Ratios and
Proportions

This
section introduces the concepts of ratio and proportion. Chunks of the material
in this section are excerpted from Ferdinand Hassler’s 1826 book, *Elements of Arithmetic, Theoretical and
Practical*. The book was written for use in American schools and in private
study.

In
the notebook, Macdonald delineates two kinds of ratios, arithmetical and geometrical.
Arithmetical ratios depend on addition and subtraction while geometrical ratios
depend on multiplication and division. Macdonald follows this with a verbatim
excerpt, with the exception of a copying error, from page 110 of Hassler:

“These
two kinds of ratio take their mark of notation from the marks applied to the
combinations or rules of arithmetic on which they depend; thus, the arithmetical
ratio of 7 to 3 [is expressed by 7 – 3 and the geometrical ratio of 7 to 3] is
expressed by 7:3 or .”

Macdonald
left out the material in square brackets [ ]. Perhaps he was distracted or just
not interested enough in the subject to pay attention to what he was doing.

This
is followed by excerpts from pages 110 and 111 of Hassler that Macdonald’s
teacher, George Baxter has slightly changed and edited. Macdonald writes:

“In
the habitual mode of writing, therefore an arithmetical ratio expresses a difference
between two quantities, and a geometrical ratio expresses the quotient arising
from the division of two quantities. It is evident that two or more such ratios
may be exactly equal to each other; such an equality of ratio is called a
proportion or equations. Hence an arithmetical proportion may be expressed thus
and a geometrical proportion thus or . The first term of
a ratio is call the antecedent, the second the consequent; the first and last
terms of a proportion are called the extremes; the second and third the means.”

A minor
change, for example, that Baxter made was to write, “It is evident” where Hassler
wrote, “The simplest reflection leads to the idea:” A less minor change is that
Baxter left out defining the index of a geometrical ratio, the number obtained
when dividing two quantities. For example, 4 is the index of the ratios and .