71)6 PROJECTION their poiut of intersection E, will coincide with their corresponding elements. The row AB has thus three points A, B, E coincident with their corresponding points, and is therefore identical with it ( 10). As there are six lines which join two and two of the four joints A, B, C, U, there are six lines such that each point in either coincides with its corresponding point. Every other line will thus have the six points in which it cuts these, and therefore all points, coincident with their corresponding jioints. The proof of the second part is exactly the same. It follows 14. If two projective figures which arc not identical lie in the same plane, then nt more than three points which are not in a line, or three lines which do n-ot 2>ass through a point, can be coincident icith their corresponding points or lines. If the figures are in perspective position, then they have in common one line, the axis, with all points in it, and one point, the centre, with all lines through it. No other point or line can there fore coincide with its corresponding point or line without the figures becoming identical. It follows also that The correspondence between two projective planes is completely determined if there are given cither to four points in the one the corresponding four points in the other provided that no three of them lie in a line, or to any four lines the corresponding lines provided that no three of them pass through a point. To show this we observe first that two jilanes IT, IT may be made projective in such a manner that four given points A, B, C, D in the oae correspond to four given points A, B , C , D in the other ; for t the lines AB, CD will correspond the lines A B and C D , and to the intersection E of the former the point E where the latter meet. The correspondence between these rows is therefore determined, as we know three j>airs of corresp aiding points. But this determines a correspondence (by 12). To prove that in this case and also in the case of 12 there is but one correspondence possible, let us suppose there were two, or that we could have in the plane ir two figures which are each projective to the figure in IT and which have each the points A B C D corresponding to the points A BCD in IT. Then these two figures will themselves be projective and have four corresponding points coincident. They are therefore identical by 13. THEOREM. Two projective planes will be in perspective position if one row coincides with its corresponding row. The line containing these rows will be the axis of projection. Proof. As in this case every point on s coincides with its corre sponding point, it follows that every row a meets its corresponding row a on s where corresponding points are united. The two rows , are therefore perspective (G. 30), and the lines joining corresjionding points will meet in a point S. If r be any one of these lines cutting a, a in the points A and A and the line s at K, then to the line AK corresponds A K, or the ray r corresponds to itself. The points B, B in which r cuts another pair b, V of corresponding rows must therefore be corresponding points. Hence the lines joining corresponding points in b and b also pass through S. Similarly all lines joining corresponding points in the two planes ir and v meet in S ; hence the planes are perspective. The following proportion is proved in a similar way : THEOREM. Two projective planes will be in perspective position if one pencil coincides with its corresponding one. The centre of these pencils will be the centre of perspective. In this case the two planes must of course coincide, whilst in the first case this is not necessary. 15. We shall now show that two planes which are projective according to definition 12 can be brought into perspective position, hence that the new definition is really equivalent to the old. We use the following property: If two coincident planes ir and ir are j>er- spective with S as centre, then any two corresjionding rows are also perspective with S as centre. This therefore is true for the rows j and j and for i and i , of which i and / are the lines at infinity in the two planes. If now the j)lane -IT be made to slide on ir so that eacli line moves parallel to itself, then the point at infinity in each line, and hence the whole line at infinity in ir , remains fixed. So does the point at infinity on j, which thus remains coincident with its corresponding point on /, and therefore the rows j and / remain perspective, that is to say the rays joining corresponding points in them meet at some point T. Similarly the lines joining cor responding points in i and i will meet in some jioint T . These two points T and T originally coincided with each other and with S. Conversely, if two projective planes are placed one on the other, then as soon as the lines j and v are parallel the two points T and T" can be found by joining corresponding points inland/, and also in i and i . If now a point at infinity is called A as a point in ir and B las a point in ir , then the point A will lie on i and B on j, so that the line AA passes through T and BB through T. These two lines are parallel. If then the j>lane IT be moved parallel to itself till T comes to T, then these two lines will coincide with each other, and with them will coincide the lines AB and A B . This line and similarly every line through T will thus now coincide with its corresponding line. The two planes arc therefore accord ing to the last