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

10 A number oddly euen (called im lattin in pariter par) is that which an odde number measureth by an euen number.* 1.1

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:* 1.2 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 euen•• number doth measure by an odde, the selfe same number doth an odde number measure by an euen. As 2. an euē number measureth 6. by 3. an odde number. Wherfore 3. an odde number doth also measure the same number 6. by 2. an euē nūber. Now if these two definiti¦ons 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. p••ouo••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 euēly euē, and euēly odde. Ye•• 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 v••apt definition before geuen, assigne this definition.

A number euenly euen, and euenly ••dde, is that which an euen number doth measure sometime by an euen number, and sometime by an odde.* 1.3

As the number 12: for 2. an euen number, measureth 12. by 6. an euen number: two times 6. i•• 12. Al∣so 4. an euen number measureth the same number 12. by 3. an odde number. Add therefore is 12. a num∣ber euenly euen, and euenly odde, and so of such others.

The cause why that Campanus and Flussates were so scrupelous in amending (as they supposed) the two definitions before,* 1.4 namely, of a number euēly euen and of a number euenly odde, the one by ad∣ding this word all, and the other by adding this word onely, was for that they were offended with the la••••eue•• and generalitie of them•• For ••ha•• by them, on and the selfe same number might be compre∣hended vnder either definition. And so, the selfe number should be both euenly euen, and also euenly odde: which they tooke for an absurditie. For that they are two distinct and diuers kindes of numbers. But all things well and iustly conceiued, it shall not be hard nor amisse to thinke, that these definitiōs were set and written by Euclide in such forme and ma••er; as they are deliuered vnto vs by Theon: and

that they neede not these corrections and amendementes by adding these wordes all and onely, for ad∣mit that they be distinct kinds of numbers, may not contraries be attributed in diuerse respectes to one thing? May not one line be sayd to be great and little, compared to diuers? Great in comparison of a lesse, and lesse in comparison of a greater? Euen so one number in diuers respects may be of diuers and contrary kindes of numbers.* 1.5 What are more diuers them a square number and a cube number. And yet •••• 64. in diuers respectes a number both square and cube. In respect of 8. to be his roote, it is a square number: for 8. times 8. is 64. and in respect of 4. to be his roote, it is a cube number for 4. times 4. fower times is 64: so in diuers respectes it is both, without any absurditie at all. Likewise this number 6. in di∣uers respectes, is a number on the on side longer, and also a triangular number: which yet are diuers and distinct kindes of numbers. For 6. described by his vnities resembling a figure

[illustration]

of leng••he and breadthe hauing two sides, namely 2. and 3. is a plaine or superfici∣all number of one side longer. And if the same 6. be so described by his vnities, that it representeth the figure of a triangle, then is it and beareth it the name of a trian∣gler ••igure: as here ye may see the forme of either. And if ye extende in description therof all his vnities in length onely, so is 6 also a lineall number. So you see 6 in di∣uers respectes is a lineall number, a number on the one side longer, and also a trigo∣nall or trianguler number, and yet therby no inconuenience at all. And why may not likewise one and the selfe number in diuers respectes be accompted a number both euenly euen, and euenly odde? Yea Euclide him selfe doth most manifes••ly proue the same, and in the ••ame wordes, if it be diligently wayed, in his ninth booke. For he sayth, that all numbers being double in continuall course from the number 2. be euenly euen numbers only: and agayne all numbers whose halues are odde, are euenly odde numbers only: and that number which neither is duple from the number two nor hath to his halfe an odde number is a number euenly euen and a number euenly odde. What in this can be spokē more playnely? So that by Euclide it is no inconuenience that on num∣ber, as 12, f••r example, in diuers respectes should be both a number euenly euen, and also a number e∣uenly odde. In respect that 6. an euen number measureth 12. by 2. an euen number, 12 is a number euen∣ly euen: and in respect that 4. an euen number measureth 12. by 3. an odde number, 12. is a number e∣uenly odde. And thus iudge ye of all others like.

There is also an other definition geuen of this kinde of number by Boetius and others commonly which is thus.

* 1.6A number eue••ly euen and euenly odde is that, which may be deuided into two equall partes, and eche of them may a••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. A∣gayne 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.

Notes

• * 1.1

  The tenth definition.
• * 1.2

  This definition not found in the Greeks.
• * 1.3

  An other defi∣nition.
• * 1.4

  Campane and Flussates with other 〈…〉〈…〉
• * 1.5

  One number in di••ers respectes may be of di∣••e•••• kindes of numbers.
• * 1.6

  Boe••ius def••ni∣tion of a num∣ber euenly euē, and euenly ••d