Line geometry, according to the principles of Grassmann s theory of extensions. By E. Müller in Vienna.

Transcription

1 De Lnengeometre nach den Prnzpen der Grassmanschen Ausdehnungslehre, Monastshefte f. Mathematk u. Physk, II (89), Lne geometry, accordng to the prncples of Grassmann s theory of extensons. By E. Müller n Venna. Translated by D. H. Delphench Although Grassmann had already expressed the dea of employng the straght lne as a spatal element n hs Ausdehnungslehre (Theory of Extensons) of 844, and to that end, exhbted systems of lne coordnates, and also ncluded many thngs that showed how smply lne geometry could be formulated usng hs method n the second edton of hs book n 86, to the best of my knowledge, no one has made such an attempt up to now. The lnes that follow contan a treatment of lne geometry that s based upon the prncples of Grassmann s lne theory of extensons. I therefore hope to gve an example of the smple applcablty of Grassmann s great creaton that has, unfortunately, stll found much too lttle crculaton. In order for me to be as bref as possble, t wll often be necessary to refer to the two edtons of Ausdehnunglehre from 844 ( * ) and 86 n order to explan the concepts that are used; as usual, they shall be denoted by A and A. The lnear ray complex.. If one let A, B,, X, denote lne segments ( ** ) of unt magntude.e., rays n whch segments of unt length and defnte drecton are found then the lne segments a A, b B,, x X, where a, b,, x, are real or complex numbers, wll represent those rays, n whch one must now thnk of the segments as havng lengths a, b,, x,, however. Any algebrac sum of lne segments can be reduced to the sum of two lne segments that have no pont n common, and can thus be represented n the form (A, no. 85): = a A + b B. () Ths expresson s tself capable of no further reducton, and has no mmedate geometrc meanng, although t plays the man role n lne geometry. ( * ) A new edton of ths verson that was edted by Grassmann appeared n 878 that was prnted by O. Wgand n Lepzg. The 86 verson s out of prnt. ( ** ) I prefer the expressons: lne segment, surface segment, space segment of H. Hankel (Theore der complexen Zahlen, Lepzg, 867) to Grassmann s expressons: lne part, surface part, body part (A ) or lne quantty, plane quantty, body space (A ).

2 Müller Lne geometry, accordng to Grassmann. Namely, f we ask what all rays (.e., lne segments) X would be that would gve zero when exteror multpled by, whch would then fulfll the equaton: [ X] = 0, () then t would be easy to see that they would all defne a lnear complex. Then, snce any ray X that goes through a pont p s representable n the form: X = x [p x], n whch x s any pont of X and x means a well-defned number, one must have: [ p x] = [ p x] = 0 for the rays that go through p, or snce [ p] = a [A P] + b [B p] s equal to a plane segment π that goes through p, one must have: [π p] = 0, whch says that every pont x, and thus, also every ray X, wll le n the plane π. The rays X that go through a pont x and satsfy equaton () wll then defne a pencl of rays. One lkewse shows that the rays X that le n the plane π and satsfy equaton () wll also defne a pencl of rays. In fact, snce every ray X that les n a plane π can be represented n the form: X = x [π ξ], n whch x means any plane (.e., plane segment) through X and x means a well-defned number, one must have: [ π ξ] = [ π ξ] = 0 for the rays that le n π, whch wll then satsfy equaton (), or snce [ π] = a [A π] + b [b π] s equal to a multple pont p that les n π, one must have: [p ξ] = 0, whch says that every plane x, and therefore, also every ray X, goes through the pont p. Wth that, we have proved: Any quantty = a A + b B determnes a unque lnear ray complex by equaton (). The fact that, conversely, any lnear ray complex determnes a quantty wll be derved later on (no. ).

