5. Given three lines and the point of contact on two of them, to construct the conic.

6. Given four lines and the line at infinity, to construct the conic.

7. Given three lines and the line at infinity, together with the point of contact at infinity, to construct the conic.

8. Given three lines, two of which are asymptotes, to construct the conic.

9. Given five tangents to a conic, to draw a tangent which shall be parallel to any one of them.

10. The lines a, b, c are drawn parallel to each other. The lines a', b', c' are also drawn parallel to each other. Show why the lines (ab', a'b), (bc', b'c), (ca', c'a) meet in a point. (In problems 6 to 10 inclusive, parallel lines are to be drawn.)



CHAPTER VI - POLES AND POLARS



*95. Inscribed and circumscribed quadrilaterals.* The following theorems have been noted as special cases of Pascal's and Brianchon's theorems:

If a quadrilateral be inscribed in a conic, two pairs of opposite sides and the tangents at opposite vertices intersect in four points, all of which lie on a straight line.

If a quadrilateral be circumscribed about a conic, the lines joining two pairs of opposite vertices and the lines joining two opposite points of contact are four lines which meet in a point.

[Figure 26]

FIG. 26



*96. Definition of the polar line of a point.* Consider the quadrilateral K, L, M, N inscribed in the conic (Fig. 26). It determines the four harmonic points A, B, C, D which project from N in to the four harmonic points M, B, K, O. Now the tangents at K and M meet in P, a point on the line AB. The line AB is thus determined entirely by the point O. For if we draw any line through it, meeting the conic in K and M, and construct the harmonic conjugate B of O with respect to K and M, and also the two tangents at K and M which meet in the point P, then BP is the line in question. It thus appears that the line LON may be any line whatever through O; and since D, L, O, N are four harmonic points, we may describe the line AB as the locus of points which are harmonic conjugates of O with respect to the two points where any line through O meets the curve.



*97.* Furthermore, since the tangents at L and N meet on this same line, it appears as the locus of intersections of pairs of tangents drawn at the extremities of chords through O.



*98.* This important line, which is completely determined by the point O, is called the polar of O with respect to the conic; and the point O is called the pole of the line with respect to the conic.



*99.* If a point B is on the polar of O, then it is harmonically conjugate to O with respect to the two intersections K and M of the line BC with the conic. But for the same reason O is on the polar of B. We have, then, the fundamental theorem

If one point lies on the polar of a second, then the second lies on the polar of the first.



*100. Conjugate points and lines.* Such a pair of points are said to be conjugate with respect to the conic. Similarly, lines are said to be conjugate to each other with respect to the conic if one, and consequently each, passes through the pole of the other.

[Figure 27]

FIG. 27



*101. Construction of the polar line of a given point.* Given a point P, if it is within the conic (that is, if no tangents may be drawn from P to the conic), we may construct its polar line by drawing through it any two chords and joining the two points of intersection of the two pairs of tangents at their extremities. If the point P is outside the conic, we may draw the two tangents and construct the chord of contact (Fig. 27).



*102. Self-polar triangle.* In Fig. 26 it is not difficult to see that AOC is a self-polar triangle, that is, each vertex is the pole of the opposite side. For B, M, O, K are four harmonic points, and they project to C in four harmonic rays. The line CO, therefore, meets the line AMN in a point on the polar of A, being separated from A harmonically by the points M and N. Similarly, the line CO meets KL in a point on the polar of A, and therefore CO is the polar of A. Similarly, OA is the polar of C, and therefore O is the pole of AC.



*103. Pole and polar projectively related.* Another very important theorem comes directly from Fig. 26.

As a point A moves along a straight line its polar with respect to a conic revolves about a fixed point and describes a pencil projective to the point-row described by A.

For, fix the points L and N and let the point A move along the line AQ; then the point-row A is projective to the pencil LK, and since K moves along the conic, the pencil LK is projective to the pencil NK, which in turn is projective to the point-row C, which, finally, is projective to the pencil OC, which is the polar of A.



*104. Duality.* We have, then, in the pole and polar relation a device for setting up a one-to-one correspondence between the points and lines of the plane—a correspondence which may be called projective, because to four harmonic points or lines correspond always four harmonic lines or points. To every figure made up of points and lines will correspond a figure made up of lines and points. To a point-row of the second order, which is a conic considered as a point-locus, corresponds a pencil of rays of the second order, which is a conic considered as a line-locus. The name 'duality' is used to describe this sort of correspondence. It is important to note that the dual relation is subject to the same exceptions as the one-to-one correspondence is, and must not be appealed to in cases where the one-to-one correspondence breaks down. We have seen that there is in Euclidean geometry one and only one ray in a pencil which has no point in a point-row perspective to it for a corresponding point; namely, the line parallel to the line of the point-row. Any theorem, therefore, that involves explicitly the point at infinity is not to be translated into a theorem concerning lines. Further, in the pencil the angle between two lines has nothing to correspond to it in a point-row perspective to the pencil. Any theorem, therefore, that mentions angles is not translatable into another theorem by means of the law of duality. Now we have seen that the notion of the infinitely distant point on a line involves the notion of dividing a segment into any number of equal parts—in other words, of measuring. If, therefore, we call any theorem that has to do with the line at infinity or with the measurement of angles a metrical theorem, and any other kind a projective theorem, we may put the case as follows:

Any projective theorem involves another theorem, dual to it, obtainable by interchanging everywhere the words 'point' and 'line.'



*105. Self-dual theorems.* The theorems of this chapter will be found, upon examination, to be self-dual; that is, no new theorem results from applying the process indicated in the preceding paragraph. It is therefore useless to look for new results from the theorem on the circumscribed quadrilateral derived from Brianchon's, which is itself clearly the dual of Pascal's theorem, and in fact was first discovered by dualization of Pascal's.



*106.* It should not be inferred from the above discussion that one-to-one correspondences may not be devised that will control certain of the so-called metrical relations. A very important one may be easily found that leaves angles unaltered. The relation called similarity leaves ratios between corresponding segments unaltered. The above statements apply only to the particular one-to-one correspondence considered.



PROBLEMS

1. Given a quadrilateral, construct the quadrangle polar to it with respect to a given conic.

2. A point moves along a straight line. Show that its polar lines with respect to two given conics generate a point-row of the second order.

3. Given five points, draw the polar of a point with respect to the conic passing through them, without drawing the conic itself.

4. Given five lines, draw the polar of a point with respect to the conic tangent to them, without drawing the conic itself.

5. Dualize problems 3 and 4.

6. Given four points on the conic, and the tangent at one of them, draw the polar of a given point without drawing the conic. Dualize.

7. A point moves on a conic. Show that its polar line with respect to another conic describes a pencil of rays of the second order.

Suggestion. Replace the given conic by a pair of protective pencils.

8. Show that the poles of the tangents of one conic with respect to another lie on a conic.

9. The polar of a point A with respect to one conic is a, and the pole of a with respect to another conic is A'. Show that as A travels along a line, A' also travels along another line. In general, if A describes a curve of degree n, show that A' describes another curve of the same degree n. (The degree of a curve is the greatest number of points that it may have in common with any line in the plane.)



CHAPTER VII - METRICAL PROPERTIES OF THE CONIC SECTIONS



*107. Diameters. Center.* After what has been said in the last chapter one would naturally expect to get at the metrical properties of the conic sections by the introduction of the infinite elements in the plane. Entering into the theory of poles and polars with these elements, we have the following definitions:

The polar line of an infinitely distant point is called a diameter, and the pole of the infinitely distant line is called the center, of the conic.



*108.* From the harmonic properties of poles and polars,

The center bisects all chords through it ( 39).

Every diameter passes through the center.

All chords through the same point at infinity (that is, each of a set of parallel chords) are bisected by the diameter which is the polar of that infinitely distant point.



*109. Conjugate diameters.* We have already defined conjugate lines as lines which pass each through the pole of the other ( 100).

Any diameter bisects all chords parallel to its conjugate.

The tangents at the extremities of any diameter are parallel, and parallel to the conjugate diameter.

Diameters parallel to the sides of a circumscribed parallelogram are conjugate.

All these theorems are easy exercises for the student.



