Trigonometry (from "trigwnon", triangle, and "metrew") is the science of the numerical relations between the sides and angles of triangles.This Treatise is intended to demonstrate, to those who have learned the principal propositions in the first six books of Euclid, so much of Trigonometry as was originally implied in the term, that is, how from given values of some of the sides and angles of a triangle to calculate, in the most convenient way, all the others. A few propositions supplementary to Euclid are premised as introductory to the propositions of Trigonometry as usually understood.What I felt the want of was a short Treatise, to be used as a Text Book after the Sixth Book of Euclid had been learned and some knowledge of Algebra acquired, which should contain satisfactory demonstrations of the propositions to be used in teaching Junior Students the Solution of Triangles, and should at the same time lay a solid foundation for the study of Analytical Trigonometry.This want I have attempted to supply by applying, in the first Chapter, Newton's Method of Limits to the mensuration of circular arcs and areas; choosing that method both because it is the strictest and the easiest, and because I think the Mathematical Student should be early introduced to the method.The succeeding Chapters are devoted to an exposition of the nature of the Trigonometrical ratios, and to the demonstration by geometrical constructions of the principal propositions required for the Solution of Triangles. To these I have added a general explanation of the applications of these propositions in Trigonometrical Surveying: concluded with a proof of the formul_ for the sine and cosine of the sum of two angles treated (as it seems to me they should be) as examples of the Elementary Theory of Projection. Having learned thus much the Student has gained a knowledge of Trigonometry as originally understood, and may apply his knowledge in Surveying; and he has also reached a point from which he may advance into Analytical Trigonometry and its use in Natural Philosophy.
About the Author: He succeeded Thomson's father James in the Chair of Mathematics. Hugh Blackburn was brought up at Killearn House, Stirlingshire, the seventh of eight children of the wealthy Glasgow merchant John Blackburn and his wife Rebecca Leslie Gillies, the daughter of a Church of Scotland minister, and a relative of Colin Maclaurin. His elder brother was the judge Colin Blackburn, Baron Blackburn. He was educated at Edinburgh Academy and Eton before entering Trinity College, Cambridge in 1840. There he met Thomson, who entered in the same year; he was also a member of the Cambridge Apostles. During this time he invented the Blackburn pendulum. In the Mathematical Tripos examinations of 1845 he graduated fifth wrangler, while Thomson graduated second wrangler. He entered the Inner Temple in 1847, but was never called to the Bar; his name was withdrawn in 1849, the year in which he became Professor of Mathematics at the University of Glasgow. Works: A treatise on trigonometry, London, 1855. Part of the Encyclopedia Metropolitana. A short sketch of the constitutional history of the University of Glasgow and of Glasgow College in that University: with remarks on the Universities (Scotland) Bill, 1858 (ed. with William Thomson ) Sir Isaac Newton's Principia, 1871 Elements of plane trigonometry for the use of the junior class of mathematics in the University of Glasgow, 1871