3 Müller Lne geometry, accordng to Grassmann. 3 If reduces to a lne segment then all rays X that satsfy equaton () wll have the property that they cut ;.e., wll then determne a specal lnear complex whose axs wll be represented by. By the consderatons above, one can lkewse fnd a smple expresson for the relatonshp n the null system that the lnear complex mples. Namely, any pont p wll correspond to the plane [ p] as the null plane. The fact that the ponts of a lne wll correspond to the planes of a pencl s deduced from that mmedately. Then, f p, p are any two ponts then any pont p on the lne [p p ] wll be representable n the form p = p p + p p ; one obtans the null plane of p by exteror multplyng by : [ p] = p [ p ] + p [ p ], whch s then derved from the null planes of p and p, and thus belongs to the pencl of these planes. The ray [p p ] s then sad to be assocated wth the ray [π π ]. One nfers the converse theorem analogously. A and B are assocated rays n the lnear ray complexes that are defned by equatons () and (); snce [A a] = 0, any pont a of A wll then correspond to the plane: [ a] = a [A a] + b [B b] = b [B a], whch goes through B. Lkewse any pont b of B wll correspond to a plane through A. If one represents n all possble ways as the sum of two lne segments then one wll obtan all pars of assocated rays. The fact that two pars of such lnes A, B; A, B wll le n a ruled famly s deduced mmedately. The equaton: a A + b B = a A + b B wll then follow from the facts that = a A + b B and = a A + b B. If one exteror multples these by a lne X that cuts three of the lnes say, A, B, A then one wll one wll also have [B X] = 0 n the resultng equaton:.e., X can also cut B. a [a X] + b [B X] = a [A X] + b [B X] [A X] = [B X] = [A X] = 0 ;. One can also represent the quantty as the sum of a lne segment and a surface space (.e., an extended quantty of rank two, a lne n the plane at nfnty), and n partcular, as the sum of a lne segment and a surface space that s perpendcular to t. If a denotes a pont, c, a segment of defnte length an drecton (.e., an nfntely-dstant pont), and k c s the extenson of that segment.e., the surface space that s perpendcular to c whose area s equal to the length of c and k s a number then Grassmann (A, no. 346, 347) has proved that can always be put nto the form:

4 Müller Lne geometry, accordng to Grassmann. 4 = [a c] + k c (3) unquely. [a c] s a lne segment that s dentcal wth the axs of the lnear complex, and k s the number that Plücker called the parameter of the lnear complex. If one bases upon ths form then t wll follow from the defnng equaton () of the lnear complex: [ X] = [ac X] + k [ c X] = 0 or k = [ ac X ]. [ c X ] Snce the exteror product of two lne segments of magntude wll be equal to the product of the shortest dstance between them wth the sne of the angle between them, the numerator of ths fracton wll be l sn c^x, f denotes the length of the segment c and l, the dstance from the lne X to the axs. As one mmedately recognzes, the denomnator wll have the value cos c^x; one wll then have: or, n words: k = l tan c^x, The product of the dstance from a ray of a complex to the axs wth the tangent of the angle that the two lnes defne s constant, and ndeed, ts absolute value s equal to the parameter of the complex when the axs and the ray of the complex are arranged n no partcular sense. Snce the latter equaton can also be employed as the defnng equaton of every lnear complex, and one can deduce [ X] = 0 from t, where has the form of equaton (3), t s thus proved that any lnear ray complex can be represented by the equaton (). The theorem above s a specal case of a more general one. Namely, f s taken to have the general form: = a A + b B then one wll have: [ X] = a [A X] + b [B X] = 0 for any complex ray X, so: b a = [ AX ] [ BX ]. If, as usual, one calls the exteror product of two rays of unt magntude vz., the product of the shortest dstance between them wth the sne of the angle between them ther moment then ths theorem wll read:

5 Müller Lne geometry, accordng to Grassmann. 5 The moments of the rays of a lnear complex relatve to any two assocated lnes n t have a constant rato ( * ). One lkewse easly arrves at another well-known theorem, as well as ts generalzaton. Let assume the general form, as before, so one wll obtan the null pont p of the plane π, from no., by exteror multplcaton of wth π: [ π] = a [A π] + b [B π] p. If one calls the smple ponts of ntersecton of π wth A and B, a and b, respectvely, when π s assumed to have magntude then: so from whch t wll follow that: or In words, ths reads: [A π] = a sn A^π, [B π] = b sn B^π, p a A^π a + b B^π b, [ a p] [ b p ] = bsn B π a sn A π, [ a p] sn A π = b [ b p]sn B π a. If the null plane of a pont p meets two assocated lnes A and B of the complex at the ponts a and b, respectvely, then the rato of the products of the dstances from the pont p to the ponts a and b wth the sne of the angle that the correspondng lnes defne wth the null plane of the pont wll have the same value for all ponts of space. If has the specal form (3) then the last equaton wll express the known theorem: The product of the dstance from a pont to the axs of a lnear ray complex wth the tangent of the angle that the null plane of the ponts makes wth the axs has the same value for all ponts n space, and s, n fact, equal to the parameter of the complex. Remark. The defnton of lnear ray complex that was gven here s only the expresson of a known mechancal property of t. The quantty, as the sum of lne quanttes, can, n fact, be consdered to be a force system that acts upon a rgd body (A, ), and [ X] s ts statc moment relatve to the axs X: [ X] = 0 then says that the force system possesses a zero statc moment relatve to all of the rays that satsfy ths equaton. The remanng equatons that appear above also have mmedate mechancal nterpretatons. ( * ) Drach, Math. Ann., Bd. II.

