3————√2 3————√2 ——————— —√18 + 2 9—√18 ————————— 9—√72 + 2 3——√2 —————————————— ——√162 + 12——√8 27——√648 + 6 ——————————————————— 27—√658—√162 + 18—√8 ====================== _A._ Why for two roots of 18 do you put the root of 72. _B._ Because 2 roots of 18 are equal to one root of four times 18, which is 72. _A._ Next we have, that the root of 2 multiplied into 2 makes the root of 8. How is that true? _B._ Does it not make 2 roots of two? And is not B R the root of 2, and 2 B R equal to the diagonal? And is not the square of the diagonal equal to 8 squares of D V? _A._ It is true. But here the root of 8 is put for the cube of the root of 2. Can a line be equal to a cube? _B._ No. But here we are in arithmetic again, and 8 is a cubic number. _A._ How does the root of 2 multiplied into the root of 72 make 12? _B._ Because it makes the root of 2 times 72, that is to say the root of 144 which is 12. _A._ How does 9 roots of 2 make the root of 162? _B._ Because it makes the root of 2 squares of 9, that is the root of 162. _A._ How does 3 roots of 72 make the root of 648? _B._ Because it makes the root of 9 times 72, that is of 648. _A._ For the total sum I see 27 and 18, which make 45. Therefore the root of 648 together with the root of 162 and of 8, which are to be deducted from 45, ought to be equal to the root of 1682. _B._ So they are. For 648 multiplied by 162 makes 104976, of which the double root is 648 and 648 and 162 added together make 810 Therefore the root of 648, added to the root of 162, makes the root of 1458 Again 1458 into 8 is 11664. The double root whereof is 216 The sum of 1458 and 8 added together 1466 The sum of 1466 and 216 is 1682, and the root, the root of 1682 _A._ I see the calculation in numbers is right, though false in lines. The reason whereof can be no other than some difference between multiplying numbers into lines or planes, and multiplying lines into the same lines or planes.