‘Pupil of Plato’
Eudoxus flourished around the middle of the 4th century BCE; he was an astronomer initiated into the Egyptian mysteries, obtaining his knowledge of the art from the priests of Isis.
His work is passed to us through Aristotle.
1684 – Germany
‘A new method for maxima and minima, as well as tangents … and a curious type of calculation’
Newton invented calculus (fluxions) as early as 1665, but did not publish his major work until 1687. The controversy continued for years, but it is now thought that each developed calculus independently.
Terminology and notation of calculus as we know it today is due to Leibniz. He also introduced many other mathematical symbols: the decimal point, the equals sign, the colon (:) for division and ratio, and the dot for multiplication.
1687 – England
‘Any two bodies attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them’
The force is known as gravitation
Expressed as an equation:
F = GmM/r^{2}
where F is Force, m and M the masses of two bodies, r the distance between them and G the gravitational constant
This follows from KEPLER’s laws, Newton’s laws of motion and the laws of conic sections. Gravitation is the same thing as gravity. The word gravity is particularly used for the attraction of the Earth for other objects.
Gravitation
Newton stated that the law of gravitation is universal; it applies to all bodies in the universe. All historical speculation of different mechanical principles for the earth from the rest of the cosmos were cast aside in favour of a single system. He demonstrated that the planets were attracted toward the Sun by a force varying as the inverse square of the distance and generalized that all heavenly bodies mutually attract one another. Simple mathematical laws could explain a huge range of seemingly disconnected physical facts, providing science with the straightforward explanations it had been seeking since the time of the ancients. That the constant of gravitation is in fact constant was proved by careful experiment, that the focus of a body’s centre of gravity appears to be a point at the centre of the object was proved by his calculus.
Calculus
The angle of curve, by definition, is constantly changing, so it is difficult to calculate at any particular point. Similarly, it is difficult to calculate the area under a curve. Using ARCHIMEDES’ method of employing polygons and rectangles to work out the areas of circles and curves, and to show how the tangent or slope of any point of a curve can be analyzed, Newton developed his work on the revolutionary mathematical and scientific ideas of RENE DESCARTES, which were just beginning to filter into England, to create the mathematics of calculus. Calculus studies how fast things change.
The idea of fluxions has become known as differentiation, a means of determining the slope of a line, and integration, of finding the area beneath a curve.
Newton’s ideas on universal gravitation did not emerge until he began a controversial correspondence with ROBERT HOOKE in around 1680. Hooke claimed that he had solved the problem of planetary motion with an inverse square law that governed the way that planets moved. Hooke was right about the inverse square law, but he had no idea how it worked or how to prove it, he lacked the genius that permitted Newton to combine Kepler’s laws of planetary motion with the assumption that an object falling towards Earth was the same kind of motion as the Earth’s falling toward the Sun.
It was not until EDMUND HALLEY challenged Newton in 1684 to show how planets could have the elliptical orbits described by Johannes Kepler, supposing the force of attraction by the Sun to be the reciprocal of their distance from it – and Newton replied that he already knew – that he fully articulated his laws of gravitation.
It amounts to deriving Kepler’s first law by starting with the inverse square hypothesis of gravitation. Here the Sun attracts each of the planets with a force that is inversely proportional to the square of the distance of the planet from the Sun. From Kepler’s second law, the force acting on the planets is centripetal. Newton says this is the same as gravitation.
In the previous half century, Kepler had shown that planets have elliptical orbits and GALILEO had shown that things accelerate at an even pace as they fall towards the ground. Newton realized that his ideas about gravity and the laws of motion, which he had only applied to the Earth, might apply to all physical objects, and work for the heavens too. Any object that has mass will be pulled towards any other object. The larger the mass, the greater the pull. Things were not simply falling but being pulled by an invisible force. Just as this force (of gravity) pulls things towards the Earth, it also keeps the Moon in its orbit round the Earth and the planets moving around the Sun. With mathematical proofs he showed that this force is the same everywhere and that the pull between two things depends on their mass and the square of the distance between them.
Newton published his law of gravitation in his magnum opus Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) in 1687. In it Newton analyzed the motion of orbiting bodies, projectiles, pendulums and free fall near the Earth.
The first book of Principia states the laws of motion and deals with the general principles of mechanics. The second book is concerned mainly with the motion of fluids. The third book is considered the most spectacular and explains gravitation.
Why do two objects attract each other?
