A History of Mathematics/Modern Europe/Descartes to Newton

by Florian Cajori

 

DESCARTES TO NEWTON.

 

Among the earliest thinkers of the seventeenth and eighteenth centuries, who employed their mental powers toward the destruction of old ideas and the up-building of new ones, ranks René Descartes (1596–1650). Though he professed orthodoxy in faith all his life, yet in science he was a profound sceptic. He found that the world's brightest thinkers had been long exercised in metaphysics, yet they had discovered nothing certain; nay, had even flatly contradicted each other. This led him to the gigantic resolution of taking nothing whatever on authority, but of subjecting everything to scrutinous examination, according to new methods of inquiry. The certainty of the conclusions in geometry and arithmetic brought out in his mind the contrast between the true and false ways of seeking the truth. He thereupon attempted to apply mathematical reasoning to all sciences. "Comparing the mysteries of nature with the laws of mathematics, he dared to hope that the secrets of both could be unlocked with the same key." Thus he built up a system of philosophy called Cartesianism.

Great as was Descartes' celebrity as a metaphysician, it may be fairly questioned whether his claim to be remembered by posterity as a mathematician is not greater. His philosophy has long since been superseded by other systems, but the analytical geometry of Descartes will remain a valuable possession forever. At the age of twenty-one, Descartes enlisted in the army of Prince Maurice of Orange. His years of soldiering were years of leisure, in which he had time to pursue his studies. At that time mathematics was his favourite science. But in 1625 he ceased to devote himself to pure mathematics. Sir William Hamilton is in error when he states that Descartes considered mathematical studies absolutely pernicious as a means of internal culture. In a letter to Mersenne, Descartes says: "M. Desargues puts me under obligations on account of the pains that it has pleased him to have in me, in that he shows that he is sorry that I do not wish to study more in geometry, but I have resolved to quit only abstract geometry, that is to say, the consideration of questions which serve only to exercise the mind, and this, in order to study another kind of geometry, which has for its object the explanation of the phenomena of nature.… You know that all my physics is nothing else than geometry." The years between 1629 and 1649 were passed by him in Holland in the study, principally, of physics and metaphysics. His residence in Holland was during the most brilliant days of the Dutch state. In 1637 he published his Discours de la Méthode, containing among others an essay of 106 pages on geometry. His Geometry is not easy reading. An edition appeared subsequently with notes by his friend De Beaune, which were intended to remove the difficulties.

It is frequently stated that Descartes was the first to apply algebra to geometry. This statement is inaccurate, for Vieta and others had done this before him. Even the Arabs sometimes used algebra in connection with geometry. The new step that Descartes did take was the introduction into geometry of an analytical method based on the notion of variables and constants, which enabled him to represent curves by algebraic equations. In the Greek geometry, the idea of motion was wanting, but with Descartes it became a very fruitful conception. By him a point on a plane was determined in position by its distances from two fixed right lines or axes. These distances varied with every change of position in the point. This geometric idea of co-ordinate representation, together with the algebraic idea of two variables in one equation having an indefinite number of simultaneous values, furnished a method for the study of loci, which is admirable for the generality of its solutions. Thus the entire conic sections of Apollonius is wrapped up and contained in a single equation of the second degree.

The Latin term for "ordinate," used by Descartes comes from the expression lineæ ordinatæ, employed by Roman surveyors for parallel lines. The term abscissa occurs for the first time in a Latin work of 1659, written by Stefano degli Angeli (1623–1697), a professor of mathematics in Rome.^[3] Descartes' geometry was called "analytical geometry," partly because, unlike the synthetic geometry of the ancients, it is actually analytical, in the sense that the word is used in logic; and partly because the practice had then already arisen, of designating by the term analysis the calculus with general quantities.

The first important example solved by Descartes in his geometry is the "problem of Pappus"; viz. "Given several straight lines in a plane, to find the locus of a point such that the perpendiculars, or more generally, straight lines at given angles, drawn from the point to the given lines, shall satisfy the condition that the product of certain of them shall be in a given ratio to the product of the rest." Of this celebrated problem, the Greeks solved only the special case when the number of given lines is four, in which case the locus of the point turns out to be a conic section. By Descartes it was solved completely, and it afforded an excellent example of the use which can be made of his analytical method in the study of loci. Another solution was given later by Newton in the Principia.

