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

¶The 2. Theoreme. The 2. Proposition. If two numbers multiplying the one the other produce a square number: those numbers are like plaine numbers.

SVppose that two number••

[illustration]

A and B multiplying the one the other do produce C a square number. Then I say, that A and B are like plaine num∣bers. For let A multiplying him selfe produce D. Wherefore D is a square number. And forasmuch as A multi∣plying him selfe produced D, and multiplying B produced C, therefore (by the 17. of the se∣••enth) as A is to B, so is D to C. And forasmuch as D is a square number, and so likewise is C, therefore D and C are like plaine numbers. Wherefore betwene D and C there is (by the 18. of the eight) one meane proportionall number. But as D is to C, so is A to B. Wherefore (by the 8. of the eight) betwene A and B there is one meane proportionall number. But if be∣twene two numbers there be one meane proportionall number, those numbers are (by the 20. of the eight) like plaine numbers. Wherefore A and B are like plaine numbers: which was required to be proued.

A Corollary added by Campane.

H•••••••• it is manifest, th••t two squar•• numbers multiplyed the one into the, other do alwayes produce a squa•••• num••••r. For they are like superficiall numbers, and therefore the num∣ber produced of them, is (by the first of this booke) a square number. But a square num∣ber mul••••plye•• into a number not square, produceth a number not square. For if they should pro∣duce a square number, they should be like superficiall numbers (by this Proposition). But they are not. Wherefore they produce a number not square. But if a square num∣ber multiplyed into an other number produce a square number, that other number shall be a square number. For by this Proposition that other number is like vnto the square number which multiplyeth it, and therefore is a square number. But if a square number multiply∣ed into an other number produce a number not square, neither shall that other number also be a square number. For if it should be a square number, then being multiplyed into the square number it should produce a square number, by the first part of this Corollary.

Notes

• This proposi∣tion is the con¦u••rse o•• t••e form••••.
• Demonstra∣tion.
• A Corollary a••ded by Campane.