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Graduate Student Theses, Dissertations, & Professional Papers Graduate School
1930
Projective geometry from 1822-1918
John B. Lennes The University of Montana
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Recommended Citation Lennes, John B., "Projective geometry from 1822-1918" (1930). Graduate Student Theses, Dissertations, & Professional Papers. 8077. https://scholarworks.umt.edu/etd/8077
This Thesis is brought to you for free and open access by the Graduate School at ScholarWorks at University of Montana. It has been accepted for inclusion in Graduate Student Theses, Dissertations, & Professional Papers by an authorized administrator of ScholarWorks at University of Montana. For more information, please contact [email protected]. PR0JECTI7E GEOMETRY from 1822 to 1918
by
J. Burr Lennea
Presented, in partial fulfillment of the requirement for the degree of Master of Arts,
State Unirer^y of Montana 1930
Approved EXAMHÎIUG COIMITTEE Chairman
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. COEfTSM?S Page Introaiiot ion 1 Chapter I. Greek Geometry with Regard to the Origin of Projective Geometry 4 Chapter II. Geometry in Europe from 1500 to 1800, with Regard to the Origin of Projective Geometry. 12 Chapter III. Projective Geometry from 1822 to 1906. Poncelet 46 Ton Stattdt 88 S tein e r 122 Reye 124 Chapter IT. The Reorganization of Pro jective Geometry in the First Part of the Present Century. 131 Chapter T. Resulting conclusions. I. The nature of Intellectual Progress 165 II. The nature, Uses, etc., of the Deductive logical Structure. 173
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. "Bn observant oe q^ne les r^sulta*ts partic-uliers avaient de oornmin entre enz, on est suocesslvement parvenu k des résultats fort étendus, et les sciences mathématiques sont a la fols devenues plus générales et plus sim ples." Laplace tOncted by Steiner on the title page of his "Systematische Bntwiclc/iielung", )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Introduction
Although many of the theorems which now form the sub ject matter of projective geometry have been known for hun dreds of years, and some of them for more than two thousand years, it was not until the nineteenth century that they became unified into a separate, well-defined branch of math ematics. It is our object to trace the origin of the methods and propositions which now constitute projective geometry, thus outlining the gradual development of the subject, and to give a picture of modern projective geometry, with special attention to the fundamental significance of the logical pro cedure. Thus, it is hoped, we may throw new light upon the early discoveries, revealing more clearly their importance in the history of thought and their position in modern geometry. The subject divides itself naturally into four parts. The first is a brief résumé of what there was in Greek geom etry that has become fundamental in projective geometry. The second part covers the period including the seventeenth and sixteenth centuries, when the French and English math ematicians were most active in this as in nearly every other branch of mathematics. The third is the account of the jrogress in the nineteenth century, when the subject assumed definite shape and developed more rapidly than in any earlier
period. In this period most of the work was done in France and Germany. The fo u rth part is an account of p ro jectiv e
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. geometry in the last thirty years. Since 1900 the whole subject has been given new, rigorous treatment and has been unified in a clear and fundamental way, with full conscious ness of the underlying logical foundation. There have been certain major changes in the develop ment of geometry which it will be well to enumerate at the o u tse t. One of the most important of these was the swing from the rigorous proofs of the Greeks to the Icgical laxness of the mathematicians of the seventeenth, eighteenth, and early nineteenth centuries, and then back to the very highly de veloped rigor of modern p ro jectiv e geometry. Another was the steady change of emphasis from the solution of individual prob lems to the development of connected unified theories - that is, the change from the particular to the general. The last Important change was the change from nstrie to non-metric geometry. Greek geometry was essentially metric, whereas mod ern projective geometry is essentially non-metric. Completely non-metric geometry is a very recent development in the history of mathematics. Such general tendencies as these give interest and sig nificance to the particular historical facts. The facts pre sented here have been selected with a view to including all those which were important, especially those which seem to have had an influence on the histoiy of the subject, and to rather rigidly excluding all which from this view point were
