**TO CAST A PEBBLE**

**Calculus – Class 12 – Main Lesson**

So, we now embark on the final mathematics main lesson for five years of high school, and the last of 36 through 12 years of schooling! The word calculus means “pebble”, as in the small stones used as counters by Roman ‘calculators’. This resonates right back to that very first maths main lesson in Class 1, Counting. Here imaginative teachers do indeed introduce the number mysteries to their 7-year-olds with pebbles and what-not as counters. In a strictly literal sense these eager children are beginning where they will end, with a kind of Calculus.

As in so many things in Steiner Education, mathematics can be seen as another aspect of the Cosmic Serpent joined in a circle with its own tail. The pebbles were igneous in Class 1, with 18-year-olds they exist, in their purest sense, in the realm of *Imaginative Cognition*, which, as Rudolf Steiner assures us, is a prelude to true spiritual *Inspiration* – the enlightenment of the 12-petal heart chakra.

Every wholesome heart is already inspired – rejoices even – when it beholds, for example, the soaring of a swallow. Our feelings are liberated, but not necessarily our thinking. For this to occur, we would have to understand the numerical dynamics of the bird’s flight – or a planet’s orbit – or an athlete’s long jump. This has only been possible since those two 17^{th} and 18^{th} Century mathematical geniuses, Isaac Newton and Gottfried Leibnitz, independently revealed the fundamental laws of *movement and change* – the mysteries of Calculus. Both the epiphany of Calculus, and the vision of the free-flying swallow, are intensely enlightening – ‘inspiring’ – experiences, the first cognitive, the second emotive. On this theme we visit the time that gave birth to this remarkable numerical breakthrough, the aptly named *Age of Enlightenment*.

The cold definition of Calculus is “the branch of mathematics dealing with the problems of defining precisely and calculating the slope of a curved line, and the area inside a region bounded by a curved line.” One begins with defining these qualities for figures involving straight lines, then approximating these quantities for figures involving straight lines, then approximating these quantities for the curved figures, then using the basic concept to render these approximations into exact quantities.

The use of ‘limits’ is central to Calculus, and is the golden heart of the subject, as well as that of analysis in general. A limit is a number concept based on the idea of closeness, and is employed to assign values to certain functions at points where no values are defined, in such a way as to be consistent with nearby values.

Limits are the method by which the *derivative*, or rate of change, of a function is calculated. They are used throughout analysis as a way of transforming approximations into exact quantities; such as when the area inside a curved region is defined to be the limit of approximations by rectangles. Finding derivatives (slopes) is the task of *differential calculus*, while finding areas, or integrals, is that of *integral calculus*. The theorem of calculus relates these two concepts by stating that finding the area enclosed by the curve of a given function is equivalent to finding a function having the given function as its derivative.

Calculus was actually anticipated by the ever-inventive, Rational Soul ancient Greeks. They used a limiting process to calculate the area of circles, cones, and spheres. However it was initiated in its present form in the 17^{th} Century by the aforementioned Newton and Leibnitz.

Partial differentiation of functions with more than one variable, as well as the theory of differential equations, emerged in the 18^{th} Century. However the essential foundations of calculus were not properly established in the mathematical fraternity till the 19^{th}. This included the fundamental theorem of calculus recognizing differentiation and integration as being inverse operations.

This is commonly seen as the area under a curve being able to be calculated by finding the *antiderivative* of the function representing the curve. This has application to any situation in which a summing process is used to approximate a quantity, such as problems involving work, volume, center of gravity and arc length.

The *calculus of variations* is the study of definite integrals that must be maximized or minimized. One of its most famous problems was proposed by Johan Bernoulli in 1696. This ‘brachistochrone problem’ required finding the equation of the path along which the fall of a particle from one point to another requires the least time. The solution involves minimizing the integral expressing this path.

The derivative in mathematics is the rate of change, or instantaneous velocity, of a function with respect to a variable. Geometrically the derivative of a function can be interpreted as the slope of the graph of the function, or more precisely, as the slope of the tangent line at a point.

Its calculation actually derives from the slope formula of a straight line, except that a limiting process must be used for curves. This ratio will depend upon where the points are chosen, reflecting the idea of the curve having a different slope at different points.

To find the slope at a desired point, the choice of the second point needed to calculate the ratio represents a difficulty, because, in general, if they are far apart, the ratio will represent more of an average slope along the portion of the curve cut off, rather than the slope at either points.

