Isaac Newton was undoubtedly a remarkable scientist. The direction of his work was set in his early twenties, when he developed his ideas about algebra and infinite series. These along with his fluxions, the approach to calculus which he alone began to develop single-handedly at that time, continued to be his main mathematical interests as long as he was involved in creative writing [4, p.63]. It has been said that the method of fluxions remains his best-known contribution to mathematics today [5, p.213]. However, not all men of science were equally impressed with this work. First, let us start with some biographical information on Newton which makes it obvious that he was a man of great accomplishments. Then fluxions will be discussed, along with some criticisms of the theory.

Isaac Newton was born into a yeoman's family on Christmas Day, 1642. His father died before his birth, and he was raised by his mother, Hannah Ayscough, in a house near Woolsthorpe, Lincolnshire, England, a rural area about one hundred miles northeast of London. When Isaac was three, Hannah remarried and the toddler was placed into the care of his maternal grandmother, with whom he remained for the next eight years.[2, p.395]

Newton was a frail and solitary child, with few friends. He invented his own diversions to pass the time, including the manufacture of kites, lanterns and windmills, making his genius apparent at an early age [2, p.395]. When her second husband died, his mother Hannah planned to live with her son once again and to make a farmer of him. Fortunately for the development of mathematics, Isaac's uncle, William Ayscough, a Cambridge graduate and the master of the nearby Grantham School, intervened and convinced Hannah that her son was much too intelligent for such a job and should be sent to a university.[1, p.91]

In 1661, Isaac Newton was admitted to Trinity College, part of the prestigious Cambridge University, as a sizar - a pupil who earned his keep by doing chores for the other more fortunate students. He was given a scholarship in 1664, no trivial matter since this meant he was eligible to become a fellow of the institution [5, p.582]. Newton received his BA in 1665.

The Great Plague (bubonic plague) spread throughout England from 1665-1667, especially in the major cities, and the universities were closed on this account, so Isaac had to remain at home during this time. While at home his attention was directed toward mathematics and physics after he read works of Euclid, Descartes, Kepler and other great mathematicians [3, p.397] He discovered the generalized form of the binomial theorem, the infinite expansion of (a + b)n, where n could be a fraction [5, p.72] Previously the theorem was only known to work in its finite form for n a positive integer. Newton invented an early form of what we call differential calculus, which he called "fluxions". A manuscript from May 1665 shows that he had developed the principles to be able to find the tangent and curvature at any point of "any" continuous curve.[1, p.187] He also performed an experiment in which he diffused white light into a spectrum of colors with a prism, and then recombined them with a second.[2, p.395] He also claimed, later, to have posited the inverse-square law of gravitational attraction.[2, p.395]

Isaac returned to Trinity in 1667 and upon completion of his masters degree the following year he was appointed a major fellow of the school, an esteemed designation [2, p.305]. He invented a reflecting telescope in 1668 which he constructed by himself. It should be noted that this year was also an important one in the history of calculus [1, p.107]: a Danish scientist and member of the Royal Society named Mercator published a work in which he stated that the series log(1+ x) = x - x2/2 + x3/3 - x4/4 + ..... could be used to calculate logarithms [5, p.355] When this was brought to Newton's attention, he discovered that the method was practically the same as his own [1, p.107]. Finally he wrote out his own much more general version, and allowed his mathematics teacher, Dr. Isaac Barrow, to send copies to some of the better known mathematicians of the day [2, p.396]. In October 1669, he was named Lucasian professor of mathematics, a very prestigious position, when Dr. Barrow stepped down in favor of his superior pupil. Then he began eighteen years of university lecturing.[3, p.398]

During the seventies, Newton continued to focus on optical research: as noted, he invented a reflecting telescope, and he published a new theory of light and colors based on his discovery of the chromatic composition of white light, which founded the science of spectroscopy.[2, p.396] However it also brought much criticism by scientists who attacked his theory of colors and certain of his deductions [3, p.398]. Robert Hooke accused Newton of stealing optical ideas from him [2, p.396]. The bitter arguments that ensued convinced Newton not to publish anything more for quite some time.[3, p.398]

