34 GRAPHICAL AND MECHANICAL AIDS TO CALCULATION 
I shall first treat briefly with graphical tables or dia- 
grams, also called graphs, and show one of the oldest graphs 
in existence: Pythagoras table of multiplication (Table [., 
Plate 1). The construction is of the simplest, on a horizontal 
line ten equal divisions are traced, numbered from 0 to 10, the 
same is repeated on a perpendicular line erected on the 
zero point, and the whole square completed. Every pro- 
duct of multiplication is indicated by the poimt where the 
horizontal line of a given number crosses the vertical line 
of another number; by connecting all products of equal 
numerical value, a system of curves will be obtained, with 
the help of which the product of any two factors may be 
read off. Every line representing the same quantity is 
called an “‘ isoplethe”’ (this term was first proposed by the 
German mathematician, Vogler, and has been universally 
adopted by others), and we find on this graph three series 
of isoplethes, two systems of straight lines representing the 
factors and a system of curves representing the products. 
To obtain the products with fractions of whole numbers, 
the values must be estimated by interpolation, which makes 
the table of little practical value. This table serves to 
illustrate the simplest of forms applied to three variables, 
in which two given values determine a third unknown. 
The celebrated French engineer, Leon Lalanne, dis- 
covered the principle of anamorphosis, by which the con- 
struction of graphs is simplified, and their utility greatly 
increased. This principle is based on the following con- 
sideration :—Each scale must be considered extensible, 
as if drawn on a sheet of rubber, each scale can be so stretched 
and transformed that the curves become straight lines, 
which not only simplifies the construction, but greatly 
facilitates the reading. Lalanne thus modified Pythagoras 
table of multiplication by stretching the horizontal and 
the vertical scales in a peculiar manner with the result shown 
on the left of Table I. that the isoplethes of products become 
straight lines running diagonally, and cutting both the 
horizontal and vertical isoplethes at the number of their 
actual value. On this improved table of multiplication, 
the squares and cubes of numbers are easily found by 
drawing diagonal lines, for the squares trom 1 to 100, and 
for the cubes from | to 3.162 (= v 10) and from 3.162 to 100, 
and reading the results at the point of intersection of ver- 
