This section is from the book "The Tinman's Manual And Builder's And Mechanic's Handbook", by Isaac Ridler Butt. Also available from Amazon: The Tinman's Manual And Builder's And Mechanic's Handbook.
To find the geometrical mean proportion between two numbers.
Set one of the numbers upon c to the same number upon D; and against the other number upon c is the mean number or side of an equal square upon D.
Required the mean proportion between 20 and 45.
Set 20 upon c to 20 upon D; and against 45 upon c is 30 upon D.
To cube any number, set the number upon c to 1 or 10 upon D; and against the same number upon d is the cube number upon c.
Required the cube of 4.
Set 4 upon c to 1 or 10 upon D; and against 4 upon D is 64 upon c.
To extract the cube root of any number, invert the slide, and set the number upon b to 1 or 10 upon d; and where two numbers of equal value coincide on the lines b d, is the root of the given number.
Required the cube root of 64. Set 64 upon b to 1 or 10 upon D; and against 4 upon B is 4 upon D, or root of the given number.
On the common rule, when 1 in the middle of the line c is set opposite to 10 on D, then c is a table of squares, and D a table of roots.
To cube any number by this rule, set the number upon c to 10 upon D-and against the same number upon D is the cube upon c.