If the number given be not a quadrate, there shall no arithmeticall side, and to be expressed by a number be found: And this figurate number is but the shadow of a Geometricall figure, and doth not indeede expresse it fully, neither is such a quadrate rationall: Yet notwithstanding the numerall side of the greatest square in such like number may be found: As in 148. The greatest quadrate continued is 144 and the side is 12. And there doe remaine 4. Therefore of such kinde of number, which is not a quadrate, there is no true or exact side: Neither shall there ever be found any so neare unto the true one; but there may still be one found more neare the truth. Therefore the side is not to be expressed by a number.
Of the invention of this there are two wayes: The one is by the Addition of the gnomon; The other is by the Reduction of the number assigned unto parts of some greater denomination. The first is thus:
17 If the side found be doubled, and to the double a unity be added, the whole shall be the gnomon of the next greater quadrate.
For the sides is one of the complements, and being doubled it is the side of both together. And an unity is the latter diagonall. So the side of 148 is 12.4/25.
The reason of this dependeth on the same proposition,
from whence also the whole side, is found. For seeing that the side of every quadrate lesser than the next follower differeth onely from the side of the quadrate next above greater than it but by an 1. the same unity, both twice multiplied by the side of the former quadrate, and also once by it selfe, doth make the Gnomon of the greater to be added to the quadrate. For it doth make the quadrate 169. Whereby is understood, that looke how much the numerator 4. is short of the denominatour 25. so much is the quadrate 148. short of the next greater quadrate. For if thou doe adde 21. which is the difference whereby 4 is short of 25. thou shalt make the quadrate 169. whose side is 13. The second is by the reduction, as I said, of the number given unto parts assigned of some great denomination, as 100. or 1000. or some smaller than those, and those quadrates, that their true and certaine may be knowne: Now looke how much the smaller they are, so much nearer to the truth shall the side found be.
Let the same example be reduced unto hundreds squared parts, thus: 1480000/10000. The side of 10000. by grant is 100. But the side of 1480000. the numerator by the former is 1216. and beside there doe remaine 1344: thus, 1216/100. that is, 12.16/100, or 4/25 which was discovered by the former way. But in the side of the numerator there remained 1344. By which little this second way is more accurate and precise than the first. Yet notwithstanding those remaines are not regarded, because they cannot adde so much as 1/100 part unto the side, found: For neither in deed doe 1344/2426. make one hundred part.
Moreover in lesser parts, the second way beside the other, doth shew the side to be somewhat greater than the side, by the first way found: as in 7. the side by the first way is 3/25. But by the second way the side of 7. reduced unto thousands quadrates, that is unto 7000000/1000000, that is, 2645/1000, and beside there doe remaine 3975. But 645/1000 are greater than 3/5.
For 3/5. reduced unto 1000. are but 600/1000. Therefore the second way, in this example, doth exceed the first by 45/1000. those remaines 3975. being also neglected.