with all vse the 47. of the first of Euclide Construction DemonstratioÌ„ Construction DemonstratioÌ„ This in maner of a Lemmâ— is presently proued Note here of Axe base soliditie more then I nede to bring any farther proofe for Note * I say halfe a circâ—lar reuolution for that suâ—â—iseth in the whole diameter ST to describe a circle by iâ— it be moued â—â—out his center Q c. Lib 2 prop 2. de Spheâ—a Cylindrâ— Note * A rectangle parallelipipedon geuâ—n equall to a Sphere geuen To a Sphere or to any part of a Sphere assigned as a third fourth fifth c to geue a parallelipipedon equall Sided Columes Pyramids and prismes to be geuen equall to a Sphere or to any certayne part thereof To a Sphere or any segment or sector of the same to geue a cone or cylinder equall or in any proportion assigned Farther vse of Sphericall Geometrie The argument of the thirtenth booke Construction Demonstration * The AssuÌ„pt proued * Because AC is supposed greater then AD therefore his residue is lesse then the residue of AD by the common sentence Wherefore by the supposition DB is greater then â—C The chieâ—e line in all Euclides Geometrie What is ment here by A section in one onely poiâ—t Construction Demonstration * Note how CE and the gnonom XOP are proued equall for it serueth in the conuerse demonstrated by M. Dee here next after This proposition â—the conuerse of the former * As we haâ—e noted the place of the peculier proâ—e there â—in the demoÌ„stration of the 3. * Therefore by my second Theoreme added vpon the second proposition DC is deuided by extreame and meane proportion in the point A. And because AC is bigger then CB therfore DA is greater then AC wherefore if a right line c. as in the proposition Which was to be demonstrated * Therefore by my second Theoreme added vpon the second proposition DC is deuided by extreame and meane proportion in the point A. And because AC is bigger then CB therfore DA is greater then AC wherefore if a right line c. as in the proposition Which was to be demonstrated Construction * Though I say perpeÌ„dicular yes you may perceue how infinite other pâ—sâ—â—ioÌ„s will serue so that DI and AD make an angle for a triangle to haue his sides proportionally cut c. Demonstration Demonstration I. Dee This is most euident of my second Theoreme added to the third propositioÌ„ For to adde to a whole line a line equall to the greater segmeÌ„t to adde to the greater segment a line equall to the whole line is all one thing in the line produced By the whole line I meane the line diuided by extreme and meane proportion This is before demonstrated most euidently and briefly by M. Dee after the 3. proposition Note Note 4. Proportional lines Note two middle proportionals Note 4. wayes of progresâ—ion in the proportion of a line deuided by extreme and middle proportion What resolution and composition is hath before bene taught in the beginning of the first booke * Proclus in the Greeke in the 58. page Construction Demonstration Two cases in this proposition Construction Thâ— first case Demonstration The second case Construction Demonstration Construction Demonstration This Corollary is the 3. proposition of the â—4 booke after Campane Demonstration of the first part Demonstration of the second part Construction Demâ—nstration Construstion Demonstration Constrâ—yction Demonstration This Corollary is the 11. propâ—sition of the 14. booke after Campane This Corollary is the 3. Corollary after the 17. proposition of the 14 booke after Campane * By the name oâ— a Pyramis both here iâ— this booke following vnderstand a Tetrahedron An other construction and demonstration of the second part after Fâ—ussas Third part of the demonstration This Corollary is the 15. proposition of the 14. booke after Campane This Corollary Campane putteth as a Corollary after