6 Müller Lne geometry, accordng to Grassmann. 6 Numercal relatonshps between lnear ray complexes. Exteror products of them. 3. We have seen that any quantty determnes a unque lnear complex by the equaton [ X] = 0, and that every lnear ray complex can be determned by an equaton of ths form. The lnear ray complex that s determned by shall be denoted by. For any ray X that does not satsfy the equaton above, one wll have [ X] 0 and [ X] shall be called the moment of complex wth the ray X or the moment of the ray X relatve to the complex. We say of n complexes that a numercal relatonshp: n a = 0 = arses f and only f the same relaton exsts between ther moments wth any ray X, so the equaton: n a [ X ] = 0 = wll be fulflled for every ray X. Due to the fact that: n n a [ X ] = [ a X], = = the assumpton above wll be fulflled only when one has: n a = 0, = so when the same numercal relatonshp exsts between the quanttes. onversely, t s easy to see that the numercal relatonshp equaton. One then has the theorem: A numercal relatonshp of the form: n a = 0 wll also follow from the last = n a = 0 =

7 Müller Lne geometry, accordng to Grassmann. 7 arses between n lnear ray complexes f and only f the same relatonshp exsts between the quanttes. 4. If a numercal relatonshp: n a A = 0 = arses between any n quanttes A then: A r = a r r + n A a A a A a A + a A a a a a a r r n r r r r r wll be numercally dervable from any of the remanng ones. If one magnes a tetrahedron as beng chosen n such a way that each pont of space can be derved from ts vertces then any ray, and therefore, also any quantty, wll be numercally dervable from the edges of the that tetrahedron (A, no. 346, A, 7). It follows mmedately from ths that any quantty can be derved from any sx quanttes that satsfy no numercal relatonshp (A, no. 4). However (no. 3), any lnear complex s then numercally dervable from any sx lnear complexes that satsfy no numercal relatonshp, or what amounts to the same thng: The lnear ray complexes defne a doman of rank sx. All lnear ray complexes that are dervable from n (n =,, 3, 4, 5) mutuallyndependent lnear ray complexes defne a complex doman of rank n. The domans of rank two, three, four, fve shall also be called pencls, sheaves, bushes, webs, respectvely. 5. From the dscusson above, the concept of exteror product can be appled to lnear ray complexes mmedately. The exteror product [ n ] (n =,, 6) of n lnear ray complexes represents the doman that they determne (A, no. 70, rem.), up to a numercal value that s equal to the determnant of the numbers by whch n can be derved from n complexes whose exteror product can be assumed to be unty. The exteror product of n complexes s zero f and only the n complexes defne a doman of rank lower than n, or what amounts to the same thng as long as a numercal relatonshp exsts between them. Snce the complexes of a pencl have a lnear congruence n common, and any such congruence determnes a pencl of complexes that goes through them, one can also say that the exteror product of two complexes represents ther common congruence. Lkewse, the exteror product of three lnear ray complexes represents ther common ruled famly, and the exteror product of four lnear ray complexes represents ther common par of rays. However, any other exteror product wll lkewse determne just

8 Müller Lne geometry, accordng to Grassmann. 8 one numercal value, and two exteror products are equal when and only when they represent the same geometrc object, as well as also determne the same numercal value. The concept of the regressve product (A, hap. 3, A, 5) can be appled here mmedately. The nner product of lnear ray complexes. 6. If a lnear ray complex s gven then the postve square root of [ ] shall be called ts numercal value. (The bass for ths term wll be gven later.) A lnear ray complex s specal whenever represents a lne segment. Ths happens f and only f [ ] = 0 (A, no. 86, A, 4). A specal complex s then one wth a numercal value of zero. Snce the lnear complex does not change when one multples ts quantty by an arbtrary number, one can always put nto a form that has the numercal value of ; to that end, one needs only to multply by / [ ]. Let, be any two lnear ray complexes, so each complex of the pencl [ ] can be represented n the form: = a + a ; snce one then has: = a + a, the condton for to be a specal complex wll be: 0 = [ ] = [(a + a ) (a + a )] = a [ ] + a a [ ] + a [ ], whch s an equaton that wll yeld two values for a / a. There are then two specal complexes n any pencl of complexes. If A, B are the specal complexes of a pencl, and, are any two complexes n t then one wll have: = u A + v B, = u A + v B. The double rato of these four complexes s then (A, 65): [ A] [ A] : [ B ] [ B ] v v = : u u v v u u. = : From the correspondng equatons for the quanttes : = u U + v V,