All the while he had been earning a reputation as the leading scientific genius of Britain [1, p.113]. Newton was elected a Fellow of the Royal Society in 1672, the highest scientific honor in England.[2, p.396]

Edmund Halley, the famous astronomer, visited with Newton in 1684 and asked him to prove that the inverse-square law of gravitational attraction agreed with Kepler's three planetary laws [2, p.396]. Much to Halley's amazement, the story goes, Isaac Newton had already done so! [1, p.105] At this time Halley convinced Isaac to submit his astronomical and physical discoveries for publication.[2, p.109] The resulting book, Philosophiae Naturalis Principia Mathematica was presented to the Royal Society in 1686 and was printed the following year at the expense of Halley [2, p.109]. This was definitely Newton's most important work - it covered his dynamics, his law of universal gravitation, and the application of both to the solar system.

E.T. Bell wrote, "After the Principia, the rest is anticlimax." Newton was elected a Whig member of Parliament in 1689. During the fall of 1692, he fell ill and seemed on the verge of total mental collapse. It has been speculated that the cause was actually mercury poisoning, since Isaac exposed himself to many dangerous chemicals when he experimented [2, p.396]. He recovered in 1693. In that same year he became aware that calculus was becoming well known in Europe and was accredited to Leibniz.[1, p.113]

In 1696, Newton was appointed Warden of the Mint when an old friend and pupil, Charles Montague, the Chancellor of the Exchequer, was able to offer the post [5, p.358] and was aware of Newton's desire for a public position. Initially the Warden had been a figure of importance in the mint's affairs, but by Newton's time it had taken on a nominal aspect [5, p.358]. The task at hand was to reform the coinage, a job that he took very seriously. He was involved in the construction of new mints and had to fight against counterfeiters [5, p.358]. Newton became Master of the Mint upon the death of the previous one in 1699 and assumed greater responsibility thereupon [5, p.358].

He was elected president of the Royal Society in 1703, a position to which he was reelected continuously until his death.[1,p.115] Queen Anne knighted Isaac in 1705. [1,p.115]

During his final years, the scientist became occupied with many mystical and humanistic concerns as well [2, p.396] The corpuscular theory of light, the nutshell theory of matter and the method of fluxions are all presented in his second major work, Opticks.[2, p.396]

The priority dispute on the origins of calculus had festered since before the turn of the century.[4, p.79] Leibniz complained to the Royal Society in 1711 about accusations of his stealing ideas from Newton, so in 1712 the Society set up a committee to investigate which man deserved priority in the invention of calculus. [4, p.79] It is quite curious that Newton himself presided over deliberations of this committee. He even wrote part of the final report, the Commercium epistolicum, which, not surprisingly, favored Newton. Kindly enough, he noted that Leibniz could not be witness in his own cause [4, p.79].

Isaac continued to study alchemy, an early form of chemistry associated with magic and the search for an elixir of perpetual youth, and theology, both of which had been other areas of interest for him since the 1670's, until the end of his life [5, p.213]. He died in 1727, at the age of 84, and was laid to rest in Westminster Abbey, a place that was previously reserved for deceased royalty.[2, p.397]

Now that we are acquainted with this complex man, let us take a look at some of his work. Newton claimed that his method of fluxions was conceived in 1665. Its fullest treatment is in De methodis fluxionum et serierum infinitorum, dated 1671.[5, p.213], which was actually a letter that was never meant to be published. In this letter Newton treated important problems such as finding the fluxional derivative of an equation - that is, the term-by-term rates of change of the elements of the equation.[4, p.73] He also attacked the inverse problem of determining the tangents to curves defined in various coordinate or reference systems, as well as finding the curvature, area and arc-length.[4, p.73] It is possible that Newton was inspired to work on "the calculus" by his mathematics professor at Trinity.[1, p.96] Dr. Isaac Barrow was a brilliant theologian and mathematician. He lectured on methods for finding areas and drawing tangent lines to curves - the main problems of integral and differential calculus. As original as he was in these fields, Barrow gladly recognized that his student, Newton, would be much greater than he would.

