Selected quad for the lemma: cause_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
cause_n	bring_v	good_a	let_v	1,459	5	4.0417	3	false

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ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A00429	The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed; Elements. English	Euclid.; Dee, John, 1527-1608.; Candale, François de Foix, comte de, 1502-1594.; Billingsley, Henry, Sir, d. 1606.	1570 (1570)	STC 10560; ESTC S106699	1,020,889	884

Againe of thinges being in like swiftnes of mouing to thinke the nerer to moue faster and the farder much slower Nay of two thinges wherof the one incomparably doth moue swifter then the other to deme the slower to moue very swift the other to stand what an error is this of our eye Of the Raynbow both of his Colours of the order of the colours of the bignes of it the place and heith of it c to know the causes demonstratiue is it not pleasant is it not necessary of two or three Sonnes appearing of Blasing Sterres and such like thinges by naturall causes brought to passe and yet neuertheles of farder matter Significatiue is it not commodious for man to know the very true cause occasion Naturall Yea rather is it not greatly against the Souerainty of Mans nature to be so ouershot and abused with thinges at hand before his eyes as with a Pecockes tayle and a Doues necke or a whole ore in water holden to seme broken Thynges farre of to seeme nere and nere to seme farre of Small thinges to seme great and great to seme small One man to seme an Army Or a man to be curstly affrayed of his owne shaddow Yea so much to feare that if you being alone nere a certaine glasse and proffer with dagger or sword to foyne at the glasse you shall suddenly be moued to giue backe in maner by reason of an Image appearing in the ayre betwene you the glasse with like hand sword or dagger with like quicknes foyning at your very eye likewise as you do at the Glasse Straunge this is to heare of but more meruailous to behold then these my wordes can signifie And neuerthelesse by demonstration Opticall the order and cause therof is certified euen so as the effect is consequent Yea thus much more dare I take vpon me toward the satisfying of the noble courrage that longeth ardently for the wisedome of Causes Naturall as to let him vnderstand that in London he may wish his owne eyes haue profe of that which I haue sayd herein A Gentleman which for his good seruice done to his Countrey is famous and honorable and for skill in the Mathematicall Sciences and Languages is the Od man of this kind c. euen he is hable and I am sure will very willingly let the Glasse and profe be sene and so I here request him for the encrease of wisedome in the honorable and for the stopping of the mouthes malicious and repressing the arrogancy of the ignorant Ye may easily gesse what I meane This Art of Perspectiue is of that excellency and may be led to the certifying and executing of such thinges as no man would easily beleue without Actuall profe perceiued I speake nothing of Naturall Philosophie which without Perspectiue can not be fully vnderstanded nor perfectly atteined vnto Nor of Astronomie which without Perspectiue can not well be grounded Nor Astrologie naturally Verified and auouched That part hereof which dealeth with Glasses which name Glasse is a generall name in this Arte for any thing from which a Beame reboundeth is called Catoptrike and hath so many vses both merueilous and proffitable that both it would hold me to long to no●● therin the principall conclusions all ready knowne And also perchaunce some thinges might lacke due credite with you And I therby to leese my labor● and you to slip into light Iudgement ● Before you haue learned sufficiently the powre of Nature and Arte. NOw to procede Astronomie is an Arte Mathematicall which demonstrateth the distance magnitudes and all naturall motions apparences and passions propre to the Planets and fixed St●rtes for any time past present and to come in respect of a certaine Horizon or without respect of any Horizon By this Arte we are certified of the distance of the Starry Skye and of eche Planete from the Centre of the Earth and of the greatnes of any Fixed starre sene or Planete in respect of the Earthes greatnes As we are sure by this Arte that the Solidity Massines and Body of the Sonne conteineth the quantitie of the whole Earth and Sea a hundred thre score and two times lesse by 1 ● one eight parte of the earth But the Body of the whole earthly globe and Sea is bigger then the body of the Mone three and forty times lesse by 1 ● of the Mone Wherfore the Sonne is