Letters on England [36]
telescope being convex on one side and flat on the
other, in case the flat side be turned towards the object, the error
which arises from the construction and position of the glass is
above five thousand times less than the error which arises from the
refrangibility; and, therefore, that the shape or figure of the
glasses is not the cause why telescopes cannot be carried to a
greater perfection, but arises wholly from the nature of light.
For this reason he invented a telescope, which discovers objects by
reflection, and not by refraction. Telescopes of this new kind are
very hard to make, and their use is not easy; but, according to the
English, a reflective telescope of but five feet has the same effect
as another of a hundred feet in length.
LETTER XVII.--ON INFINITES IN GEOMETRY, AND SIR ISAAC NEWTON'S
CHRONOLOGY
The labyrinth and abyss of infinity is also a new course Sir Isaac
Newton has gone through, and we are obliged to him for the clue, by
whose assistance we are enabled to trace its various windings.
Descartes got the start of him also in this astonishing invention.
He advanced with mighty steps in his geometry, and was arrived at
the very borders of infinity, but went no farther. Dr. Wallis,
about the middle of the last century, was the first who reduced a
fraction by a perpetual division to an infinite series.
The Lord Brouncker employed this series to square the hyperbola.
Mercator published a demonstration of this quadrature; much about
which time Sir Isaac Newton, being then twenty-three years of age,
had invented a general method, to perform on all geometrical curves
what had just before been tried on the hyperbola.
It is to this method of subjecting everywhere infinity to
algebraical calculations, that the name is given of differential
calculations or of fluxions and integral calculation. It is the art
of numbering and measuring exactly a thing whose existence cannot be
conceived.
And, indeed, would you not imagine that a man laughed at you who
should declare that there are lines infinitely great which form an
angle infinitely little?
That a right line, which is a right line so long as it is finite, by
changing infinitely little its direction, becomes an infinite curve;
and that a curve may become infinitely less than another curve?
That there are infinite squares, infinite cubes, and infinites of
infinites, all greater than one another, and the last but one of
which is nothing in comparison of the last?
All these things, which at first appear to be the utmost excess of
frenzy, are in reality an effort of the subtlety and extent of the
human mind, and the art of finding truths which till then had been
unknown.
This so bold edifice is even founded on simple ideas. The business
is to measure the diagonal of a square, to give the area of a curve,
to find the square root of a number, which has none in common
arithmetic. After all, the imagination ought not to be startled any
more at so many orders of infinites than at the so well-known
proposition, viz., that curve lines may always be made to pass
between a circle and a tangent; or at that other, namely, that
matter is divisible in infinitum. These two truths have been
demonstrated many years, and are no less incomprehensible than the
things we have been speaking of.
For many years the invention of this famous calculation was denied
to Sir Isaac Newton. In Germany Mr. Leibnitz was considered as the
inventor of the differences or moments, called fluxions, and Mr.
Bernouilli claimed the integral calculus. However, Sir Isaac is now
thought to have first made the discovery, and the other two have the
glory of having once made the world doubt whether it was to be
ascribed to him or them. Thus some contested with Dr. Harvey the
invention of the circulation of the blood, as others disputed with
Mr. Perrault
other, in case the flat side be turned towards the object, the error
which arises from the construction and position of the glass is
above five thousand times less than the error which arises from the
refrangibility; and, therefore, that the shape or figure of the
glasses is not the cause why telescopes cannot be carried to a
greater perfection, but arises wholly from the nature of light.
For this reason he invented a telescope, which discovers objects by
reflection, and not by refraction. Telescopes of this new kind are
very hard to make, and their use is not easy; but, according to the
English, a reflective telescope of but five feet has the same effect
as another of a hundred feet in length.
LETTER XVII.--ON INFINITES IN GEOMETRY, AND SIR ISAAC NEWTON'S
CHRONOLOGY
The labyrinth and abyss of infinity is also a new course Sir Isaac
Newton has gone through, and we are obliged to him for the clue, by
whose assistance we are enabled to trace its various windings.
Descartes got the start of him also in this astonishing invention.
He advanced with mighty steps in his geometry, and was arrived at
the very borders of infinity, but went no farther. Dr. Wallis,
about the middle of the last century, was the first who reduced a
fraction by a perpetual division to an infinite series.
The Lord Brouncker employed this series to square the hyperbola.
Mercator published a demonstration of this quadrature; much about
which time Sir Isaac Newton, being then twenty-three years of age,
had invented a general method, to perform on all geometrical curves
what had just before been tried on the hyperbola.
It is to this method of subjecting everywhere infinity to
algebraical calculations, that the name is given of differential
calculations or of fluxions and integral calculation. It is the art
of numbering and measuring exactly a thing whose existence cannot be
conceived.
And, indeed, would you not imagine that a man laughed at you who
should declare that there are lines infinitely great which form an
angle infinitely little?
That a right line, which is a right line so long as it is finite, by
changing infinitely little its direction, becomes an infinite curve;
and that a curve may become infinitely less than another curve?
That there are infinite squares, infinite cubes, and infinites of
infinites, all greater than one another, and the last but one of
which is nothing in comparison of the last?
All these things, which at first appear to be the utmost excess of
frenzy, are in reality an effort of the subtlety and extent of the
human mind, and the art of finding truths which till then had been
unknown.
This so bold edifice is even founded on simple ideas. The business
is to measure the diagonal of a square, to give the area of a curve,
to find the square root of a number, which has none in common
arithmetic. After all, the imagination ought not to be startled any
more at so many orders of infinites than at the so well-known
proposition, viz., that curve lines may always be made to pass
between a circle and a tangent; or at that other, namely, that
matter is divisible in infinitum. These two truths have been
demonstrated many years, and are no less incomprehensible than the
things we have been speaking of.
For many years the invention of this famous calculation was denied
to Sir Isaac Newton. In Germany Mr. Leibnitz was considered as the
inventor of the differences or moments, called fluxions, and Mr.
Bernouilli claimed the integral calculus. However, Sir Isaac is now
thought to have first made the discovery, and the other two have the
glory of having once made the world doubt whether it was to be
ascribed to him or them. Thus some contested with Dr. Harvey the
invention of the circulation of the blood, as others disputed with
Mr. Perrault