The Principia introduces differentials in Book II, Section II,
Lemma II by the name "moments," which were produced by variable
quantities called "genita". They were explained in a section dealing
with the motion of bodies that move against a resistance. Says Lemma
II: "The moment of any genitum is equal to the moments of each of the
generating sides multiplied by the indices of the powers of those
sides, and by their coefficients continually." A copy of this lemma
follows for the curious mind. *(omitted from HTML version, may be
added later)*

D. J. Struik argued that it isn't easy to understand the meaning attached to the "moments", whether they are just "nascent principles" or equivalent to "finite quantities proportional to velocities." He pointed out that a genitum is an expression of one term which is dependent upon one variable. It may be helpful to note that the Latin original is "quantitas genita", or "generated quantity." There are also differences between the two editions of the Principia. The first edition claimed "Moments, as soon as they are of finite magnitude, cease to be moments. To be given finite bounds is in some measure contradictory to their continuous increase or decrease." [7, p.300] On the other hand, the second version read "Finite particles are not moments, but the very quantities generated by moments." [7, p.300] Apparently Newton himself struggled with the meaning of it all, just as Leibniz struggled with his ideas of differences and "differentials". [7, p.300]

Reading and comprehending what Newton wrote in the Principia may be a challenge. However, a reasonable explanation of Newton's fluxions is found in The Newton Handbook, by Derek Gjertsen. It is worthwhile to examine this to get a good idea of what Newton was thinking about. The starting point was that geometrical magnitudes arise from continuous motion.[5, p.213] In other words, a line or curve is generated by the continuous motion of a point, a surface by the motion of a line and a solid by the motion of a surface.[5, p.213] Newton called the line a "fluent" because he thought of it as a flowing quantity .[5, p.213] The fluxion, or rate at which it flowed, was the point's velocity .[5, p.213] Dotted letters were used to represent the fluxions. These fluxions could, in turn, be taken as fluents and this led to higher order fluxions, which were represented by two dots. There is an obvious relation: the velocity of a point determines the nature of the curve, and a curve of a given nature can only be generated by a point with a certain velocity.[5, p.214] Two problems arose: given a relationship between fluents, how one can determine the corresponding relationship between fluxions, and the inverse, how one can determine fluents on the basis of fluxions.[5, p.214] These processes are known today as differentiation and integration.

Newton defined the "moment" to be an infinitesimally small amount by which fluents grew in infinitesimally small periods of time.[5, p.214] To make sense of this it is helpful to use Leibniz notation which is now rather common to any mathematics student. He used the notation o to represent that small change in time, or what is often seen today as dt. The moment of the fluents x and y would be xo and yo respectively. Today these are denoted as dx and dy. Note, then, that x and y must be dx/dt and dy/dt, respectively. In a minute time interval, then, the quantities x and y will become x + xo and y + yo. Gjertsen also interpreted a problem from Newton's De methodis , which follows. The similarity between this method and that of Fermat was noted by Dr. Gregory Cherlin (personal communication). One can read this in Eves beginning on page 291.

Consider two curves related by the equation: x3 + ax2 + axy - y3 = 0 Newton then substituted x + xo and y + yo for x and y, respectively, to determine how the fluxions which generated the two curves were related: (x + xo)3 - a(x + xo)2 + a(x + xo)(y + yo) - (y + yo)3 = 0 and this becomes: (x3 + 3xox2 + 3x2o2x + x3o3) - (ax2 + 2axox + ax2o2) + (axy + axoy + ayox + axyo2) - (y3 + 3yoy2 + 3y2o2y +y3o3) = 0 Next, Newton eliminated the initial equation: (*) (3xox2 + 3x2o2x + x3o3) - (2axox + ax2o2) + (axoy + ayox + axyo2) - (3yoy2 + 3y2o2y +y3o3) = 0 and divided all of the remaining terms by o: (3xx2 + 3x2ox + x3o2) - (2axx + ax2o) + (axy + ayx + axyo) - (3yy2 + 3y2oy +y3o2) = 0 Then, and only then, he argued that since o was "infinitesimally small", any term multiplied by it would be "nothing compared to the rest" and could be safely eliminated. There remained: 3xx2 - 2axx + axy + ayx - 3yy2 = 0 And then the relation between the fluxions is: y = 3x2 - 2ax + ay = dy x 3y2 - ax dx

