This is the rate of a quadrate with a rectangle & a quadrate, from whence is had the analisis or resolution of the side of a quadrate expressable by a number. For it is the same way fro Cambridge to London, that is from London to Cambridge. And this use of geometricall analysis remaineth, as afterward in a Cube, when as otherwise through the whole booke of Euclides Elements there is no other use at all of that.

Here therefore thou shalt note or marke out the severall quadrates, beginning at the right hand and so proceeding towards the left; after this manner, 144. These notes doe signifie that so many severall sides to be found, to make up

the whole side of the quadrate given. And here first, it shall not be amisse to warne thee, before thou commest to practice, that for helpe of memory and speed in working, thou know the Quadrats of the nine single numbers of figures; which are these

1.2.3.4.5.6.7.8.9. Sides.
1. 4. 9. 16. 25. 36. 49. 64. 81. Qu.

Then beginning at the left hand, as in Division, that is where we left in multiplication, and I seeke amongst the squares the greatest conteined in the first periode, which here is 1; And the side of it, which is also 1, I place with my quotient: Then I square this quotient, that is I multiply it by it selfe, and the product 1, I sect under the same first periode: Lastly, I subtract it from the same periode, and there remaineth not any thing. Then as in division I set up the figures of the next periode one degree higher.

Secondly double the side now found, and it shall be 2, which I place in manner of a Divisor, on the left hand, within the semicircle: By this I divide the 40, the two complements or Plaines, and I finde the quotient or second side 2; which I place in the quotient by 1, This side I multiply first quadrate like, that is by it selfe; and I make 4, the lesser Diagonall: And therefore I place under the last 4: Then I multiply the said Divisor 2, by the same 2 the quotient, and I make in like manner, 4 which I place under the dividend, or the first 4. Lastly I subtract these products from the numbers above them, and remaineth nothing. Therefore I say first, That 144, the number given is a quadrate: And more-over, That 12 is the true side of it.
44
1,44(12
2) 1
44

Or thus:

44
1,44(12
20) 1
40
4

Againe, let the side of 15129 be sought. First divide it into imperfect periods as before was taught; in this manner: 15129. Then I seeke amongst the former quadrates, for the side of 1, the quadrate of the first periode; and I finde it to be 1: This side I place within the quotient or lunular on the right side: Lastly I subtract 1 from 1, and nothing remaineth. Then I double the said side found; and I make 2: This 2, I place for my divisor within the lunular or semicircle on the left hand: By which I divide 5; and I finde the quotient 2, which I place by the former quotient: Then I multiply the same 2, first quadratelike by it selfe, and I make 4. Then I multiply the sayd divisour by 2, the quotient, and I make likewise 4: which I place underneath 51. Lastly, I subtract the same 44, from 51, and there remaine 7, over the head of 1; By which I place 29, the last periode remaining.

Againe I double 12, my whole quotient, and I make 24. By this double I divide 72, the double Complement remaining, and I finde 3 for the side or quotient: First this side I multiply quadratelike by it selfe, and I make 9, which I place underneath 9, the last figure of my dividende. Then againe, by the same quotient, or side 3, I multiply 24: my divisour, and I make 72; which I place under 72, the two figures of my Dividende: Lastly I subtract the under figures, from the upper, and there is likewise nothing remaining: Wherefore I say, as afore; that the figurate 15129 given, is a square: And the side thereof is 123.
7 29
51
1,51,29(123
2)1
24)44
7 29

Or thus:

7 29
51
1,51,29(123
20)1
40
4
240) 7 20
9

Sometime after the quadrate now found, in the next places, there is neither any plaine nor square to bee found: Therefore the single side thereof shall be O. As in the quadrate 366025, the whole side is 605, consisting of three severall sides, of which the middle one is o.

Sometime also the middle plaine doth containe a part ofthe quadrate next following: Therfore if the other side remainingbe greater than the side of the quadrate following,it is to be made equall unto it: As for example, Let the sideof the quadrate 784, be sought; The side of the first quadrateshall be 2, and there shall remaine 3, thus: Then the sameside doubled is 4 for the quotient; Which is found in 38, thedouble plaine remaining 9 times, for the other side: But thisside is greater than the side of the next following quadrate:Take therefore 1 out of it: And for nine take 8, and placeit in your quotient; Which 8 multiplyed by it selfe maketh64, for the Lesser quadrate: And againe the same multiplyedby 4 the divisour maketh 32; the summe of which twoproducts 384, subtracted from the remaine 384, leave nothing:Therefore 784 is a Quadrate: And the side is 28.
384
4)784 (28
4

Or thus:

384
40)784 (28
4
320
64

And from hence the invention of a meane proportionall, betweene two numbers given, (if there be any such to be found) is manifest. For if the product of two numbers given be a quadrate, the side of the quadrate shall be the meane proportionall, betweene the numbers given; as is apparent by the golden rule: As for example, Betweene 4. and 9. two numbers given, I desire to know what is the meane proportion. I multiply therefore 4 and 9. betweene themselves, and the product is 36: which is a quadrate number; as you see in the former; And the side is 6. Therefore I say, the meane proportionall betweene 4. and 9. is 6, that is, As 4. is to 6. so is 6. to 9.