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. less relevant. heedless to say, not all the original documents, especial ly in regard to the Greeks, are available. And even in the parts dealing with the suhjeot from 1800 to the present time, where the original sources have been used, not all the facts have been considered, since only the major works were studied. It is hoped, however, that nothing of really great significance has been overlooked. The last chapter of this thesis (chapter 7) contains ma terial less strictly mathematical and more general in its sig nificance. We have seen how painfhlly slow and halting was the development of a subject whiah in the finished state seems relatively simple, and that in spite of the fact that many of the world’s greatest geniuses gave it very insistent and pro longed effort. Why did tiiey fail to hit upon some of the later and much more powerful methods? Again the very existence of huge logical structure purely deductive in character is cer tainly of general philosophical significance. 7/hat is the nature of such a deductive logical structure, and is it as barren as some have claimed it to be? What is the function in thought of purely deductive logic? If final answers to some of these questions cannot be given, it is believed that they are formulated in a manner to bring out their significance.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter I
Plato (430—347 B.C.) is the first of the Greek mathe maticians In whose work we find anything related at all di rectly to oar subject. It was he who studied conic sections, which foim a considerable part of the subject matter of pro jective geometry. There is, however, no evidence that Plato developed many of the properties of these curves. He was the first to femulate the idea of geometrical loci, which is ingortant in all branches of mathematics. Euclid, about 300 B. C., is famous for his "Elements" in which he formulated rather completely the classical Greek geometry, with considerable extension and improvement of his own. This is the basis of ordinary modern elementary geometry. The method of "reductio ad absurdum" which is valuable in projective geometry as everywhere else in logic, was for mulated by Euclid. Among his works which have not come down to us there are four books on conic sections, the theory of which he had considerably augmented. Apollonius of Perga, about 247 B. C., perhaps the great est of a n the Greek mathematicians, wrote, besides other minor works, a treatise of eight books on conic sections. It was Apollonius who first considered conics as. sections of any obliq[ue cone with a circular base; until then they had
been conceived only as sections of right circular cones, and the cutting plane had always been supposed perpendicular to
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. one of the lines of the cone passing throngh the vertex. This required, three cones of different vertex angles to form the three conic sections. These curves were designated by the words "section of the acute-angled cone", "section of the right-angled cone", "section of the obtuse-angles cone"; they first took definitely the names ellipse, parabola, and hyperbola in the work of Apollonius, although the words ellipse and parabola were known to Archimedes. The whole of Apollonius’ treatise on conics was based on a unique property of conic sections: Suppose an oblique cone with a circular base to be out by a plane through its vertex and through that diameter of the circle at its base which makes this plane perpendicular to the base. The plane will contain the axis of the cone, (the line from the vertex through the center of the circle at the base) and will deter mine, with the cone and the circle at the base, a trian'^e whose base is the diam eter of the c irc le and whose two sides are the lines from the end-points of this diameter to the vertex of the cone. This is called the "triangle through the axis". Apollonius supposes the cutting plane always perpen dicular to the plane of this triangle. In the figure, YQ is the axis and A YA3 is the triangle through the axis. The plane EFQ- is perpendicular to plane YAB.
- ' G P i t I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The pointa A^and s'in which the cutting plane meets the two sides of this triangle are the vertices of the curve determined "by the cone and this plane, and the line which joins these two points is a diameter. Apollonius calls this diameter the latus tranaversum.
If through either of the two vertices of the curve, we erect a perpendicular to the plane of the triangle through the axis, and give it a certain length, to he determined as we shall say later, and if from the extremity D of this per pendicular we draw a line DB to the other vertex of the curve, and if further we erect an ordinate perpendicularly from some point 0 of the diameter of the curve, the square of this ordinate taken between the diameter and the curve will equal the rectangle 02-OA constructed on the part of the ordinate between the diameter and the line and on the part of the diam eter between the first vertex and the foot of the rodinate, for any point £, provided A D is taken such that this rela tion holds for any one point 0.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In the hands of Apollonius this relation plays much the
game part as the equation of the seooni degree in analytic
geom etry. It is seen from this that the diameter of the curve and
the perpendicular to one of its eztremities suffices to oon^-
struot the curve. The perpendicular in question was called
latus erectumj but is now called latu s^ eotum or parameter.
Working with this relation Apollonius found all of the
best-known properties of conics. These were^prinoipally, the
constant ratio of the products of the segments made by a conic
on two transversals parallel to two fixed axes and drawn
through any point, the principal properties of the foci of
the ellipse and hyperbola, which he calls "points of applica
tion", and the idea of conjugate diameters. The conception of
asymptotes likewise originated with Apollonius.
