So long as the early geometers confined their speculations to the comparison of the areas of rectilinear figures they encountered little difficulty. They readily showed that the determination of the area of any such figure can be reduced to that of a rectangle, or of a square, and thus be completely effected. This process of finding areas was named the “method of quadratures.” It failed, however, when they attempted to determine the areas bounded by curved lines, or the surfaces of the elementary solids such as the right cone and the sphere. In treating of these the ancients found it necessary to introduce new notions and modes of demonstration into geometry, and the difficulty of comparing the areas of curvilinear with those of rectilinear figures gave rise to the “method of exhaustions.” The fundamental principle of this method consists in conceiving the continual approach of two varying magnitudes to a fixed intermediate magnitude, with which they never become identical, though they may approach it to within less than any assignable difference. For example, a polygon may be inscribed in a circle, and another circumscribed to it, each differing from it by less than any assignable area; hence the ancients may have concluded that areas of circles have to each other the same ratio as the similar polygons inscribed in or circumscribed to them, —that is, the ratio of the squares of the radii. But, as this kind of proof was of a different nature from that by which the more elementary doctrines were established, the Greek geometers fortified it by a reductio ad absurdum, — proving, in the above example, that the square of the radius of one circle is to that of another as the area of the former is to a space which is neither less nor greater than the latter, and therefore exactly equal to it.

By the aid of this method Archimedes arrived at his great geometrical discoveries. He determined that the ratio of the circumference to the diameter of a circle lies between 3∣ and 3j-⅞, by considering the regular polygons of ninety-six sides which may be inscribed in or circumscribed to the circle. He proved that the area of a segment of a parabola cut off by any chord equals two-thirds of a parallelogram included between the chord and the parallel tangent to the curve. He determined the quadrature of the ellipse. In the curves named after him the “spirals of Archimedes,” he showed how to draw a tangent at any point, and also determined the area of any portion.

In space of three dimensions, Archimedes proved that the surface of a sphere equals four times that of one of its great circles, that the surface of a spherical cap is equal to the area of a circle the length of whose radius is the distance from the vertex of the cap to any point on its bounding circle; that a sphere has a volume which is two-thirds of that of a cylinder circumscribed to it, and that their surfaces are in the same ratio. Further, the same method of exhaustions furnished Archimedes with the cubature of conoids and spheroids, as he termed surfaces generated by the revolution of the parabola, the hyperbola, and the ellipse.

During nearly two thousand years no new method enabled mathematicians to rise to a higher generality than that attained in the works of the great Greek geometers. The celebrated Kepler was the first to extend the results of Archimedes. In his treatise entitled Nova Stereornetria Doliorum ; accessit stereometriae Archimedeae supplementum (1615), he discussed a number of solids of revolution,— for example, those formed by the revolution of a conic section about any ordinate, or a tangent at the vertex, or any line within or without the curve. Thus he considered some ninety new solids, and proposed problems concerning them; of these problems he resolved only a few of the most simple. In this treatise he introduced for the first time the name and notion of “infinity” into the language of geometry. Thus, he considered a circle as composed of an “infinite” number of triangles, having their common vertex at the centre, and forming the circumference by their bases. In like manner he regarded a cone as composed of an infinite number of pyramids, having their vertices at its vertex, and standing on an infinite number of triangular bases, bounded by the circular base of the cone. It may also be noted that Kepler was the first to observe that the increment of a variable—the ordinate of a curve, for example—is evanescent for values infinitely near a maximum or minimum value of the variable. This remark contains the germ of the rule for determining “maxima” and “minima,” given by Fermat about twenty years subsequently.

