Synthetic Spacetime
Axiom 1: Through two points one and only one line can be drawn
Axiom 2: Two lines intersect in one and only one point, except that-
Axiom 3: Through any point not on a given line one and only one parallel (non-intersecting) line can be drawn.
Axiom 4: The line shall be regarded as a continuous array of points in open order.
>>Axioms on translation (also applicable in Euclidean geometry):
Axiom 5: Any point P can be carried to any point P', and any two translations which carry P to P' are identical.
Axiom 6: Any line is carried into a parallel line.
Axiom 7: Any line parallel to PP' remains unchanged.
Axiom 8: The succession of two translations is a translation.
>>Universal propostions:
Proposition 1: If two intersecting lines are parallel respectively to two other intersecting lines, the corresponding angles are congruent.
>proof: By translation the points of intersection may be made to coincide, and the lines of the first pair, remaining parallel with the lines of the other pair (axiom 6), must come into coincidence with them, by axiom 3.
Proposition 2: The opposite sides of a parallelogram anre congruent.
>proof: If ABCD is a parallelogram and if A be translated to B, the line of DC remains unchanged by axiom 7, and the line of AD falls along the line of BC by proposition 1. Hence D falls on C by axiom 2.
Corollary: If two points P,P' are carried by translation into Q,Q', the figure PP'QQ' is a parallelogram.
Definition: The measure of the separation of P and P' we will call the interval PP'.
The translation that carries P to P' will carry P' to P'' on the same line. Since the segment PP' is congruent to the segment P'P'', the intervals PP' and P'P'' are said to be equal.
Axiom 9: If a sufficient number of equal intervals be laid off on a line, any point of the line may be surpassed.(Archimedian property for intervals)
>>More universal propositions:
Proposition 3: The diagonals of a parallelogram bisect eachother.
Proposition 4: If two triangles have the sides of one respectively parallel to the sides of the other, and if one side is congruent to one side of the other, then the remaining sides of the one are respectively congruent to the sides of the other.
Definition: Two triangles, with the sides of one respectively parallel to the sides of the other, will be called similar.
Proposition 5: In two similar triangles the sides of the one are respectively proportional to the sides of the other.
>>Propositions involving area:
Any arbitrarily chosen unit intervals along any selected pair of intersecting lines determine a parallelogram which may be taken as having unit area.
Proposition 6: Any parallelogram with sides parallel to those of the unit parallelogram has an area equal to the product of the intervals along two intersecting sides.
Proposition 7: The diagonal of a parallelogram divides it into two equal areas.
Proposition 8: If from any point in the diagonal of a parallelogram lines be drawn parallel to the sides, the two parallelograms formed on either side of the diagonal are equal in area.
Proposition 9: Two parallelograms between the same parallel lines and with congruent bases are equal in area.
Corollary: Two triangles having congruent bases on one line and vertices on a parallel line have equal areas.
Corollary: The diagonals divide a parallelogram into four equal triangular areas.
Proposition 10: Of all parallelograms having two sides common to two sides of a given triangle and a vertex on the third side of the triangle, the one whose vertex bisects the third side has the greatest area.
>proof uses propositions 4 and 8.
Definition Any two polygons which have their corresponding sides parallel and in proportion are similar.
Proposition 11: If on two sides of a triangle similar parallelograms are constructed, and on the third side a parallelogram with diagonals parallel to the diagonals of the other parallelograms, the area of this parallelogram will be equal to the difference of the areas of the other two.
>>Hyperbolic rotation:
This set of postulates differentiates the non-Euclidean geometry from Euclidean geometry.
Axiom 10: Any one point and only that one remains fixed. This point may be called the center of the hyperbolic rotation.
Axiom 11: Two lines through this point remain unchanged. These line may be called the fixed lines of the hyperbolic rotation.
Axiom 12: Any half-line (or ray) from the center, and lying in one of the angles determined by the fixed lines, may be turned into any other ray in the same angle, and this uniquely determines the hyperbolic rotation.
Axiom 13: The succession of two hyperbolic rotations about the same point is a hyperbolic rotation.
Axiom 14: The result of a hyperbolic rotation about O and a translation from O to O' is independent of the order in which the hyperbolic rotation and translation are carried out.
>>Classes of lines:
It follows immediately from axiom 14 that the fixed lines in a hyperbolic rotation about any point O are parallel to the fixed lines in a hyperbolic rotation about any other point O'. All the lines in the plane may now be divided into classes in such a manner that neither translation nor hyperbolic rotation can change the classification:
____(α) lines parallel to one of the fixed directions,
____(β) lines parallel to the other of the fixed directions,
____(γ) lines which lie in one of the pairs of vertical angles determined by the fixed directions,
____(δ) lines which lie in the other pair of vertical angles determined by the fixed directions.
Definitions: The lines of fixed direction, namely, (α)-lines and (β)-lines, will be called singular lines.
