E produced D wherfore A measureth D but it also measureth it not which is impossible Wherfore it is impossible to finde out a fourth number proportionall with these numbers A B C whensoeuer A measureth not D. ¶ The 20. Theoreme The 20. Proposition Prime numbers being geuen how many soeuer there may be geuen more prime numbers SVppose that the prime numbers geuen be A B C. Then I say that there are yet more prime numbers besides A B C. Take by the 38. of the seuenth the lest number whom these numbers A B C do measure and let the same be DE. And vnto DE adde vnitie DF. Now EF is either a prime number or not First let it be a prime number then are there found these prime numbers A B C and EF more in multitude then the prime numbers â—irst geuen A B C. But now suppose that EF be not prime Wherefore some prime number measureth it by the 24. of the seuenth Let a prime number measure it namely G. Then I say that G is none of these numbers A B C. For if G be one and the same with any of these A B C. But A B C measure the nuÌ„ber DE wherfore G also measureth DE and it also measureth the whole EF. Wherefore G being a number shall measure the residue DF being vnitieâ— which is impossible Wherefore G is not one and the same with any of these prime numbers A B C and it is also supposed to be a prime number Wherefore there are â—ound these prime numbers A B C G being more in multitude then the prime numbers geuen A B C which was required to be demonstrated A Corollary By thys Proposition it is manifest that the multitude of prime numbers is infinite ¶ The 21. Theoreme The 21. Proposition If euen nuÌ„bers how many soeuer be added together the whole shall be eueÌ„ SVppose that these euen numbers AB BC CD and DE be added together Then I say that the whole number namely AE is an euen number For forasmuch as euery one of these numbers AB BC CD and DE is an euen number therefore euery one of them hath an halfe Wherefore the whole AE also hath an halfe But an euen number by the definition is that which may be deuided into two equall partes Wherefore AE is an euen number which was required to be proued ¶ The 22. Theoreme The 22. Proposition If odde numbers how many soeuer be added together if their multitude be euen the whole also shall be euen SVppose that these odde numbers AB BC CD and DE being euen in multitude be added together Then I say that the whole AE is an euen number For forasmuch as euery one of these numbers AB BC CD and DE is an odde number is ye take away vnitie from euery one of them that which remayneth oâ— euery one of theÌ„ is an euen number Wherefore they all added together are by the 21. of the ninth an euen number and the multitude of the vnities taken away is euen Wherefore the whole AE is an euen number which was required to be proued ¶ The 23. Theoreme The 23. Proposition If odde numbers how many soeuer be added together and if the multitude of them be odde the whole also shall be odde SVppose that these odde numbers AB BC and CD being odde in multitude be added together Then I say that the whole AD is an odde number Take away from CD vnitie DE wherefore that which remaineth CE is an euen number But AC also by the 22. of the ninth is an euen number Wherfore the whole AE is an euen number But DE which is vnitie being added to the euen number AE maketh the whole AD aâ— odde number which was required to be proued◠¶ The 24. Theoreme The 24. Proposition If from an euen number be takeÌ„ away an euen number that which remaineth shall be an euen number SVppose that AB be an euen number and from it take away an euen number CB. Then I say that that which remayneth namely AC is an euen number For forasmuch as AB is an euen euen number it hath an halfe and by the same reason also BC hath an halfe Wherfore the residue CA hath an halfe Wherfore AC is an euen number which was required to be demonstrated ¶ The 25. Theoreme The 25. Proposition If from an euen number be taken away an odde number that which remaineth shall be an odde number SVppose that AB be an euen number and take away from it BC an odde number Then I say that the residue CA is an odde number Take away from BC vnitie CD Wherfore DB is an euen number And AB also is an euen number wherefore the residue AD is an euen number by the â—ormer proposition But CD which is vnitie being taken away from the euen nuÌ„ber AD maketh the residue AC an odde number which was required to be proued ¶ The 26. Theoreme The 26. Proposition If from an odde number be taken away an odde number that which remayneth shall be an euen number SVppose that AB be an odde number and from it take away an odde number BC. TheÌ„ I say that the residue CA is an euen number For forasmuch as AB is an odde number take away from it vnitie BD. Wherfore the residue AD is euen And by the same reason CD is an euen number wherfore the residue CA is an euen number by the 24. of this booke â— which was required to be proued ¶ The 27. Theoreme The 27. Proposition If from an odde number be taken a way an euen number the residue shall be an odde number SVppose that AB be an odde number and from it take away an euen number BC. Then I say that the residue CA is an odde number Take away froÌ„ AB vnitie AD. Wherfore the residue DB is an eueÌ„ number BC is by supposition euen Wherfore the residue CD is an euen number Wherefore DA which is vnitie beyng added vnto CD which is an euen number maketh the whole AC an â—dde number which was required to be proued ¶ The 28. Theoreme The 28. Proposition If an odde number multiplieng an euen number produce any number the number produced shall be an euen number SVppose that A being an odde number multiplieng B being an euen number do produce the number C. Then I say that C is an euen number For forasmuch as A multiplieng B produced C therfore C is composed of so many numbers equall vnto B as there be in vnities in A. But B is an euen nuÌ„ber wherfore C is composed of so many euen numbers as there are vnities in A. But if eueÌ„ numbers how many soeuâ—r be added together the whole by the 21. of the ninth is an euen number wherfore C is an euen number which was required to be demonstrated ¶ The 29. Theoreme The 29. Proposition Iâ— an odde number multiplying an