The methods of drawing tangents invented by Roberval and Fermat were noticed earlier. Descartes gave a third method. Of all the problems which he solved by his geometry, none gave him as great pleasure as his mode of constructing tangents. It is profound but operose, and, on that account, inferior to Fermat's. His solution rests on the method of Indeterminate Coefficients, of which he bears the honour of invention. Indeterminate coefficients were employed by him also in solving bi-quadratic equations.

The essays of Descartes on dioptrics and geometry were sharply criticised by Fermat, who wrote objections to the former, and sent his own treatise on "maxima and minima" to show that there were omissions in the geometry. Descartes thereupon made an attack on Fermat's method of tangents. Descartes was in the wrong in this attack, yet he continued the controversy with obstinacy. He had a controversy also with Roberval on the cycloid. This curve has been called the "Helen of geometers," on account of its beautiful properties and the controversies which their discovery occasioned. Its quadrature by Roberval was generally considered a brilliant achievement, but Descartes commented on it by saying that any one moderately well versed in geometry might have done this. He then sent a short demonstration of his own. On Roberval's intimating that he had been assisted by a knowledge of the solution, Descartes constructed the tangent to the curve, and challenged Roberval and Fermat to do the same. Fermat accomplished it, but Roberval never succeeded in solving this problem, which had cost the genius of Descartes but a moderate degree of attention.

He studied some new curves, now called "ovals of Descartes," which were intended by him to serve in the construction of converging lenses, but which yielded no results of practical value.

The application of algebra to the doctrine of curved lines reacted favourably upon algebra. As an abstract science, Descartes improved it by the systematic use of exponents and by the full interpretation and construction of negative quantities. Descartes also established some theorems on the theory of equations. Celebrated is his "rule of signs" for determining the number of positive and negative roots; viz. an equation may have as many $\scriptstyle {+}$ roots as there are variations of signs, and as many $\scriptstyle {-}$ roots as there are permanencies of signs. Descartes was charged by Wallis with availing himself, without acknowledgment, of Harriot's theory of equations, particularly his mode of generating equations; but there seems to be no good ground for the charge. Wallis also claimed that Descartes failed to observe that the above rule of signs is not true whenever the equation has imaginary roots; but Descartes does not say that the equation always has but that it may have so many roots. It is true that Descartes does not consider the case of imaginaries directly, but further on in his Geometry he gives incontestable evidence of being able to handle this case also.

In mechanics, Descartes can hardly be said to have advanced beyond Galileo. The latter had overthrown the ideas of Aristotle on this subject, and Descartes simply "threw himself upon the enemy" that had already been "put to the rout." His statement of the first and second laws of motion was an improvement in form, but his third law is false in substance. The motions of bodies in their direct impact was imperfectly understood by Galileo, erroneously given by Descartes, and first correctly stated by Wren, Wallis, and Huygens.

One of the most devoted pupils of Descartes was the learned Princess Elizabeth, daughter of Frederick V. She applied the new analytical geometry to the solution of the "Apollonian problem." His second royal follower was Queen Christina, the daughter of Gustavus Adolphus. She urged upon Descartes to come to the Swedish court. After much hesitation he accepted the invitation in 1649. He died at Stockholm one year later. His life had been one long warfare against the prejudices of men.

It is most remarkable that the mathematics and philosophy of Descartes should at first have been appreciated less by his countrymen than by foreigners. The indiscreet temper of Descartes alienated the great contemporary French mathematicians, Roberval, Fermat, Pascal. They continued in investigations of their own, and on some points strongly opposed Descartes. The universities of France were under strict ecclesiastical control and did nothing to introduce his mathematics and philosophy. It was in the youthful universities of Holland that the effect of Cartesian teachings was most immediate and strongest.

