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 work. This alteration was made to afford facility and security in the process of evolution of the root.

We pause a moment to sketch the life of Vieta, the most eminent French mathematician of the sixteenth century. He was born in Poitou in 1540, and died in 1603 at Paris. He was employed throughout life in the service of the state, under Henry III. and Henry IV. He was, therefore, not a mathematician by profession, but his love for the science was so great that he remained in his chamber studying, sometimes several days in succession, without eating and sleeping more than was necessary to sustain himself. So great devotion to abstract science is the more remarkable, because he lived at a time of incessant political and religious turmoil. During the war against Spain, Vieta rendered service to Henry IV. by deciphering intercepted letters written in a species of cipher, and addressed by the Spanish Court to their governor of Netherlands. The Spaniards attributed the discovery of the key to magic.

An ambassador from Netherlands once told Henry IV. that France did not possess a single geometer capable of solving a problem propounded to geometers by a Belgian mathematician, Adrianus Romanus. It was the solution of the equation of the forty-fifth degree:—

$\scriptstyle{45y-3795y^2+95634y^3- \cdots +945y^{41}-45y^{41}+y^{41}=C}$.

Henry IV. called Vieta, who, having already pursued similar investigations, saw at once that this awe-inspiring problem was simply the equation by which $$\scriptstyle{C=2 \sin \phi}$$ was expressed in terms of $$\scriptstyle{y=2 \sin \frac{1}{45} \phi}$$; that, cincesince [sic] $$\scriptstyle{45=3 \cdot 3 \cdot 5}$$, it was necessary only to divide an angle once into 5 equal parts, and then twice into 3,—a division which could be effected by corresponding equations of the fifth and third degrees. Brilliant was the discovery by Vieta of 33 roots to this equation, instead