Definition Of A Straight Line

Mathematicians pretend to give an exact definition of a straight line when they say that it is the shortest way between 2 points.

This is more the discovery of one of the properties of a straight line.

A person who things of a straight line immediately sees it as the shortest way between two points.

He does not see this shortest way being an accident.

A straight line can be comprehended alone.

But this definition is unintelligible without a comparison with other lines.

I object to this definition with 2 arguments.

1. In common life, it is established as a maxim, that the straightest way is always the shortest.

It would be as absurd as to say that ’the shortest way is always the shortest’, if our idea of a straight line was the same as the shortest way between two points.

2. We have no precise idea of equality and inequality, shorter and longer, more than of a straight line or a curve.

Consequently, the one can never give us a perfect standard for the other. An exact idea can never be built on such as are loose and undetermined. The idea of a flat surface also cannot have a precise standard.

We can only distinguish such a surface through its general appearance. Mathematicians say that a flat surface is created by the flowing of a straight line. We can immediately object that straight lines constrained into a plane necessarily creates a flat surface. This description explains a thing by itself, and is circular reasoning.

The ideas most essential to geometry are those of:

• equality and inequality, and
• a straight line and a plain surface.

These are far from being exact and determinate, according to our common method of conceiving them.

We are incapable of telling if the case is doubtful in any degree, when:

• such figures are equal,
• such a line is a right one, and
• such a surface is a plain one.

But we cannot form an idea of that proportion or these figures which is firm and invariable.

Our appeal is still to the weak and fallible judgment, which we: make from the appearance of the objects, and correct by a compass or common measure. If we join the supposition of any further correction, it is of such-a-one as is useless or imaginary.

We would:

• have recourse to the common topic in vain, and
• employ the supposition of a deity whose omnipotence may enable him to:
  • create a perfect geometrical figure, and
  • describe a straight line without any curve or inflexion.

The ultimate standard of these figures is only derived from the senses and imagination.

We cannot talk of any perfection beyond what these faculties can judge of.

Since the true perfection of anything is in its conformity to its standard.

Since these ideas are so loose and uncertain, I ask any mathematician: what infallible assurance he has of:

• the more intricate and obscure propositions of mathematics, and
• the most vulgar and obvious principles?

How can he prove:

• that 2 straight lines cannot have one common segment?
• that it is impossible to draw more than one straight line between any two points?

If he tells me that these opinions are absurd and repugnant to our clear ideas, I would answer that I do not deny, where two straight lines incline on each other with a sensible angle.

But it is absurd to imagine them to have a common segment.

But if these two lines approach at the rate of an inch per 100 kilometers, they will eventually meet.

By what rule do you use to assert that the line cannot make the same straight line with those two, that form so small an angle between them?

You must surely have some idea of a straight line, to which this line does not agree.

Do you therefore mean that it does not take the points in the same order and by the same rule, as is peculiar and essential to a straight line?

If so, I must inform you that:

• besides that in judging after this manner, that space is composed of indivisible points.
  • Perhaps this is more than you intend.
• this is not the standard from which we form the idea of a straight line.
  • if it were, is there any such firmness in our senses or imagination to determine when such an order is violated or preserved?

The original standard of a straight line is in reality nothing but a certain general appearance.

Straight lines may be made to concur with each other, and yet correspond to this standard, though corrected by other means.

This dilemma meets mathematicians whatever side they turn to.

If they judge of equality by the exact standard of the enumeration of the minute indivisible parts, they:

• employ a standard that is useless, and
• actually establish the indivisibility of space.

If they employ, as is usual, the inaccurate standard derived by comparing objects on their general appearance, corrected by measuring and juxtaposition, their infallible first principles are too coarse to afford any subtle inferences.

The first principles are founded on the imagination and senses.

The conclusion, therefore, can never go beyond, much less contradict the imagination.

Thus, no geometrical demonstration for the infinite divisibility of space can be supported.

This is also why geometry falls of evidence in this single point, while all its other reasonings command our fullest assent and approbation.

It seems more requisite to give the reason of this exception, than to show that we really must:

• make such an exception
• regard all the mathematical arguments for infinite divisibility as sophistical.

Since no idea of quantity is infinitely divisible, it is most absurd to try:

• to prove that quantity itself admits of such a division, and
• to prove this through directly opposite ideas.

All arguments founded on this absurdity will have a new absurdity.

For example, all mathematicians are judged by the diagrams they write on paper. They tell us these diagrams are loose drafts which only convey certain ideas which are the true foundation of all our reasoning. This is satisfactory, if they just refer to them as ideas. A circle touching a line

I ask our mathematician to think of a circle touching a straight line.

Do they intersect or touch at a precise point?

If they intersect at a precise point, then it means a precise point exists.

Or does their intersection and touch occupy the same space?

If he says that they occupy the same space, then it means:

• that shapes cannot be analyzed beyond a certain degree of minuteness, and
• that the circle and straight line creates a new shape.