*Differentiation* – calculating the derivative – seldom involves the necessity of using the basic definition, but instead can be accomplished through a knowledge of the derivative of three of four basic functions and the use of four basic rules for handling functions made up of simpler functions. Derivatives are involved in all phases of calculus and differential equations; they find many applications in problems involving velocity, maxima, curve analysis and approximations.

The differentiation process is finding the *derivative*, or rate of change, of a function. In contrast to the abstract nature of the theory behind it, the practical technique of differentiation can be carried out by purely algebraic manipulations, using the above three basic derivatives and four rules of operation, and a knowledge of how to manipulate functions.

It was late in 1678 = over a century before Captain Cook ‘discovered’ Australia! – that Leibnitz laid the foundations of both integral and differential calculus. With this discovery, probably equal to Cook’s in global significance, he ceased to consider time and space as substances – this was yet another step in this Mind Discoverer’s philosophy of *Monadology*, one based on the core concept of a simple, indestructible nonspatial element regarded as the unit of which reality exists. Rudolf Steiner referred to his as *Monadism*, and related it, of his 12 Philosophical Viewpoints, to the sign of Sagittarius (see Philosophy main lesson in my book *Tree of Time* for fuller description).

Leibnitz developed the notion that the concepts of extension and motion contained an element of the imaginary, so that the basic laws of motion could not be discovered merely from a study of their nature. Nevertheless, he continued to hold that extension and motion could provide a means for explaining and predicting the course of phenomena. Thus, contrary to Descartes, he held that it would not be contradictory to posit that this world is a well-related dream – definitely *not* Captain Cook!

If visible movement depends on the imaginary element found in the concept of extension, it can no longer be defined by simple local movement – it must be the result of a force. In contending the Cartesian formulation of the laws of motion – mechanics – Leibnitz became the founder of a new formulation – dynamics. This substituted kinetic energy for the conservation of movement. Beginning with the principle that light follows the path of least resistance, this Cook-like navigator of the Sea of Mathematika believed he could demonstrate the ordering of nature toward a final goal or cause.

Descartes? In 1650 this mind mariner published probably his greatest discovery, Analytical Geometry, in which a point can be represented by a pair of numbers giving its distances from tow lines, the axes. The resulting numbers are the ‘coordinates’ of the point.

Neither coordinates nor the graphical representations of functions was new at the time of Descartes, but he, with fellow countryman Pierre de Fermat, correlated these in the basic principle that an equation in tow unknown quantities represents a curve, and vice versa. For example; a general pint has coordinates xy. The equation y = x describes all points whose x coordinate equals their y coordinate. These lie on a straight line bisecting the angle between the axes.

Except for a few cases, like Projective Geometry, Cartesian Geometry eclipsed Pure Geometry for almost two centuries. One of the results being a huge increase in the variety of curves studied. So few curves had previously been known that no need had been felt for a general definition of tangent line; thought of as a line that touches the curve at one point only. It was Fermat who introduced the modern idea of the tangent to a curve at any given point P. Thus he was the true discover of that mist-shrouded island, *Differential Calculus*; one in which the rate of change of a function is found.

English mathematical adventurers, Pascal and Wallis, studied the *Cycloid*, the curve traced out by a fixed point on a circle as the circle rolls along a line. Because of its beauty, it is known as the “Helen of Geometry’. It would seem only mathematicians can be aroused to infinite orgasmic heights over such numerical sublimity!

And while on infinity, *Infinitesimal Calculus *forms an important part of a student’s sea chest of navigational instruments. This particular calculus requires a completely different mode of thinking form that of any other calculus without infinity. One has to actually sail to the veritable edge of the world when approaching infinity.

And it can ever only be an approach, never a destination. But this does not mean that the results of infinitesimal calculus are vague or approximate. This principle may be illustrated in the following example, in which a series of fractions, diminishing by half with each step, is added together:

When the series is continued, the sum in the bottom line, which is the addition of all the fractions up to that point in the top line, gets nearer and nearer to 1. It can be shown that, however far the sum goes, it will always be smaller than 1, and will reach this value only after an infinite number of steps. Of course this can never be achieved, but the conclusion can be reached. Thinking in such a problem sets its jib to sail swiftly right over all the definite steps, to perceive the answer against the background of infinity. It does not become lost in a numerical fog, but sails beyond the step-by-step into a high realm. Hence this branch of mathematics is rightly called *Higher Calculus*.