*110. Classification of conics.* Conics are classified according to their relation to the infinitely distant line. If a conic has two points in common with the line at infinity, it is called a hyperbola; if it has no point in common with the infinitely distant line, it is called an ellipse; if it is tangent to the line at infinity, it is called a parabola.



*111.* In a hyperbola the center is outside the curve ( 101), since the two tangents to the curve at the points where it meets the line at infinity determine by their intersection the center. As previously noted, these two tangents are called the asymptotes of the curve. The ellipse and the parabola have no asymptotes.



*112.* The center of the parabola is at infinity, and therefore all its diameters are parallel, for the pole of a tangent line is the point of contact.

The locus of the middle points of a series of parallel chords in a parabola is a diameter, and the direction of the line of centers is the same for all series of parallel chords.

The center of an ellipse is within the curve.

[Figure 28]

FIG. 28



*113. Theorems concerning asymptotes.* We derived as a consequence of the theorem of Brianchon ( 89) the proposition that if a triangle be circumscribed about a conic, the lines joining the vertices to the points of contact of the opposite sides all meet in a point. Take, now, for two of the tangents the asymptotes of a hyperbola, and let any third tangent cut them in A and B (Fig. 28). If, then, O is the intersection of the asymptotes,—and therefore the center of the curve,— then the triangle OAB is circumscribed about the curve. By the theorem just quoted, the line through A parallel to OB, the line through B parallel to OA, and the line OP through the point of contact of the tangent AB all meet in a point C. But OACB is a parallelogram, and PA = PB. Therefore

The asymptotes cut off on each tangent a segment which is bisected by the point of contact.



*114.* If we draw a line OQ parallel to AB, then OP and OQ are conjugate diameters, since OQ is parallel to the tangent at the point where OP meets the curve. Then, since A, P, B, and the point at infinity on AB are four harmonic points, we have the theorem

Conjugate diameters of the hyperbola are harmonic conjugates with respect to the asymptotes.



*115.* The chord A"B", parallel to the diameter OQ, is bisected at P' by the conjugate diameter OP. If the chord A"B" meet the asymptotes in A', B', then A', P', B', and the point at infinity are four harmonic points, and therefore P' is the middle point of A'B'. Therefore A'A" = B'B" and we have the theorem

The segments cut off on any chord between the hyperbola and its asymptotes are equal.



*116.* This theorem furnishes a ready means of constructing the hyperbola by points when a point on the curve and the two asymptotes are given.

[Figure 29]

FIG. 29



*117.* For the circumscribed quadrilateral, Brianchon's theorem gave ( 88) The lines joining opposite vertices and the lines joining opposite points of contact are four lines meeting in a point. Take now for two of the tangents the asymptotes, and let AB and CD be any other two (Fig. 29). If B and D are opposite vertices, and also A and C, then AC and BD are parallel, and parallel to PQ, the line joining the points of contact of AB and CD, for these are three of the four lines of the theorem just quoted. The fourth is the line at infinity which joins the point of contact of the asymptotes. It is thus seen that the triangles ABC and ADC are equivalent, and therefore the triangles AOB and COD are also. The tangent AB may be fixed, and the tangent CD chosen arbitrarily; therefore

The triangle formed by any tangent to the hyperbola and the two asymptotes is of constant area.



*118. Equation of hyperbola referred to the asymptotes.* Draw through the point of contact P of the tangent AB two lines, one parallel to one asymptote and the other parallel to the other. One of these lines meets OB at a distance y from O, and the other meets OA at a distance x from O. Then, since P is the middle point of AB, x is one half of OA and y is one half of OB. The area of the parallelogram whose adjacent sides are x and y is one half the area of the triangle AOB, and therefore, by the preceding paragraph, is constant. This area is equal to xy . sin α, where α is the constant angle between the asymptotes. It follows that the product xy is constant, and since x and y are the oblique cooerdinates of the point P, the asymptotes being the axes of reference, we have

The equation of the hyperbola, referred to the asymptotes as axes, is xy = constant.

This identifies the curve with the hyperbola as defined and discussed in works on analytic geometry.



[Figure 30]

FIG. 30

*119. Equation of parabola.* We have defined the parabola as a conic which is tangent to the line at infinity ( 110). Draw now two tangents to the curve (Fig. 30), meeting in A, the points of contact being B and C. These two tangents, together with the line at infinity, form a triangle circumscribed about the conic. Draw through B a parallel to AC, and through C a parallel to AB. If these meet in D, then AD is a diameter. Let AD meet the curve in P, and the chord BC in Q. P is then the middle point of AQ. Also, Q is the middle point of the chord BC, and therefore the diameter AD bisects all chords parallel to BC. In particular, AD passes through P, the point of contact of the tangent drawn parallel to BC.

Draw now another tangent, meeting AB in B' and AC in C'. Then these three, with the line at infinity, make a circumscribed quadrilateral. But, by Brianchon's theorem applied to a quadrilateral ( 88), it appears that a parallel to AC through B', a parallel to AB through C', and the line BC meet in a point D'. Also, from the similar triangles BB'D' and BAC we have, for all positions of the tangent line B'C,

B'D' : BB' = AC : AB,

or, since B'D' = AC',

AC': BB' = AC:AB = constant.

If another tangent meet AB in B" and AC in C", we have

AC' : BB' = AC" : BB",

and by subtraction we get

C'C" : B'B" = constant;

whence

The segments cut off on any two tangents to a parabola by a variable tangent are proportional.

If now we take the tangent B'C' as axis of ordinates, and the diameter through the point of contact O as axis of abscissas, calling the coordinates of B(x, y) and of C(x', y'), then, from the similar triangles BMD' and we have

y : y' = BD' : D'C = BB' : AB'.

Also

y : y' = B'D' : C'C = AC' : C'C.

If now a line is drawn through A parallel to a diameter, meeting the axis of ordinates in K, we have

AK : OQ' = AC' : CC' = y : y',

and

OM : AK = BB' : AB' = y : y',

and, by multiplication,

OM : OQ' = y2 : y'2,

or

x : x' = y2 : y'2;

whence

The abscissas of two points on a parabola are to each other as the squares of the corresponding cooerdinates, a diameter and the tangent to the curve at the extremity of the diameter being the axes of reference.

The last equation may be written

y2 = 2px,

where 2p stands for y'2 : x'.

The parabola is thus identified with the curve of the same name studied in treatises on analytic geometry.



*120. Equation of central conics referred to conjugate diameters.* Consider now a central conic, that is, one which is not a parabola and the center of which is therefore at a finite distance. Draw any four tangents to it, two of which are parallel (Fig. 31). Let the parallel tangents meet one of the other tangents in A and B and the other in C and D, and let P and Q be the points of contact of the parallel tangents R and S of the others. Then AC, BD, PQ, and RS all meet in a point W ( 88). From the figure,

PW : WQ = AP : QC = PD : BQ,

or

AP . BQ = PD . QC.

If now DC is a fixed tangent and AB a variable one, we have from this equation

_AP . BQ = _constant._

This constant will be positive or negative according as PA and BQ are measured in the same or in opposite directions. Accordingly we write

AP . BQ = +- b2.

[Figure 31]

FIG. 31

Since AD and BC are parallel tangents, PQ is a diameter and the conjugate diameter is parallel to AD. The middle point of PQ is the center of the conic. We take now for the axis of abscissas the diameter PQ, and the conjugate diameter for the axis of ordinates. Join A to Q and B to P and draw a line through S parallel to the axis of ordinates. These three lines all meet in a point N, because AP, BQ, and AB form a triangle circumscribed to the conic. Let NS meet PQ in M. Then, from the properties of the circumscribed triangle ( 89), M, N, S, and the point at infinity on NS are four harmonic points, and therefore N is the middle point of MS. If the cooerdinates of S are (x, y), so that OM is x and MS is y, then MN = y/2. Now from the similar triangles PMN and PQB we have

BQ : PQ = NM : PM,

and from the similar triangles PQA and MQN,

AP : PQ = MN : MQ,

whence, multiplying, we have

+-b2/4 a2 = y2/4 (a + x)(a - x),

where

[formula]

or, simplifying,

[formula]

which is the equation of an ellipse when b2 has a positive sign, and of a hyperbola when b2 has a negative sign. We have thus identified point-rows of the second order with the curves given by equations of the second degree.