The only prominent Frenchman who immediately followed in the footsteps of the great master was De Beaune (1601–1652). He was one of the first to point out that the properties of a curve can be deduced from the properties of its tangent. This mode of inquiry has been called the inverse method of tangents. He contributed to the theory of equations by considering for the first time the upper and lower limits of the roots of numerical equations.

In the Netherlands a large number of distinguished mathematicians were at once struck with admiration for the Cartesian geometry. Foremost among these are van Schooten, John de Witt, van Heuraet, Sluze, and Hudde. Van Schooten (died 1660), professor of mathematics at Leyden, brought out an edition of Descartes' geometry, together with the notes thereon by De Beaune. His chief work is his Exercitationes Mathematicæ, in which he applies the analytical geometry to the solution of many interesting and difficult problems. The noble-hearted Johann de Witt, grand-pensioner of Holland, celebrated as a statesman and for his tragical end, was an ardent geometrician. He conceived a new and ingenious way of generating conics, which is essentially the same as that by projective pencils of rays in modern synthetic geometry. He treated the subject not synthetically, but with aid of the Cartesian analysis. René François de Sluze (1622–1685) and Johann Hudde (1633–1704) made some improvements on Descartes' and Fermat's methods of drawing tangents, and on the theory of maxima and minima. With Hudde, we find the first use of three variables in analytical geometry. He is the author of an ingenious rule for finding equal roots. We illustrate it by the equation $\scriptstyle {x^{3}-x^{2}-8x+12=0}$. Taking an arithmetical progression 3, 2, 1, 0, of which the highest term is equal to the degree of the equation, and multiplying each term of the equation respectively by the corresponding term of the progression, we get $\scriptstyle {3x^{3}-2x^{2}-8x=0}$, or $\scriptstyle {3x^{2}-2x-8=0}$. This last equation is by one degree lower than the original one. Find the G.C.D. of the two equations. This is $\scriptstyle {x-2}$; hence 2 is one of the two equal roots. Had there been no common divisor, then the original equation would not have possessed equal roots. Hudde gave a demonstration for this rule.^[24]

Heinrich van Heuraet must be mentioned as one of the earliest geometers who occupied themselves with success in the rectification of curves. He observed in a general way that the two problems of quadrature and of rectification are really identical, and that the one can be reduced to the other. Thus he carried the rectification of the hyperbola back to the quadrature of the hyperbola. The semi-cubical parabola $\scriptstyle {y^{3}=ax^{2}}$ was the first curve that was ever rectified absolutely. This appears to have been accomplished independently by Van Heuraet in Holland and by William Neil (1637–1670) in England. According to Wallis the priority belongs to Neil. Soon after, the cycloid was rectified by Wren and Fermat.

The prince of philosophers in Holland, and one of the greatest scientists of the seventeenth century, was Christian Huygens (1629–1695), a native of the Hague. Eminent as a physicist and astronomer, as well as mathematician, he was a worthy predecessor of Sir Isaac Newton. He studied at Leyden under the younger Van Schooten. The perusal of some of his earliest theorems led Descartes to predict his future greatness. In 1651 Huygens wrote a treatise in which he pointed out the fallacies of Gregory St. Vincent (1584–1667) on the subject of quadratures. He himself gave a remarkably close and convenient approximation to the length of a circular arc. In 1660 and 1663 he went to Paris and to London. In 1666 he was appointed by Louis XIV. member of the French Academy of Sciences. He was induced to remain in Paris from that time until 1681, when he returned to his native city, partly for consideration of his health and partly on account of the revocation of the Edict of Nantes.

The majority of his profound discoveries were made with aid of the ancient geometry, though at times he used the geometry of Descartes or of Cavalieri and Fermat. Thus, like his illustrious friend, Sir Isaac Newton, he always showed partiality for the Greek geometry. Newton and Huygens were kindred minds, and had the greatest admiration for each other. Newton always speaks of him as the "Summus Hugenius."