theorem in 14 in perspective position. It will be noticed that the plane ir may be placed on ir in two different ways, viz. , if we have jdaeed ir on ir we may take it olf and turn it over in space before we bring it back to ir, so that what was its upper becomes now its lower face. For each of these positions we get one jiair of centres T, T , and only one pair, because the above process must give every perspective position. It follows In two projective plains there are in general two and only two jicncils in cither such that angles in one are equal to their correspond ing angles in the other. If one of these pencils is made coincident with its corresponding one, then the planes will be perspective. This agrees with the fact that two perspective jilanes in space can be made coincident by turning one about their axis in two different ways ( 8). In the reasoning employed it is essential that the lines j and i are finite. If one lies at infinity, say j, then i and j coincide, hence their corresj>onding lines i and/ will coincide ; that is, i also lies at infinity, so that the lines at infinity in the two planes are correspond ing lines. If the planes are now made coincident and perspective, then it may happen that the lines at infinity correspond point for point, or can be made to do so by turning the one plane in itself. In this case the line at infinity is the axis, whilst the centre may be a finite point. This gives similar figures (see 16). In the other case the line at infinity corresponds to itself without being the axis ; the lines joining corresponding points therefore all coincide with it, and the centre S lies on it at infinity. The axis will be some finite line. This gives parallel projection (see 17). For want of space we do not show how to find in these cases the perspective position, but only remark that in the first case any pair of corresponding points in ir and ir may be taken as the points T and T , whilst in the other case there is a pencil of parallels in ir such that any one line of these can be made to coincide point for point with its corre sponding line in ir , and thus serve as the axis of projection. It will therefore be possible to get the planes in perspective position by first placing any point A on its corresponding point A and then turning ir about this point till lines joining corresponding points are parallel. 16. SIMILAR FIGURES. If the axis is at infinity every line is parallel to its corresponding line. Corresponding angles are there fore equal. The figures are similar, and (10) the ratio of simili tude of any two corresponding rows is constant. If similar figures are in perspective position they are said to be similarly situated, and the centre of projection is called the centre of similitude. To place two similar figures in this position, we observe that their lines at infinity will coincide as soon as both figures are put in the same plane, but the rows on them are not necessarily identical. They are projective, and hence in general not more than two points on one will coincide with their corresponding points in the other (G. 34). To make them identical it is either sufficient to turn one figure in its plane till three lines in one are parallel to their corresponding lines in the other, or it is necessary before this can be done to turn the one plane over in space. It can be shown that in the former case all lines are, or no line is, parallel to its cor responding line, whilst in the second case there are two directions, at right angles to each other, which have the property that each line in either direction is parallel to its corresponding line. We also see that If in two similar figures three lines, of which no two are parallel, are parallel respectively to their corresponding lines, then every line has this property and the two figures are similarly situated ; or Two similar figures are similarly situated as soon as two corre sponding triangles are so situated. If two similar figures are perspective without being in the same plane, their planes must be parallel as the axis is at infinity. Hence Any plane figure is projected from any centre to a parallel plane into a similar figure. If two similar figures are similarly situated, then corresponding points may cither be on the same or on different sides of the centre. If, besides, the ratio of similitude is unity, then corresponding points will be equidistant from the centre. In the first case therefore the two figures will be identical. In the second case they will be identi cally equal but not coincident. They can be made to coincide by turning one in its plane through two right angles about the centre of similitude S. The figures are in involution, as is seen at once, and they arc said to be symmetrical witJi regard to the point S as centre. If the two figures be considered as part of one, then this is said to have a centre. Thus regular jtolygons of an even number of sides and parallelograms have each a centre, which is a centre of symmetry. 17. PARALLEL PROJECTION. If, instead of the axis, the centre be moved to infinity, all the projecting rays will be parallel, and we get what is called Parallel Projection. In this case the line at in finity passes through the centre and therefore corresponds to itself, but not point for point as in the case of similar figures. To any point I at infinity corresponds therefore a point I also at infinity but
different from the first. Hence to jiarallel lines meeting at 1 cor-