PROBLEMS

1. Draw a chord of a given conic which shall be bisected by a given point P.

2. Show that all chords of a given conic that are bisected by a given chord are tangent to a parabola.

3. Construct a parabola, given two tangents with their points of contact.

4. Construct a parabola, given three points and the direction of the diameters.

5. A line u' is drawn through the pole U of a line u and at right angles to u. The line u revolves about a point P. Show that the line u' is tangent to a parabola. (The lines u and u' are called normal conjugates.)

6. Given a circle and its center O, to draw a line through a given point P parallel to a given line q. Prove the following construction: Let p be the polar of P, Q the pole of q, and A the intersection of p with OQ. The polar of A is the desired line.



CHAPTER VIII - INVOLUTION



[Figure 32]

FIG. 32

*121. Fundamental theorem.* The important theorem concerning two complete quadrangles ( 26), upon which the theory of four harmonic points was based, can easily be extended to the case where the four lines KL, K'L', MN, M'N' do not all meet in the same point A, and the more general theorem that results may also be made the basis of a theory no less important, which has to do with six points on a line. The theorem is as follows:

Given two complete quadrangles, K, L, M, N and K', L', M', N', so related that KL and K'L' meet in A, MN and M'N' in A', KN and K'N' in B, LM and L'M' in B', LN and L'N' in C, and KM and K'M' in C', then, if A, A', B, B', and C are in a straight line, the point C' also lies on that straight line.

The theorem follows from Desargues's theorem (Fig. 32). It is seen that KK', LL', MM', NN' all meet in a point, and thus, from the same theorem, applied to the triangles KLM and K'L'M', the point C' is on the same line with A and B'. As in the simpler case, it is seen that there is an indefinite number of quadrangles which may be drawn, two sides of which go through A and A', two through B and B', and one through C. The sixth side must then go through C'. Therefore,



*122.* Two pairs of points, A, A' and B, B', being given, then the point C' corresponding to any given point C is uniquely determined.

The construction of this sixth point is easily accomplished. Draw through A and A' any two lines, and cut across them by any line through C in the points L and N. Join N to B and L to B', thus determining the points K and M on the two lines through A and A', The line KM determines the desired point C'. Manifestly, starting from C', we come in this way always to the same point C. The particular quadrangle employed is of no consequence. Moreover, since one pair of opposite sides in a complete quadrangle is not distinguishable in any way from any other, the same set of six points will be obtained by starting from the pairs AA' and CC', or from the pairs BB' and CC'.



*123. Definition of involution of points on a line.*

Three pairs of points on a line are said to be in involution if through each pair may be drawn a pair of opposite sides of a complete quadrangle. If two pairs are fixed and one of the third pair describes the line, then the other also describes the line, and the points of the line are said to be paired in the involution determined by the two fixed pairs.

[Figure 33]

FIG. 33



*124. Double-points in an involution.* The points C and C' describe projective point-rows, as may be seen by fixing the points L and M. The self-corresponding points, of which there are two or none, are called the double-points in the involution. It is not difficult to see that the double-points in the involution are harmonic conjugates with respect to corresponding points in the involution. For, fixing as before the points L and M, let the intersection of the lines CL and C'M be P (Fig. 33). The locus of P is a conic which goes through the double-points, because the point-rows C and C' are projective, and therefore so are the pencils LC and MC' which generate the locus of P. Also, when C and C' fall together, the point P coincides with them. Further, the tangents at L and M to this conic described by P are the lines LB and MB. For in the pencil at L the ray LM common to the two pencils which generate the conic is the ray LB' and corresponds to the ray MB of M, which is therefore the tangent line to the conic at M. Similarly for the tangent LB at L. LM is therefore the polar of B with respect to this conic, and B and B' are therefore harmonic conjugates with respect to the double-points. The same discussion applies to any other pair of corresponding points in the involution.

[Figure 34]

FIG. 34



*125. Desargues's theorem concerning conics through four points.* Let DD' be any pair of points in the involution determined as above, and consider the conic passing through the five points K, L, M, N, D. We shall use Pascal's theorem to show that this conic also passes through D'. The point D' is determined as follows: Fix L and M as before (Fig. 34) and join D to L, giving on MN the point N'. Join N' to B, giving on LK the point K'. Then MK' determines the point D' on the line AA', given by the complete quadrangle K', L, M, N'. Consider the following six points, numbering them in order: D = 1, D' = 2, M = 3, N = 4, K = 5, and L = 6. We have the following intersections: B = (12-45), K' = (23-56), N' = (34-61); and since by construction B, N, and K' are on a straight line, it follows from the converse of Pascal's theorem, which is easily established, that the six points are on a conic. We have, then, the beautiful theorem due to Desargues:

The system of conics through four points meets any line in the plane in pairs of points in involution.



*126.* It appears also that the six points in involution determined by the quadrangle through the four fixed points belong also to the same involution with the points cut out by the system of conics, as indeed we might infer from the fact that the three pairs of opposite sides of the quadrangle may be considered as degenerate conics of the system.



*127. Conics through four points touching a given line.* It is further evident that the involution determined on a line by the system of conics will have a double-point where a conic of the system is tangent to the line. We may therefore infer the theorem

Through four fixed points in the plane two conics or none may be drawn tangent to any given line.

[Figure 35]

FIG. 35



*128. Double correspondence.* We have seen that corresponding points in an involution form two projective point-rows superposed on the same straight line. Two projective point-rows superposed on the same straight line are, however, not necessarily in involution, as a simple example will show. Take two lines, a and a', which both revolve about a fixed point S and which always make the same angle with each other (Fig. 35). These lines cut out on any line in the plane which does not pass through S two projective point-rows, which are not, however, in involution unless the angle between the lines is a right angles. For a point P may correspond to a point P', which in turn will correspond to some other point than P. The peculiarity of point-rows in involution is that any point will correspond to the same point, in whichever point-row it is considered as belonging. In this case, if a point P corresponds to a point P', then the point P' corresponds back again to the point P. The points P and P' are then said to correspond doubly. This notion is worthy of further study.

[Figure 36]

FIG. 36



*129. Steiner's construction.* It will be observed that the solution of the fundamental problem given in 83, Given three pairs of points of two protective point-rows, to construct other pairs, cannot be carried out if the two point-rows lie on the same straight line. Of course the method may be easily altered to cover that case also, but it is worth while to give another solution of the problem, due to Steiner, which will also give further information regarding the theory of involution, and which may, indeed, be used as a foundation for that theory. Let the two point-rows A, B, C, D, ... and A', B', C', D', ... be superposed on the line u. Project them both to a point S and pass any conic κ through S. We thus obtain two projective pencils, a, b, c, d, ... and a', b', c', d', ... at S, which meet the conic in the points α, β, γ, δ, ... and α', β', γ', δ', ... (Fig. 36). Take now γ as the center of a pencil projecting the points α', β', δ', ..., and take γ' as the center of a pencil projecting the points α, β, δ, .... These two pencils are projective to each other, and since they have a self-correspondin ray in common, they are in perspective position and corresponding rays meet on the line joining (γα', γ'α) to (γβ', γ'β). The correspondence between points in the two point-rows on u is now easily traced.



*130. Application of Steiner's construction to double correspondence.* Steiner's construction throws into our hands an important theorem concerning double correspondence: If two projective point-rows, superposed on the same line, have one pair of points which correspond to each other doubly, then all pairs correspond to each other doubly, and the line is paired in involution. To make this appear, let us call the point A on u by two names, A and P', according as it is thought of as belonging to the one or to the other of the two point-rows. If this point is one of a pair which correspond to each other doubly, then the points A' and P must coincide (Fig. 37). Take now any point C, which we will also call R'. We must show that the corresponding point C' must also coincide with the point B. Join all the points to S, as before, and it appears that the points α and π' coincide, as also do the points α'π and γρ'. By the above construction the line γ'ρ must meet γρ' on the line joining (γα', γ'α) with (γπ', γ'π). But these four points form a quadrangle inscribed in the conic, and we know by 95 that the tangents at the opposite vertices γ and γ' meet on the line v. The line γ'ρ is thus a tangent to the conic, and C' and R are the same point. That two projective point-rows superposed on the same line are also in involution when one pair, and therefore all pairs, correspond doubly may be shown by taking S at one vertex of a complete quadrangle which has two pairs of opposite sides going through two pairs of points. The details we leave to the student.

