Euclidean geometry begins with the formalisation of surveyors' work, drawing straight lines and describing circles. Geometry shifts from the land to the table where the mathematician traces his diagrams, and from there to paper and books. Even the operating tools are different. What could be obtained on a large scale by pulling ropes, can now be drawn on a small scale with a ruler and compass, translated into mental objects and regulated by a complex system of definitions and axioms. The resulting reduction of scale takes the surveyor away from the land, and allows him to dominate, in all their complexity, many problems that the immensity of the horizon had excluded from his sight. Evidence for the use of geometry along the banks of the Nile dates to an extract from Herodotus, reported at the beginning, as well as a consistent tradition seeing the origins of Greek geometry in Egypt. To what extent this tradition is legendary is unclear, but it is reasonable to to expect that evidence of the activity of the Egyptian land surveyors to be found in Greek mathematics and in particular in Euclid's Elements.

Faint evidence, since the Elements is a late work, where previous elaborations - now lost - converge, and one that is organised according to the axiomatic - deductive process, typical of Greek thinking and of all western mathematics. Evidence of early Egyptian geometry, then, won't be found in the general body of the work, nor in the demonstration of theorems. It is more likely that they appear in principles, definitions and postulates which are notable for having the function of translating natural phenomena into symbols and geometric shapes. In order that the abstract geometry of classical Greece describe the most profound properties of the real world, illuminate the most obscure relationships among the objects of the external world, what needs to be done is the following: First of all, it is necessary to precisely set the definitions of the objects taken in consideration to be totally sure of what is included or excluded in the dominion of geometry. Secondly, one must identify, through postulates, a system of primary properties and possible operations, starting from which the surveyors, aided only by logic, could extract the network of consistencies and interrelations that lie - often inaccessible to material investigation - in abstract concepts of geometry and, consequently, in ordinary objects.

If the definitions and postulates translate material objects from nature and the empiric procedures of the practice into the abstract figures and operations of geometry, it is in them that the evidence of a lost tradition can and must be found. They are conceptual evidences, since the main concepts echo the procedures and operations that were not formalised. These are linguistic clues, because the choice of terms is influenced by the operations on the defined objects.

Seen in this light, Euclid's Elements show a partial, but surprisingly clear correspondence, with the operations of the 'arpedonapti'.

We have already discussed the right-angle. Early than that is the definition of the straight line -an always finite line, a segment- that recalls the operation of "pulling" a rope between two "points" ( literally "marks") that define its "limits". Its straightness is not dependent on it being the shortest distance between two points, but once again it brings us back to the uniformity of the tension, according to which "it lies uniformly in relation to its marks", a property that becomes even more evocative if read together with the definition preceding it: " the marks are the limits of the line".

Of the postulates, the first three reproduce exactly the land surveyors' operations:

to draw a line between two points:

to prolong a given line:

To describe a circumference:

While the other two testify to, so to speak, the impossibility of demonstration. The fourth of them states that

and the last is the famous "postulate of parallel lines":

The difference between these two postulates and the first three is evident. The first one, put into geometric terms, describes usual practical operations. What they ask is nothing but the translation into abstract form of the concrete process of land surveyors. The last two, on the contrary, have the nature of theorems. They do not express that which the surveyors can do, but the properties of the mathematic objects already introduced - essential, assumed properties in the demonstration of the theorems to follow.

In fact, these postulates are treated as theorems, and demonstrations attempted in different sources. Proclo, in his comment on the first book of the Elements, tried to prove the fourth postulate. The fifth was to be object of numerous attemps of demonstrations, which lead to the discovery of non Euclidean geometry at the beginning of the 19th century.

Even though they contain indications of ancient, practical origins, the Euclidean Elements are a late work where the axiomatic deductive structure is developed completely, even if not with absolute rigor. Their variety shows the legacy of generations of surveyors who, from the procedures learnt from the Egyptians, were able to create a totally new science as a theoretical system much richer than those who pulled ropes or traced circles could have imagined.

Nonetheless, some problems remained persistently beyond the scope of the line and circle, even when geometry was moved to the world of paper, ruler and compasses. Among them are three classic problems: the duplication of the cube, the trisection of the angle and the quadrature of the circle.