That Newton conceived of this on his own may be appreciated by any student of math. However, there was a point of contention which brought much criticism by other men of science, and this was the way in which Newton divided by a small nonzero finite quantity o, and then, a few lines later, set it equal to zero.[4, p.210] Today a limit is applied, and although we think of this as an extremely subtle concept, Newton felt that it could be grasped intuitively - as an extra step which could simply be omitted.[1, p.98] This division which he considered acceptable was seen as "illegal" by others: how could a quantity be there one minute and not the next? Was this, in effect, division by zero? Any mathematician knows that is not allowed!

One of Newton's greatest critics was George Berkeley (1685-1753), the dean of the Church of England in Ireland.[7, p.333] He was a vehement and highly competent critic of many aspect's of Newtonian science and sought to show that Newton's universe was constructed on quicksand. [4, p.210] Berkeley's main attack can be found in a publication entitled The Analyst. He hit the method of fluxions in its weak spot - the infinitesimals, which he called "the ghosts of departed quantities."[7, p.333] He said "He who is willing to accept the mysteries of the calculus, which nevertheless lead to true results, need not hesitate to accept the mysteries of religion" [7, p.333 as quoted in The Analyst].

Berkeley clearly did not like the method of fluxions from the beginning - he did not know what to make of the "moments." He also stated, in reference to the higher order fluxions, "the velocities of the velocities, the second, third, fourth, and fifth velocities, and so on, exceed, if I mistake not, all human understanding."[7, p.333 as quoted in the Analyst] Of course one must note his main point when considering the infinitesimally small amounts, "Now to conceive a quantity infinitely small, that is, infinitely less than any sensible or imaginable quantity, or than any the least finite magnitude is, I confess, above my capacity. But to conceive a part of such infinitely small quantity that shall be still infinitely less than it, and consequently though multiplied infinitely shall never equal the minutest finite quantity is, I suspect, an infinite difficulty to any man whatsoever; and will be allowed such by those who candidly say what they think; provided that they really think and reflect, and do not take things upon trust " [7, p.333 as quoted in The Analyst].Berkeley argues that the infinitesimal cannot be gotten rid of by legitimate reasoning: if one takes, for instance, a very small quantity ab (Newton's o), then one can't proceed from (*) until it is gotten rid of. [7, p.333] In other words, the o's cannot be simply eliminated from Newton's equation (*). Berkeley stated that the quantity cannot be both zero and nonzero, that this was largely incompatible and furthermore an inconsistent way of arguing.[4, p.210] "Men would hardly admit such a reasoning as this in other branches of knowledge which in mathematics is accepted for demonstration."[4, p.210 as quoted in the Analyst]

It appears that Newton may have completely skipped an important step in the groundwork he laid for differential and integral calculus. Perhaps if he had lived longer the necessity of defining the very subtle limit would have occurred to him - noone can say. However his work was unprecedented and clearly one cannot deny his genius.

Bibliography

1. Bell, E.T. **Men of Mathematics**. New York: Simon &
Schuster, 1965.

2. Calinger, Ronald. **Classics of Mathematics**. Oak Park,
Ill.: Moore Publishing Co., 1982.

3. Eves, Howard. **An Introduction to
the History of Mathematics**. Philadelphia:
Saunders College Publishing, 1990.

5. Gjertsen, Derek. **The Newton Handbook**.
New York: Routledge & Kegan Paul, 1986.

6. Newton, Isaac. **The Principia.**
New York: Prometheus Books, 1995.

7. Struik, D. J. ** A Source Book in
Mathematics 1200-1800**. Princeton: Princeton
University Press, 1986.