9 Müller Lne geometry, accordng to Grassmann. 9 = u U + v V, t follows by multplyng by a plane π that: or, when one has: that: [ π] = u [U π] + v [V π], [ π] = u [U π] + v [V π], [ π] = p, [ π] = p, [U π] = u, [V π] = v, p = u u + v v, p = u u + v v. The double rato of these four ponts: [ u p] [ u p] : [ v p ] [ v p ] v u v u = : then has the same value as the double rato of the correspondng complex. In partcular, f ths double rato has the value then, accordng to F. Klen s termnology, the two complexes and wll le n nvoluton. The condton for that s then: u v + u v = 0. However, from the above, t wll follow that: [ ] = [(u U + v V) (u U + v V)] = (u v + u v ) [U V], snce one has [U U] = [V V] = 0 (by assumpton). Therefore, f the two complexes, le n nvoluton then one must have: [ ] = 0. onversely, as long as ths equaton s vald, f [U V] 0 then one must have u v + u v = 0, so the two complexes must le n nvoluton. [U V] 0 corresponds to just the requrement that the axes of the two specal complexes of the pencl should not ntersect, so the pencl of complexes wll not be a pencl of specal complexes, and, wll not be specal complexes. If we assume that for the case n whch and are specal complexes they wll be sad to le n nvoluton f and only f ther axes ntersect then one can express the followng theorem n full generalty: The necessary and suffcent condton for two complexes, to le n nvoluton s that [ ] = 0.

10 Müller Lne geometry, accordng to Grassmann All complexes Y that le n nvoluton wth a complex wll be determned by the equaton: [ Y] = 0, (4) f Y refers to the sum of lne segments that belongs to Y ( * ). If one derves Y from any sx mutually-ndependent complexes Y say, Y = 6 3) Y = y Y, and the equaton above wll then read: = 6 y Y then one wll also have (no. = 6 y [ Y] = 0, = whch then expresses a lnear relaton between the dervng numbers y. However, as s easy to see, each of the complexes Y that satsfy equaton (4) can be derved from fve mutually-ndependent complexes, or belong to a doman of rank fve. Furthermore, any complex Y that belongs to ths doman of rank fve wll le n nvoluton wth. Thus, f Y, Y,, Y 5 are any fve mutually-ndependent complexes that le n nvoluton wth then Y wll be numercally dervable from them; perhaps: and therefore: Y = Y = 5 y Y, = 5 y Y. = Upon multplyng by, one wll obtan: [Y ] = 5 y [ Y ] = 0, = snce all products [Y ] are zero, by assumpton. Therefore, any complex Y les n nvoluton wth, and one can state the theorem: All lnear ray complexes that le n nvoluton wth a lnear ray complex defne a doman of rank fve. It follows from ths mmedately (A, 6) that: ( * ) It follows from ths equaton that the axes of all specal complexes that le n nvoluton wth wll defne the rays of the complex.

11 Müller Lne geometry, accordng to Grassmann. The lnear complexes that le n nvoluton wth n lnear ray complexes (n =,,, 5) defne a doman of rank 6 n. One can then also choose sx lnear ray complexes, any two of whch le n nvoluton. 8. If we choose sx lnear ray complexes,,, 6 wth numercal values of unty, any two of whch le n nvoluton, to be the orgnal unts and set ther exteror product equal to one, so: [ ] =, then the necessary and suffcent condtons for the defnton of the nner product are met, and one can all mmedately apply all of the theorems that Grassmann derved n A, hap. 4 here. However, one must establsh the meanngs that concepts lke normal, numercal value, value that were presented there for arbtrary lnear manfolds n general mght have for the geometry of lnear ray complexes. We would frst lke to prove that two complexes that le n nvoluton are dentcal wth two normal complexes. We wll have that proof when we have shown that: Any sx complexes wth numercal values of unty, any two of whch le n nvoluton, can be derved from the complexes that were chosen to be the orgnal unts by repeated crcular alteraton. Then, snce the orgnal unts are sx quanttes that are normal to each other (A, no. 6) and, n turn, only mutually-normal quanttes wll emerge by crcular alteratons (A, no. 55), any sx complexes that are recprocally n nvoluton must also be sx mutuallynormal quanttes. In order to prove the theorem we need the lemma that: Two complexes wth numercal values of unty that le n nvoluton to each other wll go to two such complexes under crcular alteraton. If, go to, under crcular alteraton then, f the complexes are assumed to have numercal values of unty and a + a =, one wll have (from A, no. 54): so m = a + a, n = a a, m = a + a, (µ) n = a a. (ν)