bigger then the Mone 7000 times lesse by 59 ●● 164 that is precisely 6940 ●5 ●● bigger then the Mone And yet the vnskillfull man would iudge them a like bigge Wherfore of Necessity the one is much farder from vs then the other The Sonne when he is fardest from the earth which now in our age is when he is in the 8. degree of Cancer is 1179 Semidiameters of the Earth distante And the Mone when she is fardest from the earth is 68 Semidiameters of the earth and 1 ● The nerest that the Mone commeth to the earth is Semidiameters 52 ¼ The distance of the Starry Skye is frō vs in Semidiameters of the earth 20081 1 ● Twenty thousand fourescore one and almost a halfe Subtract from this the Mones nerest distance from the Earth and therof remaineth Semidiameters of the earth 20029 1 ● Twenty thousand nine and twenty and a quarter So thicke is the heauenly Palace that the Planetes haue all their exercise in and most meruailously perfourme the Commaūdement and Charge to them giuen by the omnipotent Maiestie of the king of kings This is that which in Genesis is called Ha Rakia Consider it well The Semidiameter of the earth coteineth of our common miles 3436 ● ● three thousand foure hundred thirty six and foure eleuenth partes of one myle Such as the whole earth and Sea round about is 21600. One and twenty thousand six hundred of our myles Allowyng for euery degree of the greatest circle thre score myles Now if you way well with your selfe but this litle parcell of f●ute Astronomicall as concerning the bignesse Dist●nces of Sonne Mone Sterry Sky and the huge massines of Ha Rakia will you not finde your Consciences moued with the kingly Prophet to sing the confession of Gods Glory and say The Heauens declare the glory of God and the Firmament Ha-Rakia sheweth forth the workes of his handes And so forth for those fiue first staues of that kingly Psalme Well well It is time for some to lay hold on wisedome and to Iudge truly of thinges and not so to expound the Holy word all by Allegories as to Neglect the wisedome powre and Goodnes of God in and by his Creatures and Creation to be seen and learned By parables and Analogies of whose natures and properties the course of the Holy Scripture also declareth to vs very many Mysteries The whole Frame of Gods Creatures which is the whole world is to vs a bright glasse from which by reflexion reboundeth to our knowledge and perceiuerance Beames and Radiations● representing the

what manly vertues in other noble men florishing before his eyes he Sythingly aspired after what prowesses he purposed and ment to achieue with what feats and Artes he began to furnish and fraught him selfe for the better seruice of his Kyng and Countrey both in peace warre These I say his Heroicall Meditations forecas●inges and determinations no twayne I thinke beside my selfe can so perfectly and truely report And therfore in Conscience I count it my part for the honor preferment procuring of vertue thus briefly to haue put his Name in the Register of Fa●● Immortall To our purpose This Iohn by one of his actes besides many other both in England and Fraunce by me in him noted did dislose his harty loue to vertuous Sciences and his noble intent to excell in Martiall prowesse When he with humble request and instant Solliciting got the best Rules either in time past by Greke or Romaine or in our time vsed and new Stratagemes therin de●ised for ordring of all Companies summes and Number● of mē Many or few with one kinde of weapon or mo appointed with Artillery or without on horsebacke or on fote to giue or take onset to seem many being few● to s●em few being many To marche in battaile or Iornay with many such feate● to Foughten f●●ld Skarmoush or Ambushe appartaining And of all these liuely desi●nementes most curiously to be in velame parchement described with Notes peculier markes as the Arte requireth and all these Rules● and descriptions Arithmeticall inclosed in a riche Case of Gold he vsed to weare about his necke● as his Iuell most precious and Counsaylour most trusty Thus Arithmetik● of him was shryned in gold Of Numbers frute he had good hope Now Numbers therfore innumerable in Numbers prayse his sh●●ne shall finde What nede I for farder ●rofe to you of the Scholemasters of Iustice to require testimony how nedefull how frutefull how skillfull a thing Arithmetike is I meane● the Lawyers of all sortes Vndoubtedly the Ciuilians can meruaylously declare how neither the Auncient Romaine lawes without good knowledge of Numbers art can be perceiued Nor Iustice in infinite Cases without due proportion narrowly considered is hable to be executed● How Iustly with great knowledge