See also:term meaning the
See also:action of forbidding or preventing by an
See also:order, decree, &c . The word is particularly applied to the forbidding by
See also:law of the sale and manufacture of intoxicating liquors (see LIQUOR
See also:LAWS and
See also:TEMPERANCE) . In law, as defined by
See also:Blackstone, prohibition is " a writ directed to the
See also:judge and parties of a suit in any inferior
See also:court, commanding them to cease from the
See also:prosecution thereof, upon a surmise either that the cause originally or some
See also:matter arising therein does not belong to that jurisdiction, but to the cognizance of some other court." A writ of prohibition is a
See also:prerogative writ—that is to say, it does not issue as of course, but is granted only on proper grounds being shown . Before the Judicature Acts prohibition was granted by one of the
See also:superior courts at
See also:Westminster; it also issued in certain cases from the court of
See also:chancery . It is now granted by the High Court of
See also:Justice . Up to 1875 the high court of
See also:admiralty was for the purposes of prohibition an inferior court . But now by the Judicature
See also:Act 1813, s . 24, it is provided that no proceeding in the High Court of justice or the court of
See also:appeal is to be restrained by prohibition, a stay of proceedings taking its place where necessary . The admiralty division being now one of the divisions of the High Court can therefore no longer § I .
See also:Projection of
See also:Plane Figures.—Let us suppose we have in space two planes a and ar' . In the plane a a figure is given having known properties; then we have the problem to find its projection from some centre S to the plane a', and to deduce from the known properties of the given figure the properties of the new one . If a point A is given in the plane a we have to join it to the centre S and find the point A' where this ray SA cuts the plane a'; it is the projection of A .
On the other
See also:hand if A' is given in the plane then A will be its projection in a . Hence if one figure in 7r' is the projection of another in 7r, then conversely the latter is also the projection of the former . A point and its projection are therefore also called corresponding points, and similarly we speak of corresponding lines and curves, &c . § 2 . We at once get the following properties: The projection of a point is a point, and one point only . The projection of a
See also:line (straight line) is a line; for all points in a line are projected by rays which lie in the plane determined by S and the line, and this plane cuts the plane a in a line which is the projection of the given line . If a point lies 'in a line its projection lies in the projection of the line . The projection of the line joining two points A, B is the line which joins the projections A', B' of the points A, B . For the projecting plane of the line AB contains the rays SA, SB which project the points A, B . The projection of the point of intersection of two lines a, b is the point of intersection of the projections a', b' of those lines . Similarly we get The projection of a
See also:curve is a curve . The projections of the points of intersection of two curves are the points of intersection of the projections of the given curves .
If a line cuts a curve in n points, then the projection of the line cuts the projection of the curve in n points . Or The order of a curve remains unaltered by projection . The projection of a tangent to a curve is a tangent to the projection of the curve . For the tangent is a line which has two coincident points in
See also:common with a curve . The number of tangents that can be
See also:drawn from a point to a curve remains unaltered by projection . Or The class of a curve remains unaltered by projection . § 3 . Two figures of which one is a projection of the other obtained in the manner described may be moved out of the position in which they are obtained . They are then still said to be one the projection of the other, or to be projective or homographic . But when they are in the position originally considered they are said to be in perspective position, or (shorter) to be perspective . All the properties stated in §§ I, 2 hold for figures which are projective, whether they are perspective or not . There are others which hold only for projective figures when they are in perspective position, which we shall now consider .
If two planes ar and 7r' are perspective, then their line of inter-section is called the
See also:axis of projection . Any point in this line coincides with its projection . Hence All points in the axis are their own projections . Hence also—Every line meets its projection on the axis . § 4 . The
See also:property that the lines joining corresponding points all pass through a common point, that any pair of corresponding points and the centre are in a line, is also expressed by saying that the figures are co-linear or co-polar; and the fact that both figures have a line, the axis, in common on which corresponding lines meet is expressed by saying that the figures are co-axal . The connexion between these properties has to be investigated . For this purpose we consider in the plane a a triangle
See also:ABC, and let the lines BC, CA, AB be denoted by a, b, c . The projection will consist of three points A', B', C' and three lines a', b', c' . These have such a position that the lines AA', BB', CC' meet in a point, viz. at S, and the points of intersection of a and a', b and b', c and c' lie on the axis (by § 2) . The two triangles therefore are said to be both co-linear and co-axal . Of these properties either is a consequence of the other, as will now be proved .
