Calculus: Barrow is credited with developing a method that was a precursor to Isaac Newton's development of calculus. He introduced the idea of a "generalized tangent," which would later become the basis for the concept of the derivative. Barrow also came close to discovering the fundamental theorem of calculus, but he did not fully develop it.
Geometric Series: Barrow provided a significant contribution to the study of geometric series. He expanded on the work of previous mathematicians, such as John Wallis, and developed new methods for determining the sum of infinite geometric series.
Infinitesimal Calculus: Barrow's work in infinitesimal calculus laid the groundwork for Newton's more comprehensive development of the subject. He used infinitesimals to find the tangents to curves and to calculate areas and volumes.
Diophantine Analysis: Barrow made advancements in the field of Diophantine analysis, studying equations with integer solutions. He extended the work of Pierre de Fermat and helped establish the foundations of modern number theory.
Optical Contributions:
Reflection and Refraction: Barrow conducted extensive research in optics, particularly in the areas of reflection and refraction of light. He developed mathematical theories to describe these phenomena, which were later expanded by Newton.
Lens Design: Barrow's research in optics also included studies on lens design, providing insights into the behavior of light passing through various types of lenses.
Physiological Contributions:
Circulatory System: Barrow conducted anatomical and physiological studies, including investigations into the circulatory system. He provided detailed descriptions of the heart and blood vessels, contributing to the understanding of cardiovascular physiology.
Muscle Contraction: Barrow's research also delved into the study of muscle contraction. He proposed that muscles contract by a process of shortening, which was a significant departure from the prevailing theories at the time.
Theological Contributions:
Sermons: Throughout his life, Barrow was known for his eloquent and powerful sermons. His theological works continue to be studied for their insights into Christian doctrine and moral living.
Treatise on the Pope's Supremacy: In addition to his scientific and religious writings, Barrow also wrote theological treatises, such as "A Treatise on the Pope's Supremacy." He argued against the authority of the Pope and defended the Anglican position on church governance.