Brook Taylor : Methodus Incrementorum Directa & Inversa

translated and annotated by
Ian Bruce

This is an annotated translation of Brook Taylor's Methodus Incrementorum Directa & Inversa (1715; 2nd ed. 1717). This is a truly inspirational work, and gives us a fascinating glimpse of the state of the mathematical world at the start of the 18th century, as presented by someone living in the shadow of Newton, and who was himself totally versed in the ways of the calculus. However, do not expect an easy ride leading up to an exposition of Taylor's Theorem, which is presented as an afterthought to the main business of Prop. VII, which is initially concerned with the mathematics of finite differences, here invented by Taylor; indeed, the function notaton had not yet been invented, so one has to make do with the equaivalent contemporary way of setting out mathematics. This is a challenging work, in which a great deal of interesting material is presented.

This work was influential at the time, though it attracted a lot of attention of the wrong kind from Johan Bernoulli, who was publishing similar material at the time, and who accused Taylor of plagiarism, which appears to be totally unfounded; at least one can see the need for a proper set of symbols for calculus! There are theorems that require extra attention to be understood; however, if you are a casual reader, and a theorem does not make sense to you, then just go on; there are parts that are quite delightful. Taylor gave a summary of the chapters in the Philosophical Transactions of the Royal Society, Vol. 29, pp 339-352, available from JSTOR, that we now to present in abbreviated form as a Proposition guide. One should note perhaps the difficulty Taylor had with the centre of percussion, which in the modern sense is just the same as the centre of oscillation.


Introduction : A description of the method of increments is given, including a definition of the symbols used, and the relation to Newton's Fluxions. Part I : Prop.I: Explanation of the direct method and relation to fluxions (differentiation); Prop.II shows how to change an (difference) incremental equation so that the quantities flow from the right rather than from the left; Prop.III sets out to show how the dependent and independent variable in a fluxional equation can be interchanged, mindful of a function and its inverse, and useful in inverting series solutions of fluctional equations; Prop.IV and Prop.V show how to find the number of boundary conditions in a fluxional or incremental equation; Prop.VI gives the general explanation of the inverse method (integration); for a given relation between increments or fluxions, to find the relation between the fluents; Taylor considers this tentative, as there is no general method; he considers solutions in terms of infinite or finite series : these are considered in Prop. VII and VIII. Prop. IX extends the method of series solutions in an elegant manner; Prop. X and XI and XII give general series solutions arising from integration by parts in an extended form. Part II deals with applications : Prop. XIII looks at the binomial theorem; Prop. XIV sums arithmetical series; Prop. XV deals with curvature and related problems; Prop. XVI : quadrature of all sorts of curves; Prop. XVII : the isoperimetric problem; Prop. XVIII : solution of catenary curves; Prop. IXX deals with the arch; Prop. XX deals with a sail filled with water; Prop. XXI deals with an arch supporting a fluid; Prop. XX II and XXIII deal with the motion of a musical string and gives a formula for the frequency in terms of the tension, density, and length of the string; these are new results of which Taylor is rightly proud; Prop. XXIV deals with centre of oscillation; Prop. XXV : centre of percussion; Prop. XXVI : the density of air at any distance from the centre of the earth; this establishes the use of calculus as a means of deriving a formula by considering the forces on a small element; Prop. XXVII : refraction of light passing through the atmosphere; this last prop. attributes unrealistic mechanical properties to light beams in the earth's gravity; yet from a calculus point of view it shows how difficult problems can be solved by series expansions.

Methodus Incrementorum Part One.

Part 1A : Methodus Incr....Prop.I-VI.

Part IB : Methodus Incr....Prop.VII-XII.

Methodus Incrementorum Part Two.

Part IIA : Methodus Incr....Prop.XIII-XVII.

Part IIB : Methodus Incr....Prop.XVIII-XXIII.

Part IIC : Methodus Incr....Prop.XXIV-XXV.

Part IID : Methodus Incr....Prop.XXVI-XXVII.

Ian Bruce. Sept. 2007 latest revision. Copyright : I reserve the right to publish this translated work in book form. However, if you are a student, teacher, or just someone with an interest, you can copy part or all of the work for legitimate personal or educational use. If you wish to send me a message, then click on my name.