If two triangles, whether in the same plane or not, are co-linear they are co-axal . Or— _ If the lines AA', BB', CC' joining the vertices of two triangles meet in a point, then the intersections of the sides BC and B'C', CA and C'A', AB and A'B' are three points in a line . Conversely If two triangles are co-axal they are co-linear . Or If the intersection of the sides of two triangles ABC and A'B'C', viz. of BC and B'C', of CA and C'A', and of AB and A'B', lie in a line, then the lines AA', BB', and CC' meet in a point . Proof—Let us first suppose the triangles to be in different places . By supposition the lines AA', BB', CC' (fig . I) meet in a point S . But three intersecting lines determine three planes, SCB, SCA and SAB . In the first lie the points B, C and also B', C' . Hence the lines BC and B'C' will intersect at some point P, because any two lines in the same plane intersect . Similarly CA and C'A' will intersect at some point Q, and AB and A'B' at some point R . These points P, Q, R lie in the plane of the triangle ABC because they are points on the sides of this triangle, and similarly in the plane of the triangle A'B'C' .
Hence they lie in the intersection of two planes—that is, in a line . This line (PQR in fig . I) is calledthe axis of perspective or homology, and the intersection of AA', BB', CC', i.e . S in the figure, the centre of perspective . Secondly, if the triangles ABC and A'B'C' lie both in the same plane the above
See also:proof does not hold . In this case we may consider the plane figure as the projection of the figure in space of which we have just proved the theorem . Let ABC, A'B'C' be the co-linear triangles with S as R centre, so that AA', BB', CC' meet at S . Take now any point in space, say your
See also:eye E, and from it draw the rays projecting the figure . In the line ES take any point SI, and in EA, EB, EC take points Al, BI, Cl respectively, but so that Si, Ai, BI, Cl are not in a plane . In the plane ESA which projects the line SIAI lie then the line SIAI and also EA'; these will therefore meet in a `point AI', of which A' will be the projection . Similarly points BI , Cl' are found . Hence we have now in space two triangles AIBICI and AI'BI'CI' which are co-linear .
They are therefore coaxal, that is, the points PI, QI, RI, where AIBI, &c., meet will lie in a line . Their projections therefore lie in a line . But these are the points P, Q, R, which were to be proved to lie in a line . This proves the first
See also:part of the theorem . The second part or converse theorem is proved in exactly the same way . For another proof see (G . § 37) . § 5 . By aid of this theorem we can now prove a fundamental property of two projective planes . Let s be the axis, S the centre, and let A, A' and B, B' be two pairs of corresponding points which we suppose fixed, and C, C' any other pair of corresponding points . Then the triangles ABC and A'B'C are co-axal, and they will remain co-axal if the one plane ar be turned relative to the other about the axis . They will therefore, by Desargue's theorem, remain co-linear, and the centre will be the point S', where AA' meets BB' .
Hence the line joining any pair of corresponding points C, C' will pass through the centre S' . The figures are therefore perspective . This will remain true if the planes are turned till they coincide, because Desargue's theorem remains true . If two planes are perspective, then if the one plane be turned about the axis through any
See also:angle, especially if the one plane be turned till it coincides with the other, the two planes will remain perspective; corresponding lines will still meet on a line called the axis, and the lines joining corresponding points will still pass through a common centre S situated in the plane . Whilst the one plane is turned this point S will move in a circle whose centre lies in the plane 7r, which is kept fixed, and whose plane is perpendicular to the axis . The last part will be proved presently . As the plane a' may be turned about the axis in one or the opposite sense, there will be two perspective positions possible when the planes coincide . § 6 . Let (fig . 2) it, ar' be the planes intersecting in the axis s whilst S is the centre of projection . To project a point A in ar we join A to S and see where this line cuts 7r' . This gives the point A' .