[Figure 37]

FIG. 37

[Figure 38]

FIG. 38



*131. Involution of points on a point-row of the second order.* It is important to note also, in Steiner's construction, that we have obtained two point-rows of the second order superposed on the same conic, and have paired the points of one with the points of the other in such a way that the correspondence is double. We may then extend the notion of involution to point-rows of the second order and say that _the points of a conic are paired in involution when they are corresponding _ points of two projective point-rows superposed on the conic, and when they correspond to each other doubly._ With this definition we may prove the theorem: _The lines joining corresponding points of a point-row of the second order in involution all pass through a fixed point _U_, and the line joining any two points _A_, _B_ meets the line joining the two corresponding points _A'_, _B'_ in the points of a line _u_, which is the polar of _U_ with respect to the conic._ For take _A_ and _A'_ as the centers of two pencils, the first perspective to the point-row _A'_, _B'_, _C'_ and the second perspective to the point-row _A_, _B_, _C_. Then, since the common ray of the two pencils corresponds to itself, they are in perspective position, and their axis of perspectivity _u_ (Fig. 38) is the line which joins the point _(AB', A'B)_ to the point _(AC', A'C)_. It is then immediately clear, from the theory of poles and polars, that _BB'_ and _CC'_ pass through the pole _U_ of the line _u_.



*132. Involution of rays.* The whole theory thus far developed may be dualized, and a theory of lines in involution may be built up, starting with the complete quadrilateral. Thus,

The three pairs of rays which may be drawn from a point through the three pairs of opposite vertices of a complete quadrilateral are said to be in involution. If the pairs aa' and bb' are fixed, and the line c describes a pencil, the corresponding line c' also describes a pencil, and the rays of the pencil are said to be paired in the involution determined by aa' and bb'.



*133. Double rays.* The self-corresponding rays, of which there are two or none, are called double rays of the involution. Corresponding rays of the involution are harmonic conjugates with respect to the double rays. To the theorem of Desargues ( 125) which has to do with the system of conics through four points we have the dual:

The tangents from a fixed point to a system of conics tangent to four fixed lines form a pencil of rays in involution.



*134.* If a conic of the system should go through the fixed point, it is clear that the two tangents would coincide and indicate a double ray of the involution. The theorem, therefore, follows:

Two conics or none may be drawn through a fixed point to be tangent to four fixed lines.



*135. Double correspondence.* It further appears that two projective pencils of rays which have the same center are in involution if two pairs of rays correspond to each other doubly. From this it is clear that we might have deemed six rays in involution as six rays which pass through a point and also through six points in involution. While this would have been entirely in accord with the treatment which was given the corresponding problem in the theory of harmonic points and lines, it is more satisfactory, from an aesthetic point of view, to build the theory of lines in involution on its own base. The student can show, by methods entirely analogous to those used in the second chapter, that involution is a projective property; that is, six rays in involution are cut by any transversal in six points in involution.



*136. Pencils of rays of the second order in involution.* We may also extend the notion of involution to pencils of rays of the second order. Thus, the tangents to a conic are in involution when they are corresponding rays of two protective pencils of the second order superposed upon the same conic, and when they correspond to each other doubly. We have then the theorem:



*137.* The intersections of corresponding rays of a pencil of the second order in involution are all on a straight line u, and the intersection of any two tangents ab, when joined to the intersection of the corresponding tangents a'b', gives a line which passes through a fixed point U, the pole of the line u with respect to the conic.



*138. Involution of rays determined by a conic.* We have seen in the theory of poles and polars ( 103) that if a point P moves along a line m, then the polar of P revolves about a point. This pencil cuts out on m another point-row P', projective also to P. Since the polar of P passes through P', the polar of P' also passes through P, so that the correspondence between P and P' is double. The two point-rows are therefore in involution, and the double points, if any exist, are the points where the line m meets the conic. A similar involution of rays may be found at any point in the plane, corresponding rays passing each through the pole of the other. We have called such points and rays conjugate with respect to the conic ( 100). We may then state the following important theorem:



*139.* _A conic determines on every line in its plane an involution of points, corresponding points in the involution _ being conjugate with respect to the conic. The double points, if any exist, are the points where the line meets the conic._



*140.* The dual theorem reads: A conic determines at every point in the plane an involution of rays, corresponding rays being conjugate with respect to the conic. The double rays, if any exist, are the tangents from the point to the conic.



PROBLEMS

1. Two lines are drawn through a point on a conic so as always to make right angles with each other. Show that the lines joining the points where they meet the conic again all pass through a fixed point.

2. Two lines are drawn through a fixed point on a conic so as always to make equal angles with the tangent at that point. Show that the lines joining the two points where the lines meet the conic again all pass through a fixed point.

3. Four lines divide the plane into a certain number of regions. Determine for each region whether two conics or none may be drawn to pass through points of it and also to be tangent to the four lines.

4. If a variable quadrangle move in such a way as always to remain inscribed in a fixed conic, while three of its sides turn each around one of three fixed collinear points, then the fourth will also turn around a fourth fixed point collinear with the other three.

5. State and prove the dual of problem 4.

6. Extend problem 4 as follows: If a variable polygon of an even number of sides move in such a way as always to remain inscribed in a fixed conic, while all its sides but one pass through as many fixed collinear points, then the last side will also pass through a fixed point collinear with the others.

7. If a triangle QRS be inscribed in a conic, and if a transversal s meet two of its sides in A and A', the third side and the tangent at the opposite vertex in B and B', and the conic itself in C and C', then AA', BB', CC' are three pairs of points in an involution.

8. Use the last exercise to solve the problem: Given five points, Q, R, S, C, C', on a conic, to draw the tangent at any one of them.

9. State and prove the dual of problem 7 and use it to prove the dual of problem 8.

10. If a transversal cut two tangents to a conic in B and B', their chord of contact in A, and the conic itself in P and P', then the point A is a double point of the involution determined by BB' and PP'.

11. State and prove the dual of problem 10.

12. If a variable conic pass through two given points, P and P', and if it be tangent to two given lines, the chord of contact of these two tangents will always pass through a fixed point on PP'.

13. Use the last theorem to solve the problem: Given four points, P, P', Q, S, on a conic, and the tangent at one of them, Q, to draw the tangent at any one of the other points, S.

14. Apply the theorem of problem 9 to the case of a hyperbola where the two tangents are the asymptotes. Show in this way that if a hyperbola and its asymptotes be cut by a transversal, the segments intercepted by the curve and by the asymptotes respectively have the same middle point.

15. In a triangle circumscribed about a conic, any side is divided harmonically by its point of contact and the point where it meets the chord joining the points of contact of the other two sides.



CHAPTER IX - METRICAL PROPERTIES OF INVOLUTIONS



[Figure 39]

FIG. 39

*141. Introduction of infinite point; center of involution.* We connect the projective theory of involution with the metrical, as usual, by the introduction of the elements at infinity. In an involution of points on a line the point which corresponds to the infinitely distant point is called the center of the involution. Since corresponding points in the involution have been shown to be harmonic conjugates with respect to the double points, the center is midway between the double points when they exist. To construct the center (Fig. 39) we draw as usual through A and A' any two rays and cut them by a line parallel to AA' in the points K and M. Join these points to B and B', thus determining on AK and AN the points L and N. LN meets AA' in the center O of the involution.



*142. Fundamental metrical theorem.* From the figure we see that the triangles OLB' and PLM are similar, P being the intersection of KM and LN. Also the triangles KPN and BON are similar. We thus have

OB : PK = ON : PN

and

OB' : PM = OL : PL;

whence

OB . OB' : PK . PM = ON . OL : PN . PL.