According to legend, when Greece was victim of a great epidemic, a delegation went to the oracle of Delo to ask respite. The response of the oracle was that the rage of the Gods would be placated only when the altar dedicated to them, which had the shape of a cube, was be replaced by one of the same shape but twice the size. The messengers, industrious but not very wise, ordered an altar to be built, again in the shape of a cube, with the side twice as long as that of the original. The epidemic didn't stop and the oracle was consulted again. On that occasion it was discovered that the new altar was not twice as big as the former one but eight times bigger, being, in fact, the volume of the cube with side a equal ³the one with side 2a will have volume (2a)³ If, on the other hand, the cube with side a needs to be doubled, it is necessary to build another one with side , so that its volume is .

Leaving aside the legend and taking an exclusively geometrical point of view, the problem consists in, given a segment of length a, drawing a second segment, with length . Now, while it is easy to build a segment with length  with ruler and compasses (it is sufficient to build a square of side a, the diagonal of which is long), and then double a square (the square built on the diagonal of the former one is suitable), to double the cube is not as simple. Instead, it was demonstrated, even in relatively recent times, that the problem cannot be solved using exclusively a ruler and compasses. The problem of Delo requires the employment of more complex tools and techniques.

The same occurs with the other problems. For those, the surveyors invent new curves, pushing themselves beyond the limits of lines and circles. Ippia from Elide (V century B.C.) invented the quadratrix, a curve that he used to obtain the quadrature of the circle and the trisection of the angle.

Later, the problem of the duplication of the cube was solved by Archita from Taranto (or maybe by Menecmo from Proconneso, who also lived around the 400 B.C.) through the intersection of two parabolas, and by Diocle (II century B.C.) through a new curve, the cissoid. The method of Archita is very ingenious and easy to describe, using the modern system of Cartesian geometry. Considering two parabolas with equation y² = 2ax e x² = ay. If the point with coordinates (x,y) belongs to both curves, we'll have x⁴ = a²y² =2a³x, from which, excluding the solution x=0, is found x³=2a³, and therefore . In conclusion, by intersecting the two parabolas, a point with the side of the double cube as abscissas can be obtained.

Naturally, we have simplified the description of the process of Archita, who, not knowing cartesian geometry and in particular the use of coordinates, employed purely geometric methods. What is most interesting is that certain curves appear in the demonstration for the first time. These were to be the object of study for many centuries to come: conic sections.

If the torch is perpendicular to the wall (if the axis of the cone of light is perpendicular to the wall), the shape formed is a circle, which gets bigger as the distance from the bulb to the wall increases. If we start to tilt the torch, the circle is deformed until it takes a shape delimited by a curve, at first almost circular, then more and more oblong: this is an ellipse. This becomes more and more oblong (eccentric), until the outer ray of the cone of light becomes parallel to the wall. Now we have a parabola. Just by tilting a bit more, the outer ray diverges from the wall, and we have an hyperbole.

These four curves are called conic sections , since they appear as sections of a cone (the cone of light) with a plane (the wall). As a matter of fact - at least in the case of the hyperbole - the experiment of the torch only gives us half of the curve. The complete hyperbole is obtained by considering the whole cone, which means that is formed by two cones joined at the origin.

If the plane of the section then passes through the vertex of the cone the section will be a point in case 1 and 2, a straight line in case 3 and a couple of lines in case 4. Thus, by intersecting a cone with a plane, straight lines, circles and even three new curves, the ellipse, the parabola and the hyperbole can be obtained.

The most extensive antique study regarding conic sections comes to us from Apollonius of Perga (III-II centuries B. C.). Among other things, Apollonius demonstrates a series of properties which lead to important applications in the fields of Science and Technology.

In the ellipse there are two points, called focuses, placed on the longer diameter so that the sum of the distances of any point of the curve from the focuses is the same. This factor can be used to trace an ellipse which is very approximate, but accurate enough to make, for example, elliptic flowerbeds (in fact it is known as the gardener ellipse).

A second property of the focuses of an ellipse consists in the fact that the perpendicular to the ellipse in any point divides the angle formed by the segments joining this point to the two focuses in half. Consequently, a ray of light originating from one of the focuses, and reflected on the ellipse, passes through the other focus.