of Arte did Pap●●●●anus institute a law of partition and allowance betwene man and wife a●●● a ●●●orce But how Acc●rsius Bald●s Bartolus Iason Alex●nder and finally Alciat●● bei●g o●●●rwise notably well learned do iumble gesse and erre from the ●quity●●rt ●nd I●●ent of the lawmaker Arithmetike can detect and conuince and clerely make the truth to shine Good Bartolus tyred in the examining proportioning of the matter ●nd with Accursius Glosse much cumbred burst out and sayd Nulla est i● 〈◊〉 libr● 〈◊〉 glossa difficili●r Cui●s c●mputationem nec Scholas●ici ●ec Doct●res intellig●●t c. That is● In the whole booke there is no ●losse harder then thi●● Whose accoump● or reckenyng neither the Scholers nor the Doctours vnderstand c. What can they say of I●lianus law Si ita S●ript●m c. Of the Testators will iustly performing betwene the wife Sonne and daughter How can they perceiue the ●●●●●tie of Aphricanus Arithmeticall Reckening where he treateth of Lex Falcid●a ● How ●●n they deliuer him from his Reprouers and their maintainers as I●●●●es Accursius Hypolitus and Alciatus How ●ustly and artificially was African●s reckening made Proportionating to the Sommes bequeathed the Contributions of eche part Namely for the hundred presently receiued 17 1 7. And for the hundred receiued after ten monethes 12 6 7 which make the 30 which were to be cōtributed by the legatari●s to the heire For what proportion 100 hath to 75 the same hath 17 1 7 to 12 6 7 Which is Sesquitertia that is as 4 to 3. which make 7. Wonderfull many places in the Ciuile law require an expert Arithmeticien to vnderstand the deepe Iudgemēt Iust determinatiō of the Auncient Romaine Lawmakers But much more expert ought he to be who should be hable to decide with aequitie the infinite varietie of Cases which do or may happen vnder euery one of those lawes and ordinances Ciuile Hereby easely ye may now coniecture that in the Canon law and in the lawes of the Realme which with vs beare the chief Authoritie Iustice and equity might be greately preferred and skilfully executed through due skill of Arithmetike and proportions appertainyng The worthy Philosophers and prudent lawmakers who haue written many bookes De Republica How the best state of Common wealthes might be procured and mainteined haue very well determined of Iustice which not onely is the Base and foundacion of Common weales but also the totall perfection of all our workes words and thoughtes defining it to be that vertue by which to euery one is rendred that to him appertaineth God challengeth this at our handes to be honored as God to beloued as a father to be feared as a Lord master Our neighbours proportiō is also prescribed of the Almighty lawmaker which is to do to other euen as we would be done vnto These proportions are in Iustice necessary● in duety commendable and of Common wealthes the life strength stay and florishing Aristotle in his Ethikes to fatch the sede of Iustice and light of direction to vse and execute the sam● was fayne to fly to the perfection and power of Numbers for proportions Arithmeticall and Geometricall Plato in his booke called Epinomis which boke is the Threasury of all his doctrine where his purpose is to seke a Science which when a man had it perfectly he might seme and so be in dede Wise. He br●efly of other Sciences discoursing findeth them not hable to bring it to passe But of the Science of Numbers he sayth Illa qua numerum mortalium generi d●●n id profecto efficiet● Deum antem aliquem magis quam fortunam ●d sa●●tem nostram hoc m●nus nobis arbitror contulisse c. Nam ipsum ●onorum omnium Authorem cur non maximi boni Prudentiae dico causam arbitramur That Science verely which hath taught mankynde number shall be able to bryng it to passe And I thinke a certaine God rather then fortune to haue giuen vs this gift for our blisse For why should we not Iudge him who is the Author of all good things to be also the cause of th● greatest good thyng namely Wisedome There at length he proueth Wisedome to be atteyned by good Skill of N●mbers With which great Testimony and the manifold profes and reasons before expressed● you may be sufficiently and fully persuaded of the perfect Science of Arithmetike to make this accounte That of all Sciences next to Theologie it is most diuine most pure most ample and generall most profounde most subtile most commodious and most necessary Whos 's next Sister is the Absolute Science of Magnitudes of which by the Direction and aide of him whose Magnitude is Infinite and of vs Incomprehensible I now