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
in that sort of vnderstaÌ„ding there is no number but that it may be deuided into two equall partes as this number 7 may be deuided into 3 partes namely 3. 1. and 3. of which two namely 3 and 3 are equal yet iâ— not 7 an euen number because 3 and 3 added together make not 7. Boetius therfore in the first booke of his Arithmetike for the more playnes after this maner defineth an euen number Numâ—ruâ— paâ— est qui potest in â—quâ—lia due diuidi vna mediâ—â—â—n intercedenta that is An euen number is that which may be deuided into two equall partes without an vnitie comming betwene them As 8 is deuided into 4 and 4 two equall partes without anâ— vnitie comming betwene them which added together make 8 so that the sence of this definition is that an euen number is that which is deuided into two such equall partes which are his two halfe partes Here is also to be noted that a part is taken in this deâ—inition and in certaine definitions following not in the signification as it was before deâ—ined namely for such a part asâ— measureth the whole number but for any part which helpeth to the making of the whole and into which the whole may be râ—solued so are 3 and 5 partes of 8 in this sence but not in the other sence For neyther 3 not 5 measureth â— Pithagoras and his scholers gaue an other definition of an euen number which definition 〈◊〉 also hath after this maner An euen number is that which in one and the same deuision is deuided into the greatest and into the least into the greatest as touching space and into the least as touching quantitie As 10. is deuided into 5 5. which 〈◊〉 his greatest partes which greatnes of partes he calleth space and in the same diuision the number 10 is deuided but into two partes but into lesse theÌ„ two partes nothing can be deuided which thing he calleth quantitie so that 10 deuided into 5 and 5. in that one deuision is deuided into the greatest namely into his haluesâ— and into the least namely into two partes and no mo There is also another deâ—inition more auncient which is thus An euen number is that which may be deuided into two equall partes and into two vnequall partes but in neyther diuision to the constitucion of the whole to the euen part is added the odde neither to the odde is added the euen As 8 may be deuided diuersly partly into euen partes as into 4 and 4. likewise into 6 and 2. and partly into odde partes as into 5 and 3. also into 7 and 1. In all which deuisions ye see no odde parte ioyned to an euen nor an euen part ioyned to an odde but if the one be euen the other is euen and if the one be odde the other is odde In the two first deuisions both partes were euen and in the two last both partes were odde It is to be considered that the two partes added together must make the whole This diffinition is generall and common to all euen numbers except to the number 2. which can not be deuided into two vnequall partes but onely into two vnities which are equall There is yet geueÌ„ an other diffinition of an euen number namely thus An euen number is that which onely by an vnitie either aboue it or vnder it differeth from an odde As 8 being an euen number differeth from 9 an odde number being aboue it but by one And also from 7. vnder it it differeth likewise by one and so oâ— others 7 An odde number is that which cannot be deuided into two equal partes or that which onely by an vnitie differeth from an euen number As the number 5 can be by no meanes deuided into two equall partes namely such two which addâ—d together shall make 5. Or by the second definition 5 an odde number differeth from 6 an euen number aboue â—t by 1. And the same 5 differeth from 4 an euen number vnder it likewise by 1. 〈…〉 number after this maner An odde number is that which cannot be deuided into two equall partes but that an vnitie shall be betwene them As if ye deuide 5 into 2 and â— which are two equall partes there remaineth one or an vnitie betwene them to make the whole number 5. There is yet an other definition of an odde number An odde number is that which being deuided into two vnequall partes howsoeuer the one is euer euen and the other odde As if 9 be deuided into two partes which added together maketh the whole namely into 4 and 5. which are vnequall ye see the one is euen namely 4 and the other is odde namely 5. so if ye deuide 9 into 6 and 3. or into 8 and 1. the one part is euer euen and the other odde 8 A number euenly euen called in latine pariter par is that number which an euen number measureth by an euen number As 8 is a number euenly euen For 4 an euen number measureth 8 by 2 which is also an euen number This definition hath much troubled many and seemeth not a true definition for that there are many numbers which euen numbers do measure and that by euen numbers which yet are not euenly eueÌ„ numbers after most mens minds as 24. which 6 an eueÌ„ number doth measure by foure which is also an euen number and yet as they thinke is not 24 an euenly euen number for that 8 an euen nuÌ„ber doth measure also â—4 by 3. an odde number Wherfore Campane to make his sentence plaine after this maner setteth forth this definition Pariter par est quem cuncti pares â—um numerantes paribus vicibuâ— numerane that is An euenly euen number is when all the euen numbers which measure it doo measure it by euen times that is by euen numbers as 16. All the euen numbers whith measure 16 as are 8. 