in fewer termes then two, when as it is a relation of one thing to an other thing: wherefore 2. is not the number of such feete, of which there are 12. in the square. Agayne, if the number 2. shoulde be the side of the square 12, so that that side should be 2, then of the multiplication of 2. into it selfe, shoulde not be made that square 12, but an other square which should be 4. foote: as of the number 2. multiply∣ed into him selfe is produced the square number 4. Neither also if any other number. measure the sid•• of the square 12, and the sayd number be multiplyed into him selfe shall it euer make the number 12. When yet all numbers denominating the side of any square number, if they be multiplyed into them selues, they make the number which denominateth the square, whose sides they denominate. As 2. multiplyed into him selfe maketh 4 : 3. maketh 9 : 4. maketh 16: and so likewise of all others. Where∣fore it is not all one, magnitudes to be in proportion the one to the other, as number is to number, and euery one of them to be measured b•• him selfe without any respect had of the proportion. As here the side of the square 12. can of it selfe by no meanes be measured, but being compared to any other magni∣tude, namely, to the side of the square 3, the proportion thereof is expressed by number. So also the side of the square 3, and of all other square figures, whose areas yet can not be expressed by square numbers. And that which we here say, is manifest euen by the wordes of Euclide in the 5.6. 7. and 8. Theoremes of this booke. Where he sayth not, that magnitudes commensurable and incommensurable are of thē selues or of their owne nature expressed by numbers, but that either they haue or haue not that propor∣tion which number hath to number. Which thing not being well considered, it should seme hath cau∣sed many to erre as hereafter shall be made manifest. And in deede they which haue demonstrated this Theoreme, may seme to some rather to haue demōstrated it particularly & not vniuersally. And doubt∣les I iudge there are some which vnderstand their sayinges otherwise then they ment: when as they thinke, that they suppose certayne lines not onely commensurable in length, as they are supposed to be in the Propositiō, but also such, that ech of
them apa
•••• may be expressed by some cer∣tayne number. Wherfore for want of right vnderstāding, this mought they say of their demonstrations: that wheras they thought that they had concluded that generally, which is in this theoreme of
Euclide, Squares described of lines cōmensurable in length, are in proportiō the one to the other that a square nū∣ber is to a square number: they conclude par∣ticularly, thys onely:
Squares described of lines which may by them selues be expressed by some certaine number, are in proportion. &c. which yet is otherwise, and their demōstra∣tions are right & agreable with the Theoreme. Onely the picture of the figures which the Greeke boke hath, may seeme to bring some doubt. For the squares are so described with certayn litle areas, that the number of them may be denominated by a square number: whereby it mought seeme that the lines A & B which describe the squares, ought to
be such that they may be expressed by some certaine nūber. As the line A to be 5. foote, and the line B 3. foote. As the two former fi¦gures here set declare. Which thyng yet
Eu∣clide supposeth not, but only requireth that they be commensurable in length, as in the former example of the two squares, the whole area of one of which is 12, and the whole area of the other is 3. For although their sides cā not by them selues be expres∣sed by some certayne number, yet are they commensurable in length. Moreouer thys describing of the squares of the lines A and B, deuided by certaine litle areas, may cause this error, that a man should thinke that it is all o
••e two numbers to be square numbers, and to be in proportion the one to the other as a square number is to a square number. For the number of the areas in the square of the line A is a square number, namely, 25. produced of the roote 5, which is the length of the line A
•• Likewise the number of the areas of the square of the line B, is a square nūber, namely, 9, which is pro∣duced of the roote 3, which is the length of the line B. But we haue before declared that it is not all one, numbers to be called square numbers, a
••d to be in proportion the one to the other, as a square nū∣ber is to a square number. Wherefore as touching those areas contayned in the greater square, which is of the line A, and which are in number 25, they do expresse that square number 25, which is produced of the number 5. multiplyed into him selfe, which number 5. is the greater extreme of the proportion betwene 5. and 3, which is the proportion of the lines A and B. And this proportion, namely, of 5. to 3. causeth that the lines A and B are commensurable in length (by the 6. of this booke). The same may be sayd also of the areas of the lesse square. Neither is it of necessitie that you vnderstand those area
•• to be squares, as either fe
••te square or pases square which make the whole square, although in deede they