**Continued** **fractions**, **one** **of** **the** **earliest** **examples** **of** **which** **is** **Lord** **Brouncker'****s** **expression** **for** **the** **ratio** **of** **the** **circumference** **to** **the** **diameter** **of** **a** **circle** (**see** **Circle**), **were** **elaborately** **discussed** **by** **John** **Wallis** **and** **Leonhard** **Euler**; **the** **convergency** **of** **series** **treated** **by** **Newton**, **Euler** **and** **the** **Bernoullis**; **the** **binomial** **theorem**, **due** **originally** **to** **Newton** **and** **subsequently** **expanded** **by** **Euler** **and** **others**, **was** **used** **by** **Joseph** **Louis** **Lagrange** **as** **the** **basis** **of** **his** **Calcul** **des** **Fonctions**.

**As** **a** **mathematician**, **he** **was** **the** **only** **Englishman** **after** **Sir** **Isaac** **Newton** **and** **Roger** **Cotes** **capable** **of** **holding** **his** **own** **with** **the** **Bernoullis**; **but** **a** **great** **part** **of** **the** **effect** **of** **his** **demonstrations** **was** **lost** **through** **his** **failure** **to** **express** **his** **ideas** **fully** **and** **clearly**.

**BERNOULLI**, **or** **Bernouilli**, **the** **name** **of** **an** **illustrious** **family** **in** **the** **annals** **of** **science**, **who** **came** **originally** **from** **Antwerp**. **Driven** **from** **their** **country** **during** **the** **oppressive** **government** **of** **Spain** **for** **their** **attachment** **to** **the** **Reformed** **religion**, **the** **Bernoullis** **sought** **first** **an** **asylum** **at** **Frankfort** (1583), **and** **afterwards** **at** **Basel**, **where** **they** **ultimately** **obtained** **the** **highest** **distinctions**.

**De** **Maupertuis** **and** **Alexis** **Claude** **Clairaut**, **whom** **the** **fame** **of** **the** **Bernoullis** **had** **attracted** **to** **Basel**.

**Epicycloids** **also** **received** **attention** **at** **the** **hands** **of** **Edmund** **Halley**, **Sir** **Isaac** **Newton** **and** **others**; **spherical** **epicycloids**, **in** **which** **the** **moving** **circle** **is** **inclined** **at** **a** **constant** **angle** **to** **the** **plane** **of** **the** **fixed** **circle**, **were** **studied** **by** **the** **Bernoullis**, **Pierre** **Louis** **M**.

**Leibnitz**, **the** **Bernoullis**, **Roger** **Cotes** **and** **others** - **and** **so** **assiduously** **was** **it** **studied** **that** **it** **was** **sometimes** **named** **the** "**Helen** **of** **Geometers**."