4. and 2. do measure it by euen numbers As 8 by 2 twise 8 is 16 4 by 4 foure times 4 is 16 and 2 by 8 8 times 2 is 16. Which particle all euen number added by Campane maketh 24 to be no euenly euen number For that some one euen number measureth it by an odde number as 8 by 3. Flussates also is plainly of this minde that Euclide gaue not this definition in such maner as it is by Theon written for the largenes generalitie therof â—or that it extendeth it to inâ—inite numbers which are not euenly euen as he thinketh for which cause in place therof he geueth this definition A number euenly euen is that which onely euen numbers do measure As 16 is measured of none but of euen numbers and therfore is euenly euen There is also of Boetius geuen an other definition of more facilitie including in it no doubt at all which is most commonly vsed of all writers and is thus A
number euenly euen is that which may be diuided into two euen partes and that part agayne into two euen partes and so continually deuiding without stay 〈◊〉 come to vnitie As by exampleâ— 64. may be deuided into 31 and. 32. And either of these partes may be deuided into two euen partes for 32 may be deuided into 16 and 16. Againe 16 may be deuided into 8 and 8 which are euen partes and 8 into 4 and 4. Againe 4 into â— and â— and last of all may â— be deuided into one and one 9 A number euenly odde called in latine pariter impar is that which an euen number measureth by an odde number As the number 6 which 2 an euen number measureth by 3 an odde number thre times 2 is 6. Likewise 10. which 2. an euen number measureth by 5 an odde number In this diffinition also is found by all the expositors of Euclide the same want that was found in the diffinition next before And for that it extendeth it selfe to large for there are infinite numbers which euen numbers do measure by odde numbers which yet after their mindes are not eueÌ„ly odde nuÌ„bers as for example 12. For 4 an eueÌ„ nuÌ„ber measureth 12 by â— an odde numberâ— three times 4 is 12. yet is not 12 as they thinke an euenly odde number Wherfore Campane amendeth it after his thinking by adding of this worde all as he did in the first and defineth it after this maner A number euenly odde is when all the euen numbers which measure it do measure it by vneuen tymes that is by an odde number As 10. is a number euenly odde for no euen number but onely 2 measureth 10. and that is by 5 an odde number But not all the euen numbers which measure 12. do measure it by odde numbers For 6 an euen number measureth 12 by 2 which is also euen Wherfore 12 is not by this definition a number euenly odde Flussates also offended with the ouer large generalitie of this definition to make the definition agree with the thing defined putteth it after this maner A number euenly odde is that which an odde number doth measure onely by an euen number As 14. which 7. an odde number doth measure onely by 2. which is an euen number There is also an other definition of this kinde of number commonly geuen of more plaines which is this A number euenly odde in that which may be deuided into two equall partes but that part cannot agayne be deuided into two equall partes as 6. may be deuided into two equall partes into 3. and 3. but neither of them can be deuided into two equall partes for that 3. is an odde number and suffereth no such diuision 10 A number oddly euen called im lattin in pariter par is that which an odde number measureth by an euen number As the number 12 for 3. an odde number measureth 12. by 4. which is an euen number three times 4. is 12. This definition is not founde in the greeke neither was it doubtles euer in this maner written by Euclide which thing the slendernes and the imperfection thereof and the absurdities following also of the same declare most manifestly The definition next before geuen is in substance all one with this For what number soeuer an euenâ— number doth measure by an odde the selfe same number doth an odde number measure by an euen As 2. an eueÌ„ number measureth 6. by 3. an odde number Wherfore 3. an odde number doth also measure the same number 6. by 2. an eueÌ„ nuÌ„ber Now if these two definitions be definitions of two distinct kindes of numbers then is this number 6. both euenly euen and also euenly odde and so is contayned vnder two diuers kindes of numbers Which is directly agaynst the authoritie of Euclide who playnely pâ—ouoâ—h here after in the 9. booke that euery nomber whose halfe is an odde number is a number euenly odde onely Flussates hath here very well noted that these two euenly odde and oddely euen were taken of Euclide for on and the selfe same kinde of nomber But the number which here ought to haue bene placed is called of the best interpreters of Euclide numerus pariter par nupar that is a number eueÌ„ly eueÌ„ and eueÌ„ly odde Yeâ— and it is so called of Euclide him selfe in the 34. proposition of his 9. booke which kinde of number Campanus and Flussates in steade of the insufficient and vâ—apt definition before geuen assigne this definition A number euenly euen and euenly â—dde is that which an euen number doth measure sometime by