12 Müller Lne geometry, accordng to Grassmann. In order to fnd the value of m, one exteror multples each sde of equaton (µ) by tself. One wll then get: [ ] = m and snce, by assumpton, one wll have: a [ ] + a a [ ] + a [ ], one wll then have: [ ] = [ ] = [ ] =, [ ] = 0, m = a + a =. One wll smlarly obtan from equaton (ν) the fact that: n =. The complexes that emerge from and by crcular alteraton wll then possess the numercal value of unty, and equatons (µ) and (ν) wll then read: = a + a, = a a. If one exteror multples them wth each other then t wll follow that: [ ] = ( a a ) [ ] a a ([ ] [ ]), so, snce [ ] = 0 and [ ] = ([ ] =, one wll have: [ ] = 0;.e., the complexes, le n nvoluton. The man theorem vz., that any sx complexes,,, 6 wth numercal values of unty, any two of whch le n nvoluton, can be derved from sx complexes,,, 6 that are chosen to be the orgnal unts by repeated crcular alteraton can now be proved n the followng way: From A, no. 60, the system of orgnal unts can be changed crcularly n such a way that one of them say, concdes wth an arbtrary quantty of rank one say,. Snce, from the lemma that was just proved, any two complexes wth numercal values of unty that are found to be n nvoluton wll agan go to other such complexes under any crcular alteraton, the orgnal unts wll go to sx complexes that mutually le n nvoluton, so the orgnal fve wll le n nvoluton wth. From no. 7, these fve complexes then belong to the doman [ 3 6 ], and, from A, no. 60, they can therefore be once more crcularly altered n such a way that one of

13 Müller Lne geometry, accordng to Grassmann. 3 them wll concde wth say, whereby the four remanng ones wll le n the doman [ ]. If one repeats ths argument then one wll ultmately arrve at the realzaton that the orgnal unts,, 6 can actually go to,, 6 under crcular alteraton. Snce the numercal values of the complexes reman unchanged by that so they wll reman equal to one we can say: Any sx ray complexes wth numercal values of unty that mutually le n nvoluton defne a complete, smple, normal system (n Grassmann s termnology). Snce any two complexes wth numercal values of unty that le n nvoluton can be regarded as a part of a complete, smple, normal system, they wll defne two mutuallynormal quanttes of rank one, and conversely, any two normal quanttes n the doman that we spoke of wll defne two complexes that le n nvoluton wth each other, snce (A, no. 6) they can be derved from the orgnal unts by crcular alteraton. The concept of normal, lnear ray complexes thus overlaps wth that of complexes that le n nvoluton. In the sequel, we would ordnarly lke to speak of normal, lnear complexes. It follows from the foregong that: The nner product of two normal complexes s zero, and conversely. 9. The general defnton of numercal value that was gven by Grassmann overlaps wth the one n no. 6 for lnear ray complexes. If one lets [ ] =, frst of all, then (from no. 8) the complex wll emerge from the orgnal unts by crcular alteraton, so, from Grassmann (A, no. 55), t wll lkewse have the numercal value of unty;.e., one wll then have = ( * ). If, moreover, [ ] then one can set: = [ ], where one now has [ ] =, and then, from no. 3, the complexes and that correspond to the quanttes and wll then have the relatonshp: so or = [ ], = [ ] = [ ], snce =, ( * ) We apply the smple square sgn wthout a prme as the sgn of the nner square, snce no confuson s possble here.

