Child J.M. The Geometrical lectures of Isaac Barrow

Chicago ; London: The Open Court Publishing Company, 1916. — 218 p.Isaac Barrow (October 1630 – 4 May 1677) was an English Christian theologian and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for proof of the fundamental theorem of calculus. His work centered on the properties of the tangent; Barrow was the first to calculate the tangents of the kappa curve. He is also notable for being the inaugural holder of the prestigious Lucasian Professorship of Mathematics, a post later held by his student, Isaac Newton.
Mr. Child begins by laying down the startling thesis that “Isaac Barrow was the first inventor of the Infinitesimal Calculus; Newton got the main idea of it from Barrow by personal communication; and Leibniz also was in some measure indebted to Barrow's work“. To interpret this according to the writer's intention we must use the term “calculus“ to mean a set of analytical rules applied to analytical expressions; with this restriction, Mr. Child has made out a case that is convincing enough in this sense, that if Barrow had been given any function likely to be constructed in his time, he would have been able to differentiate it by applying a few standard rules.Introduction.
The work of Barrow's great predecessors.
Life of Barrow, and his connection with Newton.
The works of Barrow.
Estimate of Barrow's genius.
The sources of Barrow's ideas.
Mutual influence of Newton and Barrow.
Description of the book from which the translation has been made.
The prefaces.
How Barrow made his constructions.
Analytical equivalents of Barrow's chief theorems.
Translation.
Lecture I.—Generation of magnitudes. Modes of motion and the quantity of the motive force. Time as the independent variable. Time, as an aggregation of instants, compared with a line, as an aggregation of points.
Lecture II.—Generation of magnitudes by "local movements." The simple motions of translation and rotation.
Lecture III.—Composite and concurrent motions. Composition of rectilinear and parallel motions.
Lecture IV.—Properties of curves arising from composition of motions. The gradient of the tangent. Generalization of a problem of Galileo. Case of infinite velocity.
Lecture V.—Further properties of curves. Tangents. Curves like the cycloid. Normals. Maximum and minimum lines.
Lecture VI.—Lemmas; determination of certain curves constructed according to given conditions; mostly hyperbolas.
Lecture VII.— Similar or analogous curves. Exponents or Indices. Arithmetical and Geometrical Progressions. Theorem analogous to the approximation to the Binomial Theorem for a Fractional Index, Asymptotes.
Lecture VIII.—Construction of tangents by means of auxiliary curves of which the tangents are known. Differentiation of a sum or a difference. Analytical equivalents.
Lecture IX. —Tangents to curves formed by arithmetical and geometrical means. Paraboliforms. Curves of hyperbolic and elliptic form. Differentiation of a fractional power, products and quotients.
Lecture X.—Rigorous determination of ds/dx. Differentiation as the inverse of integration. Explanation of the "Differential Triangle" method; with examples. Differentiation of a trigonometrical function.
Lecture XI.—Change of the independent variable in integration. Integration the inverse of differentiation. Differentiation of a quotient. Area and centre of gravity of a paraboliform. Limits for the arc of a circle and a hyperbola. Estimation of π.
Lecture XII.—General theorems on rectification. Standard forms for integration of circular functions by reduction to the quadrature of the hyperbola. Method of circumscribed and inscribed figures. Measurement of conical surfaces. Quadrature of the hyperbola. Differentiation and Integration of a Logarithm and an Exponential. Further standard forms.
Lecture XIII.—These theorems have not been inserted.
Postscript.
Extracts from Standard Authorities.
Appendices.
Solution of a test question by Barrow's method.
Graphical integration by Barrow's method.
Reduced facsimiles of Barrow's pages and figures.
Index.