where none of the indices is equal to −1, was used by John
Wallis in his Arithmetica infinitorum (1655) as well as by Fermat
(1659). The case in which m = −1 was that of the
ordinary rectangular hyperbola; and Gregory of
St Vincent in his Opus geometricum quadraturae
Various Integrations.
circuli et sectionum coni (1647) had proved by the
method of exhaustions that the area contained between the
curve, one asymptote, and two ordinates parallel to the other
asymptote, increases in arithmetic progression as the distance
between the ordinates (the one nearer to the centre being kept
fixed) increases in geometric progression. Fermat described
his method of integration as a logarithmic method, and thus
it is clear that the relation between the quadrature of the
hyperbola and logarithms was understood although it was not
expressed analytically. It was not very long before the relation
was used for the calculation of logarithms by Nicolaus Mercator
in his Logarithmotechnia (1668). He began by writing the
equation of the curve in the form y = 1/(1 + x), expanded this
expression in powers of x by the method of division, and integrated
it term by term in accordance with the well-understood
rule for finding the quadrature of a curve given by such an
equation as that written at the foot of p. 325.
By the middle of the 17th century many mathematicians could perform integrations. Very many particular results had been obtained, and applications of them had been made to the quadrature of the circle and other conic sections, and to various problems concerning the Integration before the Integral Calculus. lengths of curves, the areas they enclose, the volumes and superficial areas of solids, and centres of gravity. A systematic account of the methods then in use was given, along with much that was original on his part, by Blaise Pascal in his Lettres de Amos Dettonville sur quelques-unes de ses inventions en géométrie (1659).
16. The problem of maxima and minima and the problem of
tangents had also by the same time been effectively solved.
Oresme in the 14th century knew that at a point where
the ordinate of a curve is a maximum or a minimum
its variation from point to point of the curve is slowest;
Fermat’s methods
of Differentiation.
and Kepler in the Stereometria doliorum remarked
that at the places where the ordinate passes from a smaller
value to the greatest value and then again to a smaller value,
its variation becomes insensible. Fermat in 1629 was in possession
of a method which he then communicated to one Despagnet of
Bordeaux, and which he referred to in a letter to Roberval of
1636. He communicated it to René Descartes early in 1638 on
receiving a copy of Descartes’s Géométrie (1637), and with it
he sent to Descartes an account of his methods for solving the
problem of tangents and for determining centres of gravity.
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| Fig. 6. |
Fermat’s method for maxima and minima is essentially our method. Expressed in a more modern notation, what he did was to begin by connecting the ordinate y and the abscissa x of a point of a curve by an equation which holds at all points of the curve, then to subtract the value of y in terms of x from the value obtained by substituting x + E for x, then to divide the difference by E, to put E = 0 in the quotient, and to equate the quotient to zero. Thus he differentiated with respect to x and equated the differential coefficient to zero.
Fermat’s method for solving the problem of tangents may be explained as follows:—Let (x, y) be the coordinates of a point P of a curve, (x′, y′), those of a neighbouring point P′ on the tangent at P, and let MM′ = E (fig. 6).
From the similarity of the triangles P′TM′, PTM we have
y′ : A − E = y : A,
where A denotes the subtangent TM. The point P′ being near the curve, we may substitute in the equation of the curve x − E for x and (yA − yE)/A for y. The equation of the curve is approximately satisfied. If it is taken to be satisfied exactly, the result is an equation of the form φ(x, y, A, E) = 0, the left-hand member of which is divisible by E. Omitting the factor E, and putting E = 0 in the remaining factor, we have an equation which gives A. In this problem of tangents also Fermat found the required result by a process equivalent to differentiation.
Fermat gave several examples of the application of his method; among them was one in which he showed that he could differentiate very complicated irrational functions. For such functions his method was to begin by obtaining a rational equation. In rationalizing equations Fermat, in other writings, used the device of introducing new variables, but he did not use this device to simplify the process of differentiation. Some of his results were published by Pierre Hérigone in his Supplementum cursus mathematici (1642). His communication to Descartes was not published in full until after his death (Fermat, Opera varia, 1679). Methods similar to Fermat’s were devised by René de Sluse (1652) for tangents, and by Johannes Hudde (1658) for maxima and minima. Other methods for the solution of the problem of tangents were devised by Roberval and Torricelli, and published almost simultaneously in 1644. These methods were founded upon the composition of motions, the theory of which had been taught by Galileo (1638), and, less completely, by Roberval (1636). Roberval and Torricelli could construct the tangents of many curves, but they did not arrive at Fermat’s artifice. This artifice is that which we have noted in §10 as the fundamental artifice of the infinitesimal calculus.
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| Fig. 7. |
17. Among the comparatively few mathematicians who before 1665 could perform differentiations was Isaac Barrow. In his book entitled Lectiones opticae et geometricae, written apparently in 1663, 1664, and published in 1669, 1670, he gave a method of tangents like that Barrow’s Differential Triangle. of Roberval and Torricelli, compounding two velocities in the directions of the axes of x and y to obtain a resultant along the tangent to a curve. In an appendix to this book he gave another method which differs from Fermat’s in the introduction of a differential equivalent to our dy as well as dx. Two neighbouring ordinates PM and QN of a curve (fig. 7) are regarded as containing an indefinitely small (indefinite parvum) arc, and PR is drawn parallel to the axis of x. The tangent PT at P is regarded as identical with the secant PQ, and the position of the tangent is determined by the similarity of the triangles PTM, PQR. The increments QR, PR of the ordinate and abscissa are denoted by a and e; and the ratio of a to e is determined by substituting x + e for x and y + a for y in the equation of the curve, rejecting all terms which are of order higher than the first in a and e, and omitting the terms which do not contain a or e. This process is equivalent to differentiation. Barrow appears to have invented it himself, but to have put it into his book at Newton’s request. The triangle PQR is sometimes called “Barrow’s differential triangle.”
The reciprocal relation between differentiation and integration (§ 6) was first observed explicitly by Barrow in the book cited above. If the quadrature of a curve y = ƒ(x) is known, so that the area up to the ordinate x is given by F(x), the curve y = F(x) can be drawn, and Barrow showed that the Barrow’s Inversion-theorem. subtangent of this curve is measured by the ratio of its ordinate to the ordinate of the original curve. The curve y = F(x) is often called the “quadratrix” of the original curve; and the result has been called “Barrow’s inversion-theorem.” He did not use it as we do for the determination of quadratures, or indefinite integrals, but for the solution of problems of the kind which were then called “inverse problems of tangents.” In these problems it was sought to determine a curve from some property of its tangent, e.g. the property that the subtangent is proportional to the square of the abscissa. Such problems are now classed under “differential equations.” When Barrow wrote, quadratures were familiar and differentiation unfamiliar, just as hyperbolas were trusted while logarithms were strange. The functional notation was not invented till long afterwards (see Function), and the want of it is felt in reading all the mathematics of the 17th century.
18. The great secret which afterwards came to be called the “infinitesimal calculus” was almost discovered by Fermat, and still more nearly by Barrow. Barrow went farther than Fermat in the theory of differentiation, though not in the practice, for he compared two increments; he went farther in the theory of integration, for he obtained the inversion-theorem. The great discovery seems to consist partly in the
