Therefore this is the Analysis or manner of finding the side of a quadrate, by the first rate of a quadrate, equall to a double rectangle and quadrate.
The Geodesy or measuring of a Triangle.
There is one generall Geodesy or way of measuring any manner of triangle whatsoever in Hero, by addition of the sides, halving of the summe, subduction, multiplication, and invention of the quadrates side, after this manner.
18 If from the halfe of the summe of the sides, the sides be severally subducted, the side of the quadrate continually made of the halfe, and the remaines shall be the content of the triangle.
As for example, Let the sides of the triangle aei, be 6. 8. 10: The summe 24. the halfe of the summe 12. From which halfe subduct the sides 6. 8. 10. and let the remaines be 6. 4. 2. Now multiply continually these foure numbers 12. 6. 4. 2. and thou shalt make first of 12. and 6. 72. Of 72. and 4. 288. Lastly, of 288. and 2. 576. And the side of 576. by the [16. e.] shall be found to be 24. for the content of the triangle: which also here will be found to be true, by multiplying the sides ae and ei, containing the right angle, the one by the other; and then taking the halfe of the product.
This generall way of measuring a triangle is most easie and speedy, where the sides are expressed by whole numbers.
The speciall geodesy of rectangle triangle was before taught (at the [9 e xj].) But of an oblique angle it shall hereafter be spoken. But the generall way is farre more
excellent than the speciall; For by the reduction of an obliquangle many fraudes and errours doe fall out, which caused the learned Cardine merrily to wish, that hee had but as much land as was lost by that false kinde of measuring.