an euen number and sometime by an odde As the number 12 for 2. an euen number measureth 12. by 6. an euen number two times 6. iâ— 12. Also 4. an euen number measureth the same number 12. by 3. an odde number Add therefore is 12. a number euenly euen and euenly odde and so of such others There is also an other definition geuen of this kinde of number by Boetius and others commonly which is thus A number eueâ—ly euen and euenly odde is that which may be deuided into two equall partes and eche of them may aâ—ayne be deuided into two equall partes and so forth But this deuision is at lenghth stayd and continueth not till it come to vnitie As for example 48 which may be deuided into two equall partes namely into 24. and 24. Agayne 24. which is on of the partes may be deuided into two equall partes 12. and 12. Agayne 12. into 6. and 6. And agayne 6 may be deuided into two equall partes into 3. and 3 but 3. cannot be deuided into two equall partes Wherefore the deuision there stayeth and continueth not till it come to vnitie as it did in these numbers which are euenly euen only 11 A number odly odde is that which an odde number doth measure by an odde number As 25 which 5. an odde number measureth by an odde number namely by 5. Fiue times fiue is 25 Likewise 21. whom 7. an odde number doth measure by 3 which is likewise an odde number Three times 7. is 21. Flussatus geueth this definition following of this kinde of number which is all one in substance with the former definition A number odly odde it that which onely an odde number doth measure As 15. for no number measureth 15. but onely 5. and 3 also 25 none measureth it but onely 5. which is an odde number and so of others 12 A prime or first number is that which onely vnitie doth measure As 5.7.11.13 For no number measureth 5 but onely vnitie For v. vnities make the number 5. So no number measureth 7 but onely vnitie .2 taken 3. times maketh 6. which is lesse then 7 and 2. taken 4. times is 8 which is more then 7. And so of 11.13 and such others So that all prime numbers which also are called first numbers and numbers vncomposed haue
odde number produce any number the number produced shal be an odde number SVppose that A being an odde number multiplying B being also an odde number doo produce the number C. Then I say that C is an odde number For forasmuch as A multiplying B produced C therefore C is composed of so many numbers equall vnto B as there be vnities in A. But either of these numbers A and B is an odde number Wherefore C is composed of odde numbers whose multitude also is odde Wherfore by the 23. of the ninth C is an odde nuÌ„ber which was required to be demonstrated A proposition added by Campane If an odde number measure an euen number it shall measure it by an euen number For if it should measure it by an odde number then of an odde number multiplyed into an odde number should be produced an odde number which by the former proposition is impossible An other proposition added by him If an odde number measure an odde number it shall measure it by an odde number For if it should measure it by an euen number then of an odde number multiplyed into an euen number should be produced an odde number which by the 28. of this booke is impossible ¶ The 30. Theoreme The 30. Proposition If an odde number measure an euen number it shall also measure the halfe thereof SVppose that A being an odde number doo measure B being an euen number Thâ—â— I say that it shall measure the halfe thereof For forasmuch as A measureth B let iâ— measure it by C. TheÌ„ I say that C is an euen number For if not then if it be possible leâ— iâ— be odde And forasmuch as A measureth B by C therfore A multiplying C produceth B. Wherfore B is composed of odde numbers whose multitude also is odde Wherfore B is an odde number by the 29. of this booke which is absurdâ— for it is supposed to be euen wherefore C is an euen numâ—er Wherefore A measureth B by an euen number and C measureth B by A. But either of these numbers C and B hath an halfe part wherfore as C is to B so is the halfe to the halfe But C measureth B by A. Wherefore the halfe of C measureth the halfe of B by A wherfore A multiplying the halfe of C produceth the halfe of B. Wherfore A measureth the halfe of B and it measureth it by the halfe of C. Wherefore A measureth the halfe of the number B which was required to be demonstrated ¶ The 31. Theoreme The 31. Proposition If an odde number be prime to any number it shal also be prime to the double thereof SVppose that A being an odde number be prime vnto the number B and let the double of B be C. Then I say that A is prime vnto C. For if A and C be not prime the one to the other some one number measureth them both Let there be such a number which measureth them both and let the same be D. But A is an odde number Wherefore D also is an odde number For if D which measureth A should be an euen number then should A also be an euen number by the 21. of this booke which is coÌ„trary to the supposition For A is supposed to be an odde nuÌ„ber therefore D also is an odde number And forasmuch as D being an odde number measureth C but C is an eueÌ„ number for that it hath an halfe namely B Wherfore by the Proposition next going before D measureth the halfe of C. But the halfe of C is B. Wherefore D measureth B and it also measureth A. Wherefore D measureth A and B being prime the one to the other which is absurde Wherefore no number measureth the numbers A C. VVherfore A is a prime number vnto C. VVherefore these numbers A and C are prime the one to the other which was required to be proued ¶ The 32. Theoreme The 32. Proposition Euery nuÌ„ber produced by the doubling of two vpward is euenly euen onely SVppose that A be the number two and from A vpward double numbers how many soeuer as B C D. Then I say that B C D are numbers euenly euen onely That euery one of them is euenly euen it is manifest for euery one of them is produced by the doubling of two I say also that euery one of them is euenly euen onely Take vnitie E. And forasmuch as from vnitie are certaine numbers in continuall proportion A which followeth next after vnitie is a prime number therefore by the 13. of the third no number measureth D being the greatest number of these numbers A B C D besides the selfe same numbers in proportion But euery one of these numbers A B C is euenly euen VVherefore D is euenly euen onely In like sort may we proue that euery one of these numbers A B C is euenly euen onely which was required to be proued ¶ The 33. Theoreme The 33. Proposition A number whose halfe part is odde is euenly odde onely SVppose that A be a number whose halfe part is odde Then I say that A is euenly od onely That it is euenly odde it is manifest for his halfe being odde measureth him by an eueÌ„ number namely by 2. by the definition I say also that it is euenly odde onely For if A be euenly euen his halfe also is euen For by the definition an euen number measureth him by an euen number Wherefore that euen number which measureth him by an euen number shall also measure the halfe thereof being an odde number by the 4. common sentence of the seuenth which is absurd Wherfore A is a number euenly odde onely which was required to be proued An other demonstration to proue the same Suppose that the number A haue to his halfe an od nuÌ„ber namely B. TheÌ„ I say that A is eueÌ„ly od onely That it is euenly odde needeth no profe forasmuch as the number 2. an euen number measureth it by the halfe thereof which is an odde number Let C be the number 2. by which B measureth A for that A is supposed to be double vnto B And let an euen number namely D measure A which is possible for that A is an euen number by the definition by F. And forasmuch as that which is produced of C into B is equall to that which is produced of D into F therefore by the 19. of the seuenth as C is to D so is B to F. But C the number two measureth D being an euen number wherfore F also measureth B which is the halfe of A. Wherfore F is an odde number For if F were an euen number then should it in the B whome it measureth an odde number also by the 21. of this booke which is contrary to the supposition And in like maner may we proue that all the eueÌ„ nuÌ„bers which measure the number Aâ— do measure it by odde numbers Wherefore A is a number euenly odde onely
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
that superficies is rationall SVppose that a parallelogramme be contained vnder a residuall line AB and a binomiall line CD and let the greater name of the binomiall line be CE and the lesse name be ED and let the names of the binomiall line namely CE and ED be commensurable to the names of the residuall line namely to AF and Fâ— and in the selfe same proportion And let the line which containeth in power that parallelograÌ„me be G. TheÌ„ I say that the line G is rational Take a rational line namely H. And vnto the line CD apply a parallelograÌ„me equal to the square of the line H and making in breadth the line KL Wherefore by the 112. of the tenth KL is a residuall line whose names let be KM and ML which are by the same coÌ„mensurable to the names of the binomiall line that is to CE and ED and are in the selfe same proportioÌ„ But by position the lines CE and ED are coÌ„mensurable to the lines AF and FB and are in the selfe same proportion Wherfore by the 12. of the tenth as the line AF is to the line FBâ— so is the line KM to the line ML Wherfore alternately by the 16. of the fift as the line AF is to the line KM so is the line BF to the line LM Wherfore the residue AB is to the residue KL as the whole AF is to the whole KM But the line AF is commensurable to the line KM for either of the lines AF and KM is commensurable to the line CE. Wherfore also the line AB is commensurable to the line KL And as the line AB is to the line KL so by the first of the sixt is the parallelogramme contained vnder the lines CD and AB to the parallelogramme contained vnder the lines CD and KL Wherfore the parallelogramme contained vnder the lines CD and AB is commensurable to the parallelogramme contained vnder the lines CD and KL But the parallelogramme contained vnder the lines CD and KL is equall to the square of the line H. Wherfore the parallelograÌ„me coÌ„tained vnder the lines CD AB is coÌ„mensurable to the square of the line H. But the parallelograÌ„me contained vnder the lines CD and AB is equall to the square of the line G. Wherfore the square of the line H is commensurable to the square of the line G. But the square of the line H is rationall Wherfore the square of the line G is also rationall Wherfore also the line G is rational and it containeth in power the parallelogramme contained vnder the lines AB and CD If therfore a parallelogramme be contained vnder a residuall line and a binomiall line whose names are commensurable to the names of the residuall line and in the selfe same proportion the line which containeth in power that superficies is rationall which was required to be proued ¶ Corollary Hereby it is manifest that a rationall parallelogramme may be contained vnder irrationall lines ¶ An otâ—â—r 〈…〉 Flussas 〈…〉 line â—D whosâ— names Aâ— and â—D let be