This textbook presents a theoretical and practical elementary algebra course for schools and colleges. Chapters cover topics such as: addition, subtraction, multiplication, and division of algebraic quantities; algebraic fractions; simple equations with one unknown quantity; simple equations with two or more unknown quantities; involution and evolution of numbers and algebraic quantities; irrational and imaginary numbers; pure equations; quadratic equations; proportion and progression; logarithms; equations of the third and higher degrees; indeterminate coefficients, vanishing fractions, and figurate and polygonal numbers; and interdeminate and diophantine analysis. An appendix by Robert Adrain presents an algebraic method of demonstrating propositions in the fifth book of Euclid's elements.