In the same way, from the similar triangles OAL and PKL, and also OA'N and PMN, we obtain

OA . OA' : PK . PM = ON . OL : PN . PL,

and this, with the preceding, gives at once the fundamental theorem, which is sometimes taken also as the definition of involution:

OA . OA' = OB . OB' = constant,

or, in words,

The product of the distances from the center to two corresponding points in an involution of points is constant.



*143. Existence of double points.* Clearly, according as the constant is positive or negative the involution will or will not have double points. The constant is the square root of the distance from the center to the double points. If A and A' lie both on the same side of the center, the product OA . OA' is positive; and if they lie on opposite sides, it is negative. Take the case where they both lie on the same side of the center, and take also the pair of corresponding points BB'. Then, since OA . OA' = OB . OB', it cannot happen that B and B' are separated from each other by A and A'. This is evident enough if the points are on opposite sides of the center. If the pairs are on the same side of the center, and B lies between A and A', so that OB is greater, say, than OA, but less than OA', then, by the equation OA . OA' = OB . OB', we must have OB' also less than OA' and greater than OA. A similar discussion may be made for the case where A and A' lie on opposite sides of O. The results may be stated as follows, without any reference to the center:

Given two pairs of points in an involution of points, if the points of one pair are separated from each other by the points of the other pair, then the involution has no double points. If the points of one pair are not separated from each other by the points of the other pair, then the involution has two double points.



*144.* An entirely similar criterion decides whether an involution of rays has or has not double rays, or whether an involution of planes has or has not double planes.

[Figure 40]

FIG. 40



*145. Construction of an involution by means of circles.* The equation just derived, OA . OA' = OB . OB', indicates another simple way in which points of an involution of points may be constructed. Through A and A' draw any circle, and draw also any circle through B and B' to cut the first in the two points G and G' (Fig. 40). Then any circle through G and G' will meet the line in pairs of points in the involution determined by AA' and BB'. For if such a circle meets the line in the points CC', then, by the theorem in the geometry of the circle which says that if any chord is drawn through a fixed point within a circle, the product of its segments is constant in whatever direction the chord is drawn, and if a secant line be drawn from a fixed point without a circle, the product of the secant and its external segment is constant in whatever direction the secant line is drawn, we have OC . OC' = OG . OG' = constant. So that for all such points OA . OA' = OB . OB' = OC . OC'. Further, the line GG' meets AA' in the center of the involution. To find the double points, if they exist, we draw a tangent from O to any of the circles through GG'. Let T be the point of contact. Then lay off on the line OA a line OF equal to OT. Then, since by the above theorem of elementary geometry OA . OA' = OT2 = OF2, we have one double point F. The other is at an equal distance on the other side of O. This simple and effective method of constructing an involution of points is often taken as the basis for the theory of involution. In projective geometry, however, the circle, which is not a figure that remains unaltered by projection, and is essentially a metrical notion, ought not to be used to build up the purely projective part of the theory.



*146.* It ought to be mentioned that the theory of analytic geometry indicates that the circle is a special conic section that happens to pass through two particular imaginary points on the line at infinity, called the circular points and usually denoted by I and J. The above method of obtaining a point-row in involution is, then, nothing but a special case of the general theorem of the last chapter ( 125), which asserted that a system of conics through four points will cut any line in the plane in a point-row in involution.

[Figure 41]

FIG. 41



*147. Pairs in an involution of rays which are at right angles. Circular involution.* In an involution of rays there is no one ray which may be distinguished from all the others as the point at infinity is distinguished from all other points on a line. There is one pair of rays, however, which does differ from all the others in that for this particular pair the angle is a right angle. This is most easily shown by using the construction that employs circles, as indicated above. The centers of all the circles through G and G' lie on the perpendicular bisector of the line GG'. Let this line meet the line AA' in the point C (Fig. 41), and draw the circle with center C which goes through G and G'. This circle cuts out two points M and M' in the involution. The rays GM and GM' are clearly at right angles, being inscribed in a semicircle. If, therefore, the involution of points is projected to G, we have found two corresponding rays which are at right angles to each other. Given now any involution of rays with center G, we may cut across it by a straight line and proceed to find the two points M and M'. Clearly there will be only one such pair unless the perpendicular bisector of GG' coincides with the line AA'. In this case every ray is at right angles to its corresponding ray, and the involution is called circular.



*148. Axes of conics.* At the close of the last chapter ( 140) we gave the theorem: _A conic determines at every point in its plane an involution of rays, corresponding rays _ being conjugate with respect to the conic. The double rays, if any exist, are the tangents from the point to the conic._ In particular, taking the point as the center of the conic, we find that conjugate diameters form a system of rays in involution, of which the asymptotes, if there are any, are the double rays. Also, conjugate diameters are harmonic conjugates with respect to the asymptotes. By the theorem of the last paragraph, there are two conjugate diameters which are at right angles to each other. These are called axes. In the case of the parabola, where the center is at infinity, and on the curve, there are, properly speaking, no conjugate diameters. While the line at infinity might be considered as conjugate to all the other diameters, it is not possible to assign to it any particular direction, and so it cannot be used for the purpose of defining an axis of a parabola. There is one diameter, however, which is at right angles to its conjugate system of chords, and this one is called the _axis_ of the parabola. The circle also furnishes an exception in that every diameter is an axis. The involution in this case is circular, every ray being at right angles to its conjugate ray at the center.



*149. Points at which the involution determined by a conic is circular.* It is an important problem to discover whether for any conic other than the circle it is possible to find any point in the plane where the involution determined as above by the conic is circular. We shall proceed to the curious problem of proving the existence of such points and of determining their number and situation. We shall then develop the important properties of such points.



*150.* It is clear, in the first place, that such a point cannot be on the outside of the conic, else the involution would have double rays and such rays would have to be at right angles to themselves. In the second place, if two such points exist, the line joining them must be a diameter and, indeed, an axis. For if F and F' were two such points, then, since the conjugate ray at F to the line FF' must be at right angles to it, and also since the conjugate ray at F' to the line FF' must be at right angles to it, the pole of FF' must be at infinity in a direction at right angles to FF'. The line FF' is then a diameter, and since it is at right angles to its conjugate diameter, it must be an axis. From this it follows also that the points we are seeking must all lie on one of the two axes, else we should have a diameter which does not go through the intersection of all axes—the center of the conic. At least one axis, therefore, must be free from any such points.

[Figure 42]

FIG. 42



*151.* Let now P be a point on one of the axes (Fig. 42), and draw any ray through it, such as q. As q revolves about P, its pole Q moves along a line at right angles to the axis on which P lies, describing a point-row p projective to the pencil of rays q. The point at infinity in a direction at right angles to q also describes a point-row projective to q. The line joining corresponding points of these two point-rows is always a conjugate line to q and at right angles to q, or, as we may call it, a conjugate normal to q. These conjugate normals to q, joining as they do corresponding points in two projective point-rows, form a pencil of rays of the second order. But since the point at infinity on the point-row Q corresponds to the point at infinity in a direction at right angles to q, these point-rows are in perspective position and the normal conjugates of all the lines through P meet in a point. This point lies on the same axis with P, as is seen by taking q at right angles to the axis on which P lies. The center of this pencil may be called P', and thus we have paired the point P with the point P'. By moving the point P along the axis, and by keeping the ray q parallel to a fixed direction, we may see that the point-row P and the point-row P' are projective. Also the correspondence is double, and by starting from the point P' we arrive at the point P. Therefore the point-rows P and P' are in involution, and if only the involution has double points, we shall have found in them the points we are seeking. For it is clear that the rays through P and the corresponding rays through P' are conjugate normals; and if P and P' coincide, we shall have a point where all rays are at right angles to their conjugates. We shall now show that the involution thus obtained on one of the two axes must have double points.

[Figure 43]

FIG. 43



*152. Discovery of the foci of the conic.* We know that on one axis no such points as we are seeking can lie ( 150). The involution of points PP' on this axis can therefore have no double points. Nevertheless, let PP' and RR' be two pairs of corresponding points on this axis (Fig. 43). Then we know that P and P' are separated from each other by R and R' ( 143). Draw a circle on PP' as a diameter, and one on RR' as a diameter. These must intersect in two points, F and F', and since the center of the conic is the center of the involution PP', RR', as is easily seen, it follows that F and F' are on the other axis of the conic. Moreover, FR and FR' are conjugate normal rays, since RFR' is inscribed in a semicircle, and the two rays go one through R and the other through R'. The involution of points PP', RR' therefore projects to the two points F and F' in two pencils of rays in involution which have for corresponding rays conjugate normals to the conic. We may, then, say:

There are two and only two points of the plane where the involution determined by the conic is circular. These two points lie on one of the axes, at equal distances from the center, on the inside of the conic. These points are called the foci of the conic.