commensurable in length vnto the names of the residuall line Aâ— which let be AF and FB And let the liâ—e AEâ— be to the line EDâ— in the same proportion that the line AF is to the line Fâ— And let the right line â— contayne in power the superficies Dâ— Then I say thaâ— the liâ—e â— is a rationall lin◠〈…〉 lâ—ne which lâ—â— bâ—â— And vpon the line â—â— describe by the 4â— of the first a parallelogramme eqâ—all to the squarâ— of the line â—â— and making in breadth the line DC Wherefore by the â—12 of this booke CD is a residuâ—ll lineâ— whose names Which let be â—â— and OD shall be coâ—mensurablâ— in leâ—gth vnto the names Aâ— and â—D and the line C o shall be vnto the line OD in the same proporâ—ion that the line AE is to the line EDâ— But as the line Aâ— is to the line â—D so by supposition is the line AF to the line FE Wherfore as the line CO is to the line OD so is the line AF to the line Fâ—â— Wherefore the lines CO and OD are commensurable with the lines Aâ— and â—â— by the â—â— of this boke Wherfore the residue namely the line CD is to the residue namely to the line Aâ— as the line CO is to the line AF by the 19. of the fifth But it is proued that the line CO is coÌ„mensurable vnto the line AF. Wherefore the line CD is commensurable vnto the line AB Wherefore by the first of the sixth the parallelogramme CA is commensurable to the parallelogramme Dâ— But the parallelogramme â—â— iâ— by construction rationall for it is equall to the square of the rationall line â— Whâ—refore the parallelogramme â—D â—s also ratâ—â—nâ—llâ— Wherâ—fore the line â— which by supposition coÌ„tayneth in power the superficies â—Dâ— is also rationall If therfore a parallelograÌ„me be contayned c which was required to be proued ¶ The 91. Theoreme The 115. Proposition Of a mediall line are produced infinite irrationall lines of which none is of the selfe same kinde with any of those that were before SVppose that A be a mediall line Then I say that of the line A may be produced infinite irrationall lines of which none shall be of the selfe same kinde with any of those that were before Take a rationall line B. And vnto that which is contained vnder the lines A and B let the square of the line C be equall by the 14. of the second â— Wherefore the line C is irrationall For a superficies contained vnder a rationall line and an irrationall line is by the Assumpt following the 38. of the tenth irrationall and the line which containeth in power an irrationall superficies is by the Assumpt going before the 21. of the tenth irrationall And it is not one and the selfe same with any of those thirtene that were before For none of the lines that were before applied to a rationall line maketh the breadth mediall Againe vnto that which is contained vnder the lines B and C let the square of D be equall Wherefore the square of D is irrationall Wherefore also the line D is irrationall and not of the self same kinde with any of those that were before For the square of none of the lines which were before applied to a rationall line maketh the breadth the line C. In like sort also shall it so followe if a man proceede infinitely Wherefore it is manifest that of a mediall line are produced infinite irrationall lines of which none is of the selfe same kinde with any of those that were before which was required to be proued An other demonstratioâ— Suppose that AC be a mediall line Then I say that of the line AC may be produced infinite irrationall lines of which none shall be of the selfe same kinde with any of those irrationall lines before named Vnto the line AC and from the point A
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
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
the residue or of this excesse But a pyramis is to the same cube inscribed in it nonecuple by the 30. of this booke Wherefore the Dodecahedron inscribed in the pyramis and containing the same cube twise taking away the selfe same third of the lesse segment and moreouer the lesse segment of the lesse segment of halfe the residue shall containe two ninth partes of the solide of the pyramis of which ninth partes eche is equall vnto the cube taking away this selfe same excesse The solide therefore of a Dodecahedron containeth of a Pyramis circumscribed about it two ninth partes taking away a third part of one ninth part of the lesse segment of a line diuided by an extmere and meane proportion and moreouer the lesse segment of the lesse segment of halfe the residue ¶ The 36. Proposition An Octohedron exceedeth an Icosahedron inscribed in it by a parallelipipedon set vpon the square of the side of the Icosahedron and hauing to his altitude the line which is the greater segment of halfe the semidiameter of the Octohedron SVppose that there be an Octohedron ABCFPL in which let there be inscribed an Icosahedron HKEGMXNVDSQTâ— by the â—6 of the fiuetenth And draw the diameters AZRCBROIF and the perpendicular KO â—arallel to the line AZR Then I say that the Octohedron ABCFPL is greater thâ—n the Icosahedron inscribed in it by a parallelipipedon set vpon the square of the side HK or GE and hauing to his altitude the line KO or RZ which is the greater segment of the semidiameter AR. Forasmuch as in the same 16. it hath bene proued that the triangles KDG and KEQ are described in the bases APF and ALF of the Octohedron therefore about the solide angle there remaine vppon the base FEG three triangles KEG KFE and KFG which containe a pyramis KEFG Vnto which pyramis shall be equall and like the opposite pyramis MEFG set vpon the same base FEG by the 8. definition of the eleuenth And