Actually it was Isaac Newton who discovered the Infinite Series; sums of an infinite number of terms of some sequence; one of is examples is: 1 + ½ = ½ squared = ½ to the power of 3, and so on and on. The value of a series is obtained by the limit of partial sums. The first term here has a value of 1, the sum of the first two is 1 ½, of the first three, 1 and 3 quarters. As the number of terms increases, the partial sum approaches a limit of 2, this being the value of the series. Such a series is *convergent*, as opposed to a *divergent* series which does not have a limit.

Area measurements have been found by summations by number masters from Archimedes to Wallis, and differentiations had already been carried out by Fermat; but it was Newton and Leibnitz who discovered the fundamental principle of the calculus that integrations can be performed far more easily by inverting the process of differentiation. Yet neither was able to establish the calculus on a sound logical basis. Newton at first adopted a band-aid explanation in terms of infinitely small ‘moments’.

Later he based his theory on rates of change; this hinged on two concepts: *fluents*, changing quantities, and *fluxions*, rates of change of magnitudes. He first published his discoveries in 1687 in that greatest scientific treatise ever, the *Philosophiae Naturalis Principia Mathematica*. This was written in Latin, the synthetic language of the ancients, though the discoveries in the book were made through the new analytic devices that the author helped to develop.

Newton’s aim was to understand nature, whereas the motive of Leibnitz was to find a general pathway to knowledge. The latter’s method of differentials, the formal algorithmic nature of which was in accord with the aims of its inventor, became the wellspring of mathematical development during the whole of the 18^{th} and 19^{th} Centuries. Isaac Newton was a giant in the English-speaking world, but Leibnitz was more highly thought of on the Continent; especially by the number-talented Bernoulli family, who were fired with enthusiasm for the enigmatic differential calculus (not the kind of thing that would fire *my* family!).

The most influential pupil of Johann Bernoulli was Leonhard Euler. Though the 18^{th} Century saw few spectacular discoveries, this prosaic period generated more new mathematics than any other; most of it orbiting around Euler’s sun. His “Introduction to Infinitesimal Analysis” (a riveting read!) of 1748 is considered the foremost maths textbook of modern times.

Euler’s treatises on the differential and integral calculus remain the source from which generations of number-nauts since have been inspired. He was one of the founders of two important branches of mathematics; the calculus of variations, and differential geometry. The former is an extension of calculus applied to cases in which a function depends on another function or a curve.

The Bernoulli family? Eight members of the Bernoullis in three generations during the 17^{th} and 18^{th}Centuries were consummate mathematicians. After forced re-locations from their Antwerp home due to Catholic persecution by the Huguenots, they finally settled in Basle. This is the city which services Dornach; spiritual home of the movement given birth to by another talented mathematician, Rudolf Steiner! It must be something in the – Inspiration enhancing – mountain air!

The most celebrated Bernoullis was brothers Jakob I and Johann I. They championed the cause of Leibnitz’s version of calculus by identifying the fundamental problems to which it was best adapted, like particle and fluid dynamics, optics and probability.

Jakob I was the first adventurer on the Mathematics Main to use the term integral in analyzing a curve of descent. His study of the catenary curve, as seen in a hanging chain, was soon applied to the building of suspension bridge. Johann I used calculus for the determination of lengths and areas of curves, such as the isochrone, along which a body will fall at constant speed, and the tautochrone, which is used in clock construction. He also made input to the theory of differential equations and the mathematics of ship sails; as well as developing a rule for solving problems involving limits that would apparently be expressed by the ratio 0:0; now unfairly called L’Hospital’s Rule on Indeterminate Forms.

Dinner-table talk at the Bernoullis must have been a blast; the story goes that wild arguments would erupt over the sauerkraut on subjects like finding the equation for the path followed by a particle from one point to another in the shortest possible time if the particle is acted upon by gravity alone. Jakob I issued a challenge to anyone solving this vexing problem. It was brother Johann I who came up with the answer, the above-mentioned cycloid, the path of a point on a moving wheel.

At the same time he described the relation this curve bears to the path described by a ray of light passing through strata of variable density. “More strudel anyone?”! Alas the dispute raged on, resulting in a new discipline, the *Calculus of Variations*. Who said family conflict has no positive outcomes?! The bellicose Bernoullis excelled themselves when, in 1735, Johann I’s son Daniel shared with his father a coveted Paris Academy of Sciences prize for their work on planetary orbits. But Bernoulli Senior threw his son out of the house, feeling it should be his alone! Most dads would throw their sons out for *not* winning the prize! So it was with a whole Medusa’s raft of Bernoullis, maths masters all.