The same happens with sound waves. If one speaks standing in one of the focuses of an elliptic vault room, the sound waves will be reflected by the vault and will gather in the other focus.

In the circle both focuses are in the centre. As the ellipse gets longer the focuses get further apart. The parabola has only one focus, the other one has (so to speak) gone to infinity. The rays that come from this focus and go to infinity are parallel lines. Reflecting on the parabola these rays will gather in the remaining focus.

Therefore, if we want to gather some parallel lines (or practically parallel, like sun rays, for example) in a certain point, we'll need to use a glass with the shape of a parabola. In so doing it is possible to construct a burning glass, capable of burning a piece of paper or wood put in its focus. The legend - for that is how it should be considered - according to which Archimedes (III century B.C.) burnt Roman ships with a burning mirror gave rise to a considerable amount of reseach in this direction until the late 17th century.

The big radio telescopes and parabolic antennas used to receive television transmissions from satellites, work according to the same principle. The practically parallel signals (considering the great distance they come from), reverberate on the antenna and are gathered in the receiver placed in its focus, thus increasing considerably the input capacity. In other words, the parabolic antenna works as an amplifier, or even better as a condenser of signals, otherwise too feeble, coming from satellites.

What happens with the hyperbole is slighly more complicated. If we stand on the outside, a ray directed to one focus is reflected in the direction of the other focus. On the inside, a ray provening from from one focus, after a reflection on the hyperbole will seem to provene from the other focus.

The interest in conic sections is not limited to these properties, even if they are important. In fact, they were involved in the solution of the scientific problems which determined what was called the "Scientific Revolution".

In the Mathematic Discourse and Demonstrations on the two New Sciences, G. Galileo (1564-1642) demonstrated that the trajectory of a bullet is a parabola.

The reasoning of Galileo is more or less as follows. First of all, analyse what happens when a body falls vertically. At first, the body is motionless. In the first instant of motion, the gravity of the body provokes a certain speed. In the second instant, the body receives a second grade of speed equal to the first one, which is added to it, in the third another grade of speed, and so on. Consequently, the body acquires many grades of speed during the free fall, as many as the instants spent since the beginning of the fall. In other words, speed is proportional to time.

If we use a diagram to illustrate the trend of speed, at the time t = AB the body will have gained a speed v = BC proportional to t: v = gt. The distance y covered in the time t will be represented by the area of the triangle ABC, with base AB and height BC, and therefore will be equal to the product of the base by half of the height: y=1/2 gt².

Let's now see what happens if the body is dropped at a certain initial speed, and suppose for simplicity that it is thrown horizontally at speed v.

Since the gravity force is directed vertically, it will not influence the horizontal motion, which will take place with constant speed v. In the time t the body will cover a horizontal distance x = vt. On the other hand, the force of gravity will produce a vertical motion according to the law y = 1/2 gt². If t = x/v is drawn from the first equation and replaced to the second, the equation

y=(g/v²) x² is obtained

representing a parabola.

If then, we want to understand better why the area of the triangle ABC gives the distance covered in time t, we could reason as follows.

Let us fix in AB an interval of time DE, and consider a body that during this interval moves at minimum speed DF. The distance covered by this body is less than the one covered by the first one during the same time and it is given by the product of the speed DF by the time DE, and therefore by the area of the rectangle DEGF.

In the same manner a body moving at a maximum speed EH will cover a distance equal to the area of the rectangle DEHI in the time DE - longer than the distance covered by the free-falling body. Let's now divide the time AB in many intervals. The distance covered by the falling body will be more than the area of the scale drawing within the triangle ABC and less than the exterior. Increasing the number of the little intervals, we will be able to see the two figures getting closer and closer to the triangle, which must always included between the areas of the two figures. The distance covered by the body must be equal to the area of the triangle ABC.

To conclude, a bullet that is cast at a certain speed describes a parabolic trajectory, at least until the initial speed is sufficiently slight to be able to ignore the resistance of the air. This is true, for example, when a stone is thrown by hand or with a slingshot and - with some approximation - for bullets expelled by a trench gun. If a cannon is used instead, the trajectory will be significantly altered by the resistance of the air, assuming a much stockier shape, well known to the 16th century's bombardiers.

