Silt) G E O M axial pencil is, of course. infinite aml indefinite too, but the same in all. For a moment we shall treat it as being a definite number vhich we denote by :1. Then :1 plane contains a‘-' points and as many lines. To see this, take a fiat pencil in a plane. It cotttains a lines, and each line contains a points, whilst each point in the plane lies on one of these lines. Similarly, in :1 plane each line cuts a fixed line in a point. Ilnt this line is cut at cach point by a lines and contains a points ; hence there are a’-' lines in a plane. A pencil in space contains as many lines as :1 plane contains points and as litany planes as a plane contains lines, for any plane cuts the pencil in a field of points and lines. Hence a pencil eon- tains a‘-’ lines and a‘3 planes. The field and the pencil arc qf two cIimcnsz'ons. - To count the number of points in space we observe that each point lies on some line in a pencil. But the pencil contains a" lines, and each line a points; hence space eontains a3 points. Eacli plane cuts any fixed plane in a line. But a plane contains a" lines, and through cach pass :1 planes; therefore space contains a” planes. Ilcnce space contains as many planes as points, but it contains an infinite number of times more lines than points or planes. 'I‘o count them, notice that every line cttts :1 fixed plane in one point. Ilut :12 lines pass through each point, and tl1erc are a” points in the plane. Hence there are a‘ lines in space. plants is of three dimensions, but the space of lines is of four (linzcn- I sions-. A field of points or lines contains an infinite nmnber of rows and flat pencils; :1 pencil contains an infinite number of fiat pencils The spncc of points and , and of axial pencils: space contains a triple infinite number of ' pencils and of fields, a‘ rows and axial pencils, and a" flat pencils- or, in other words, each point is a centre of a‘-' fiat pencils. § 7. The .bove enumeration allows a classification of figures. Figures in :1 row consist of groups of points only, and figures in the fiat or axial pencil consist of groups of lines or pl:mes. In the pl me we may draw polygons; and in the pencil or in the point, solid angles, and so on. We may also distinguish the diflercnt mcasurentcnts. Vel1avc— In the rov, length of segment ; In the flat pencil, angles ; In the axial pencil, dihedral angles between tvo planes ; In the plane, areas ; In the pencil, solid angles ; In the space of points or planes, volttmcs. SI-1(:.l r-..'Ts or A I.t.'i:. § S. Atty two points A and I} in space determine on the line through them :1 finite part, which may be considered as being described by
- 1 point; moving from A to I}. This we shall denote by Al}, and
distinguish it from BA, wl.icl& is supposed as being described by a mint moving from B to A, an hcnce in :1 direction or in a “ sense ” hpposite to AB. Such a finite line, which has a dcfinite sense, we shall call a " segment,” so that Al} and BA dcnote different seg- ments, which are said to be equal in length but of opposite sense. The one sense is often called positive and the other negative. In introducing the word “ sense” for direction in :1 line, we have the word direetion reserved for direction of the line itself, so that -dilI‘crent lines have different directions, unless they be parallel, whilst in each line we have :1 positive and negative sense. 'e may also say, with Professor Clilford, that Al’; denotes the "step" of going from A to B. §9. If we now have three points A, B, C in a line (fig. 2:‘, the step Ali will bring us from A to I}, and the step BF‘ from B to C. Hence both A 13 C steps are equivalent to the one stcp * ' At‘. This is expressed by saying that AC is the “sum" of AB and BC; in symbols—— C A 13 where account is to be taken of the sense. ____,____?,__A P’ This equation is true whatever he the pi... r) position of the three points on the line. i" As :1 special case we have Al’»+ l}A=0 . (I), and similarly A1:+]}C+CA—o . . (2), which again is true for any three points in a line. We further wiite AB = -— BA , 11 here — denotes negative sense. We can then, just as in algebra, change subtraction of segments into addition by changing the sense, so that A l5—CB is the same as AB+(—Cl}) or A I5 + IEL‘. A figure will at once show the truth of this. The sense is, in fact, in every l'c.