To the two curves (cubical parabola and cycloid) previously rectified he added a third,—the cissoid. He solved the problem of the catenary, determined the surface of the parabolic and hyperbolic conoid, and discovered the properties of the logarithmic curve and the solids generated by it. Huygens' De horologio oscillatorio (Paris, 1673) is a work that ranks second only to the Principia of Newton and constitutes historically a necessary introduction to it.^[13] The book opens with a description of pendulum clocks, of which Huygens is the inventor. Then follows a treatment of accelerated motion of bodies falling free, or sliding on inclined planes, or on given curves,—culminating in the brilliant discovery that the cycloid is the tautochronous curve. To the theory of curves he added the important theory of "evolutes." After explaining that the tangent of the evolute is normal to the involute, he applied the theory to the cycloid, and showed by simple reasoning that the evolute of this curve is an equal cycloid. Then comes the complete general discussion of the centre of oscillation. This subject had been proposed for investigation by Mersenne and discussed by Descartes and Roberval. In Huygens' assumption that the common centre of gravity of a group of bodies, oscillating about a horizontal axis, rises to its original height, but no higher, is expressed for the first time one of the most beautiful principles of dynamics, afterwards called the principle of the conservation of vis viva.^[32] The thirteen theorems at the close of the work relate to the theory of centrifugal force in circular motion. This theory aided Newton in discovering the law of gravitation.

Huygens wrote the first formal treatise on probability. He proposed the wave-theory of light and with great skill applied geometry to its development. This theory was long neglected, but was revived and successfully worked out by Young and Fresnel a century later. Huygens and his brother improved the telescope by devising a better way of grinding and polishing lenses. With more efficient instruments he determined the nature of Saturn's appendage and solved other astronomical questions. Huygens' Opuscula posthuma appeared in 1703.

Passing now from Holland to England, we meet there one of the most original mathematicians of his day—John Wallis (1616–1703). He was educated for the Church at Cambridge and entered Holy Orders. But his genius was employed chiefly in the study of mathematics. In 1649 he was appointed Savilian professor of geometry at Oxford. He was one of the original members of the Royal Society, which was founded in 1663. Wallis thoroughly grasped the mathematical methods both of Cavalieri and Descartes. His Conic Sections is the earliest work in which these curves are no longer considered as sections of a cone, but as curves of the second degree, and are treated analytically by the Cartesian method of co-ordinates. In this work Wallis speaks of Descartes in the highest terms, but in his Algebra he, without good reason, accuses Descartes of plagiarising from Harriot. We have already mentioned elsewhere Wallis's solution of the prize questions on the cycloid, which were proposed by Pascal.

The Arithmetic of Infinites, published in 1655, is his greatest work. By the application of analysis to the Method of Indivisibles, he greatly increased the power of this instrument for effecting quadratures. He advanced beyond Kepler by making more extended use of the "law of continuity" and placing full reliance in it. By this law he was led to regard the denominators of fractions as powers with negative exponents. Thus, the descending geometrical progression $\scriptstyle {x^{3},~x^{2},~x^{1},~x^{0}}$, if continued, gives $\scriptstyle {x^{-1},~x^{-2},~x^{-3}}$, etc.; which is the same thing as $\scriptstyle {{\tfrac {1}{x}},~{\tfrac {1}{x^{2}}},~{\tfrac {1}{x^{3}}}}$. The exponents of this geometric series are in continued arithmetical progression, $\scriptstyle {3,~2,~1,~0,~-1,~-2,~-3}$. He also used fractional exponents, which, like the negative, had been invented long before, but had failed to be generally introduced. The symbol $\scriptstyle {\infty }$ for infinity is due to him.