first set they also must needes be at the least commensurable in power the one to the other For forasmuch as their squares are rationall they shall bee commensurable to the square of the rationall line first set Wherfore by the 12. of this booke they are also commensurable the one to the other Wherefore their lines are at the least commensurable in power the one to the other And it is possible also that they may be commēsurable in lēgth the one to the other For suppose that A be a rationall li●e first set and let the line B be vnto the same rationall line A commensurable in power onely that is incommensurable in length vnto it Let there be also an other line C commensurable in length to the lyne B which is possible by the principles of this booke Now by the 13. of the tenth it is manifest that the line C is incommensurable in length vnto the line A. But the square of the line A is cōmēsurable to the square of the line B by supposition and the square of the line C is also commensurable to the square of the line B by supposition Wherefore by the 12. of this booke the square of the line C is commensurable to the square of the line A. Wherfore by the definition the line C shall be rationall commensurable in power onely to the line A as also is the line B. Wherefore there are geuen two rationall lines commensurable in power onely to the rationall line first set and commēsurable in length the one to the other Here is to be noted which thing also we before noted in the definitions that Campane and others which followed him brought in these phrases of speaches to call some lynes rationall in power onely and other some rationall in length and in power which we cannot finde that Euclide euer vsed For these wordes in length and in power are neuer referred to rationalitie or irrationalitie but alwayes to the commensurabilitie or incommensurablitie of lines Which peruerting of wordes as was there declared hath much increased the difficulty and obscurenes of this booke And now I thinke it good agayne to put you in minde that in these propositions which follow we must euer haue before our eyes the rationall line first set vnto which other lines compared are either rationall or irrationall according to their commensurability or incommensurabilitie ¶ The 16. Theoreme The 19. Proposition A rectangle figure comprehended vnder right lines commensurable in lengthe being rationall according to one of the foresaide wayes is rationall SVppose that this rectangle figure AC be comprehended vnder these right lines AB and BC being commensurable in length and rationall according to one of the foresaid wayes Then I say that the superficies AC is rationall describe by the 46. o● the first vpon the line AB a square AD. Wherfore that square AD is rationall by the definition And forasmuch as the line AB is commensurable in length vnto the line BC and the line AB is equall vnto the lyne BD therefore the lyne BD is commensurable in length vnto the line BC. And as the line BD is to the line BC so is the square DA to the superficies AC by the first of the sixt but it is proued that the line BD is commensurable vnto the line BC wherfore by the 10. of the tenth the square DA is commensurable vnto the rectangle superficies AC But the square DA is rationall wherfore the rectangle superficies AC also is rationall by the definition A rectangle figure therfore comprehended vnder right lines commensurable in length beyng rationall accordyng to one of the foresayd wayes is rationall which was required to be proued Where as in the former demonstration the square was described vpon the lesse line we may also demonstrate the Proposition if we describe the square vpon the greater line and that after thys maner Suppose that the rectangle superficies BC be contayned of these vnequall lines AB and AC which let be rationall commensurable the one to the other in length And let the line AC be the greater And vpon the line AC describe the square DC Then I say that the parallelogramme BC is rationall For the line AC is commensurable in length vnto the line AB by supposition and the line DA is equall to the line AC Wherefore the line DA is commensurable in length to the line AB But what proportion the line DA hath to the line AB the same hath the square DC to the para●lelogramme C● by the first of the sixt Wherefore by the 10. of this booke the square DC is commensurable to the parallelogramme CB. But it is manifest that the square DC is rationall for that it is the square of a rationall line namely AC Wherefore by the definition the parallelogrāme also CB is rationall Moreouer forasmuch as those two former demonstrations seeme to speake of that parallelogrāme which is made of two lines of which any one may be the li●e