~1oL-ct equivalent to the “ sign" ofa number in alg-_-b;,t_ E '1‘ 11 Y § 10. Of the many formuhe which exist between points in a line we shall have to use only one more, which connects the segments between any four points A, I}. C. I) in a line. We have- l}C—-lll) —I-Ilt‘ , (‘ «'11 4 in, ABrAD+DB; or multiplying these by AI), 13]), [cr.oJ1:cTtvE. (_'l) rcspt-ctivcl_v, vc gt_t—- l§C . AI) = II). All + IIC. All = l‘»ll. Al) — (‘IL All (‘A . Ill) =— (‘II . Ill) + DA. Ill) ~ I'll. Ill) —All. l’.ll AI}. CI) = AI). (‘D + l)ll . ('I)= All. (‘D —— III). I'll. It will be seen that the sum of the right han l sides vanishes, hence that lit‘. Al) + CA . 111) + All. (ID = for any four points on a line. §11. If C is any point in the line Al}, then 'c say that I‘ 0 .. . G) divides the segtncnt Al} in the ratio , account being lill{C'll of ) the sense of the two segments A(‘ and Cl}. If C lies bctwten A and C the ratio is positive, as AC‘ and (‘B have the s:une sense. But if C lies without the segment AP», i.c., if C divides Ali ex- ternally, then the ratio is negative. To see how the value of this ratio changes with t‘, we. will move 0 1,5‘ “F ll’ - along the whole ,. ,, ‘ 113. u. line(fig. 3), whilst A and II rcmain fixed. If C lies at the point A, then A(‘=o, hence the ratio AC‘ : (QB vanishes. As O moves towards I}, At.‘ increases and CB decreases, so that our ratio increases. At the middle point M of All it assumes the value + I, and then in«_-rca.~t-s till it reaches an inlinitely large value, when U arrives at ll. Hn passittg beyond B the ratio becomes negative. If U is at 1’ we have AC =A l'=A I} + Iii’, ltcncc AC Al) l}P_ All ('|}_I’l} l‘l3_ 151’ In the last expression the ratio Al} : III’ is positive, has its greatest value so when (.5 coincides with I}, and vanishes when lit? becomes infinite. Ilencc, as 0 moves from I} to the right to the point at infinity, the ratio AC : CI} varies from ——-/2 to ——I. If on the other hand C is to the left of A, say at Q, we have I I Ac=AQ=An+nQ=An_QnJmmm9(=§P_ Cl} Qli Ilere Al} <Ql‘», hence the ratio All : Ql’. is positive and always less than one, so that the whole is negative and <1. If C is at the point at in linity it is -1, :111d then increases as U movss to the right, till for C at A wc get the ratio =0. Ilence—- “ As C moves along the line from an inlinite distance to the left to an infinite distance at the right, the ratio always increases ; it starts with the value -1, reaches 0 at A, +1 at M, on at 13, changes now sign to -03, and increases till at an infinite d1st:mce it reaches again the value -1. It assumes I/acrcfurc (Ill possible t'(:Im's from -09 to +09 , and each crzlztc only once, so that not only does cr:-7'_1/ position of C (lctcrminc a (Ir-_/initc twhlc of the ratio At,‘ : (‘l3, but also, con 1.-rrscI_a/, I0 crery posilz'1'-' or 7u'gr.z!iz'c mlzzc (fl/u'.s ratio belongs one single point in the [Inc A I}. 1 . 1‘t:oJiacr1o.' A.'I) Cnoss—I‘..trtos. § 12. If we join a point A to a point S, then the point wln-re the linc SA cttts a fixed )lllll(.' 7r is called the n'o'cction of A on the . , 1 . . . ,1 J plane 1r lrotn .8 as centre of 1Il‘0,)t't'l,lUll. II we have two planes 1r | and 1r’ and a point S, we may project every point A in 11' to the other plane. If A’ is the projection of A, then A is also the projection of A’, so that the relations are reciprocal. ’I‘0 every figure in 7r we get as its projection a corresponding figure in 7r’. It will be our business to find such properties of figures as remain trite for the projection, :n1d which are called pro_jcctive properties. For this purpose it will be sullicicnt to consider at first only construc- tions in one plane. I .et; us suppose we have given in
- 1 plane two lines 3) and 7)’ and :1
centre S (fig. 4) ; we may then pro- jcct the points in 1) from S top‘. Let A’, 3’... be the projections of A, 15..., the point at infinity in 1) which we shall denote by J will be projected into :1 finite point I’ in 1)’, viz., into the point where the parallel to 1) through S cuts I I . II’ at infinity in 1:’. .’) Hg4. Similarly one point J in )2 will be projected into the point
This point J is of course the point where