Several years after Kepler had given his method of determining volumes of revolution, another celebrated theory, of a similar kind,—the “geometry of indivisibles” (1635) of Cavalieri, professor of geometry at Bologna,— marks an epoch in the progress which science has made in modern times. In this work lines were considered as composed of an infinite number of points, surfaces of lines, and solids of surfaces. For example, if the perpendicular of a triangle be divided into an infinite number of equal parts, and through each point of division a line be drawn parallel to the base and terminated by the sides, then, according to Cavalieri, we may consider the area of the triangle as the sum of all these parallel lines, regarded as its elements. Again, as these parallels form a series in arithmetical progression, of which the first term is zero, this sum is represented by half the product of the last term and the number of terms. Now the base js the last term, and the altitude measures the number of terms; thus he deduced the ordinary expression for the area of a triangle. Cavalieri applied his method to a number of problems, such as finding the volumes of pyramids, the areas of certain simple curves of the parabolic species, the determination of centres of gravity, &c.; and it is remarkable that he was the first who gave an accurate demonstration of the well-known properties of the centre of gravity, originally announced by Pappus, but commonly called Guldin’s theorems. It is accordingly to Cavalieri, and not to Guldin, that the credit is due of having made the first advance beyond Pappus. Cavalieri’s method is analogous to that employed in the integral calculus, tl∣e “indivisible being that which has since been styled the “differential element” of the integral.

The method of Cavalieri was severely criticized "by some of his contemporaries, more especially by Guldin. They alleged that, since a line has no breadth, no number of right lines, however great, when taken together, could make up a plane area. This objection was answered by Cavalieri; but the reply was stated in the clearest form by Pascal, who observed (letter to M. de Carcavi, 1658) that the method of indivisibles possessed all the rigour of that of exhaustions, from which it differed only in the manner of expression, and that, when we conceive an area as a sum of a system of parallel ordinates, we mean in reality an indefinite number of rectangles under the several ordinates, and the small equal portions into which we conceive the common perpendicular to these ordinates to be divided. This passage is remarkable—as was well observed by Carnot—as it shows that the notion of mathematical infinity, as now employed, was not strange to the geometers of that time ; for it is clear that Pascal employed the word “indefinite” in the same signification as we now attach to the word “infinite,” and that he called “small” that which is now called “infinitely small,” also that he neglected these small quantities in comparison with finite quantities—thus he regarded as simple rectangles the small portions of the area of the curve comprised between two consecutive ordinates, neglecting the small triangles which have for their bases the differences of these ordinates. Carnot adds that no person attempted to reproach Pascal with want of rigour in his demonstrations.

Pascal applied the method of Cavalieri with eminent success to the investigation of properties of the cycloid, and other problems. His researches, according to D’Alembert, closely approach to the integral calculus, and form the connexion between the methods of Archimedes and of Newton.

The most important application, however, of Cavalieri’s method was that of Wallis, Savilian professor at Oxford, who, in 1655, gave an admirable specimen of this method in his Arithmetica Infinitorum, sive nova Methodus inquirendi in Curvilinearum Quadraturam. Pursuing Cavalieri’s views, he reduced the problem of finding the areas of a large number of curves, and the volumes of solids of revolution, to the summation of the powers of the terms of arithmetical series, consisting of an infinite number of terms,—or rather to the determination of the ratio of the arithmetical mean of all such powers of the terms to the like power of the last term.

For example, in the series of square numbers 0, 1, 4, 9, 16, &c., the ratio of the mean to the last is, for the first , κ , 0+1+4 i 1 1 f . 1 n . f 0+l+4 + 9

three terms, = ⅜ + ⅛ for the first four, 9+9+9+9 ι,ι r 4-i /·4.c 0 + 1 + 4+9 + 16 , ,

= ⅞ + ⅛; for the tot five, nj + i6- l -18 + 16 + i6 = * + ⅛ in like manner the next fraction is ⅜ + ⅛. Hence Wallis noticed that the fractions approach nearer and nearer to ⅜; and, as the denominators in the fractions ∙⅛, -j⅛-, -f- ι , π ⅛ form an arithmetical series, with a common difference 6^ it follows that, when the number of terms is indefinitely increased, the resulting fraction becomes ultimately ⅜.