Cavalieri and the French geometers had ascertained the formula for squaring the parabola of any degree, $\scriptstyle {y=x^{m}}$, m being a positive integer. By the summation of the powers of the terms of infinite arithmetical series, it was found that the curve $\scriptstyle {y=x^{m}}$ is to the area of the parallelogram having the same base and altitude as 1 is to $\scriptstyle {m+1}$. Aided by the law of continuity, Wallis arrived at the result that this formula holds true not only when m is positive and integral, but also when it is fractional or negative. Thus, in the parabola $\scriptstyle {y={\sqrt {px}}}$, $\scriptstyle {m={\tfrac {1}{2}}}$; hence the area of the parabolic segment is to that of the circumscribed rectangle as $\scriptstyle {1:1{\tfrac {1}{2}}}$, or as $\scriptstyle {2:3}$. Again, suppose that in $\scriptstyle {y=x^{m}}$, $\scriptstyle {m=-{\tfrac {1}{2}}}$; then the curve is a kind of hyperbola referred to its asymptotes, and the hyperbolic space between the curve and its asymptotes is to the corresponding parallelogram as $\scriptstyle {1:{\tfrac {1}{2}}}$. If $\scriptstyle {m=-1}$, as in the common equilateral hyperbola $\scriptstyle {y=x^{-1}}$ or $\scriptstyle {xy=1}$, then this ratio is $\scriptstyle {1:-1+1}$, or $\scriptstyle {1:0}$, showing that its asymptotic space is infinite. But in the case when m is greater than unity and negative, Wallis was unable to interpret correctly his results. For example, if $\scriptstyle {m=-3}$, then the ratio becomes $\scriptstyle {1:-2}$, or as unity to a negative number. What is the meaning of this? Wallis reasoned thus: If the denominator is only zero, then the area is already infinite; but if it is less than zero, then the area must be more than infinite. It was pointed out later by Varignon, that this space, supposed to exceed infinity, is really finite, but taken negatively; that is, measured in a contrary direction.^[31] The method of Wallis was easily extended to cases such as $\scriptstyle {y=ax^{\frac {m}{n}}+bx^{\frac {y}{z}}}$ by performing the quadrature for each term separately, and then adding the results.

The manner in which Wallis studied the quadrature of the circle and arrived at his expression for the value of $\scriptstyle \pi$ is extraordinary. He found that the areas comprised between the axes, the ordinate corresponding to x, and the curves represented by the equations $\scriptstyle {y=(1-x^{2})^{0},~y=(1-x^{2})^{1},y=(1-x^{2})^{2},~y=(1-x^{2})^{3},}$ etc., are expressed in functions of the circumscribed rectangles having x and y for their sides, by the quantities forming the series

${\begin{aligned}&\scriptstyle {x,}\\&\scriptstyle {x-{\frac {1}{8}}x^{3},}\\&\scriptstyle {x-{\frac {2}{8}}x^{3}+{\frac {1}{5}}x^{5},}\\&\scriptstyle {x-{\frac {3}{8}}x^{3}+{\frac {3}{5}}x^{5}-{\frac {1}{7}}}x^{7},~{\text{etc.}}\end{aligned}}$

When $\scriptstyle {x=1}$, these values become respectively $\scriptstyle {1,~{\frac {2}{3}},~{\frac {8}{15}},~{\frac {48}{105}},}$ etc. Now since the ordinate of the circle is $\scriptstyle {y=(1-x^{2})^{\frac {1}{2}},}$ the exponent of which is $\scriptstyle {\frac {1}{2}}$ or the mean value between 0 and 1, the question of this quadrature reduced itself to this: If 0, 1, 2, 3, etc., operated upon by a certain law, give $\scriptstyle {1,~{\frac {2}{3}},~{\frac {8}{15}},~{\frac {48}{105}},}$ what will $\scriptstyle {\frac {1}{2}}$ give, when operated upon by the same law? He attempted to solve this by interpolation, a method first brought into prominence by him, and arrived by a highly complicated and difficult analysis at the following very remarkable expression:

$\scriptstyle {{\frac {\pi }{2}}={\frac {2.3\cdot 4.4\cdot 6.6\cdot 8.8\cdots }{1\cdot 3.3\cdot 5.5\cdot 7.7\cdot 9\cdots }}}$