14 Müller Lne geometry, accordng to Grassmann. 4 = [ ] ;.e., the numercal value of, accordng to Grassmann s defnton, agrees wth the defnton n no. 6. Snce a specal complex has a numercal value of zero, one can also call t a null complex. Snce t follows from A, no. 68 that all theorems that are true for the orgnal unts wll also reman vald for the quanttes of a complete, smple, normal system, the numercal value of an exteror product of n (n =, 3,, 6) complexes of numercal value unty that are normal to each other wll by unty, n any case. 0. Another mportant concept s that of extenson. If,, 6 are the complexes of a smple, normal system so [ 6 ] = then the extenson of a product A of α of these complexes s equal to the product B of the remanng (6 α) complexes, endowed wth a sgn that wll make [A B] = + (A, no. 67). In order to obtan the extenson of any other quantty A of rank α, one assumes that one has a complete, smple, normal system such that α of ts unts come to le n the doman of A. If A s ther exteror product, and a s the numercal value of A then one wll have: so A = a A, A = a A (A, no. 9), n whch A denotes the extenson of A, so, from the above, t wll be the product of (6 α) complexes that are normal to each other and to all complexes of A, and therefore also A. If one calls the doman of rank (6 α) that s defned by all complexes that are normal to all complexes of a doman A of rank α the extended doman of A then one can say: The extenson of a quantty A s a quantty that belongs to the extended doman of A whose numercal value s equal to that of A. Snce any quantty n one of two extended domans s normal to any quantty of the other one, (from no. 7, remark) all complexes of two extended complex domans must go through the axes of the specal complexes n the other one. The extenson of an exteror product of two lnear ray complexes, whch from no. 5, wll be a lnear congruence, represents the complex doman of rank four that s normal to t, so, from no. 5, t wll be a lne par. Snce, from what we just sad, all complexes n ths doman of rank four wll go through the two drectrces of that congruence, the par of drectrces wll then be represented by the exteror product of any four complexes of the doman, and one can say:

15 Müller Lne geometry, accordng to Grassmann. 5 The extenson of a lnear ray congruence s ts par of drectrces. One lkewse fnds that: The extenson of a ruled famly s ts gudng lne. These two theorems can also be expressed n the followng form: The axes of the specal complexes of a complex bush defne a ray congruence, and The axes of the specal complex of a sheaf of complexes defne a ruled famly. ( * ). We ask what the number would be that the nner product of two complexes, determnes. If, are two complexes wth numercal values of unty that belong to the pencl [ ] then one wll have the relatons: = a + a, = a + a, so f one nteror multples (A, no. 43) the two equatons then one wll get: snce = = and [ ] = 0. However, one fnds the equatons: [ ] = a a + a a, = a + a, = a + a between the quanttes, n any case. If one exteror multples them then one wll obtan [ ] = a a + a a, snce [ ] = [ ] = and [ ] = 0. Therefore: or [ ] = [ ], ( * ) Fve complexes determne a web of complexes and a complex that s normal to them that s defned by the axes of the specal complexes of that web.

16 Müller Lne geometry, accordng to Grassmann. 6 The nner product of two lnear ray complexes s equal to the exteror product of ther quanttes. If, are any two lnear ray complexes then [ ] shall be called the moment of the two complexes relatve to each other ( ** ). One can then also say: The nner product of two lnear ray complexes s equal to ther moment relatve to each other. Two complexes are therefore normal as long as ther moment relatve to each other s zero.. The angle between two lnear ray complexes. Grassmann (A, no. 95) understood the angle AB between two quanttes A and B of equal rank, whose numercal values were a and b, to mean the angle between 0 and π whose cosne was equal to the nner product of those quanttes, dvded by the numercal values;.e., he set: cos AB = [ A B] ab. Therefore, for two lnear ray complexes, one wll have: [ ] cos ^ = = [ ] [ ][ ]. Ths expresson for the angle between two lnear complexes agrees wth the other defntons that are gven for t. In partcular, one can prove that arc cos ^ s equal to the logarthm of the double rato, multpled by /, that the two complexes defne wth the specal complexes n ther pencl. The proof of ths s acheved n the followng way: If, are two complexes wth unty numercal values that defne the angle γ wth each other then a + a wll represent a specal complex as long as one has (a + a ) = 0. Ths gves the equaton: or, snce: = a + a a [ ] + a = 0, = and [ ] = cos ^ = cos γ a + a a cosγ + a = 0. ( ** ) See F. Klen, Math. Ann., Bd. II, page 368.