*153. The circle and the parabola.* The above discussion applies only to the central conics, apart from the circle. In the circle the two foci fall together at the center. In the case of the parabola, that part of the investigation which proves the existence of two foci on one of the axes will not hold, as we have but one axis. It is seen, however, that as P moves to infinity, carrying the line q with it, q becomes the line at infinity, which for the parabola is a tangent line. Its pole Q is thus at infinity and also the point P', so that P and P' fall together at infinity, and therefore one focus of the parabola is at infinity. There must therefore be another, so that

A parabola has one and only one focus in the finite part of the plane.

[Figure 44]

FIG. 44



*154. Focal properties of conics.* We proceed to develop some theorems which will exhibit the importance of these points in the theory of the conic section. Draw a tangent to the conic, and also the normal at the point of contact P. These two lines are clearly conjugate normals. The two points T and N, therefore, where they meet the axis which contains the foci, are corresponding points in the involution considered above, and are therefore harmonic conjugates with respect to the foci (Fig. 44); and if we join them to the point P, we shall obtain four harmonic lines. But two of them are at right angles to each other, and so the others make equal angles with them (Problem 4, Chapter II). Therefore

The lines joining a point on the conic to the foci make equal angles with the tangent.

It follows that rays from a source of light at one focus are reflected by an ellipse to the other.



*155.* In the case of the parabola, where one of the foci must be considered to be at infinity in the direction of the diameter, we have

[Figure 45]

FIG. 45

A diameter makes the same angle with the tangent at its extremity as that tangent does with the line from its point of contact to the focus (Fig. 45).



*156.* This last theorem is the basis for the construction of the parabolic reflector. A ray of light from the focus is reflected from such a reflector in a direction parallel to the axis of the reflector.



*157. Directrix. Principal axis. Vertex.* The polar of the focus with respect to the conic is called the directrix. The axis which contains the foci is called the principal axis, and the intersection of the axis with the curve is called the vertex of the curve. The directrix is at right angles to the principal axis. In a parabola the vertex is equally distant from the focus and the directrix, these three points and the point at infinity on the axis being four harmonic points. In the ellipse the vertex is nearer to the focus than it is to the directrix, for the same reason, and in the hyperbola it is farther from the focus than it is from the directrix.

[Figure 46]

FIG. 46



*158. Another definition of a conic.* Let P be any point on the directrix through which a line is drawn meeting the conic in the points A and B (Fig. 46). Let the tangents at A and B meet in T, and call the focus F. Then TF and PF are conjugate lines, and as they pass through a focus they must be at right angles to each other. Let TF meet AB in C. Then P, A, C, B are four harmonic points. Project these four points parallel to TF upon the directrix, and we then get the four harmonic points P, M, Q, N. Since, now, TFP is a right angle, the angles MFQ and NFQ are equal, as well as the angles AFC and BFC. Therefore the triangles MAF and NFB are similar, and FA : FM = FB : BN. Dropping perpendiculars AA and BB' upon the directrix, this becomes FA : AA' = FB : BB'. We have thus the property often taken as the definition of a conic:

The ratio of the distances from a point on the conic to the focus and the directrix is constant.

[Figure 47]

FIG. 47



*159. Eccentricity.* By taking the point at the vertex of the conic, we note that this ratio is less than unity for the ellipse, greater than unity for the hyperbola, and equal to unity for the parabola. This ratio is called the eccentricity.

[Figure 48]

FIG. 48



*160. Sum or difference of focal distances.* The ellipse and the hyperbola have two foci and two directrices. The eccentricity, of course, is the same for one focus as for the other, since the curve is symmetrical with respect to both. If the distances from a point on a conic to the two foci are r and r', and the distances from the same point to the corresponding directrices are d and d' (Fig. 47), we have r : d = r' : d'; (r +- r') : (d +- d'). In the ellipse (d + d') is constant, being the distance between the directrices. In the hyperbola this distance is (d - d'). It follows (Fig. 48) that

In the ellipse the sum of the focal distances of any point on the curve is constant, and in the hyperbola the difference between the focal distances is constant.



PROBLEMS

1. Construct the axis of a parabola, given four tangents.

2. Given two conjugate lines at right angles to each other, and let them meet the axis which has no foci on it in the points A and B. The circle on AB as diameter will pass through the foci of the conic.

3. Given the axes of a conic in position, and also a tangent with its point of contact, to construct the foci and determine the length of the axes.

4. Given the tangent at the vertex of a parabola, and two other tangents, to find the focus.

5. The locus of the center of a circle touching two given circles is a conic with the centers of the given circles for its foci.

6. Given the axis of a parabola and a tangent, with its point of contact, to find the focus.

7. The locus of the center of a circle which touches a given line and a given circle consists of two parabolas.

8. Let F and F' be the foci of an ellipse, and P any point on it. Produce PF to G, making PG equal to PF'. Find the locus of G.

9. If the points G of a circle be folded over upon a point F, the creases will all be tangent to a conic. If F is within the circle, the conic will be an ellipse; if F is without the circle, the conic will be a hyperbola.

10. If the points G in the last example be taken on a straight line, the locus is a parabola.

11. Find the foci and the length of the principal axis of the conics in problems 9 and 10.

12. In problem 10 a correspondence is set up between straight lines and parabolas. As there is a fourfold infinity of parabolas in the plane, and only a twofold infinity of straight lines, there must be some restriction on the parabolas obtained by this method. Find and explain this restriction.

13. State and explain the similar problem for problem 9.

14. The last four problems are a study of the consequences of the following transformation: A point O is fixed in the plane. Then to any point P is made to correspond the line p at right angles to OP and bisecting it. In this correspondence, what happens to p when P moves along a straight line? What corresponds to the theorem that two lines have only one point in common? What to the theorem that the angle sum of a triangle is two right angles? Etc.



CHAPTER X - ON THE HISTORY OF SYNTHETIC PROJECTIVE GEOMETRY



*161. Ancient results.* The theory of synthetic projective geometry as we have built it up in this course is less than a century old. This is not to say that many of the theorems and principles involved were not discovered much earlier, but isolated theorems do not make a theory, any more than a pile of bricks makes a building. The materials for our building have been contributed by many different workmen from the days of Euclid down to the present time. Thus, the notion of four harmonic points was familiar to the ancients, who considered it from the metrical point of view as the division of a line internally and externally in the same ratio(1) the involution of six points cut out by any transversal which intersects the sides of a complete quadrilateral as studied by Pappus(2); but these notions were not made the foundation for any general theory. Taken by themselves, they are of small consequence; it is their relation to other theorems and sets of theorems that gives them their importance. The ancients were doubtless familiar with the theorem, Two lines determine a point, and two points determine a line, but they had no glimpse of the wonderful law of duality, of which this theorem is a simple example. The principle of projection, by which many properties of the conic sections may be inferred from corresponding properties of the circle which forms the base of the cone from which they are cut—a principle so natural to modern mathematicians—seems not to have occurred to the Greeks. The ellipse, the hyperbola, and the parabola were to them entirely different curves, to be treated separately with methods appropriate to each. Thus the focus of the ellipse was discovered some five hundred years before the focus of the parabola! It was not till 1522 that Verner(3) of Nuernberg undertook to demonstrate the properties of the conic sections by means of the circle.



*162. Unifying principles.* In the early years of the seventeenth century—that wonderful epoch in the history of the world which produced a Galileo, a Kepler, a Tycho Brahe, a Descartes, a Desargues, a Pascal, a Cavalieri, a Wallis, a Fermat, a Huygens, a Bacon, a Napier, and a goodly array of lesser lights, to say nothing of a Rembrandt or of a Shakespeare—there began to appear certain unifying principles connecting the great mass of material dug out by the ancients. Thus, in 1604 the great astronomer Kepler(4) introduced the notion that parallel lines should be considered as meeting at an infinite distance, and that a parabola is at once the limiting case of an ellipse and of a hyperbola. He also attributes to the parabola a "blind focus" (caecus focus) at infinity on the axis.