Imagine slinging stones with a catapult, or shooting bullets with a trench gun (always shooting with the same force). The trajectories of the bullets will be different according to the direction of the throw, but all will have a parabolic shape. By varying the inclination of the device, we can hit different targets, both on the ground and in the air, as long as they are not too far away. The maximum distance is the one that can be reached with an inclination of 45 degrees. One might then ask: which points can be reached? Or rather, seen from the side - where does one have to stand to be certain not to be hit?

The reachable area is represented by the points of the plane crossed by at least one of the curves covered by the bullets shot at different angles. The curve that delimits it is called the envelope of the given curves. In our case it is once again a parabola, and it is called a security parabola.

Conic sections represent the key to the solution to another problem: the orbit of planets. In ancient times a system was imagined whereby the earth was at the centre of the universe, with the sun, the moon and the five known planets (Mercury, Venus, Mars, Jupiter and Saturn) rotating around it. In such a system, a circular orbit is not compatible with the observations and therefore a system of epicycles was thought out - a system of circles rotating above other circles. This system permitted the reasonably precise prediction of celestial movement. The introduction of the Copernican system, with the sun at the centre of the universe and the earth and the planets rotating in circular orbits, didn't significantly improve the description of the phenomena, which still needed the consideration of the epicycles. The astronomical advantages that derived from the adoption of the new system weren't powerful enough to overcome the philosophical prejudices that supported the old ones.

On the other hand both factions remained rooted in the idea of circular orbits, that seemed evident for a series of reasons that are no longer valid today, but that at the beginning of the 17th century seemed very solid. One of those was the argument by Aristotle: simple bodies have simple motions. The heavenly bodies are simple bodies and therefore they must move in the simplest way, which is the circular orbit. And even if, as in the case of Galileo, arguments of this kind were rejected, there were other reasons not to reject the notion of circularity.

Galileo started with the premise that the motion of a body on an inclined plane increases its speed if it is descending, which means as the body approaches the centre of gravity, while it slows down if it is ascending, and is constant on a horizontal plane since the body gets neither closer nor further away from the centre of the earth. In reality, said Galileo, this is true because the plane is very small compared to the earth's diameter. If the reasoning is shifted on a much larger scale, the surface of inertia - where there is no acceleration - is not a plane but a sphere with the centre in the centre of attraction, since only here the body remains at the same distance from the centre.

Besides, since the movement of planets repeats itself constantly in the same manner without significant acceleration or deceleration, it follows that their orbit occurs on a circular line, with its centre in the sun. In fact, only thus can the stability and uniformity of the universe be preserved.

One can easily understand how difficult it must have been even to imagine movements different from circular ones, and what incredible intellectual effort was required to change such a point of view, like passing from a circle to an ellipse. This step was made, not without effort, by G. Keplero (1571-1630) who discovered that the orbit of Mars is elliptical. This later became one of his most famous laws:"Planets cover elliptical orbits with the sun as one of the focuses".

Fifty years later, I. Newton demonstrated the three laws by Keplero on the basis of his dynamics, in the hypothesis that the force of attraction is inversely proportional to the square of the distance. It can be stated that, only after the demonstration by Newton, the Copernican hypothesis and the laws by Keplero were accepted by all scientists. Another century had to pass before the Dialogue of the maximum systems by Galileo was struck from the index.

Both ruler and compasses, and conic sections belong to the scientific heritage of classic Greece and to some extent we embody this heritage. More than a few other lines can be found in the works by Greek mathematicians - spirals, quadratrices, conchoids, cissoids and in some cases even special curves like the helices of Pappo (III century DA.D.) on the surface of the sphere. But, in any case, these particular curves derive more from the imagination of this or that surveyor, than from an internal mathematical internal dynamic, which was, instead, slowly withering. The appearance of these particular curves was not a sign of scientific progress, but rather a symptom of a lack of inspiration and increasing levels of self-indulgence.

Apart, perhaps, from conic sections, all the curves of Greek geometry are "nominated" curves. All of them are defined through characteristic properties valid only for themselves. Each one needs specific methods that cannot be applied in other situations.

In order to leave this limited world, a radically different point of view was required - a method that could applied to all curves without being specific to any of them. A method by which the general gained at the expense of the minute details of procedure.