17 Müller Lne geometry, accordng to Grassmann. 7 Ths equaton gves the two values e γ, e γ for a / a ; the two null complexes A, B of the pencl can thus be represented n the form: from whch, t wll follow that: A = e γ, B = e γ, [ A] [ ] A = eγ, [ B] [ B ] = [ e γ A], so [ A ] : [ B] [ B ] = eγ, If δ s the value of the double rato, so δ = e γ, then ( * ): γ = log δ. One also arrves very easly at the expresson for cos ^ that F. Klen found. If we assume that the two complexes, are gven by: = a A + a A, = a A + a A, n whch (see no. ) A, A should denote the axes of the two complexes wth magntude unty and A, A should denote the surface spaces that are normal to them, whch lkewse have magntude, so a / a = k, a / a = k wll be the parameters of the two complexes, then, due to the fact that [ A A ] = 0, one wll obtan: [ ] = [ ] = a a [A A ] + a a [A A ] + a a [A A ]. If the angle between the two axes s denoted by ϕ and the shortest dstance between them by then one wll have: so [A A ] = sn ϕ, [A A ] = [A A ] = cos ϕ, [ ] = a a sn ϕ + (a a + a a ) cos ϕ. Snce one further has: one wll have ( ** ): = [ ] = a a, = [ ] = a a, ( * ) f., Lndemann, Math. Ann., VII, pp. 66.

18 Müller Lne geometry, accordng to Grassmann. 8 [ ] cos ^ = = aa sn ϕ + ( aa + aa)cosϕ a a a a = a a snϕ + + cosϕ a a a a a a sn ϕ + ( k + k ) cosϕ. k k = 3. If one chooses sx mutually-normal complexes,, 6 wth unty numercal values to be the orgnal unts then any complex can be derved from them numercally. Let x,, x 6 be the dervng numbers, so one has: = x + + x 6 6 = 6 x. = The numbers x are nothng but F. Klen s complex coordnates. multples the equaton above by then, snce [ k ] = 0 ( k) and have: [ ] = x ; If one then nner =, one wll.e., the numbers x wll be equal to the moments of the complex relatve to the sx fundamental complexes. By squarng the equaton above, t follows that: = x + + x. 6 s therefore dentcal wth the nvarant of a lnear ray complex. By nner multplyng the two complexes: t follows that: = = 6 x, = 6 x, = [ 6 ] = x x. = ( ** ) f., Segre, Borch. J., Bd. I.

19 Müller Lne geometry, accordng to Grassmann. 9 The quantty [ ] s then dentcal wth the smultaneous nvarant of the two lnear ray complexes. Metrc relatons between lnear ray complexes ( * ): Every complex n the pencl that s determned by the two complexes, can be represented n the form a + a. The two specal complexes of the pencl wll then be determned by the equaton: (a + a ) = a + a a [ ] + a = 0. They wll concde whenthe equaton above for a / a gves two equal roots, so when one has: [ ] = 0. Snce, from A, no. 77, one has: [ ] = [ ], one can also wrte the condton equaton for the pencl [ ] to possess two concdent specal complexes as: [ ] = 0, whch, from no. 9, says that the numercal value of the product [ ], or the lnear congruence [ ], s zero. Snce the congruence possesses two concdent drectrces n ths case so t s a specal congruence one can say: A lnear congruence s a specal one f and only f ts numercal value s equal to zero. If and are themselves specal then, due to the fact that have: [ ] = [ ]. = = 0, one wll In ths case, the congruence wll be specal for [ ] = 0;.e., when the axes of and cut each other. All complexes of the pencl wll be specal then, and ther axes wll defne a pencl of rays; the common congruence wll be a decomposable one ( ** ). ( * ) See: F. Klen, Über gewsse n der Lnengeometre auftretende Dfferentalglechungen, Math. Ann., Bd. V. ( ** ) See loc. ct.

20 Müller Lne geometry, accordng to Grassmann. 0 Every complex n the sheaf of complexes that s determned by the three complexes,, 3 can be represented n the form a + a + a 3 3. The specal complexes of the sheaf, whose axes defne a ruled famly, from no. 0, are determned by the equaton: (a + a + a 3 3 ) = a + a + a + a a [ ] + a a 3 [ 3 ] + a a 3 [ 3 ] = The polynomal n ths equaton can be represented as a product of two lnear factors as long as the determnant satsfes: [ ] [ ] [ ] [ ] [ ] [ ] = 0 Snce, from A, no. 75, ths determnant s dentcal wth [ 3 ], and thus represents the square of the numercal value of [ 3 ], one can also say that the polynomal n the equaton above can be decomposed nto two lnear factors as long as the numercal value of [ 3 ] s zero. The equaton can then be wrtten n the form: (a m + a m + a 3 m 3 ) (a n + a n + a 3 n 3 ) = 0, and the values of a, a, a 3 wll the satsfy one of the two equatons: a m + a m + a 3 m 3 = 0, (µ) a n + a n + a 3 n 3 = 0. All dervng numbers a, a, a 3 that satsfy (µ) determne specal complexes a + a + a 3 3 wth the property that any of them s numercally dervable from two of them, and s then a pencl of complexes that conssts of only specal complexes whose axes therefore defne a pencl of rays. Equaton (ν) wll also determne such a pencl of rays. Snce equatons (µ) and (ν) have a par of solutons a / a 3, a / a 3 n common, the two pencls of rays must possess a common ray. The axes of the specal complexes of the pencl [ 3 ] then defne a ruled famly that decomposes nto two pencls of rays. Snce ts gudng lne then decomposes nto two pencls of rays, n any case, one can say: The necessary and suffcent condton for three complexes,, 3 that do not belong to the same pencl to have a ruled famly n common that decomposes nto two pencls of rays s [ 3 ] = 0. (ν)