*163. Desargues.* In 1639 Desargues,(5) an architect of Lyons, published a little treatise on the conic sections, in which appears the theorem upon which we have founded the theory of four harmonic points ( 25). Desargues, however, does not make use of it for that purpose. Four harmonic points are for him a special case of six points in involution when two of the three pairs coincide giving double points. His development of the theory of involution is also different from the purely geometric one which we have adopted, and is based on the theorem ( 142) that the product of the distances of two conjugate points from the center is constant. He also proves the projective character of an involution of points by showing that when six lines pass through a point and through six points in involution, then any transversal must meet them in six points which are also in involution.



*164. Poles and polars.* In this little treatise is also contained the theory of poles and polars. The polar line is called a traversal.(6) The harmonic properties of poles and polars are given, but Desargues seems not to have arrived at the metrical properties which result when the infinite elements of the plane are introduced. Thus he says, "When the traversal is at an infinite distance, all is unimaginable."



*165. Desargues's theorem concerning conics through four points.* We find in this little book the beautiful theorem concerning a quadrilateral inscribed in a conic section, which is given by his name in 138. The theorem is not given in terms of a system of conics through four points, for Desargues had no conception of any such system. He states the theorem, in effect, as follows: Given a simple quadrilateral inscribed in a conic section, every transversal meets the conic and the four sides of the quadrilateral in six points which are in involution.



*166. Extension of the theory of poles and polars to space.* As an illustration of his remarkable powers of generalization, we may note that Desargues extended the notion of poles and polars to space of three dimensions for the sphere and for certain other surfaces of the second degree. This is a matter which has not been touched on in this book, but the notion is not difficult to grasp. If we draw through any point P in space a line to cut a sphere in two points, A and S, and then construct the fourth harmonic of P with respect to A and B, the locus of this fourth harmonic, for various lines through P, is a plane called the polar plane of P with respect to the sphere. With this definition and theorem one can easily find dual relations between points and planes in space analogous to those between points and lines in a plane. Desargues closes his discussion of this matter with the remark, "Similar properties may be found for those other solids which are related to the sphere in the same way that the conic section is to the circle." It should not be inferred from this remark, however, that he was acquainted with all the different varieties of surfaces of the second order. The ancients were well acquainted with the surfaces obtained by revolving an ellipse or a parabola about an axis. Even the hyperboloid of two sheets, obtained by revolving the hyperbola about its major axis, was known to them, but probably not the hyperboloid of one sheet, which results from revolving a hyperbola about the other axis. All the other solids of the second degree were probably unknown until their discovery by Euler.(7)



*167.* Desargues had no conception of the conic section of the locus of intersection of corresponding rays of two projective pencils of rays. He seems to have tried to describe the curve by means of a pair of compasses, moving one leg back and forth along a straight line instead of holding it fixed as in drawing a circle. He does not attempt to define the law of the movement necessary to obtain a conic by this means.



*168. Reception of Desargues's work.* Strange to say, Desargues's immortal work was heaped with the most violent abuse and held up to ridicule and scorn! "Incredible errors! Enormous mistakes and falsities! Really it is impossible for anyone who is familiar with the science concerning which he wishes to retail his thoughts, to keep from laughing!" Such were the comments of reviewers and critics. Nor were his detractors altogether ignorant and uninstructed men. In spite of the devotion of his pupils and in spite of the admiration and friendship of men like Descartes, Fermat, Mersenne, and Roberval, his book disappeared so completely that two centuries after the date of its publication, when the French geometer Chasles wrote his history of geometry, there was no means of estimating the value of the work done by Desargues. Six years later, however, in 1845, Chasles found a manuscript copy of the "Bruillon-project," made by Desargues's pupil, De la Hire.



*169. Conservatism in Desargues's time.* It is not necessary to suppose that this effacement of Desargues's work for two centuries was due to the savage attacks of his critics. All this was in accordance with the fashion of the time, and no man escaped bitter denunciation who attempted to improve on the methods of the ancients. Those were days when men refused to believe that a heavy body falls at the same rate as a lighter one, even when Galileo made them see it with their own eyes at the foot of the tower of Pisa. Could they not turn to the exact page and line of Aristotle which declared that the heavier body must fall the faster! "I have read Aristotle's writings from end to end, many times," wrote a Jesuit provincial to the mathematician and astronomer, Christoph Scheiner, at Ingolstadt, whose telescope seemed to reveal certain mysterious spots on the sun, "and I can assure you I have nowhere found anything similar to what you describe. Go, my son, and tranquilize yourself; be assured that what you take for spots on the sun are the faults of your glasses, or of your eyes." The dead hand of Aristotle barred the advance in every department of research. Physicians would have nothing to do with Harvey's discoveries about the circulation of the blood. "Nature is accused of tolerating a vacuum!" exclaimed a priest when Pascal began his experiments on the Puy-de-Dome to show that the column of mercury in a glass tube varied in height with the pressure of the atmosphere.



*170. Desargues's style of writing.* Nevertheless, authority counted for less at this time in Paris than it did in Italy, and the tragedy enacted in Rome when Galileo was forced to deny his inmost convictions at the bidding of a brutal Inquisition could not have been staged in France. Moreover, in the little company of scientists of which Desargues was a member the utmost liberty of thought and expression was maintained. One very good reason for the disappearance of the work of Desargues is to be found in his style of writing. He failed to heed the very good advice given him in a letter from his warm admirer Descartes.(8) "You may have two designs, both very good and very laudable, but which do not require the same method of procedure: The one is to write for the learned, and show them some new properties of the conic sections which they do not already know; and the other is to write for the curious unlearned, and to do it so that this matter which until now has been understood by only a very few, and which is nevertheless very useful for perspective, for painting, architecture, etc., shall become common and easy to all who wish to study them in your book. If you have the first idea, then it seems to me that it is necessary to avoid using new terms; for the learned are already accustomed to using those of Apollonius, and will not readily change them for others, though better, and thus yours will serve only to render your demonstrations more difficult, and to turn away your readers from your book. If you have the second plan in mind, it is certain that your terms, which are French, and conceived with spirit and grace, will be better received by persons not preoccupied with those of the ancients.... But, if you have that intention, you should make of it a great volume; explain it all so fully and so distinctly that those gentlemen who cannot study without yawning; who cannot distress their imaginations enough to grasp a proposition in geometry, nor turn the leaves of a book to look at the letters in a figure, shall find nothing in your discourse more difficult to understand than the description of an enchanted palace in a fairy story." The point of these remarks is apparent when we note that Desargues introduced some seventy new terms in his little book, of which only one, involution, has survived. Curiously enough, this is the one term singled out for the sharpest criticism and ridicule by his reviewer, De Beaugrand.(9) That Descartes knew the character of Desargues's audience better than he did is also evidenced by the fact that De Beaugrand exhausted his patience in reading the first ten pages of the book.



*171. Lack of appreciation of Desargues.* Desargues's methods, entirely different from the analytic methods just then being developed by Descartes and Fermat, seem to have been little understood. "Between you and me," wrote Descartes(10) to Mersenne, "I can hardly form an idea of what he may have written concerning conics." Desargues seems to have boasted that he owed nothing to any man, and that all his results had come from his own mind. His favorite pupil, De la Hire, did not realize the extraordinary simplicity and generality of his work. It is a remarkable fact that the only one of all his associates to understand and appreciate the methods of Desargues should be a lad of sixteen years!



