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The world of ideas which it [mathematics] discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connexion of its parts, the infinite hierarchy and absolute evidence of the truths with which it is concerned, these, and such like, are the surest grounds of the title of mathematics to human regard, and would remain unimpeached and unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance.
Presidential Address to British Association, 1869.
Loci: Convergence
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Mathematical Treasures
Christian Wolff's Treatise of Algebra
This is the title page of the Treatise of Algebra by Christian Wolff (1679-1754). Wolff was a student of Leibniz and is most famous for his work in philosophy. His school of philosophy, in fact, was the most prominent in Germany prior to Kant. This book was originally written in Latin in 1713. It first appeared in English in 1739, though this copy is of the second edition on 1765.
On pp. 140-141 of his book, Wolff presents a derivation of the formula for the sums of the integers as well as the sums of the squares and cubes of the integers, a derivation which is generalizable to higher powers as well.
On pages 142-143, Wolff generalizes his earlier results and shows how to derive formulas for the sums of higher powers of the integers.
Here, on page 144, Wolff completes the derivation with the complete formula for the sums of third and fourth powers.
On pages 202-203, Wolff discusses some elements of the theory of equations. Note that he mentions Descartes' rule of signs, without attribution to Descartes. In fact, he attributes it to Thomas Harriot and claims further that no one had yet proven it. The first published proof of the result was due to Jean Paul de Gua de Malves (1713 - 1785), who gave two proofs in 1741 in a paper in the Memoires of the Paris Academy.