“He gave more elegant rules for the sum of the squares and cubes of an initial segment of the positive integers. The sixth part of the product of three quantities consisting of the number of terms, the number of terms plus one, and twice the number of terms plus one is the sum of the squares. The square of the sum of the series is the sum of the cubes.”

Out of Control: The New Biology of Machines, Social Systems and the Economic World (1995), New Rules for the New Economy: 10 Radical Strategies for a Connected World (1999)

Theoria motus corporum coelestium in sectionibus conicis solem ambientum (1809) Tr. Charles Henry Davis as Theory of the Motion of the Heavenly Bodies moving about the Sun in Conic Sections http://books.google.com/books?id=cspWAAAAMAAJ&amp; (1857) <br class="br">Context: The principle that the sum of the squares of the differences between the observed and computed quantities must be a minimum may, in the following manner, be considered independently of the calculus of probabilities. When the number of unknown quantities is equal to the number of the observed quantities depending on them, the former may be so determined as exactly to satisfy the latter. But when the number of the former is less than that of the latter, an absolutely exact agreement cannot be obtained, unless the observations possess absolute accuracy. In this case care must be taken to establish the best possible agreement, or to diminish as far as practicable the differences. This idea, however, from its nature, involves something vague. For, although a system of values for the unknown quantities which makes all the differences respectively less than another system, is without doubt to be preferred to the latter, still the choice between two systems, one of which presents a better agreement in some observations, the other in others, is left in a measure to our judgment, and innumerable different principles can be proposed by which the former condition is satisfied. Denoting the differences between observation and calculation by A, A&amp;rsquo;, A&amp;rsquo;&amp;rsquo;, etc., the first condition will be satisfied not only if AA + A&amp;rsquo; A&amp;rsquo; + A&amp;rsquo;&amp;rsquo; A&amp;rsquo;&amp;rsquo; + etc., is a minimum (which is our principle) but also if A4 + A&amp;rsquo;4 + A&amp;rsquo;&amp;rsquo;4 + etc., or A6 + A&amp;rsquo;6 + A&amp;rsquo;&amp;rsquo;6 + etc., or in general, if the sum of any of the powers with an even exponent becomes a minimum. But of all these principles ours is the most simple; by the others we should be led into the most complicated calculations.

The Triumph of Numbers: How Counting Shaped Modern Life (2005)

Source: The Alphabet of Grace

However, negative numbers gained acceptance slowly.
Source: Mathematical Thought from Ancient to Modern Times (1972), p. 185.

Nicomachus of Gerasa: Introduction to Arithmetic (1926)

As quoted by James Gow, A Short History of Greek Mathematics https://books.google.com/books?id=9d8DAAAAMAAJ (1884) <br class="br">Arithmetica (c. 250 AD)

Contributions of Jane Wagner

Other material for Lily Tomlin