Steiner has even used his regional predecessors as examples of inherited physiognomic features aiding the number talents. My assessment from studying their portraits, is a rise in the astral (3^{rd}) region of the eyebrow; maths being the astral (number ether) subject par excellence.

Another mathematical genius of the French Enlightenment was Jean Le Ron D’Alembert. His Christian name derives from the church of Saint -Jean-le-Rond, upon whose steps he was dumped as an illegitimate child of a Parisian ‘hostess’. Although trained as a doctor, he moved to pure mathematics: “the only occupation which really interested me” as he claimed. Astonishingly he was almost entirely self-taught!

Among many lofty achievements, in 1743 at the tender age of 26, he published a fundamental treatise on dynamics, which included the famous and eponymous *d’Alembert’s Principle*. This states that Newton’s 3^{rd}law of motion (for every action there is an equal and opposite reaction) is true for bodies that are free to move as well as for bodies rigidly fixed.

In 1744 he applied his principle to the theory of equilibrium and motion of fluids. This discovery was followed by the development of partial differential equations. In 1747 he elevated his research to an ever-higher level of numerical sophistication when he applied his new calculus to the problem of vibrating chords; then followed up my furnishing a method of applying his principle to the motion of any body of a given shape – then capped it all with an explanation of the precession of the equinoxes, a subject close to Rudolf Steiner’s heart. Here, since 1413, to be true to the heavens, we all have to put our star sign back one sign.

In the same vein, Alembert explained the phenomena of the nutation of the earth’s axis. The art of mathematics is greatly indebted to him, not the least for his research on integral calculus. Here he devised relationships of variables by means of rates of change of their numerical value.

Calculus even supports the arcane area of economics. Many money problems take the form of maximizing some variable (such as profits) subject to ta constraint (such as the production function), for which calculus supplies the simplest technique. Traditionally it was applied to problems in comparative statics. These include ‘endogenous variables or parameters. The object is to discover the effects of changes in one or more of the parameters upo9n the equilibrium situation. The latter is a situation in which all of the endogenous variables are simultaneously in a state of rest. If the value of some of the parameters is changed, the result is a new equilibrium state.

Just as differential calculus is the mathematics of comparative statics, *Difference* and *Differential* equations are the ideal tools for handling dynamic problems. Difference equations deal with time as a discrete variable – changing only from period to period – whereas differential equations treat time as a continuous variable. We may ask: If the system is pushed out of equilibrium, perhaps due to one of the parameters of the model changing, will economic forces drive it towards a new equilibrium position, or away from one?

An ancient Greek forerunner of integral calculus was the work of Eudoxus. His definition of equal ratios is the principle source of the modern view of irrational numbers – not bad for a habitué of *Rational Soul*Greece! Especially in the area of finding the volume of solids, Eudoxus dealt, by means of rational approximations, with measurements that involved irrational numbers on straight lines.

More difficult was the calculation of areas and volumes bounded by curves; his method was ‘exhaustion;’ no, not burning the number candle at both ends, rather a modern term to describe including not only the irrational, but also the concept of the infinitesimally small quantity. He showed how to subdivide continuously a known magnitude until it closely approached that of an unknown, such as the properties of a curve.

Eudoxus used this method to prove that the volumes of pyramids and cones are one third the volume of prisms and cylinders respectively, with the same bases and heights. He also demonstrated that the areas of circles are proportional to the squares of their diameters. His technique was akin to (but much more rigorous) inscribing polygons with increasing numbers of sides within a circle in order to find its area. To cap this Eudoxus was the first to calculate the length of the solar year.

Soon after moving to Russia after an invitation from Catherine the Great (following a rift with his patron, Frederick the Greater!), Euler sadly went blind. Not so sadly perhaps, all the better to behold those night-borne number ether vision! It must have been so, as his productivity waxed undiminished. As such, he helped establish an advanced mathematical culture in Russia.

Euler was obsessed with the problem of developing a more perfect theory of lunar motion, which was particularly troublesome as it involved the “the three-body problem”, the interactions of sun, moon and earth. This is still unsolved. My guess is that if they expanded to a “nine body problem”, which includes the other (astrological) planets, they may have greater success. Steiner taught of the powerful psychic and physical influences on all solar system bodies by all others.