by the â—ame reason shall there at euery solide angle of the Octohedron remayne two pyramids equall and like namely two vpon the base AHK two vpon the base BNV two vpon the base DPS and moreouer two vpon the base QLT. Now theÌ„ there shal be made twelue pyramids set vpon a base contained of the side of the Icosahedron and vnder two leâ—â—e segmentes of the side of the Octohedron containing a right angle as for example the base GEF And forasmuch as the side GE subteÌ„ding a right angle is by the 47. of the â—irst in power duple to either of the lines EF and FG and so the â—â—deâ— KH is in power duple to either of the sides AH and AK and either of the lines AH AK or EF FG is in power duple to eyther of the lines AZ or ZK which coÌ„tayne a right angle made in the triangle or base AHK by the perpendicular AZ Wherfore it followeth that the side GE or HK is in power quadruple to the triangle EFG or AHK But the pyramis KEFG hauing his base EFG in the plaine FLBP of the Octohedron shall haue to his altitude the perpendicular KO by the 4. definition of the sixth which is the greater segment of the semidiameter of the Octohedron by the 16. of the fiuetenth Wherfore three pyramids set vnder the same altitude and vpon equall bases shall be equall to one prisme set vpon the same base and vnder the same altitude by the 1. Corollary of the 7. of the twelfth Wherefore 4. prismes set vpon the base GEF quadrupled which is equall to the square of the side GE and vnder the altitude KO or RZ the greater segment which is equall to KO shall containe a solide equall to the twelue pyramids which twelue pyramids make the excesse of the Octohedron aboue the Icosahedron inscribed in it An Octohedron therefore excedeth an Icosahedron inscribed in it by a parallelipipedon set vpon the square of the side of the Icosahedron and hauing to his altitude the line which is the greater segment of halfe the semidiameter of the Octohedron ¶ A Corollary A Pyramis exceedeth the double of an Icosahedron inscribed in it by a solide set vpon the square of the side of the Icosahedron inscribed in it and hauing to his altitude that whole line of which the side of the Icosahedron is the greater segmeÌ„t For it is manifest by the 19. of the fiueteÌ„th that an octohedroÌ„ an IcosahedroÌ„ inscribed in it are inscribed in one the self same pyramis It hath moreouer bene proued in the 26. of this boke that a pyramis is double to an octohedroÌ„ inscribed in it Wherfore the two excesses of the two octohedrons vnto which the pyramis is equal aboue the two Icosahedrons inscribed in the said two octohedrons being brought into an solide the said solide shal be set vpon the selfe same square of the side of the Icosahedron and shall haue to his altitude the perpendicular KO doubled whose double coupling the opposite sides HK and XM maketh the greater segment the same side of the Icosahedron by the first and second corollary of the 14. of the fiuâ—â—enâ—h The 37. Proposition If in a triangle hauing to his base a rational line set the sides be commensurable in power to the base and from the toppe be drawn to the base a perpendicular line cutting the base The sections of the base shall be commensurable in length to the whole base and the perpendicular shall be commensurable in power to the said whole base And now that the perpendicular AP is commensurable in power to the base BG iâ— thus proued Forasmuch as the square of AB is by supposition commensurable to the square of BG and vnto the rational square of AB is commensurable the rational square of BP by the 12. of the eleuenth Wherfore the residue namely the square of PA is commensurable to the same square of BP by the 2. part of the 15. of the eleuenth Wherefore by the 12. of the tenth the square of PA is commensurable to the whole square of BG Wherefore the perpendicular AP is commensurable in power to the base BG by the 3. diffinition of the tenth which was required to be proued In demonstrating of this we made no mention at all of the length of the sides AB and AG but only of the length of the base BG for that the line BG is the rational line first set and the other lines AB and AG are supposed to be commensurable in power only to the line BG Wherefore if that be plainely demonstrated when the sides are commensurable in power only to the base much more easily wil it follow if the same sides be supposed to be commensurable both in length and in power to the base that is if their lengthes be expressed by the rootes of square nombers ¶ A Corollary 1. By the former things demonstrated it is manifest that if from the powers of the base and of one of the sides be taken away the
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
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
in the poynt E. And vnto the line CG put the line CL equall Now forasmuch as the lines AG and GC are the greater segâ—â—â—tes of halfe the line AB for â—che of them is the halfe of the greater segment of the whole line AB the lines EB and EC shall be the lesse segmentes of halfe the line AB Wherefore the whole line Câ— is the greater segment and the line CE is the lesse segment But as the line CL is to the line CE so is the line CE to the residue EL. Wherfore the line EL is the greater segment of the line CE or of the line EB which is equall vnto it Wherfore the residue LB is the lesse segment of the same EB which is the lesâ—e segment of halfâ— the side of the cube But the lines AG GC and CL are three greater segmentes of the halfe of the whole line AB which thre greater segmentes make the altitude of the foresayd solide wherefore the