Wallis applied the same method to the series 0, 1 3 , 2 3 , 3 3 , &c., and found without difficulty that the aforesaid ratio is I in this case; and so generally. He also introduced into analysis the notation of fractional indices instead of radicals, and extended his method of summation to series proceeding by fractional powers of the natural numbers 1, 2, 3, &c. Wallis was enabled by these principles to obtain the areas of many curves, and the volumes of solids which had not been previously found. He also, by aid of this method, combined with the principles of “interpolation,” arrived at his well-known expression for τr, viz. :—

π 2+ £6 fi;8

4 - 3∙3 ' 5∙5 ‘ 7∙7

Again, in his treatise De Curν. rectif. (1659), Wallis showed that certain curves were capable of being “rectified,” or that straight lines might be found to which they were exactly equal, a remark which was very soon verified by a young English mathematician William Neil, who, by Wallis’s method, obtained in 1660 the length of any arc of a semicubical parabola. This is the first curve that was rectified. The cycloid is the second j its rectification was effected by Sir C. Wren (Phil. Trans., 1673). The methods we have thus far considered were more especially precursory to the integral calculus, having mainly reference to the quadrature of curves and cubature of solids. We now propose to consider the question of tangents to curves, in which the differential calculus may be said to have originated.

The great discovery of Descartes in his application of algebra to geometry (1637) imparted to the latter science the character of abstraction and generality which distinguishes modern from ancient geometry. By it the study of curves was brought under the domain of analysis, and instead of investigation being restricted to particular properties of a few isolated curves, as it had been hitherto, general views and methods applicable to all curves were introduced.

Hence the general problem of drawing tangents to curved lines started immediately into prominence. It was found necessary to depart from the definition of tangents given by the ancient geometers, and to consider them in other points of view. A tangent, accordingly, came to be regarded either (1) as a secant of which the points of intersection became coincident; or (2) as the prolongation of the element of the curve, regarded as a polygon of an infinite number of sides; or (3) as the direction of the resultant motion by which the curve may be described. The first view was that of Descartes and Fermat; the second was introduced by Barrow, who thus simplified the method of Fermat; and the third was that of Roberval.

Descartes’s method of drawing a tangent consisted in supposing a circle (whose centre he placed on the axis of x) to cut the curve in two points; then, if the radius of the circle be supposed to decrease, its centre remaining fixed, so that the points of section approach nearer and nearer and finally coincide, the circle will touch the curve; thus, by aid of the equation of the curve, the problem was reduced to one of finding the condition of equal roots in an equation. This method is remarkable as being the first general process of applying analysis to the problem of tangents; at the same time it is only capable of practical application in a small number of simple cases.

Many years subsequently (Act. Erud. Lips., 1691) John Bernoulli extended Descartes’s method with success to the problem of finding the centre of curvature and the equation of the evolute of an algebraic curve. In his application he supposed the centre of a circle taken on the normal to a curve, and the centre to vary until three of the points of intersection of the circle with the curve became coincident, i.e., so that the resulting equation should have three equal roots. Thus, for example, he showed, without difficulty, that the evolute of a parabola was a semicubical parabola. He also remarked that, when four roots coincide, the centre of curvature becomes a cusp on the evolute.

It should also be noticed that we owe to Descartes the general method of drawing a tangent to a roulette. This was given by him in a letter to Mersenne (Aug. 23, 1638), from which we take the following extracts :—“I have been very glad to see the questions which you say that the geometers, even Μ. Roberval, whom you esteem the principal of them, confess that they cannot solve; for in investigating them I may discover whether my analysis is better than theirs. The first of these questions is that of drawing tangents to curves described by a roulette motion. My solution is as follows. If a rectilinear polygon be conceived to roll on a right line, the curve described by any one of its points will be composed of a number of arcs of circles, and the tangent at any point on one of these arcs is perpendicular to the line drawn from the point to that in which the polygon is in contact with the base, when describing the arc. Consequently, if we consider a rolling curve as a polygon of an infinite number of sides, we see clearly that the roulette traced by any point must possess the same property; that is to say, the tangent at any of its points is perpendicular to the right line connecting it with the point of contact of the rolling curve and its base.” In this we perceive that Descartes gave a genuine and most important application of the infinitesimal method.