‘I frame no hypotheses’, said Newton
It was Newton’s acceptance of the possibility that there are mysterious forces in the world, his passions for alchemy and the study of the influence of the Divine that led him to the idea of an invisible gravitational force – something that the more rationally minded Galileo had not been able to accept.
Newton’s use of mathematical expression of physical occurrences underlined the standard for modern physics and his laws underpin our basic understanding of how things work on an everyday scale. The universality of the law of gravitation was challenged in 1915 when EINSTEIN published the theory of general relativity.
1670-71 Newton composes ‘Methodis Fluxionum‘, his main work on calculus, which is not published until 1736. His secrecy meant that in the intervening period, the German mathematician LEIBNIZ could publish his own independently discovered version – he gave it the name calculus, which stuck.
1687 – England
First Law: An object at rest will remain at rest and an object in motion will remain in motion at that velocity until an external force acts on the object
Second Law: The sum of all forces (F) that act on an object is equal to the mass (m) of the object multiplied by the acceleration (a), or F = ma
Third Law: To every action, there is an equal and opposite reaction
introduces the concept of inertia, the tendency of a body to resist change in its velocity. The law is completely general, applying to all objects and any force. The inertia of an object is related to its mass. Things keep moving in a straight line until they are acted on by a force. The Moon tries to move in a straight line, but gravity pulls it into an orbit.
Weight is not the same as mass.
explains the relationship between mass and acceleration, stating that a force can change the motion of an object according to the product of its mass and its acceleration. That is, the rate and direction of any change depends entirely on the strength of the force that causes it and how heavy the object is. If the Moon were closer to the Earth, the pull of gravity between them would be so strong that the Moon would be dragged down to crash into the Earth. If it were further away, gravity would be weaker and the Moon would fly off into space.
shows that forces always exist in pairs. Every action and reaction is equal and opposite, so that when two things crash together they bounce off one another with equal force.
1672 – New Theory about Light and Colours is his first published work and contains his proof that white light is made up of all colours of the spectrum. By using a prism to split daylight into the colours of the rainbow and then using another to recombine them into white light, he showed that white light is made up of all the colours of the spectrum, each of which is bent to a slightly different extent when it passes through a lens – each type of ray producing a different spectral colour.
Newton also had a practical side. In the 1660s his reflecting telescope bypassed the focusing problems caused by chromatic aberration in the refracting telescope of the type used by Galileo. Newton solved the problem by swapping the lenses for curved mirrors so that the light rays did not have to pass through glass but reflected off it.
At around the same time, the Dutch scientist CHRISTIAAN HUYGENS came up with the convincing but wholly contradictory theory that light travels in waves like ripples on a pond. Newton vigorously challenged anyone who tried to contradict his opinion on the theory of light, as Robert Hooke and Leibniz, who shared similar views to Huygens found out. Given Newton’s standing, science abandoned the wave theory for the best part of two hundred years.
1704 – ‘Optiks’ published. In it he articulates his influential (if partly inaccurate) particle or corpuscle theory of light. Newton suggested that a beam of light is a stream of tiny particles or corpuscles, traveling at huge speed. If so, this would explain why light could travel through a vacuüm, where there is nothing to carry it. It also explained, he argued, why light travels in straight lines and casts sharp shadows – and is reflected from mirrors. His particle theory leads to an inverse square law that says that the intensity of light varies as the square of its distance from the source, just as gravity does. Newton was not dogmatic in Optiks, and shows an awareness of problems with the corpuscular theory.
In the mid-eighteenth century an English optician John Dolland realized that the problem of coloured images could largely be overcome by making two element glass lenses, in which a converging lens made from one kind of glass was sandwiched together with a diverging lens made of another type of glass. In such an ‘achromatic’ lens the spreading of white light into component colours by one element was cancelled out by the other.
During Newton’s time as master of the mint, twenty-seven counterfeiters were executed.
1755 – Switzerland
‘Analytical calculus – the study of infinite processes and their limits’
Swiss mathematician. His notation is even more far-reaching than that of LEIBNIZ and much of the mathematical notation that is in use to-day may be credited to Euler.
The number of theorems, equations and formulae named after him is enormous.
Euler made important discoveries in the analytic geometry of surfaces and the theory of differential equations.
Euler popularised the use of the symbols ‘Π‘ (Pi); e , for the base of the natural logarithm; and i , for the imaginary unit.
Euler is credited with contributing the useful notations f (x) , for the general function of x ; and Σ , to indicate a general sum of terms.