The following theorem is particularly interesting from
our view-point:
F ig . 3 "If from the point of intersection of two tangents to a
conic section one draws a transversal meeting the curve in
two pointy and the chord joining the points of contact
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of the two tangents In a third point, this third point and
the point of intersection of the two tangents w ill be harmon ic conjugates with respect to the first two."
(Apollonius did not use the word "harmonic" but gave the
ratio, which in the case of the above figure, is
Although Apollonius was one of the most original and
prolific of all the ancient geometers, it seems that all of
his work that has any direct interest to us here is that
which has been outlined above.
The period after Archimedes and Apollonius, whose works
marked the most brilliant epoch of ancient geometry, was one
of great progress in astronomy, and mathematicians were prin
cipally concerned with the problems it presented, w4*h the
result^that plane and spherical trigonometry were discovered
and developed. Results of importance in geometry were only
occasional and isolated.
Ptolemy, an astronomer and mathematician of about 125
A. P., discovered two theorems of interest to us in this con
nection. They were: # "The product of the two diagonals of a quadrilateral in
scribed in a circle equals the sum of the products of the op
posite sides." "A transversal drawn in the plane of a triangle cuts six
segments on its three aides such that the product of three of
these segments having no common extremities equals the product
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of the other three."
F ' l t That I s , AL*BM*C]? =
This latter theorem formed the basis for Ptolemy's
trigonometry. (Ptolemy, after Hipparohns, whose work has
been lost, was the great originator of trigonometry.)
Here the period of Greek geometry ended, to have a
short revival nearly three hundred years later in the figure
of Pappus, before it became completely a thing of the past.
Pappus of Alexandria, towards the last part of the third
century A. D., gathered together in his "Mathematical Collec
tions" various isolated discoveries of the greatest mathe
maticians and stated a multitude of curious propositions and
lemmas designed to facilitate the reading of their works.
These collections contained many of the propositions of Pappus
him self, whom Descartes considered one of the most excellent
geometers of antiquity. It is through his writings that we
have obtained much of our knowledge of the works of the
Greek mathematicians.
Of these original theorems there are three that are of
interest to us, two of them in particular. Proposition 1S9 isi
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”Wh«tt fottr lin es are drawn from
a single point they form on a trans
versal drawn arbitrarily in their
plane four segments which have among them a certain constant ratio, what
ever the transversal."
Thnm, if A, B, C, B are the
points, and AC, jü), BC, BD the seg
m en ts, I ' / / AC BC A G B C jjy ‘ ^ " '£' W * etc., whatever the transversal, Proposition 131 is
"In every quadrilateral, a diagonal is cut harmonically
by the second diagonal and the line which joins the point of
intersection of opposite sides." ,
A
6 F that on a line through the two points of intersection of op posite sides of a quadrilateral the points formed by the two diagonals divide these two first points harmonically, it is, however, its exact equivalent, as w ill be seen if we oon^ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 slder ABCD in figura 6 to be the «quadrilateral in the first ease. Pappus* theorem then says that A are harmonic points. If, in the other theorem, which is the basis of the modern definition of the harmonic set of points, the quadri lateral is taken to be AÇBP, then the second theorem also, using the same lines, says that A E C^F are harmonic points. 1 3 9 th i s ; "When a hexagon has its six vertices placed, three by three, on two straight lines, the three points of intersec tion of its opposite sides are in a straight line." F '& 7 This is known as the theorem of Pappus. It is, as w ill be seen later, a special case of Pascal's thaorem. It is interesting to know that Pappus, in completing the work of Apollonius, found the focus of the parabola (it w ill be remanbered that Apollonius found those of the ellipse and hy perbola only) and discovered the property of a conic having a d i r e c t r i x . These are the principal contributions which Pappus made to the evolution of the subject of projective geometry, and in turning from this last figure in the ranks of the Greek math ematicians we leave the first great stage in the history of mathematics — the Greek geometry. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. IS C hapter I I ruin of the magnifioent libraries of Alexandria, due to neglect and to depredations from many quarters, the progress of geometry came to a standstill* For nearly ten centuries, while Europe was in the ignorant darkness of the Middle Ages, the Arabs were the custodians of the ancient Greek mathematics. Although they did practically nothing to extend or to improve upon geometry as the Greeks had left it, they at least saved it from oblivion by their copies and translations of the more in^ortant works* The first sign of an awakening interest in geometry was the publication in 1482 of Ratdolt's edition of Euclid's Elements,. From th is time on geometry was taken over by the Europeans, although not a great deal was done at first* In the time between 1480 and 1600 there were several European geometricians, the more notable ones being Terner, Paciuolo, Messina, leonardo da Vinci, Oordano, Albrecht Durer^ and K epler* Kepler was important in geometry because he originated the conception of infinity. He stated that one focus of a parabola is at Infinity. This contribution opened a vast new field of speculation# in geometry* Girard Desargues, born at lyon in 1593, was the first great geometrician of this renaissance of mathematics, which followed somewhat behind the great Renaissance of art and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 letters. Sis work, thoiigh. far from volxualnous, was extraor dinarily rioh. in new conceptions and new theorems. Desargnes oonoeired a straight line as extended to infin ity and tkns its "opposite ends" nnited. He considered two parallel lines as two straight lines intersecting at in finity, and, conversely, if two sneh straight lines intersec ted at infinity, he considered them parallel. He showed that a line and a circle were two oases of the same kind of curve, and that their construction could be stated in the same words. Desargues was the first to make explicit statements about points and lines at infinity, among other things to con sider asymptotes as tangents through the points at infinity on the curves in question* Desargues* major work was a short treatise on conics, in which the following was the central theorem: "Given a conic and a quadrilateral inscribed in it, and a transversal drawn in the plane of the curve. The product of the segments on the transversal between a point on the conic and >fee two opposite sides of the quadrilateral is to the product of the segments between the other point of the conic and the two other opposite sides of the quadrilateral, in a ratio which is equal to that of the similar products with the second point of the curve located on the transversal." Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 F' ê s That is, in Fig. 8, wm _ scnffi*Hl'HB The relation of these siz points on the transversal Dgs- argnes called, an involution of siz points. His original déf inition of involution which he showed to he equivalent to the above relation was: If on a straight line a point A is taken arbitrarily, and. if siz points B, G, _H, F are taken on the line so that AB*AH = AC'AG « AD'AF, then these six points constitute an involution of which B and_H, 0^ and. G, and D and F are pairs o f p o in t s . The following very important propositions on poles and polars were for a long time attributed to Be la Hire, but are now known to have originated with Besargues. (1) "*If about a fixed point one turns a transversal which meets a conic in two points, the tangents at these points w ill always meet on a fixed lin e,” and reciprocally, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 "If from each point of a given line one draws two tan gents to a oonio, the line which joins the two points of contact w ill pass through a fixed point." (2) "If through a fixed point one draws several trans versals which meet a oonio in two points each, the lines which join the points in which each of two of the transversals cut the oonio meet on the polar of the fixed point." (3) "The point where each transversal meets the polar of the fixed point la the harmonic conjugate of this fixed point with respect to the two points where the transversal meets the curve." This last; as may be remembered, was known to Apollonius. Probably Desargues» greatest single achievement was re garding a conic section as the curve determined by cutting any cone with any conic for base by any plane, thus definitely doing away with the triangle throngh the axis and including the point, the line, and the system of two oopointal lines under the general category of conic sections. Easing shown that any conic section is the projection of a circle, ard noticing that the involution of six points is a projective relation, he proved his famous theorem {stated a- bove) by proving it for the case of the circle by means of ordinary Euclidian methods and then extending the theorem to all conic sections, by projection and section. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 In oonclxid.ing otir discussion of Desargues we w ill state the famous theorem which bears his name. It never appeared in any of his own works, but was communicated by him to his friend Bosse, who published it in his work on perspective. "If two triangles in space or in the same plane have their vertices placed two by two on three straight lines intersecting in the same point, their sides w ill meet two by two in three points on a straight line, and reciprocally." Blaise Pascal, next to Descartes, was the most influen tia l figure of his time in mathematical thought, although his work in geometry was not very significant from our view-point, except that it was based on Desargues procedure of regarding any conic as the projection of a circle or of any other conic. Pascal's one real contribution was the theorem which he called the "hexagrammum mystioum", and^is now known as Pascal's theorem. In its original form it read; "Every ohord—hexagon in a circle has the property that the three interseetion-points of each two opposite sides lie in one and