This section is from the book "The American House Carpenter", by R. G. Hatfield. Also available from Amazon: The American House Carpenter.

In all problems to be solved there are certain conditions or quantities given, by means of which an unknown quantity is to be evolved. For example, in the problem in Art. 397, there were three certain lines given: to find a fourth, based upon the condition that the four lines were four proportionals. Now, it has been found that the relation between quantities and the conditions of a question can better be stated by letters than by numerals; and it is the office of algebra to present by letters a concise statement of a question, and by certain processes of comparison, substitution and elimination, to condense the statement to its smallest compass, and at last to present it in a formula or rule, which exhibits the known quantities on one side as equal to the unknown on the other side. Here algebra ends, at the completion of the rule. To use the rule is the office of arithmetic. For, in using the rule, each quantity in numerals must be substituted for the letter representing it, and the arithmetical processes indicated performed, as was done in Art. 397.

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