*172. Pascal and his theorem.* One does not have to believe all the marvelous stories of Pascal's admiring sisters to credit him with wonderful precocity. We have the fact that in 1640, when he was sixteen years old, he published a little placard, or poster, entitled "Essay pour les conique,"(11) in which his great theorem appears for the first time. His manner of putting it may be a little puzzling to one who has only seen it in the form given in this book, and it may be worth while for the student to compare the two methods of stating it. It is given as follows: "If in the plane of M, S, Q we draw through M the two lines MK and MV, and through the point S the two lines SK and SV, and let K be the intersection of MK and SK; V the intersection of MV and SV; A the intersection of MA and SA (A is the intersection of SV and MK), and μ the intersection of MV and SK; and if through two of the four points A, K, μ, V, which are not in the same straight line with M and S, such as K and V, we pass the circumference of a circle cutting the lines MV, MP, SV, SK in the points O, P, Q, N; I say that the lines MS, NO, PQ are of the same order." (By "lines of the same order" Pascal means lines which meet in the same point or are parallel.) By projecting the figure thus described upon another plane he is able to state his theorem for the case where the circle is replaced by any conic section.



*173.* It must be understood that the "Essay" was only a resume of a more extended treatise on conics which, owing partly to Pascal's extreme youth, partly to the difficulty of publishing scientific works in those days, and also to his later morbid interest in religious matters, was never published. Leibniz(12) examined a copy of the complete work, and has reported that the great theorem on the mystic hexagram was made the basis of the whole theory, and that Pascal had deduced some four hundred corollaries from it. This would indicate that here was a man able to take the unconnected materials of projective geometry and shape them into some such symmetrical edifice as we have to-day. Unfortunately for science, Pascal's early death prevented the further development of the subject at his hands.



*174.* In the "Essay" Pascal gives full credit to Desargues, saying of one of the other propositions, "We prove this property also, the original discoverer of which is M. Desargues, of Lyons, one of the greatest minds of this age ... and I wish to acknowledge that I owe to him the little which I have discovered." This acknowledgment led Descartes to believe that Pascal's theorem should also be credited to Desargues. But in the scientific club which the young Pascal attended in company with his father, who was also a scientist of some reputation, the theorem went by the name of 'la Pascalia,' and Descartes's remarks do not seem to have been taken seriously, which indeed is not to be wondered at, seeing that he was in the habit of giving scant credit to the work of other scientific investigators than himself.



*175. De la Hire and his work.* De la Hire added little to the development of the subject, but he did put into print much of what Desargues had already worked out, not fully realizing, perhaps, how much was his own and how much he owed to his teacher. Writing in 1679, he says,(13) "I have just read for the first time M. Desargues's little treatise, and have made a copy of it in order to have a more perfect knowledge of it." It was this copy that saved the work of his master from oblivion. De la Hire should be credited, among other things, with the invention of a method by which figures in the plane may be transformed into others of the same order. His method is extremely interesting, and will serve as an exercise for the student in synthetic projective geometry. It is as follows: Draw two parallel lines, a and b, and select a point P in their plane. Through any point M of the plane draw a line meeting a in A and b in B. Draw a line through B parallel to AP, and let it meet MP in the point M'. It may be shown that the point M' thus obtained does not depend at all on the particular ray MAB used in determining it, so that we have set up a one-to-one correspondence between the points M and M' in the plane. The student may show that as M describes a point-row, M' describes a point-row projective to it. As M describes a conic, M' describes another conic. This sort of correspondence is called a collineation. It will be found that the points on the line b transform into themselves, as does also the single point P. Points on the line a transform into points on the line at infinity. The student should remove the metrical features of the construction and take, instead of two parallel lines a and b, any two lines which may meet in a finite part of the plane. The collineation is a special one in that the general one has an invariant triangle instead of an invariant point and line.



*176. Descartes and his influence.* The history of synthetic projective geometry has little to do with the work of the great philosopher Descartes, except in an indirect way. The method of algebraic analysis invented by him, and the differential and integral calculus which developed from it, attracted all the interest of the mathematical world for nearly two centuries after Desargues, and synthetic geometry received scant attention during the rest of the seventeenth century and for the greater part of the eighteenth century. It is difficult for moderns to conceive of the richness and variety of the problems which confronted the first workers in the calculus. To come into the possession of a method which would solve almost automatically problems which had baffled the keenest minds of antiquity; to be able to derive in a few moments results which an Archimedes had toiled long and patiently to reach or a Galileo had determined experimentally; such was the happy experience of mathematicians for a century and a half after Descartes, and it is not to be wondered at that along with this enthusiastic pursuit of new theorems in analysis should come a species of contempt for the methods of the ancients, so that in his preface to his "Mechanique Analytique," published in 1788, Lagrange boasts, "One will find no figures in this work." But at the close of the eighteenth century the field opened up to research by the invention of the calculus began to appear so thoroughly explored that new methods and new objects of investigation began to attract attention. Lagrange himself, in his later years, turned in weariness from analysis and mechanics, and applied himself to chemistry, physics, and philosophical speculations. "This state of mind," says Darboux,(14) "we find almost always at certain moments in the lives of the greatest scholars." At any rate, after lying fallow for almost two centuries, the field of pure geometry was attacked with almost religious enthusiasm.



*177. Newton and Maclaurin.* But in hastening on to the epoch of Poncelet and Steiner we should not omit to mention the work of Newton and Maclaurin. Although their results were obtained by analysis for the most part, nevertheless they have given us theorems which fall naturally into the domain of synthetic projective geometry. Thus Newton's "organic method"(15) of generating conic sections is closely related to the method which we have made use of in Chapter III. It is as follows: If two angles, AOS and AO'S, of given magnitudes turn about their respective vertices, O and O', in such a way that the point of intersection, S, of one pair of arms always lies on a straight line, the point of intersection, A, of the other pair of arms will describe a conic. The proof of this is left to the student.



*178.* Another method of generating a conic is due to Maclaurin.(16) The construction, which we also leave for the student to justify, is as follows: _If a triangle _C'PQ_ move in such a way that its sides, _PQ_, _QC'_, and _C'P_, turn _ around three fixed points, _R_, _A_, _B_, respectively, while two of its vertices, _P_, _Q_, slide along two fixed lines, _CB'_ and _CA'_, respectively, then the remaining vertex will describe a conic._



*179. Descriptive geometry and the second revival.* The second revival of pure geometry was again to take place at a time of great intellectual activity. The period at the close of the eighteenth and the beginning of the nineteenth century is adorned with a glorious list of mighty names, among which are Gauss, Lagrange, Legendre, Laplace, Monge, Carnot, Poncelet, Cauchy, Fourier, Steiner, Von Staudt, Moebius, Abel, and many others. The renaissance may be said to date from the invention by Monge(17) of the theory of descriptive geometry. Descriptive geometry is concerned with the representation of figures in space of three dimensions by means of space of two dimensions. The method commonly used consists in projecting the space figure on two planes (a vertical and a horizontal plane being most convenient), the projections being made most simply for metrical purposes from infinity in directions perpendicular to the two planes of projection. These two planes are then made to coincide by revolving the horizontal into the vertical about their common line. Such is the method of descriptive geometry which in the hands of Monge acquired wonderful generality and elegance. Problems concerning fortifications were worked so quickly by this method that the commandant at the military school at Mezieres, where Monge was a draftsman and pupil, viewed the results with distrust. Monge afterward became professor of mathematics at Mezieres and gathered around him a group of students destined to have a share in the advancement of pure geometry. Among these were Hachette, Brianchon, Dupin, Chasles, Poncelet, and many others.



*180. Duality, homology, continuity, contingent relations.* Analytic geometry had left little to do in the way of discovery of new material, and the mathematical world was ready for the construction of the edifice. The activities of the group of men that followed Monge were directed toward this end, and we now begin to hear of the great unifying notions of duality, homology, continuity, contingent relations, and the like. The devotees of pure geometry were beginning to feel the need of a basis for their science which should be at once as general and as rigorous as that of the analysts. Their dream was the building up of a system of geometry which should be independent of analysis. Monge, and after him Poncelet, spent much thought on the so-called "principle of continuity," afterwards discussed by Chasles under the name of the "principle of contingent relations." To get a clear idea of this principle, consider a theorem in geometry in the proof of which certain auxiliary elements are employed. These elements do not appear in the statement of the theorem, and the theorem might possibly be proved without them. In drawing the figure for the proof of the theorem, however, some of these elements may not appear, or, as the analyst would say, they become imaginary. "No matter," says the principle of contingent relations, "the theorem is true, and the proof is valid whether the elements used in the proof are real or imaginary."