the same straight line." / W / \ \ / Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 This he immediately extended to any conic, use of Desargues* proof that all oonios are projections of a circle* With this property of conies as a basis, Pascal wrote a treatise on oonio sections, in which four hundred corollaries to the above theorem were proved. This treatise, which, un fortunately, was entirely lost, was written in 1640, when the author was sixteen years of age. Pascal seems to have no other importance for us except as he encouraged his contemporaries, particularly Desargues, from whom he learned a great deal of what he knew about geom etry* Descartes, a contemporary of Desargues and Pascal, by his conception of the application of algebra to the theory of curves created at once an entirely new geometry, and provided ne ans for overcoming obstacles which had hitherto baffled the greatest geometricians* The distinguishing characteristics doyc T.‘'ie 4 of analytical geometry are too w ell known to be djfi-muewed here, The importance of Descartes in regard to "pure geometry" was indirect but great, in that he created a completely new branch of mathematics whose advantages and novelties attracted the attention of mathematicians to the exclusion of the older geometry. This phenomenon was observed before, when in the first and second centuries A* D. astronomy and trigonometry superseded geometry in the interest of the Greek mathemati c ia n s . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 Howerer, there were a few mathematicians who continued to work with the classical geometry, and it was usually the greatest innovators in mathematics who were the greatest masters of the Greek geometry» Jan de Witt (1625—1672)^ although no very famous mathe matician, stated one theorem which is very interesting to us. It is generalization of a theorem of Cavalieri: “Given an angle, draw transversals parallel among them selves; from the points where each transversal outs the sides of the angle, draw two lines through two fixed points, respectively. These two lines w ill intersect in a point which w ill have for geometric locus a conic passing throngh the fixed points." This is a special case of the construction of a conic by two projective pencils, which is fundamental in recent projective geometry. De la Hire, although he did not, as has been supposed, discover the theorems on poles and polars which were above attributed to Desargues, was, according to several authori ties, the first to solve the problem; "Given three points of a harmonic set; to construct the fourth by means of the ruler alone. Pappus certainly had the means to solve this problem with ease, had it ever come to his attention, so this dls- covery of De la Hire may have been very clever, butait may Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 hare been a veiry simple use of the proposition whieh Pappus had already stated. ]Jewton, in his "Enumeration of lines of the third order" (1700) stated the following two propositions, which he pre sented as extensions of the principal properties of conics. "Given, in the plane of a geometric curve, transversals parallel among themselves, and taking on each the center of mean distances of all the points in which it meets the curve, all these centers lie on a straight line which w ill be the diameter of the curve conjugate to the direction of the transversals." "If through some point taken in the plane of a geometric curve one draws two transversals parallel to two fixed axes, the product of the segments on these two lines between the point through which they are drawn and the curve, are in a constant ratio, whatever the point taken." Eewton means by "geometric curve" an algebraic curve of any o r d e r . Similar to these theorems, and at least as inportant, are the two following due to Maclaurin, somewhat newton's ju n io r . (l) "If, about a fixed point, one turns a transversal which meets a geometric curve in as many points A, B, ... as it has dimensions, and if one takes on this transversal, in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. so eaola of its positions, a point M so that the reciprocal of its distance to the fixed point is the arithmetic mean be tween the reciprocals of the distances of A, B to this fixed point, the point g w ill have a straight line for geometric locos.” (This first theorem was suggested to Maclaurin by his friend Cotes.) (2) ”Iet a transversal be drawn through a fixed point in the plane of the geometric curve, meeting the curve in as many points as it has dimensions; let tangents be drawn to the curve through these points; and draw, through the fixed point, a second line of arbitrary direction, but which w ill remain fixed; the segments on this line, between the fixed point and a ll the tangents to the curve, w ill have the sum of their reciprocals constant, whatever the first trans versal drawn through the fixed point; this sum w ill be equal to that of the reciprocals of the segments on the same fixed line, between the same point and those in which the line in tersects the curve." In these two theorems the "dimensions" of a curve ne an its degree. Thus a curve of two dimensions is one of the second degree. These theorems represent considerable that is new and that is of importance and interest. In the first place, they are the basis for an immediate extension of the theory