He did not succeed in making the interpolation itself, because he did not employ literal or general exponents, and could not conceive a series with more than one term and less than two, which it seemed to him the interpolated series must have. The consideration of this difficulty led Newton to the discovery of the Binomial Theorem. This is the best place to speak of that discovery. Newton virtually assumed that the same conditions which underlie the general expressions for the areas given above must also hold for the expression to be interpolated. In the first place, he observed that in each expression the first term is x, that x increases in odd powers, that the signs alternate + and —, and that the second terms $\scriptstyle {{\frac {0}{3}}x^{3},~{\frac {1}{3}}x^{3},~{\frac {2}{3}}X^{3},{\frac {3}{3}}X^{3},}$ are in arithmetical progression. Hence the first two terms of the interpolated series must be $\scriptstyle {x-{\frac {{\frac {1}{2}}x^{3}}{3}}}$. He next considered that the denominators 1, 3, 5, 7, etc., are in arithmetical progression, and that the coefficients in the numerators in each expression are the digits of some power of the number 11; namely, for the first expression, $\scriptstyle {11^{0}}$ or 1; for the second, $\scriptstyle {11^{1}}$ or 1, 1; for the third, $\scriptstyle {11^{2}}$ or 1, 2, 1; for the fourth, $\scriptstyle {11^{3}}$ or 1, 3, 3, 1; etc. He then discovered that, having given the second digit (call it m), the remaining digits can be found by continual multiplication of the terms of the series $\scriptstyle {{\frac {m-0}{1}}\cdot {\frac {m-1}{2}}\cdot {\frac {m-2}{3}}\cdot {\frac {m-3}{4}}\cdot }$etc. Thus, if $\scriptstyle {m=4}$, then $\scriptstyle {4\cdot {\frac {m-1}{2}}}$ gives 6; $\scriptstyle {6\cdot {\frac {m-2}{3}}}$ gives 4; $\scriptstyle {4\cdot {\frac {m-3}{4}}}$ gives 1. Applying this rule to the required series, since the second term is $\scriptstyle {{\frac {{\frac {1}{2}}x^{2}}{3}},}$ we have $\scriptstyle {m={\frac {1}{2}},}$ and then get for the succeeding coefficients in the numerators respectively $\scriptstyle {-{\frac {1}{8}}}$, $\scriptstyle {+{\frac {1}{16}}}$, $\scriptstyle {-{\frac {5}{128}}}$, etc.; hence the required area for the circular segment is $\scriptstyle {x-{\frac {{\frac {1}{2}}x^{3}}{3}}-{\frac {{\frac {1}{8}}x^{5}}{5}}-{\frac {{\frac {1}{16}}x^{7}}{7}}-}$etc. Thus he found the interpolated expression to be an infinite series, instead of one having more than one term and less than two, as Wallis believed it must be. This interpolation suggested to Newton a mode of expanding $\scriptstyle {(1-x^{2})^{\frac {1}{2}}}$, or, more generally, $\scriptstyle {(1-x^{2})^{m}}$, into a series. He observed that he had only to omit from the expression just found the denominators 1, 3, 5, 7, etc., and to lower each power of x by unity, and he had the desired expression. In a letter to Oldenburg (June 13, 1676), Newton states the theorem as follows: The extraction of roots is much shortened by the theorem

$\scriptstyle {(P+PQ)^{\frac {m}{n}}=P^{\frac {m}{n}}+{\frac {m}{n}}AQ+{\frac {m-n}{2n}}BQ+{\frac {m-2n}{3n}}CQ+}$ etc.,

where A means the first term, $\scriptstyle {P^{\frac {m}{n}}}$, B the second term, C the third term, etc. He verified it by actual multiplication, but gave no regular proof of it. He gave it for any exponent whatever, but made no distinction between the case when the exponent is positive and integral, and the others.

It should here be mentioned that very rude beginnings of the binomial theorem are found very early. The Hindoos and Arabs used the expansions of $\scriptstyle {(a+b)^{2}}$ and $\scriptstyle {a+b)^{3}}$ for extracting roots; Vieta knew the expansion of $\scriptstyle {(a+b)^{4}}$; but these were the results of simple multiplication without the discovery of any law. The binomial coefficients for positive whole exponents were known to some Arabic and European mathematicians. Pascal derived the coefficients from the method of what is called the "arithmetical triangle." Lucas de Burgo, Stifel, Stevinus, Briggs, and others, all possessed something from which one would think the binomial theorem could have been gotten with a little attention, "if we did not know that such simple relations were difficult to discover."