A decisive step consisted in the introduction of the cartesian coordinates, which are called after the philosopher and mathematician Réné Descartes (1596-1650). Any point P of the plane can be found through two numbers (x, y) that are the distances from two perpendicular lines. these last ones are the cartesian axis, and the numbers x and y the cartesian coordinates of the point P. The x is called abscissa, and y ordinate of the point P. The abscissa is positive on the right and negative on the left, the ordinate positive high up and negative low down.

When its coordinates x and y vary in all possible ways, the point P describes the whole plane. If, instead, the coordinates of P are tied up by an equation, the corresponding point will not be able to move arbitrarily on the plane, but will be forced to follow a curve. For instance, the equation y=2 describes a straight horizontal line, x²+y²=1 is the equation of a circle and y=x² of a parabola.

Generally, an algebraic equation F(x,y)=0, where F(x,y) is a polinomy in variables x and y, can be connected to a curve that is the place of points whose coordinates satisfy the equation under consideration. The latter can be relatively simple, like the one expressing a line or a circle, but also quite complex, like for example x²⁷+y⁴-x³y¹³=1. The most complex curves can be described through equations. Bernoully claimed that he could write the equation of the face of a man!

A method to obtain the points of the curves consists in fixing the value of one of its coordinates, for example x=1, thus obtaining the equation F(1,y)=0, only in the y. By solving the latter, one or more values of y can be obtained, individuating one or more points of the curve, its correspondence to the abscissa x=1. Taking different values for x, various points of the curve are obtained and the curve can be drawn with the necessary precision.

For example, to draw the points of the ellipse x² + 4y² = 1, we make x = a, and we solve the equation a²+4y² = 1. The solutions of this equation are given by 4y² = 1-a² and, therefore, if -1<a< there are two solutions    e   , while if a = 1 meaning a = -1 there is only the solution y = 0, and if a>-1 there are no solutions. Attributing finally, to a, different values comprised between -1 and 1, and reporting the corresponding points on the diagram, we obtain a series of points of the curve.

It should be noted that this is not the best procedure to trace this or that particular curve. For example, a circle can be drawn better and more easily with a compass, rather than by constructing its points verifying the equation x²+y² = 1. The same applies to the ellipse, conic sections and practically to all particular curves. For each, it is possible to make an instrument to trace it accurately and faster than can be done employing the method of successive determination of its points. This has, however, an important decisive advantage - that of being a general method. It is true that for each curve a tool can be invented to trace it, but, changing the curve we will have to abandon the mechanism used to trace it and employ another one intended for the new one. However, the method of tracing points is independent from the curve in question, which only enters into its equation to differenciate the otherwise uniform procedure. It is precisely the possibility of study methods and general procedures that makes the new cartesian formulation more adaptable and powerful than the constructing techniques of classic geometry.

Curves can, therefore, be drawn by points, solving some equations, and, conversely, equations can be solved through the intersection of two curves.

If F(x,y) = 0 and G(x,y) = 0 are the equations of two curves, the points belonging to the intersection of these will have to verify both equations. Now, one of the two variables can be derived, for example y in function of x, from the first equation, and inserting the value y(x) found in the second, one obtains an equation P(x) = 0 only in x, the solutions of which are the abscissas of the points of intersection of the two curves. If, therefore, one wants to solve the equation P(x) = 0,two curves F(x,y) and G(x,y) must be found so that the given equation is the one resulting from the elimination of y. The abscissas of the points of intersection of the two curves give the solution to the equation.

Naturally, the same equation P(x) = 0 can be obtained through different choices of curves, or of the functions F(x,y) and G(x,y). Doing so for example, the solutions x₁ and x₂ of the second grade equation

x²+x=4 can be constructed intersecting the line y = x with the parabola x²+y = 4,or with the hyperbole xy+x = 4, or with the circle (x+1)²+y² = 9, that have the centre in the point (-1,0) and radius 3. In fact, replacing the value x into y in the equation of the circle, (x+1)²+x² = 9, or 2x²+2x+1 = 9 can be obtained and then x²+x = 4. Among all of them the one corresponding to the best criteria of simplicity can be chosen for example the last one which only has the intervention line and circle and can, therefore, be drawn with a ruler and compass.