first set which is called the first rationall line from which we sayd ought to be taken the measures of the other lines compared vnto it and the other is commēsurable in length to the same first rationall line which is the first kinde of rationall lines cōmensurable in length I thinke it good here to set an other case of the other kinde of rationall lines of lines I say rationall cōmensurable in length compared to an other rationall line first set to declare the generall truth of this Theoreme and that we might see that this particle according to any of the foresayd wayes was not here in vaine put Now then suppose first a rationall line AB Let there be also a parallelogrāme CD contayned vnder the lines CE and ED which lines let be rationall that is commensurable in length to the ●irst rationall line propounded AB Howbeit let those two lines CE and ED be diuers and vnequall lines vnto the first rationall line AB Then I say that the parallelogramme CD is rationall Describe the square of the line DE which let be DF. First it is manifest by the 12. of this booke that the lines CE ED are commensurable in lēgth the one to the other For either of them is supposed to be commensurable in length vnto the line AB But the line ED is equall to the line EF. Wherefore the line CE is commensurable in length to the line BF But 〈◊〉 the line CE is ●o the line ● F ●o is the parallelogramme CD to the square DF by the first of the sixt Wh●refore by the 10. of this booke the parallelogramme CD shall be commensurable to the square DF. But the square DF is commensurable to the square of the line AB which is the first rationall line propounded Wherfore by the 12. of this booke the parallelogramme CD is commensurable to the square of the line AB But the square of

of BE. But that which is produced of AB into BC together with the square number of CE is supposed to be equal to the square number of BE wherfore that which is produced of GB into BC together with the square number of CE is equall to that which is produced of AB into BC together with the square number of CE. Wherefore taking away the square number of CE which is common to them both the number AB shall be equall to the number GB namely the greater to the lesse which is impossible Wherfore that which is produced of AB into BC together with the square number of CE is not equall to the square number of BE I say also that that which is produced of AB into BC together with the square number of CE is not lesse then the square number of BE. For if it be possible thē shall it be equ●l to some square number lesse then the square number of BE. Wherfore let the number produced of AB into BC together with the square of the number CE be equal to the square number of BF And let the number HA be double to the number DF. Thē also it followeth that the number HC is double to the number CF so that HC also is deuided into two equall partes in F and therfore also the number which is produced of HD into BC together with the square number of FC is equall to the square number of the number BF But by supposition the number which is produced of AB into BC together with the square number of CE is equall to the square number of BF Wherfore it followeth that the number produced of AB into BC together with the square number of CE is equall to that which is produced of HB into BC together with the square number CF which is impossible For if it should be equall then forasmuch as the square of CF is lesse then the square of CE the number produced of HB into BC should be greater then th● number produced of AB into BC. And so also should the number HB be greater then the number AB when yet it is lesse then it Wherfore the number produced of AB into BC together with the square number of CE is not lesse then the square nūber of ● E. And it is also proued that it cannot be equall to the square number of BE neither greater then it Wherfore that which is produced of AB into BC added to the square number of CE maketh not a square number And although it be possible to demonstrate this many other wayes yet this semeth to vs suffici●n● least the matter beyng ouer long should seeme to much tedious ¶ The 6. Probleme The 29. Proposition To finde out two such rationall right lynes commensurable in power only that the greater shall be in power more then the lesse by the square of a right line commensurable in length vnto the greater LEt there be put a rational line AB and take also two such square numbers CD and DE that their excesse CE be not a square number by the corolary of the first assumpt of the 28. of the tenth And vpon the line AB describe a semicircle AFB And by the