of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 polea and polars to cxurves of any degree. In the second place, they involTe the idea of a continuous harmonic scale, and the center of mean harmonics with respect to such a scale. In the third place, they involre a new metric statement of the harmonic relation, peculiarly adapted for the theory of c e n te r s o f mean h arm on ics. In these last fonr theorems the tendency toward gener alization in geometry becomes quite noticeable. This ten dency should be kept in mind in coB^aring the theorems and methods of different epochs. It has been the unvaiying di rection of progress in geometry that whatever the changes in method or subject-matter^ the underlyizg change has been away from the solution of particular problems toward greater and greater generality. Before discussing the next major development in geom etry, which took place toward the end of the eighteenth cen tury, namely the descriptive geometry of Monge, let us pause to trace the really rather gradual evolution of this branch of mathematics that seemed to spring into being so su d d en ly . Descriptive geometry seems to have owed its first de velopment to architecture. Vitruvius, the famous architect and builder of the time of Julius Caesar and Augustus, speaks in the first book of his "Archltectura", of the "ichono- graphia" and "orthographia", that is, the "ground plan" and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. s s th# "elevation" of a bnllâlng, as aids to the description of it. These methods had quite possibly originated with the Greeks, but it was he who made them well-known and widely used. However, although this prototype of a very well-known method of descriptive geometry gave a very easy and convenient way of representing three-dimensional fig ures, it did not point out the way to apply the ordinary methods of geometry to the problem. Those who strove with this difficulty were notably Desargues (aside from being a great mathematician he was a well-known architect and civ il engineer). Bosse, the engraver, who published a work on per spective, Pater Derand, and M illiet—Dechales. Hone of these were very successful, but each by his interest and contri butions kept the subject before the minds of at least a few mathematicians. Andrée Francois Frezier, an engineer in the French army, was born in 1682. It was he who finally achieved what Desargues and Bosse had been unable to do — to give "Ster- eotomy" a comparatively firm rational foundation. In 1737 he published his work "la théorie et la prati- tue de la coupe des pierres et des bois,..., on Traité de stéréotomie a l ’usage de l ’architecture". The first book consists of two parts. The first deals with the plane sections of various bodies as the sphere, cone, cylinder, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 armtilar surface, etc. The second part deals with Tarions curves and surfaces of the fourth degree. The second part of the second book considers curves formed by the intersec tion of two curved surfaces. The first part of the third book is given to projection. It seems that Frezier was the first to use two projections of a figure. This, in regard to both projective and descriptive geometry, was a very im portant innovation. The great founder of descriptive geometry was Gaspard Monge (1746—1818}. Perhaps the greatest single achievement of Monge was the application of analytic methods to the representation of surfaces. This, as w ill be seen, has an insert ant bearing on his descriptive gome try. The characteristic procedure in the geometry of Monge is to project the three-dimensional figure in question by means of parallel lines onto a plane perpendicular to these lines. The same figure is projected twice by this "orthog onal projection" — first onto one plane and then onto an o th e r . The first volume of Monge»s great work "Geometrie des criptive^"' (1800) is devoted to an explanation of the method of double orthogonal projection (representation of theee dimensional figures on a plane, and the derivation of the properties of these figures from this representation.) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 Here the author shows the usefulness of taking the two pro ject iour-planes at right angles to each other, one horizontal and the other vertical, and then, leaving the figures on them invariable, turning them about their line of intersection until they coincide. The really novel thing about this method was not so much the description of space curves but of surfaces. In view of the fact that a curved surface is developed by the motion of a curve, whose form is variable subject to certain restric tions, and whose motion is likewise subject to certain restric tions, the procedure is to represent a system of its curves. It is more convenient to describe two such systems, chosen so that a curve of each system goes through each point of the surface. The conical, cylindrical^ and rotational surfaces make interesting examples of this method of description; the same is true of the ruled surfaces, whose determination by the motion of a straight line which constantly intersects three fixed directrices is given in the first part of the first volume. The plane is described by the motion of a straight line constantly intersecting another straight line and parallel to a third; as determining lines of a plane it is convenient to choose those in which the described plane cuts the projection- plane s ; these are called the "traces" of the plane, a name c o in e d by Monge and now in common u s a g e . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. These methods allow the author to solve a large number of important problems; some of these are: through a point to construct the line which is parallel to another, or perpendic ular to a plane, or to construct the plane which is parallel to another or cuts another line perpendicularly; to find the line of intersection of two planes; and to determine the an^e between two planes, two lines, or a plane and a line. With these problems the f ir s t part ends. The second part deals with the tangent planes and the normals to surfaces. The tangent plane at g. given point of a surface is determined by the tangents to the two projections of the surface in the two points corresponding to this point. With this equipment the tangent planes to cylindrical, conical, and rotational surfaces are constructed. The tangent planes to ruled sur faces are treated in the third part of the first volume. Then follows the determination of the shortest distance be tween two skew lines, vhich is accomplished by use of the tan gent plane to a circular cylinder. Monge then tums to the determination of tangent planes when the point of tangency is not given. The first problem given is to find the tangent plane to a given sphere passing through a given line. He gives two solutions; 1. by passing a plane through the center of the sphere and perpendicular to the line in question, or 2. by use of the cone determined by this sphere and a point of the line. With th is second solu- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 6 tion are inoluded the principal properties of polara with respect to circles and conics, and also the characteristic properties of the center of similitude of circles or spheres. "The intersect ion-line s of curved surfaces" is the sub ject of the fourth part of the work. Here Monge gives the general method that is s till used to construct these curves. To determine the intersection-line of two surfaces F'^and F^, point by point, he makes use of an auxiliary surface F. This determines two curves FF^and FF" that can supposedly be represented by his method of description. The intersec- tion-point of these two curves is a point of the desired curve, so if we move F we also move th is point FF^^F" along the curve. In this manner the whole curve can be described. XTsually_F is taken as a horizontal or vertical plane, but if F^^and ^ are cones or cylinders some other plane is taken, and if they are two surfaces of rotation whose axes inter sect F is taken as a sphere. In this way Monge investigated the cases of the intersection of cones, cylinders, and sur faces of rotation which either intersect with each other or with planes. In the case of the intersection with planes not only were the projections of the points and tangents found, but also the real form and dimensions of the curve, by composition from the projections. The usefulness of the above constructions for the solur- tion of problems of theoretical and practical interest is shown in the fourth part, by means of several examples. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 Ike first problem is the determination of the center of the sphere inscribed in or oironmsoribed about a tetra hedron, and the determination of the point which has given distances from three other points. The other problems have to do mainly with cartography. The following bit of theory is interesting mostly for its curiosity, but also as a proof of the steadily irt- oreasing freedom of the use of projection. If the point 0,is a source of light, and if all bodies with which we are dealing are opaque, then, if these bodies are polyhedrons, the problem of finding the shows they w ill oast on a plane of projection is easily solved by the fun damental method of descriptive geometry. The illuminated part of the body is separated from the dark part by a skew polygon DDhioh can be determined by the position of the body with respect to 0. If, however, we deal with a curved sur face, the dividing-line is a very important curve, which Monge called "ligne de contour apparent d'une surface". To construct this curve he passed a p^ane throu^ 0, normal to the plane of projection; this cuts the surface in a curve r~ which can be constructed and described; the point of tan gency of the tangent drawn to£” from £ belongs to the re quired line. If ^ is allowed to turn about 0 in such a way that it is always normal to the plane of projection, this Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 line is desorilïed point "by point. 