Though Wallis had obtained an entirely new expression for $\scriptstyle {\pi }$, he was not satisfied with it; for instead of a finite number of terms yielding an absolute value, it contained merely an infinite number, approaching nearer and nearer to that value. He therefore induced his friend, Lord Brouncker (1620?-1684), the first president of the Royal Society, to investigate this subject. Of course Lord Brouncker did not find what they were after, but he obtained the following beautiful equality:

$\pi =\scriptstyle {\frac {4}{1+{\frac {1}{2+{\frac {9}{2+{\frac {25}{2+{\frac {49}{2+{\text{etc.}}}}}}}}}}}}$

Continued fractions, both ascending and descending, appear to have been known already to the Greeks and Hindoos, though not in our present notation. Brouncker's expression gave birth to the theory of continued fractions.

Wallis' method of quadratures was diligently studied by his disciples. Lord Brouncker obtained the first infinite series for the area of an equilateral hyperbola between its asymptotes. Nicolaus Mercator of Holstein, who had settled in England, gave, in his Logarithmotechnia (London, 1668), a similar series. He started with the grand property of the equilateral hyperbola, discovered in 1647 by Gregory St, Vincent, which connected the hyperbolic space between the asymptotes with the natural logarithms and led to these logarithms being called hyperbolic. By it Mercator arrived at the logarithmic series, which Wallis had attempted but failed to obtain. He showed how the construction of logarithmic tables could be reduced to the quadrature of hyperbolic spaces. Following up some suggestions of Wallis, William Neil succeeded in rectifying the cubical parabola, and Wren in rectifying any cycloidal arc.

A prominent English mathematician and contemporary of Wallis was Isaac Barrow (1630–1677). He was professor of mathematics in London, and then in Cambridge, but in 1669 he resigned his chair to his illustrious pupil, Isaac Newton, and renounced the study of mathematics for that of divinity. As a mathematician, he is most celebrated for his method of tangents. He simplified the method of Fermat by introducing two infinitesimals instead of one, and approximated to the course of reasoning afterwards followed by Newton in his doctrine on Ultimate Ratios.

He considered the infinitesimal right triangle $\scriptstyle {ABB^{\prime }}$ having for its sides the difference between two successive ordinates, the distance between them, and the portion of the curve intercepted by them. This triangle is similar to $\scriptstyle {BPT}$, formed by the ordinate, the tangent, and the sub-tangent. Hence, if we know the ratio of $\scriptstyle {B^{\prime }A}$ to $\scriptstyle {BA}$, then we know the ratio of the ordinate and the sub-tangent, and the tangent can be constructed at once. For any curve, say $\scriptstyle {y^{2}=px}$, the ratio of $\scriptstyle {B^{\prime }A}$ to $\scriptstyle {BA}$ is determined from its equation as follows: If x receives an infinitesimal increment $\scriptstyle {PP^{\prime }=e}$, then y receives an increment $\scriptstyle {B^{\prime }A=a}$, and the equation for the ordinate $\scriptstyle {B^{\prime }P^{\prime }}$ becomes $\scriptstyle {y^{2}+2ay+a^{2}=px+pe}$. Since $\scriptstyle {y^{2}=px}$, we get $\scriptstyle {2ay+a^{2}=pe}$, neglecting higher powers of the infinitesimals, we have $\scriptstyle {2ay=pe}$, which gives

$\scriptstyle {a:e=p:2y=p:2{\sqrt {px}}}$.

But $\scriptstyle {a:e=}$the ordinate: the sub-tangent; hence

$\scriptstyle {p:2{\sqrt {px}}={\sqrt {px}}:{\text{sub-tangent}}}$,

giving 2x for the value of the sub-tangent. This method differs from that of the differential calculus only in notation.^[31]