Again, Descartes first introduced the method of indeterminate coefficients into analysis,—a principle, as was ably shown by Carnot, which is of itself sufficient to establish, by ordinary algebra, the fundamental principles of the infinitesimal calculus.

The method of Fermat for drawing tangents was based on his method of maxima and minima. This latter was founded, as already observed, on a principle of Kepler’s, viz., that, whenever a magnitude attains a maximum or minimum, its increment or diminution, for a very small change in the variable on which it depends, becomes evanescent.

Accordingly, to determine the maximum or minimum of any function of x, Fermat substituted x + e instead of x, and equated the two consecutive values of the function; then, removing the common terms, and dividing by e, , he made e = 0, and obtained an equation for determining the maximum or minimum value.

Thus, adopting the modern notation, let y = fix), and y 1 = f(x + e), then f(x + é) -f(x) = 0. Dividing by e, f(X + e)-fX) (y. c ’ hence f'(x) = 0.

Thus the roots of the derived equation, f'(x) = 0, correspond to the maximum or minimum values of f(x). Consequently we see that Fermat’s rule agrees with that of the differential calculus, and in fact is the method of the calculus as applied to such cases.

In consequence of Fermat’s both having introduced the conception of an infinitely small difference, and also having arrived at the principle of the calculus for determining maxima and minima, it was maintained by Laplace, Lagrange, Fourier, and other eminent French mathematicians that Fermat ought to be regarded as the first inventor of the differential calculus. In reply to this we need but introduce the remark of their distinguished countryman Poisson, “that this calculus consists in a system of rules proper for finding the differentials of all functions, rather than in the use which may be made of these infinitely small variations in the solution of one or two isolated problems” ( Mém.de l’Àcad. des Sci., 1831).

Fermat seems to have given no general demonstration of his method, but contented himself with giving particular applications of it to some problems of maxima and minima, as well as to finding the tangents to and the centres of gravity of a few curves.

Fermat applied his method to drawing a tangent, as follows :—

Suppose CD (fig. 1) the ordinate, and CF the tangent at the point C in a curve, meeting the axis AB in F; from E, a near point on CF, draw an ordinate EG; then CD FC HC

τ - τi = FΓr, ≥ tvs > *f the curve be concave to the axis, l√f Gr trΓ

CD HG and —- < —£ , if the curve be convex.

JDK Gr

Hence, in either case, the ratio of the ordinate CD to the subtangent DF is a maximum or a minimum relatively to the ratio for a near ordinate HG to GF, the abscissa measured from F, the foot of the tangent.

Accordingly, if CD = y, HG = y 1 , and DF = i, we have, by the method of maxima and minima, t tic

It is easily seen that this method furnishes the ordinary value for the subtangent, as obtained by infinitesimals; for, denoting by x, y the coordinates of C, let Z = FD, DG = <⅛ y 1 =y + dy, and we have y y+dy . t≈yt t + dx ’ ’ j dy from which the subtangent t can be obtained.

The method of Fermat was improved and extended by an Italian, Cardinal Ricci, in his Geometrica exercitatio (1666). Ricci was the first who showed that, if (<ι - ar) m ir n is a maximum, we must have nα x= .

m+n

This he easily established when m and n are integers, from the principle that if a magnitude be divided into r equal parts, their continued product is greater than that obtained by dividing it into r parts in any other manner

The following application, as given by him, to the curve y m =2⅛r n , m>n, will help to illustrate this method of drawing tangents. To draw the tangent at C (fig. 1 ) take AF: AD = m - n: n, and join FC; then FC touches the curve at C.

For the product AF m ^ n AD n is a maximum by the preceding lemma; hence the product AF" -,, AG" is not a maximum for the line FG; consequently

AF ,ft - n AD" AF m -"AG n . ∕ADV∙ ∕FD∖"∙ .

FD” ^ FG” ’ ’ ∖AG√ ∖FG∕ ’

but, from the equation of the curve,

/ADV ∕CD∖ m

(ag∕ (hg∕ ’

a . FD CD CD CD λ ,, p ∩ pττ also FS-Bδ∙ H3>eS' oteγ1 =∙ gπ ' i.β., the...