Euler’s partial solution to the lunar problem, published in 1753, assisted the British Admiralty in calculating lunar tables. This was of great importance in determining longitude at sea. One of the feats of his blind years was to perform all the elaborate calculations in his head for his second theory of lunar motion (1772). He ‘saw’ the problem much like the deaf Beethoven ‘heard’ the music; their respective sense impairments enhancing access to the same metaphysical realm, the Astral, Number of Tone Ether – after all, maths and music *are* natural brothers.

A salutary exercise for students is to attempt their own calculations or number problem-solving with their eyes closed. This can be done with increasing degrees of difficulty. Both teacher and taught might be pleasantly surprised by the results!

Although Euler died in St. Petersburg in 1783; he is recognized as one of the greatest number masters of the 18^{th} Century. Could this pioneer of Imaginative Cognition be a harbinger of its future-age complement, Spirit Self – said by Dr. Steiner to incubate in Russia (circa 3573)?

England’s brightest pre-Newton maths luminary was John Wallis, who also contributed greatly to the origins of calculus. He shot to prominence as a mathematician when he deciphered a number of cryptic messages from Royalist partisans that had dropped off the back of a wagon into the hands of the Parliamentarians. Wallis maintained an abiding interest in the age-old problem of the quadrature of a circle, i.e. finding a square that has an area equal to that of a given circle. He extended Cavalieri’s law of quadrature by devising a way to include negative and fractional exponents. He also assigned numerical values to spatial indivisibles. By means of a complex logical sequence, he established the following relationship:

Much of Newton’s work on calculus, he conceded, arose from a study of Wallis’s *Arithmetica Infinitorum*(Thomas Hobbs described it as “a scab of symbols” – vilification, as usual, is the natural resort of the ignorant!). Wallis even introduced the symbol for infinity . This is an esoteric cryptograph of the astral body, and its particular expression (of the 4 operations), multiplication. A parabolic lemniscate emerges when a series of multiplication algorithms are performed (see my book *What is X? Why is Y?*). The parabola is one of the classic four sections of a cone, thought so important in Steiner maths. Again it was Wallis who first described the four cross sections of cutting a cone with a plane as properties of algebraic coordinates. He also astonished his peers by asserting that the gravity of the earth is localized at its center.

In Wallis’s case, the infinity symbol found use in a less spiritual area, in treating a series of squares of indivisibles. He was also the first to demonstrate the utility of the exponent, such as 10 squared, particularly by his negative and fractional exponents. By applying algebraic techniques rather than those of traditional geometry, Wallis contributed to solving problems involving infinitesimals; thereby mathematics, eventually through the differential and integral calculus, became the most powerful tool of research in astronomy and theoretical physics. Without brilliant number navigators like John Wallis, we may never have reaped the bumper harvest of things like the electric light – or the atomic bomb! This following are examples taken from a student’s Calculus main lesson book:

Integral calculus rinds the areas and volumes under curves.

The ‘primitive function’ is the reverse of the differential process.

Differential calculus expressing a ‘rate of change’ of a function. An equation for distance (displacement) is known, and an equation for velocity can be found by differentiating to find the first derivative. If the procedure is repeated, we find the acceleration equation, or ‘second derivative’.

If there is a certain force at distance D, then applying an inverse square relationship, if the distance is doubled the force is quartered – treble the distance gives 1/9^{th} of the force, etc.

*Right Image:* Uses: Finding rates of change. Maximum and minimum values of functions.

*Left Image:* Uses: Finding areas and volumes. Collecting and summing data.

An example of *Differentiation* on previous page; given a function which relates distance to time, we can find the speed at any point by differentiating the function in order to find the gradient function A vehicle travels so that its distance from the start (S) at a time (t) in seconds is given by the equation: S = 1/2t squared.

Given a function which relates speed to time, for example, we can find distance traveled in a given time by integrating and working out the distance by calculating the area under the curve. Calculus can help find the speed of the vehicle at various intervals. But first we find the slopes geometrically.

Like other areas of maths, calculus has a special language. Unfortunately the language of calculus has several forms; such as that developed by Newton, which uses a simple dot to indicate a derivative, and the modern functional notation of f. Then there are the following:

The Battling Bernoulli Boys (Jakob I left, Johann I right);

astral eyebrows raised in skepticism about being included in a maths book due to their looks!

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