Either of the shankes is proportionall betweene the summe, and the difference of the base and the other shanke. And contrariwise. If one side be proportionall betweene the summe and the difference of the others, the triangle given is a rectangle. M. H. Brigges.
This is a consectary arising likewise out of the [4 e.] of very great use.
In the triangle ead, the shanke ad, 12. is the meane proportionall betweene bd, 18. (the summe of the base ae, 13. and the shanke ed, 5.) and 8. the difference of the said base and shanke: For if thou shalt draw the right lines ba, and ac, the angle bac, shall be by the [6. e], a rectangle; (because it is equall to the angles at b, and c, seeing that the triangles bea, and eac, are equicrurall.) And by the [9 e], bd, da, and dc, are continually proportionall.
If a quadrate of a number, given for the first shanke, be divided of another, the halfe of the difference of the divisour, and quotient shall be the other shanke, and the halfe of the summe shall be the base. Or thus, The side of divided number doubled, and the difference of the divisour and quotient, shall be the two shankes, and the summe of them shall be the base.
Let the number given for the first shanke be 4. And let 8. divide 16. the quadrate of 4. by 2. The halfe of 8 - 2, that is 3. shall be the other shanke: And the halfe of 8 + 2, that is 5. shall be the base.
Therefore
If any one number shall divide the quadrate of another, the side of the divided, and the halfe of the difference of the divisour and the quotient, shall be the two shankes of a rectangled triangle, and the halfe of the summe of them shall be the base thereof.
Let the two numbers given be 4. and 6. The square of
6. let it be 36. and the quotient of 36. by 4. be 9: And the side is 6. for the one shanke. Now 9 - 4. that is, 5. is the difference of the divisour and quotient, whose halfe 2.½, is the other shanke. And 9 + 4. that is 13. is the summe the said devisour and quotient, whose halfe 6.½, is the base.
Againe let 4. and 8. be given. The quadrate of 8. is 64. And the quotient of 64 is 16. and the side of 64. is 8. for the one shanke. The halfe 16 - 4. that is 6. is the other shanke. And the halfe of 16 + 4. that is 10, is the base.