But if we draw through S any line parallel to 7r, then this line will cut a' in some point I', and if all lines through S be drawn which are parallel to a these will
See also:form a plane parallel to ar which will cut the plane ar' in a line i' parallel to the axis s .. If we say that a line parallel to a plane cuts the latter at an infinite distance, we may say that all points at an infinite distance in a are projected into points FIG . 2 . which lie in a straight line i', and conversely all points in the line are projected to an infinite distance in ar, whilst all other points are projected to finite points . We say therefore that all points in the plane a at an infinite distance may be considered as lying in a straight line, because their projections lie in a line . Thus we are again led to consider points at infinity in a plane as lying in a line (cf . G . §§ 2-4) . Similarly there is a line j in ar which is projected to infinity in 7r'; this projection will be denoted by j' so that i and j' are lines at infinity . § 7 . If we suppose through S a plane drawn perpendicular to the axis s cutting it at T, and in this plane the two lines SI' parallel to it and SJ parallel to ar', then the lines through I' and J parallel to the axis will be the lines i' and j . At the same
See also:time a parallelogram SJTI'S has been formed .
If now the plane r' be turned about the axis, then the points I' and J will not move in their planes; hence the lengths TJ and TI', and therefore also SI' and SJ, will not
See also:change . If the plane r is kept fixed in space the point J will remain fixed, and S describes a circle about J as centre and with SJ as
See also:radius . This proves the last part of the theorem in § 5 . § S . The plane may be turned either in the sense indicated by the arrow at Z or in the opposite sense till r' falls into rr . In the first case we get a figure like fig . 3; i' and j will be on the same side of the axis, and on this side will also lie the centre S; and T f J S s T 1 S then ST=SJ+SI' or SI'=JT, SJ=I'T . In the second case (fig . 4) i' and j will be on opposite sides of the axis, and the centre S will lie between them in such a position that I'S=TJ and I'T =SJ If I'S=SJ, the point S will lie on the axis . It follows that any one of the four points S, T, J, I' is completely determined by the other three: if the axis, the centre, and one of the lines i' or j are given the other is determined; the three lines s, i', j determine the centre; the centre and the lines i', j determine the axis . § 9 . We shall now suppose that the two projective planes r, r' are perspective and have been made to coincide .
If the centre, the axis, and either one pair of corresponding points on a line through the centre or one pair of corresponding lines
See also:meeting on the axis are given, then the whole projection is determined . Proof.—If A and A' (fig . I) are given corresponding points, it has to be shown that we can find to every other point B the corresponding point B' . Join AB to cut the axis in R . Join RA'; then B' must lie on this line . But it must also lie on the line SB . Where both meet is B' . That the figures thus obtained are really projective can be seen by aid of the theorem of § 4 . For, if for any point C the corresponding point C' be found, then the triangles ABC and A'B'C' are, by construction, co-linear, hence co-axal; and s will be the axis, because AB and AC meet their corresponding lines A'B' and A'C' on it . BC and B'C' therefore also meet on s . If on the other hand a, a' are given corresponding lines, then any line through S will cut them in corresponding points A, A' which may be used as above . § Io .
Rows and pencils which are projective or perspective have been considered in thearticle
See also:GEOMETRY (G . §§ 12-40) . All that has been said there holds, of course, here for any pair of corresponding rows or pencils . The centre of perspective for any pair of corresponding rows is at the centre of projection S, whilst the axis contains coincident corresponding elements . Corresponding pencils on the other hand have their axis of perspective on the axis of projection whilst the coincident rays pass through the centre . We mention here a few of those properties which are
See also:independent of the perspective position: The
See also:correspondence between two projective rows or pencils is completely determined if to three elements in one the corresponding ones in the other are given . If for instance in two projective rows three pairs of corresponding points are given, then we can find to every other point in either the corresponding point (G . §§ 29-36) . If A, B, C, D are four points in a
See also:row and A', B', C', D' the corresponding points, then their
See also:cross-ratios are equal (AB, CD) = (A'B', C'D')—where (AB, CD) =AC/CB :AD/DB . If in particular the point D be at infinity we have (AB, CD) = —AC/CB =AC/BC . If therefore the points D and D' are both at infinity we have AC/BC=AD/BD, and the rows are similar (G . § 39) .
