714 



SCIENCE. 



[N. S. Vol. XIV. No. 358. 



lieved that this method of proof for the 

 equality of area, simple and standing upon 

 a sharp definition, must be renounced, and 

 it has been replaced, as is known, by this, 

 that each of the two parallelograms is rep- 

 resented as the difference between the 

 same trapez and one of two congruent 

 triangles. 



" But before the measurement of plane 

 surfaces by sects has been attained, which 

 just first becomes possible through the 

 theorem to be proven, the application of 

 the above magnitude-theorem is justified 

 by nothing. 



" We must therefore throw away this 

 method of proof, and that so much the 

 more, as in every case each of two parallel- 

 ograms with common base and equal alti- 

 tude in very simple way comprehensible to 

 every scholar can be so cut into a number 

 of parts that to each part of the one 

 parallelogram corresponds a part congruent 

 to it of the other. 



" One may find that, e. g., set forth in 

 ' Stoltz's Vorlesungen ' ueber allgemeine 

 Arithmetik, I. Theil (Leipzig, 1885), S. 

 75 fi". 



" We can still somewhat simplify this 

 method, and lessen the number of parts. 

 Draw, namely, through each of the two 

 end-points next one another of the sides 

 lying opposite the common base, parallels 

 to the sides of the other parallelogram, and 

 prolong these to the two outer of the sides 

 not parallel to the base. The join of the 

 two end-points so obtained is then parallel 

 to the base, and cuts from the two parallel- 

 ograms two new parallelograms which 

 without anything further are divided into 

 triangles every two congruent to one 

 another. 



"If then the sides opposite the common 

 base of the remaining parallelograms again 

 have no common point, then we proceed 

 just so with them, and come thus, after a 

 finite number of repetitions, to a pair of 



parallelograms, to which the customary 

 procedure can be applied. 



" If the distance of those two neighbor- 

 ing end-points of the sides opposite the 

 base is greater than the tifold of the base, 

 on the other hand at the highest equal to 

 the {n + l)fold of the base, then is each 

 parallelogram cut into a trapez (respect- 

 ively triangle), three triangles and n paral- 

 lelograms, and each of such parts of the 

 one parallelogram corresponds to a part 

 congruent to it of the other." Now it so 

 happens that I myself had reached this 

 method and published it seven years before 

 Schur in my ' Elements of Geometry ' (John 

 Wiley & Sons, New York). It may be 

 described more concisely as taking away 

 pairs of congruent triangles each with base 

 equal to the common base of the two paral- 

 lelograms and sides respectively parallel to 

 their other pairs of sides, until we have 

 left two parallelograms to which the cus- 

 tomary dissection into a triangle and trap- 

 ezoid will apply, to finish with congruent 

 parts. 



But this demonstration, though the very 

 simplest possible, yet postulates the Archi- 

 medes axiom, though neither I myself, in 

 1885, nor Schur, seven years later, in 1892, 

 said a word about this assumption. How- 

 ever, before 1898 Schur became conscious 

 that elementary geometry can be built up 

 without the Archimedes axiom. He states 

 this in the preface to his remarkable ' Lehr- 

 buch der analytischen Geometric ' (Leipzig, 

 Veit& Comp., 1898), referring to his article 

 ' Ueber den Fandamentalsatz der projec- 

 tiven Geometric' Math. Annalen, Bd. 51), 

 where he proves the theorems of Desargues 

 and of Pascal without using either the 

 parallel postulate or the axiom of Archi- 

 medes, proving that the ordinary sect- cal- 

 culus can be built up independently of 

 number-measure and the Archimedean pos- 

 tulate. 



Professor Anne Bosworth, of Ehode 



