Three Hypotheses of Giovanni Saccheri and Revised Three Hypotheses of Johann Lambert - Report Example

Three Hypotheses of Giovanni Saccheri and Revised Three Hypotheses of Johann Lambert

Giovani Saccheri Girolamo was an Italian Jesuit and a great mathematician. Saccheri’s interest in geometrical mathematics was by one of his teachers Tommaso Ceva. Ceva had lots of interest in geometry of triangles. This interest was also sparked in Saccheri who published his first books Quaesita geometrica where he gave elementary and coordinate geometry solutions to several problems. Saccheri also wrote Logica demonsstrativa where he dealt with questions relating to compatibility of definitions. Logica Demonstrativa was founded on a given form of reasoning as used by Euclid where, through the assumption as hypothesis that the proposition that is to be proved is false, one could conclude that it was true. A major assumption by Euclid is that straight line is infinite. This assumption is crucial to note prior to the exposition of Saccheri’s work.

Saccheri also wrote the book on Neo-statica that was inspired by Ceva’s De natura gravium. As a precursor to non-Euclidean geometry, Saccheri wrote Euclides ab omni naevo vindicates that contains classic notes on Euclidean geometry. Throughout his life, Saccheri was intrigued by Euclid’s fifth postulate. It is for that reason that he wrote the book Logica where he investigated the nature of definitions and another book Euclides where he attempted to apply his logic to prove the extent of correctness of the fifth postulate. Among the consequences of the assumption that the Fifth Postulate is false, Saccheri sought for some propositions that would enable him to confirm that the postulate itself was true.

Today, much of Saccheri’s logical reasoning in mathematics is part of non-Euclidean geometry and mathematical logic. In Euclides Saccheri used his three “Blemishes” logical principle in the book Elements. With his interest on Euclid’s first twenty-six propositions, Saccheri assumed that the Fifth principle was false, while all the rest were true. Through the assumption that the fifth principle was false, Saccheri developed a hypothesis to work with. Saccheri sought a proposition to examine the hypothesis. The testing of the postulate was only possible in the now called Quadrilateral. This paper seeks to evaluate the three hypotheses of Saccheri and investigate the revised three hypothesis by Lambert.

Saccheri’s three hypotheses

In testing the fifth postulate, Saccheri used an isosceles birectangular quadrilateral. He stated that, the quadrilateral constituting of side AB, and two sides of equal length, AD and BC that are perpendicular to AB. Without the fifth postulate, there lacks proof that the angles at D and C are right. However, there is proof that the two angles are equal if when a line MP is drawn through the midpoint M of AB perpendicular to AB, the line DC is intersected at its midpoint, P. Therefore, there are three possibilities that give rise to three hypotheses:

• Right angle: angle C = angle D = 1 right angle
• Obtuse angle: angle C = angle D > 1 right angle
• Acute angle: angle C = angle D < 1 right angle

Saccheri went ahead to prove that when each of these hypothesis is true in just one case, it is true in only case, and it is true in every other case. That is, in the first case, the sum of the angles of a triangle is equal to, in the second, the sum is greater than, and in the third case, and it is less than two right angles. In order to proof the three hypotheses, Saccheri required the axiom of Archimedes and the principle of continuity. The crucial finding from Saccheri was the proof that for both the hypothesis of the right angle and that for both the hypothesis of the right angle and that of the obtuse angle, the fifth postulate holds. However, the fifth postulate implied that the hypothesis of the right angle, thus the hypothesis of the obtuse angle is false. Consequently, Saccheri could not dispose the hypothesis of the acute angle in such a manner. However, Saccheri was not able to show that it led to the existence of asymptotic straight lines and he concluded, was repugnant to the form of a straight line. From the findings, Saccheri believed he had discovered the truth regarding the hypothesis of the right angle, and consequently, the fifth postulate of the Euclidean geometry as a whole.

Revised Three Hypotheses of Johann Lambert

Lambert was one of the many geometors influenced by Saccheri’s work in the 18th century. His work on Theorie der Parallellinien was published in after the death of the author and was divided into three. The first part was philosophical and critical in nature and dealt with the two-fold question resulting from the fifth postulate. First question was whether the fifth postulate could be proved with the aid of preceding propositions only. Second question was whether there was need for the assistance of some hypothesis. The second part of would deal with the discussion of various attempts where the Euclidean postulate reduces to very simple propositions that need to be proved. Thirdly, Lambert’s work contained an investigation similar to that of Saccheri.

Like Saccheri, Lambert used a fundamental figure that was a quadrilateral with three right angles. Lambert then made three hypotheses regarding the nature of the fourth angle. The first hypothesis regards the right angle, the second regards the obtuse angle, and the third regards the acute angle. Lambert also made use of Saccheri’s method in his treatment of the hypotheses. His proof of the fourth hypothesis led to the Euclidean system. Lambert worked by rejecting the second hypothesis. The rejection of the second hypothesis led to the reliance of a figure formed by two straight lines a, b, perpendicular to a third line AB as shown in the figure below.

Figure 1: Lambert’s figure

Starting from points B, B1, B2….Bn in succession upon point b, the perpendiculars, BA, B1A1, B2A2… BnAn meet line a. Lambert proved that the perpendiculars disappear beginning with BA while the difference between each and the one succeeding it increase continually, again beginning with perpendicular BA. Additionally, the difference between one perpendicular and the one succeeding it increase continually as below.

BA - BnAn > n(BA - B1A1)

However, where n is sufficiently large, the value BnAn becomes great enough forming the postulate of Archimedes, and BA is always greater than B1A1. This contradiction makes it possible for Lambert to proof that the second hypothesis was false.

Lambert used figure 1 above to exam the third hypothesis. His prove showed that the perpendiculars BA, B1A1, B2A2… BnAn continued to increase while the variation between each perpendicular and the one proceeding it continued to increase. Consequently, there was no contradiction for the third hypothesis, and unlike Saccheri, Lambert proceeded with his argument. Lambert’s further analysis of the third hypothesis led to the finding that on the third hypothesis the sum of the angles of a triangle was less than two right angles. Lambert’s findings also included the defect of a polygon where the variation between n(n-2) right angles and the summation of the angles were equivalent to the area of the polygon. Other major discoveries by Lambert were with reference to the measurement of geometrical magnitudes. Other major discoveries by Lambert were the relative and absolute measurements of geometric magnitude. Lambert discovered that each segment of measurement could be associated with definite angle whose construction was easy. The absolute measure of the areas of polygons is given once by the defect of polygons, and the denial of the existence of the absolute unit for segments leads to the rejection of the third hypothesis. However, the findings did not succeed in proving the fifth postulate. In order to obtain the wished-for proof, Lambert proceeded with investigating the outcomes of the third hypothesis, only to transform his questions into more complex and difficult to answer questions.

In summary, Saccheri and Lambert’s investigations did not qualify as proof for the impossibility of demonstrating the Euclidean hypothesis. The two would neither offer proof for the Euclidean hypothesis if research proceeded along those lines. Nevertheless, no geometer can proceed to proof the Euclidean Hypothesis without the presumptions by Saccheri since contradictions would be found there. The contradiction is that Saccheri’s presumptions led to the discovery that the Euclid Postulate could not be proved, while resulting in the formation of the non-Euclidean geometries.