Conic sections in the scientific revolution   

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.