21 Müller Lne geometry, accordng to Grassmann. F. Klen called such a ruled famly specal. If one s gven four complexes,, 3, 4 whose exteror product s not equal to zero, but whose numercal values are equal to zero, then the equaton: [ 3 4 ] = 0 wll be true for them, so t wll follow mmedately from the concept of nner product that: [ 3 4 ] = 0. The extended congruence of the par of rays [ 3 4 ] s therefore a specal one, so, from no. 0, the ray-par that s represented by [ 3 4 ] wll consst of two concdent rays, or a specal ray-par. If,, 3, 4 are specal complexes then: [ 3 4 ] = 0 wll be the condton for there to be just one lne that cuts ts axes. For fve complexes,, 5 whose exteror product s not equal to zero, the equaton: [ ] = 0 lkewse represents the condton for one to have [,, 5 ] = 0;.e., for the complex that s normal to all fve complexes to be a specal one, or, snce the fve complexes must go through ts axs, for the fve complexes to have a common ray. All complexes of the web of complexes that they determne then lkewse contan ths ray; such a web shall be called specal. If the fve complexes are all specal then [ 5 ] = 0 wll represent the condton for ts axs to be cut by that ray. If one wrtes ths expresson, from A, no. 75, as the determnant: [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] and remarks that = 0 and [ k ] = [ k ] = m k s the moment of the axes of the specal complexes relatve to each other then one wll obtan the condton equaton for the fve rays to determne a specal complex n the form: = 0

22 Müller Lne geometry, accordng to Grassmann. 0 m m m 3 5 m 0 m m 3 5 m m m = 0. The exteror product of sx arbtrary complexes,, 6 represents a number x, whle the equaton: [ 6 ] = 0 then means nothng but x = 0, or: x = [ 6 ] = 0;.e., the sx complexes belong to the same bush. If the complexes are all specal then [ 6 ] = 0 wll gve the condton for ther axes to belong to those lnear complexes; f one wrtes ths equaton, as above, n the form: 0 m m m 3 6 m 0 m m 3 6 m m m then t wll represent the condton for the sx rays to belong to a lnear complex. Snce the exteror product of seven complexes,, 7 s always zero, one wll also always have the equaton: [ 7 ] = 0. If the complexes are specal then one wll obtan the equaton that exsts between the moments of any seven rays n space n the form: 0 = 0 0 m m m m 7 7 m m 0 = 0. Only one of the other easly-obtaned equatons shall be derved n order to represent the method. From no. 0, any fve lnear ray complexes determne a complex that s normal to them. Now, let two groups of any fve lnear ray complexes,, 5,,, 5 be gven, so one seeks the condton for the normal complexes that are determned by the two groups to be normal to each other.

23 Müller Lne geometry, accordng to Grassmann. 3 Snce the two normal complexes are gven by [ 5 ] and [ 5 ], from no. 8, the desred condton equaton wll read: [ ] [ 3 4 5] = 0 = [ 5 ], 5 or n determnant form: [ ] [ ] [ 5] [ ] [ ] [ ] 5 [ ] [ ] [ ] = 0. If one assumes that all complexes are specal and sets [ ] = m k here then the equaton: m m m 5 m m m 5 m m m wll gve the condton for the lnear complexes that are determned by those two groups of fve rays to be normal to each other (.e., to le n nvoluton). These, and smlar, equatons are analogous to the known equatons between spheres and ponts that were found by Darboux, for the most part. Ths analogy s a complete one, n that t allows one to treat sphere geometry by the same prncples that were appled here to ray geometry, and every formula n the theory of extensons can then be nterpreted mmedately n one or the other realm. = 0 k