This can only happen in
See also:special cases . For the line joining corresponding points passes through the centre; the latter must therefore lie at infinity if D, D' are different points at infinity . But if D and D' coincide they must lie on the axis, that is, at the point at infinity of the axis unless the axis is altogether at infinity . Hence In two perspective planes every row which is parallel to the axis is similar to its corresponding row, and in general no other row has this property . But if the centre or the axis is at infinity then every row is similar to its corresponding row . In either of these two cases the metrical properties are particularly
See also:simple . If the axis is at infinity the ratio of similitude is the same for all rows and the figures are similar . If the centre is at infinity we get parallel projection; and the ratio of similitude changes from row to row (see §§ 16, 17) . In both cases the
See also:mid-points of corresponding segments will be corresponding points . § ii . Involution.—If the planes of two projective figures coincide, then every point in their common plane has to be counted twice, once as a point A in the figure it, once as a point B' in the figure r' . The points A' and B corresponding to them will in general be different points; but it may happen that they coincide .
Here a theorem holds similar to that about rows (G . §§ 76 seq.) . If two projective planes coincide, and if to one point in their common plane the same point corresponds, whether we consider the point as belonging to the first or to the second plane, then the same will happen for every other point—that is to say, to every point will correspond the same point in the first as in the second plane . In this case the figures are said to be in involution . Proof.—Let (fig . 5) S be the centre, s the axis of projection, and let a point denoted by A in the first plane and by B' in the second have the property that the points A' and B corresponding to them again coincide . Let C and D' be the names which some other point has in the two planes . If the line AC cuts the axis in X, then the point where the line XA' cuts SC will be the point C' corresponding to C (§ 9) . The line B'D' also cuts the axis in X, and therefore the point D corresponding to D' is the point where XB cuts SD' . But this is the same point as C' . This point C' might also be got by
See also:drawing CB and joining its intersection Y with the axis to B' . Then C' must be the point where B'Y meets SC .
This figure, which now forms a
See also:quadrilateral, shows that in order to get involution the corresponding points A and A' have to be
See also:harmonic conjugates with regard to S and the point T where AA' cuts the axis . If two perspective figures be in involution, two corresponding points are harmonic conjugates with regard to the centre and the point where the line joining them cuts the axis . Similarly Any two corresponding lines are harmonic conjugates with regard to the axis and the line from their point of intersection to the centre . Conversely If in two perspective planes one pair of corresponding points be harmonic conjugates with regard to the centre and the point where the line joining them cuts the axis, then every pair of corresponding points has this property and the planes are in involution . § 12 . Projective Planes which are not in perspective position.—We return to the case that two planes it and r' are projective but not in perspective position, and state in some of the more important cases the conditions which determine the correspondence between them . Here it is of
See also:advantage to start with another definition which, though at first it may seem to be of far greater generality, is in reality
See also:equivalent to the one given before . We
See also:call two planes projective if to every point in one corresponds a point in the other, to every line a line, and to a point in a line a point in the corresponding line, in such a manner that the cross-ratio of four points in a line, or of four rays in a pencil, is equal to the cross-ratio of the corresponding points or rays . The last part about the equality of cross-ratios can be proved to be a consequence of the first . As space does not allow us to give an exact proof for this we include it in the definition . If one plane is actually projected to another we get a correspondence which has the properties required in the new definition . This shows that a correspondence between two planes conform to this definition is possible .
That it is also definite we have to show . It follows at once that Corresponding rows, and likewise corresponding pencils, are projective in the old sense (G . §§ 25, 30) . Further If two planes are projective to a third they are projective to each other . The correspondence between two projective planes it and it is deter-
See also:mined if we have given either two rows u, v in it and the corresponding rows u', v' in the point where u and v meet corresponding to the points where u' and v' meet, or two pencils U, V in it and the corresponding pencils U', V' in r', the ray UV joining the centres of the pencils in it corresponding to the ray U'V' . It is sufficient to prove the first part . Let any line a cut u, v in the points A and B . To these will correspond points A' and B' in u' and v' which are known . To the line a corresponds then the line A'B' . Thus to every line in the one plane the corresponding line in the other can be found, hence also to every point the corresponding point . § 13 . If the planes of two projective figures coincide, and if either four points, of which no three lie in a line, or else four lines, of which no three pass through a point, in the one coincide with their corresponding points, or lines, in the other, then every point and every line coincides with its corresponding point or line so that the figures are identical .
