next up previous contents index
Next: Gauss Up: The Work of Saccheri Previous: The Work of Saccheri

Saccheri

Girolamo Saccheri was a Jesuit priest living from 1667 to 1733. Before he died he published a book entitled Euclides ab omni nævo vindicatus ( Euclid Freed of Every Flaw). It sat unnoticed for over a century and a half until rediscovered by the Italian mathematician Beltrami.

He wished to prove Euclid's Fifth Postulate from the other axioms. To do so he decided to use a reductio ad absurdum argument. He assumed the negation of the Parallel Postulate and tried to arrive at a contradiction. He studied a family of quadrilaterals that have come to be called Saccheri quadrilaterals . Let S be a convex quadrilateral in which two adjacent angles are right angles. The segment joining these two vertices is called the base . The side opposite the base is the summit  and the other two sides are called the sides . If the sides are congruent to one another then this is called a Saccheri quadrilateral. The angles containing the summit are called the summit angles .

Theorem 11.1:  In a Saccheri quadrilateral

figure2857

Proof: Let M be the midpoint of AB and let N be the midpoint of CD.

  1. We are given that
    displaymath15324
    Now, tex2html_wrap_inline15336 and tex2html_wrap_inline15338, so that by SAS tex2html_wrap_inline15340, which implies that tex2html_wrap_inline15342. Also, since tex2html_wrap_inline15344 then we may apply the SSS criterion to see that tex2html_wrap_inline15346. Then, it is clear that tex2html_wrap_inline15348.
  2. We need to show that tex2html_wrap_inline15350 is perpendicular to both tex2html_wrap_inline12754 and tex2html_wrap_inline14474. Now tex2html_wrap_inline15356, tex2html_wrap_inline15336, and tex2html_wrap_inline15348. Thus by SAS tex2html_wrap_inline15362. This means then that tex2html_wrap_inline15364. Also, tex2html_wrap_inline15366 and tex2html_wrap_inline15368. By SSS tex2html_wrap_inline15370 and it follows that tex2html_wrap_inline15372. They are supplementary angles, hence they must be right angles. Therefore tex2html_wrap_inline15350 is perpendicular to tex2html_wrap_inline12754.

    Using the analogous proof and triangles tex2html_wrap_inline15378 and tex2html_wrap_inline15380, we can show that tex2html_wrap_inline15350 is perpendicular to tex2html_wrap_inline14474.

Thus, we are done.

Saccheri considered all possible cases of such a quadrilateral. They are:
tex2html_wrap_inline15122 is a right angle HRA, the hypothesis of the right angle.
tex2html_wrap_inline15122 is an obtuse angle HOA, the hypothesis of the obtuse angle.
tex2html_wrap_inline15122 is an acute angle HAA, the hypothesis of the acute angle.

We shall see that HRA is equivalent to Euclid's Postulate V, so we may take HOA or HAA as negations of Postulate V. The Three Musketeers Theorem implies that if one of HRA, HOA, or HAA holds for one quadrilateral, then it holds for all.

Theorem 11.2:  In a Saccheri quadrilateral tex2html_wrap_inline15392 on the base AB under the assumption HRA, HOA, or HAA we have tex2html_wrap_inline15396, AB>CD, or AB<CD, respectively, and the angle sum of a triangle is equal to, greater than, or less than two right angles, respectively.

Proof: Let M and N denote the midpoints of AB andCD, respectively. We will work with the assumption HAA, since the others are similar or already known. We wish to show that AB<CD.

Suppose CD<AB, then CN<BM. There is a point tex2html_wrap_inline15416 so that tex2html_wrap_inline15418 and tex2html_wrap_inline15420. Thus, tex2html_wrap_inline15422 must be greater than a right angle, since tex2html_wrap_inline13316 is between tex2html_wrap_inline15426 and tex2html_wrap_inline15428.

tex2html_wrap_inline15430 is a Saccheri quadrilateral on the base NM, which implies that tex2html_wrap_inline15434, which implies that tex2html_wrap_inline15436 is greater than a right angle. In tex2html_wrap_inline15438 tex2html_wrap_inline15440 is an exterior angle and hence is greater than tex2html_wrap_inline15436. Recall that by hypothesis, tex2html_wrap_inline15440 is an acute angle. Thus, we have an acute angle greater than an angle that is greater than a right angle, our contradiction. Therefore, AB<CD.

Now, suppose we have a right triangle tex2html_wrap_inline11270 with right angle at B. Construct AD perpendicular to AB, with tex2html_wrap_inline15456. Assuming HAA, CD>AB and tex2html_wrap_inline15460 since the greater side subtends the greater angle. Now, tex2html_wrap_inline15462 is a right angle. Thus, the angle sum of tex2html_wrap_inline11270 is less than two right angles.

Consider two parallel lines k and tex2html_wrap_inline11154, and let tex2html_wrap_inline15470 and tex2html_wrap_inline15472. Let tex2html_wrap_inline15474 and construct lines perpendicular to tex2html_wrap_inline12754 through X and Y. Call these lines tex2html_wrap_inline15482 and tex2html_wrap_inline15484.

figure2922

We are interested in the angles tex2html_wrap_inline15502 and tex2html_wrap_inline15504. If both angles are acute, then the two lines tex2html_wrap_inline12754 and tex2html_wrap_inline14474 have a common perpendicular between the segments UX and VY. If one of the angles is a right angle and the other is acute, then the two lines already have a common perpendicular. Suppose that tex2html_wrap_inline15502 is acute and tex2html_wrap_inline15504 is obtuse. If we move V away from U along tex2html_wrap_inline12754 then tex2html_wrap_inline15504 may change to a right angle or remain obtuse.

Theorem 11.3: As above and assuming HAA, if as V moves away from U along tex2html_wrap_inline12754, tex2html_wrap_inline15504 remains obtuse, then tex2html_wrap_inline14474 is asymptotic to tex2html_wrap_inline12754.

Saccheri now shows the following:

Theorem 11.4: Given a point P not on a line tex2html_wrap_inline11154, there are three classes of lines through P:

  1. lines meeting tex2html_wrap_inline11154,
  2. lines with a common perpendicular to tex2html_wrap_inline11154, and
  3. lines withour a common perpendicular to tex2html_wrap_inline11154 and hence asymptotic to tex2html_wrap_inline11154.

Saccheri now concludes ``the hypothesis of the acute angle is absolutely false; because it is repugnant to the nature of straight lines.''

The flaw in Saccheri's work was observed by the Swiss mathematician J.H. Lambert in 1786. He himself contributed to the work in non-euclidean geometries with the following.

A convex quadrilateral three of whose angles are right angles is called a Lambert quadrilateral .

Under the HAA assumption the following are true.

Theorem 11.5:  The fourth angle of a Lambert quadrilateral is acute.

Theorem 11.6:  The side adjacent to the acute angle of a Lambert quadrilateral is greater than its opposite side.

Theorem 11.7:  In a Saccheri quadrilateral the summit is greater than the base and the sides are greater than the altitude.

Proof: Using Theorem 12.1 if M is the midpoint of AB and N is the midpoint of CD, then tex2html_wrap_inline15560 is a Lambert quadrilateral. Thus, AB>MN and, since tex2html_wrap_inline15564, both sides are greater than the altitude.

Also, applying Theorem 12.1 DN>AM. Since tex2html_wrap_inline15568 and tex2html_wrap_inline15570 it follows that CD>AB, so that the summit is greater than the base.

next up previous contents index
Next: Gauss Up: The Work of Saccheri Previous: The Work of Saccheri

droyster@math.uncc.edu