corollary of the 6. of the tenth as the number DC is to the number CE so let the square of the lyne BA be to the square of the line AF. And draw a line from F to B. Now for that as the square of the line BA is to the square of the line AF so is the number CD to the number CE therfore the square of the line BA hath to the square of the line AF that proportion that the nūber CD hath to the number CE. Wherfore the square of the line BA is cōmēsurable to the square of the line AF by the 6. of the tēth But the square of the line AB is rational Wherfore also the square of the line AF is rational Wherfore also the line AF is rationall And forasmuch as the number CD hath not vnto the number CE that proportion that a square number hath to a square number therfore neither also hath the square of the line AB to the square of the line AF that proportion that a square number hath to a square number Wherfore by the 9. of the tēth the line AB is vnto the line AF incommensurable in length Wherfore the lines AF and AB are rationall commensurable in power onely And for that as the number DC is to the number CE so is the square of the line AB to the square of the line AF therfore by conuersion or euerse proportiō which is demonstrated by the corollary of the 19. of the fifth as the number CD is to the number DE so is the square of the line AB to the square of the line BF which is the excesse of the square of the line AB aboue the square of the line AF by the assumpt put before the 14. of this booke But the number CD hath to the number DE that proportion that a square number hath to a square number wherfore the square of the line AB hath to the square of the line BF that proportion that a square num●er hath to a square number Wherefore by the 9. of the tenth the line AB is commensurable in length vnto the line BF And by the 47. of the first the square of the line AB is equall to the squares of the lines AF and FB Wherfore the line AB is in power more then the line AF by the square of the line BF which is commensurable in length vnto the line AB Wherefore there are found out two such rationall lines commensurable in power onely namely AB and AF so that the greater line AB is in power more then the lesse line AF by the square of the line FB which is commensurable in length vnto the line AB which was required to be done ¶ The 7. Theoreme The 30. Proposition To finde out two such rationall lines commensurable in power onely that the greater shal be in power more then the lesse by the square of a right line incommensurable in length to the greater LEt there be put a rationall line AB and take also by the 2. assumpt of the 28. of the tenth two square numbers CE and ED which being added together make not a square number and let the numbers CE and ED added together make the number CD And vpon the line AB describe a sencircle AFB And by the corollary of the 6. of the tenth as the number DC is to the number CE so let the square of the line AB be to the square of the line AF and draw a line from F to B. And we may in like sort as we did in the former proposition proue that the lines BA and AF are rationall commensurable in power onely And for that as the number DC is

this Probleme it is now easie to execute and that two wayes I meane to A the sphere geuen to make an vpright cone in any proportion geuen betwen two right lines For let the proportion geuen be that which is betwene X and Y. By the order of my additions vpon the 2. of this twelfth booke to the circle EKG make an other circle in that proportion that X is to Y which let be Z. Vpon the center of Z reare a line perpendicular and equall to FL. I say that the cone whose base is Z and the height equall to FL is to A in the proportion of X to Y. For the cone vppon Z by construction hath height equall to the height of the cone LEKG and Z by construction is to EKG● as X is to Y Wherefore by the 11. of this twelfth the cone vpon Z is to the cone LEKG as X is to Y. But the cone LEKG is proued equall to the sphere A. Wherfore the cone vpon Z is to A as X is to Y by the 7. of the fift To a Sphere geuen therefore we haue made a cone in any proportion geuen betwene two right lines Secondly as X is to Y so to FL let there be a fourth line by the 12. of the sixt and suppose it to be W. I say that a cone whose base is equall to EKG and height the line W is to A as X is to Y. For by the 14. of this twelfth cones being set on equall bases are one to the other as their heightes are But by construction the height W is to the height FL a● X is to Y. Wherefore the cone which hath his base equall to EKG and height the line W is to the cone LEKG as X is to Y. And