If the four points A, B, C, D coincide with their corresponding points, then every line joining two of these points will coincide with its corresponding line . Thus the lines AB and CD, and therefore also their point 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 (§ io) . As there are six lines which join two and two of the four points A, B, C, D, 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 points . The proof of the second part is exactly the same . It follows § 14 . If two projective figures, which are not identical, lie in the same plane, then not more than three points which are not in a line, or three lines which do not pass through a point, can be coincident with 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—either 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 planes r, x' may be made projective in such a manner that four given points A, B, C, D in the one correspond to four given points A', B , C', D' in the other; for to 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 pairs of corresponding 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 two figures which are each projective to the figure in a and which have each the points A'B'C'D' corresponding to the points
See also:ABCD in ir . Then these two figures will themselves be projective and have four corresponding points coincident . They are therefore identical by § 13 . Two projective planes will be in perspective if one row coincides with its corresponding row . The line containing these rows will be the axis of projection . As in this case every point on s coincides with its corresponding point, it follows that every row a meets its corresponding row a' on s where corresponding points are
See also:united . The two rows a, a' are therefore perspective (G . § 30), and the lines' joining corresponding 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, b' 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 r and a' meet in S; hence the planes are perspective . The following proposition is proved in a similar way: 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 . § ig . 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 fclllowing property: If two coincident planes a and tr' are perspective with S as centre, then any two corresponding rows are also perspective with S as centre . This therefore is true for the row) and j' and for i and i', of which i and j' are the lines at infinity in the two planes . If now the plane 7f be made to slide on tr so that each line moves parallel to itself, then the point at infinity in each line, and hence the whole line at infinity in jr', remains fixed . So does the point at infinity on j, which thus remains coincident with its corresponding point on j', and therefore the rows j and j' remain perspective, that is to say the rays joining corresponding points in them meet at some point T . Similarly the lines joining corresponding points in i and i will meet in some point 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 i are parallel the two points T and T' can be found by joining corresponding points in j and j', and also in i and i' . If now a point at infinity is called A as a point in ir and B' as a point in ,r', 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 plane tr' 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 are therefore according to the last theorem in § 14 in perspective position . It will be noticed that the plane tr' may be placed on it in two different ways, viz. if we have placed on a we may take it off and turn it over in space before we bring it back to r, so that what was its upper becomes now its
See also:face . For each of these positions we get one pair of centres T, T', and only one pair, because the above
See also:process must give every perspective position . It follows In two projective planes there are in general two and only two pencils in either such that angles in one are equal to their corresponding 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 planes 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 corresponding lines i' and j' will coincide; that is, i' also lies at infinity, so that the lines at infinity in the two planes are corresponding 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 § i6) . 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 it and ow' may be taken as the points T and T', whilst in the other case there is a pencil of
See also:parallels in ir such that any one line of these can be made to coincide point for point with its corresponding 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 7r' about this point till lines joining corresponding points are parallel . § i6 . 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 (§ io) the ratio of similitude of any two corresponding rows is
See also: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 corresponding 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 corresponding 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 either 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 there-fore the two figures will be identical . In the second case they will be identically 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 are said to be symmetrical with 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
See also:regular polygons 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 iet what is called parallel projection . In this case the line at innity 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 parallel lines meeting at I correspond parallel lines of another direction meeting at I . Further, in any two corresponding rows the two points at infinity are corresponding points; hence the rows are similar .
This gives the
See also:principal properties of parallel projection: To parallel lines correspond parallel lines; or To a parallelogram corresponds a parallelogram . The correspondence of parallel projection is completely determined as soon as for any parallelogram in the one figure the corresponding parallelogram in the other has been selected, as follows from the general case in § 14 . [Corresponding rows are similar (§ Io).] The ratio of similitude for these rows changes with the direction: If a row is parallel to the axis, its corresponding row, which is also parallel to the axis, will be equal to it, because any two pairs AA' and BB' of corresponding points will form a parallelogram . Another important property is the following: The areas of corresponding figures have a constant ratio . We prove this first for parallelograms . Let ABCD and EFGH be any two parallelograms in a, A'B'C'D' and E'F'G'H' the corresponding parallelograms in . Then to the parallelogram KLMN which lies (fig . 6) between the lines AB, CD and EF, GH will correspond a parallelogram K'L'M'N' formed in exactly the same manner . As ABCD and KLM N are between the same parallels their areas are as the bases .
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