altitude of the sayd solide wanteth of AB the side of the cube by the line LB which is the lesse sâ—gment of the line BE. Which line BE agayne is the lesse segment of halfe the side AB of the cube Wherefore the foresayd solide consisting of the sixe solides whereby the dodecahedron exceedeth the cube inscribed in it is set vpon a base which wanteth of the base of the cube by a third part of the lesse segment and is vnder an altitude wanting of the side of the cube by the lesse segment of the lesse segment of halfe the side of the cube The solide therefore of a dodecahedron exceedeth the solide of a cube inscribed in it by a parallelipipedon whose base wanteth of the base of the cube by a third part of the lesse segment and whose altitude wanteth of the altitude of the cube by the lesse segment of the lesse segment of halfe the side of the cube ¶ A Corollary A Dodecahedron is double to a Cube inscribed in it taking away the third part of the lesse segment of the cube and moreouer the lesse segment of the lesse segment of halfe of that excesse For if there be geuen a cube from which is cut of a solide set vpon a third part of the lesse segment of the base and vnder one and the same altitude with the cube that solide taken away hath to the whole solide the proportion of the section of the base to the base by the 32. of the eleuenth Wherefoâ—e from the cube is taken away a third â—art of the lesse segment Farther forasmuch as the residue wanteth of the altitude of the cube by the lesse segment of the lesse segment of halfe the altitude or side and that residue is a parallelipipedon if it be cut by a plaine superficies parallel to the opposite plaine superficieces cutting the altitude of the cube by a point it shall take away from that parallelipipedon a solide hauing to the whole the proportion of the section to the altitude by the 3. Corollary of the 25. of the eleuenth Wherefore the excesse wanteth of the same cube by the thiâ—d part of the lesse segment and moreouer by the lesse segment of the lesse segment of halfe of that excesse ¶ The 34. Proposition The proportion of the solide of a Dodecahedron to the solide of an Icosahedron inscribed in it consisteth of the proportion tripled of the diameter to that line which coupleth the opposite bases of the Dodecahedron and of the proportion of the side of the Cube to the side of the Icosahedron inscribed in one and the selfe same Sphere SVppose that AHBCK be a Dodecahedronâ— whose diametâ—r let be AB and let the line which coupleth the ceÌ„tres of the opposite bases be KHâ— and let the Icosahedron inscribed in the Dodecahedron ABC be d ee whose diameter let be DE. Now forasmuch aâ— oâ—e and the selfe same circle coÌ„taineth the pentagon of a Dodecahedron the triangle of an Icosahedroâ— described in one and the selfe same Sphere by the 14. of the fourtenth Let that circle be IGO. Wherfore IO is the side of the cube and IG the side of the Icosahedron by the same TheÌ„ I say that the proportion of the Dodecahedron AHBCK to the Icosahedron DEF inscribed in it coÌ„sisteth of the proportioÌ„ tripled of the line AB to the line KH and of the proportion of the line IO to the line IG For â—oâ—asmuch as the Icosahedron DEF is inscribed in the DodecahedroÌ„ ABC by suppositioÌ„ the diameter DE shal be equal to the line KH by the 7. of the fiuetenth Wherefore the Dodecahedron set vpoÌ„ the diameter KH shall be inscribed in the same Sphere wherein the Icosahedron DEF is inscribed but the Dodecahedron AHBCK is to the Dodecahedron vpon the diameter KH in triple proportion of that in which the diameter AB is to the diameter KH by the Corollary of the 17. of the twelfth and the same Dodecahedron which is set vpon the diameter KH hath to the Icosahedron DEF which is set vpon the same diameter or vpon a diameter equall vnto it namely DE that proportion which IO the side of the cube hath toâ— IG the side of the Icosahedron inscribed in one the selfe same Sphere by the 8 of the fouretenth Wherefore the proportion of the Dodecahedron AHBCK to the Icosahedron DEF inscribed in it consisteth of the proportion tripled of the diameter AB to the line KH which coupleth the centres of the opposite bases of the Dodecahedron which proportion is that which the Dodecahedron AHBCK hath to the Dodecahedron set vpon the diameter KH and of the proportion of IO the side of the cube to IG the side of the Icosahedron which is the proportion of the Dodecahedron set vpon the diameter KH to the Icosahedron DEF described in one and the selfe same Sphere by the 5. definition of the sixth The proportion therefore of the solide of a Dodecahedron to the solide of an Icosahedron inscribed in it conâ—isteth of the proportion tripled of the diameter to that line which coupleth the opposite bases of the Dodecahedron and of the propâ—â—tion of the side of the cube to the side of the Icosahedron inscribed in one and the selfe same Sphere The 35. Proposition The solide of a Dodecahedron containeth of a Pyramis circumscribed about it two ninth partes taking away a third part of one ninth part of the lesse segment of a line diuided by an extreme and meane proportion and moreouer the lesse segment of the lesse segment of halfe the residue IT hath bene proued that the Dodecahedron together with the cube inscribed in it is contained in one and the selfe same pyramis by the Corollary of the first of this booke And by the Corollary of the 33. of this booke it is manifest that the Dodecahedron is double to the same cube taking away the third part of the lesse segment and moreouer the lesse segment of the lesse segment of halfe