it is proued that to the cone LEKG the Sphere A is equall Wherfore by the 7. of the fift the cone whose base is equall to EKG and height the line W is to A as X is to Y. Therefore a Sphere being geuen we haue made an vpright cone in any proportion geuen betwene two right lines And before we made an vpright cone equall to the Sphere geuen Wherfore a Sphere being geuen we haue made an vpright cone equall to the same or in any other proportion geuen betwene two right lines I call that an vpright cone whose axe is perpendicular to his base ¶ A Corollary Of the first part of the demonstration it is euident A Sphere being propounded that a Cone whose base hath his semidiameter equall to the diameter therof and height equall to the semidiameter of the same Sphere is equall to that sphere propounded ¶ A Probleme 2. A Sphere being geuen and a circle to re●re an vpright Cone vpon that circle as a base equall to the Sphere geuen or in any proportion betwene two right lines assigned And the second part of this Probleme is thus performed Suppose the proportion geuen to be that which is betwene X Y. Then as X is to Y so let an other right line found be to the h●ight of F which line let be G. For this G the found height by construction being to the height of F as X is to Y doth cause this cone which let be M vpon C the circle geuē or an other to it equall duely reared to be vnto the cone F as X is to Y by the 14. of this twelfth But F is proued equall to the Sphere geuen Wherfo●e M is to the Sphere geuen as X is to Y. And M is ●eared vpon the circle geuen or his equall Wherfo●e a Sphere being geuen a circle we haue reared an vp●ight cone vpon that geuen circle as a base equall to the Sphere geuen or in any proportion betwene two right lines assigned which was required to be done ¶ A Probleme 3. A Sphere being geuen and a right line to make an vpright cone equall to the Sphere geuen or in any other proportion geuen betwene two right lines which made cone shall haue his height equall to the right line geuen For the second part sinde a circle which shall haue to the base of L any proportion appointed in ●ight lines as the proportion of X to Y which by my additions vpō the second of this booke ye haue l●arned to do Then with the height equall to the heigth of L reared vpon this last found circle which l●t be T as a base you shall satisfie the Probleme L●t that Cone be V. For this last cone V is to L as his base is to the base of L by the 11. of this twelfth But L is proued equall to the Sphere geuen Wherfore by the 7. of the fift this l●●t cone V hath to R the Sphere geuen that proportion which is betwene X and Y assigned and forasmuch as the height of this cone V is equall to the height of L and the height of L equall to S the right line geuen by construction it is euident that a Sphere being geuen a right line we haue made an vpright cone equall to the Sphere geuen or in any other proportion geuen betwene two right lines which made cones haue their height equall to the right line geuen which ought to be done Vuwilling I am to vse thus many wordes in matters so plain● and ease But this I thinke can not hinder them that by nature are not so quicke of inuention as to lead euery thing gen●rally spoken to a particular execution ¶ A Theoreme 3. Euery Cylinder which hath his base the greatest Circle in a Sp●er● heith equall to the diameter of that Sphere is Sesquialtera to that Sphere Also the superficies of that Cylinder with his two bases is Sesquilatera to the superficies of the Sphere and without his two bases is equall to the superficies of that Spher● Suppose a sphere to be signified by A and an vpright cylinder hauing to his base a circle equall to the greatest circle in A contayned and his heith equall to the diameter of A let be signified by FG. I say that FG is sesquialter to A Secondly I say that the croked cylindricall super●icies of FG together with the superfici●ces of his two opposit bases is sesquialtera to the whole superficies sphericall of A. Thirdly I say that the cylindricall superficies of FG omitting his two opposite bases is equall to the superficies of the spere A. Let the base of FG be the circle FLB whose center sup●ose M and diameter FB And the axe of the same FG let be MH Which is his heith for we suppose the cylinder to be vpright and suppose H to be his toppe or vertex Forasmuch as by supposition MH is equall to the diameter of A. Let MH be deuided into two equall partes in the point N by a playne superficies passing by the point N and being parallell to the opposit bases of FG. By the thirtenth of this twelfth booke it then foloweth that the cylinder FG is