If the grade of the equation is higer than 2, lines and circles are no longer sufficient, and it is necessary to resort to curves such as conic sections or even others less familiar. For example, the solutions of the equation x⁴+x²+3x = 1 can be obtained by intersecting the circle x²+y²+3x =1 with the parabola y = x². As the grade of the equation increases, more complex curves become necessary.

The possibility of considering curves "generic"sheds a different light on many classic problems, especially those regarding squaring and tangents.

To square a figure literally means finding an equivalent square to the given figure. Obviously, the solution to a squaring problem depends on the tools considered acceptable for the construction of the required square. For example, when speaking about the classical problem of the squaring of the circle, a ruler and compass are considered appropriate. The problem thus formulated was only solved in the last century, with the demonstration of the impossibility of such a construction. If on the other hand, more tools are accepted, such as the "quadratrice of Ippia", a positive answer is found to the problem - one that has been well known since Greek times.

If limited to the use of ruler and compasses, few positive results can be obtained. Hippocrates managed to square lunes, the first example of exact squaring of a curvilinear figure. Archimedes discovered the squaring of parabolas. Similarly, classic surveyors solved, in some cases, the problem of tangents, or rather that of determining the straight line best approaching a given curve when close to one of its points. The Greeks determined the tangents to circles or conic sections and also to other particular curves.

In the new Cartesian formulation the two problems assume different aspects: rather than squaring this or that figure, or finding the tangent to this or that line, what must be found is a uniform method which allows one to trace the tangent to an arbitrary curve or to provide the procedure to square a figure delimited by any curve.

The first of these problems,which was partly solved by Descartes, was to lead to the discovery of differential calculus by Newton (1643-1727) and Leibniz (1646-1716). The second was to be the object of integral calculus.

More difficult is the so called inverse problem of the tangent, or in modern terms, the integration of a differential equation.

From the geometrical point of view, the problem consists in finding a curve, knowing a relationship between its points and the relative tangents. Analytically, it translates in an equation which binds the variables x and y to their differentials. These equations lead to the solutions not only of geometrical problems (such as the so called Baune's problem , the first to be formulated in terms of differential equations ) but also to a number of physics problems - especially mechanical ones - first of all, finding the trajectory of a body subject to a given force. Such are the falls of bodies and in general the motion of a body in the void or in a resistant medium and the defining of the orbits of planets, subject to an attractive force proportional to the inverse of the squared distance from the sun. Both problems are looked into by Newton in his Principia. A few years later Leonhard Euler (1707-1783) submitted all mechanics to calculus.

Differential equations bring to the fore a new class of curves, which only emerged sporadically in the course of the 17th century - transcendental curves. These are curves which cannot be expressed through algebraic expressions but which, in order to be described analytically, require the introduction of new functions, among which are trigonometric equations, logarithms and exponentials.

The first problem that can be traced back to a differential equation, the De Baune's problem, has the exponential curve of equation y = e^x as solution. The same curve, of equation x = log y,if seen inverting the two axes interferes in the problem of the squaring of the hyperbola. As new problems, impossible to solve before the invention of calculus, come to be faced, new transcendent curves are discovered. The first problem faced and successfully solved is that of a chain hanging from the ends: its solution is a curve, the catenary, of equation

y = (e^x+e^-x)/2.

A second success of new calculus was the definition of the brachistochrone, the curve which reduces to the minimum the falling time from one of its ends to the other.

More precisely, suppose we fix two points P and Q, the first higher than the second but not on the vertical line, and let a body fall from P to Q sliding on a curve which joins the two points. The problem now is: among all the curves joining P to Q, which is the one that reduces the falling time to the minimum? It isn't the straight line joining the two points as it might seem at first sight. In fact, in order to reduce falling time it is expedient to start almost vertically, so to gain speed straight away, even if risking a longer path.

In this case the solution is a cycloid, the curve described by one point of a circle which rotates without grazing.

The same curve represents the solution to another problem too. Suppose you put a little ball onto a profile shaped like a semicircle and let it go. The ball will start swinging back and forth, while the time it'll take to complete one oscillation will decrease as the magnitude of the oscillation gets smaller, remaining almost constant for small oscillations. One should then ask: is there a curve on which any oscillation, big or small, takes the same time? The answer is affirmative: the isochrone curve is, again, the cycloid.

The curvature   

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The shortest way   

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Length and dimension   

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