By Dr. Wen-tsün Wu (auth.)

There looks doubtless that geometry originates from such sensible activ ities as climate remark and terrain survey. yet there are diverse manners, tools, and how one can increase some of the stories to the extent of concept so they ultimately represent a technology. F. Engels acknowledged, "The aim of arithmetic is the learn of house kinds and quantitative kin of the true global. " Dur ing the time of the traditional Greeks, there have been various tools facing geometry: one, represented by means of the Euclid's "Elements," only pursued the logical family between geometric entities, apart from thoroughly the quantita tive family, as to set up the axiom process of geometry. this technique has turn into a version of deduction equipment in arithmetic. the opposite, represented by means of the suitable paintings of Archimedes, considering the learn of quantitative re lations of geometric items in addition to their measures equivalent to the ratio of the circumference of a circle to its diameter and the world of a round floor and of a parabolic quarter. although those techniques fluctuate widespread, have their very own good points, and replicate various viewpoints within the improvement of geometry, either have made nice contributions to the improvement of arithmetic. the advance of geometry in China used to be all alongside considering quanti tative relations.

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**Example text**

Q B' B are parallel to each other, by Desargues' axiom Dl the lines OQ, AB', A'B are also parallel to each other or are concurrent at a point P. In the former case, C = C' = 0 = 0'. Hence, a' + b' = 0' = (a + b), and the conclusion is proved. In the latter case, the connecting lines of the corresponding vertices of 44 Fig. 32 Desarguesian geometry Fig. PAA' are concurrent at point 0, and QC II PA, QC' I PA', so CC' II AA'. Therefore CC' I I I' and (a + b)' = a' + b' still holds. Next, we prove that (ab)' = a'b'.

B --"*--4r--:--------l\--- 11 Fig. 6 Number system associated with a plane If II, h 13 lie on the same line, then 14 also lies on this line. In this case, is quite evident. In what follows we assume that II, h 13 are not collinear. Applying Oesargues' axiom 02 to LOllI14 and L02hh one has 1I14 II hh Now draw through a point XI (# 01, II) on line iI, XI X2 II 0102, meeting i2 at X2, through X2, X2X3 I hh meeting i3 at X3 and, through X3, X3X4 II 1)14, meeting i4 at X4. Applying Oesargues' axiom 02 to L04XIX4 and L03X2X3 yields XIX4 I X2X3, and they are parallel to hI) and IIk By definition, we have FII(Xj) = X2, F; (XI) = X4, F[ (X2) = X3, F;[(X4) = X3· In the case XI = 01, II, the above formula is more obvious (X3 = 01, II), so Similarly, if there are two successive isomorphism correspondences F[[ F[ of different kinds, then there are F; and F; [ such that F[[ F[ = F; F; [.

3. In case il is distinct from i3, the action of two successive isomorphism correspondences and 50 Desarguesian geometry ~=--------lk--:---1r-----11 Fig. 40 Fig. 41 of the second kind can be merged into one isomorphism correspondence F" II = F'II FII·. of the second kind. For the proof, see Fig. 41. 4. , 11 = 13. Then the intermediate base line 12 and h can be arbitrarily alternated to another base line 14 and ]4. 6 Number system associated with a plane Fig. 42 where 04 = 01, one should have F'F / /- F"'F" / /.