>>Angular magnitudes:
A succession of hyperbolic rotations may be used in the same manner as the succession of translations was used to establish the measure of interval along a line. Thus if a line a is carried into a line a' and at the same time the line a' is carraied into the line a'', the angles between a and a' and between a' and a'' are congruent and the measures of the angles are said to be equal. Now as the hyperbolic rotation may be repeated any number of times without reaching the fixed line, it is possible to find an angle aa(n) which will be n times the angle aa'.
Axiom 15: If a sufficient number of equal angles be laid off about a point from any initial ray, any ray of that class may be surpassed.(Archimedean property for angles)
Axiom 16: In hyperbolic rotation, an area becomes an equal area.
Definition: Given point P there is a Q on the line of OP, on the other side of O from P, such that OP = OQ. The locus to the points P and Q under hyperbolic rotations is a pseudo-circle.
In ordinary geometry the psuedo-circle is identified to be a hyperbola.
Proposition 12: The tangent to a pseudo-circle lies between the curve and its center, and the portion of the tangent intercepted between the two fixed lines is bisected at the point of tangency.
>>Hyperbolic orthogonality:
Definition: In a pseudo-circle the radius and the tangent at its extremity are hyperbolic-orthogonal.
In virtue of proposition 12 we may say the hyperbolic-orthogonal line from any point O to any non-singular line is the line from O to the middle point of that segment of the line which is intercepted by the fixed lines through O.
Proposition 13: If line a is hyperbolic-orthogonal to line b, then b is hyperbolic-orthogonal to line a.
Proposition 14: Through any point one and only one hyperbolic-orthogonal line can be drawn to any line.
Proposition 15: All lines hyperbolic-orthogonal to the same line are parallel.
Proposition 16: The singular line of one class which is drawn through the intersection of any two hyperbolic-orthogonal lines will bisect the segment intercepted by these lines upon any singular line of the other class.
Proposition 17: The hyperbolic-orthogonal line to a (γ)-line is a (δ)-line, and vice-versa.
>>Intervals of opposite class:
Intervals along lines of class (δ) cannot be compared by congruence with intervals along lines of the (γ) class. For resolution, the following convention is established: If two hyperbolic-orthogonal lines are drawn from any point and terminate on a singular line, the intervals of these lines will be said to be equal.
Note that under hyperbolic rotation two hyperbolic-orthogonal lines approach eachother, and the fixed line between them, scissor-wise.
Definition: A triangle of which two sides are hyperbolic-orthogonal will be called a right triangle, and the third side will be called the hypotenuse.
Definition A parallelogram of which the two adjacent sides are hyperbolic-orthogonal and of equal interval will be called a square.
Proposition 18: One diagonal of every square is a singular line and the other diagonal is a singular line of the other class.
Proposition 19: (cf. Pythagorean Theorem) The area of the square on the hypotenuse of a right triangle is equal to the difference of the areas of the squares on the other two sides.
Proposition 20: Any two squares whose sides are of unit interval are equal in area.
Corollary: The area of any rectangle is the product of the intervals of two adjoining sides.
Proposition 21:The square of the interval of the hypotenuse of a right triangle is equal to the difference in the squares of the intervals of the other two sides.
Corollary: The hyperbolic-orthogonal line from a point to a line has a greater interval than any other line of the same class drawn from the given point to the given line.
Definition: The unit angle will be taken as that angle which, in a pseudo-circle of unit radius, encloses a sectorial area of one-half the unit area.
The authors of the 1912 report excerpted here are Gilbert N. Lewis and Edwin Bidwell Wilson. Two significant changes in terminology have been made: rotation --> hyperbolic rotation and perpendicular --> hyperbolic-orthogonal. The tension of adapting a well-worn term to new use seems too much to expect of contemporary readers. The text has been contracted, proofs omitted, diagrams have yet to be supplied. Original text: Proceedings of the American Academy of Arts and Sciences 48:387-507. The above passage occurs between pages 393 and 405. The article title is "The space-time manifold of relativity: The non-Euclidean geometry of mechanics and electromagnetics."
Sir James Cockle Homepage James Cockle as inventor of tessarines produced an analytic tool for geometry, a tool so intriguing that foundational questions arise about this geometry. The non-Euclidean process given in synthetic spacetime settles some of the quandry.