0 ma^ of oonTse, be taken at infinity if desired. Monge also made the re- gtLired modification to extend this theory to the case where the source of li^ t is a surface. y*1rt This concludes^the work of Monge which we shall out line here, except for the following two theorems which seem worth quoting. (1) "If one has in a plane two triangles ABC and A^B* whose corresponding sides intersect two by two in three points I, M, n in the same straight line 1, and if through a point 0 taken arbitrarily one draws the three lines OA, OB, and extends them until they cut the line 1 in th e three points Mf if one joins these points, respectively, with the three vertices of the second triangle by three lines f these lines w ill intersect in a single point ^ . a 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 2>esargties* theorem is ohTiotusly a special case of th is, namely, when 0 and. ^ ooinci&e. Then ^ and l ' , M and M^, H and w ill coincide# (£) "If one draws any transversal throngh the inter section point* of the two tangents common to any two conics in a plane, cutting each of these curves in two points, and if one draws their tangents at these points, the tangents to the first w ill meet the tangents to the second in four points which w ill be, two by two, on two lines which remain fixed, whatever the transversal drawn through the intersec- tion-point of the two tangents common to the two conics." If much of this material seems detailed and trivial for inclusion in a work of this scope, it should be remembered AS yi«U that here, on the eve of the birth of projective geometry, it is important to know what was occupying the attention of the greatest geometricians in Europe, and that mathematics, as well as every other field of human thought, was much more extensive and complicated than ever before. To the end that we sh ^ l get a more complete picture of geometry in the eighteenth century, inasfar as it has a bearing on our subject, we sh& i now turn our attention to the development of what was known as "perspective", or, in its later and more developed stages, "perspective geometry". Although perspective was not in teelf a very highly organized Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 branoh of geometry, it doubtless had an effect on the origin and development of projective geometry equal to that of des criptive geometry, although apparently not so direct, Mong^ by his technique of studying space figures, added immensely to the subject matter of geoumtry^ ^ obtaining figures in space analogous to the already well-known plane figures and demonstrating that their properties were also analogous. He also set the stage for the discovery of the principle of duality. In perspective geometry, on the other hand, we see not so much a great body of subjeot—yatter in the form of the descriptions and properties of new figures as the con scious use and development of a method which is fundamental in projective geometry, Wilhelm Jacob s'Graves&nde (1688—1742) was the author of a famous work "Essai de Perspective" (1711), He begins with a discussion of the usefulness of the theory of per spective and a statement of the fundamental methods in its use. Then he builds the mathematical foundation of perspec tive, He first shows that a line parallel to the "table" (plane on which projections are made) is also parallel to its projection, and that "a figure whose plane is parallel to the table is similar to its perspective," Then he makes the following important observation: "If a line cuts the table in a single point, its perspective is the line connecting this point with that in which the table is cut by the line Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 parallel to the first line and drawn throng the center of projection .... The position of the perspective of a line undergoes no change if the center of projection moves along this parallel line." These principles are then applied to the construction of the “perspectives" of figures which lie in a horizontal plane, the table beiqg placed vertically upon it. In fig. It T is the center of projection ^d 0^ the per pendicular dropped from V onto the table; P is an arbitrary point of the horizontal plane and S its vertical projection on the table. Then the "perspective" of ^ is the point p' where TP and OH intersect; therefore p'divides the segment OH in the ratio TO:PH. From these considerations s'Graves- ande derives the first of his methods of perspective. . II Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 In fig. n ie t t be the "ligne de terre" (intersection of the horizontal plane and the table) and £ the line parallel to it drawn through Oj the segment 07 is perpendionlar to SL and equal to the segment 70 in fig. f * ; the lines OH and 7? will then intersect in the point If one draws the cir cles with P and 7, respectively, as centers and ^ and 70 as radii, then P^ is the center of aymmsti^ of the two circles; therefore P^ can also be found as the intersection-point of the two tangents conmon to the two circles. How if one considers a second point Q of the horizontal plane and wishes to find o f , one can make use of the fact that ^ and P ^ intersect on the line jt; one determines the point P€l*t and connects it with P^ ; this connecting line w ill be cut by the line 7Q, at How, after one has determined P^and cf, if one considers a third point R of the horizontal plane, in order to find R'^use can be made o f th e fa c t t h a t PQR and P^ ^ R* are perspective triangles, with _7 as center and t as axis of symmetry. It must be noticed that in the above cases the entire figure of fig ,// is collapsed so as to lie in a plane. We determined the loœation of perspectives ^ of points ^ by the considera tions, metric and otherwise, derived from fig.// . In des cribing the last case it suffices to say that the table is first ta]œn perpsedioular to the horizontal plane, as in fig.lj , and then is laid upon it by simply turning it about Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 tj therefore, what la this ooimeotion s^G-ravesande calls the projection of a fignre F we would call the superposition (F), g)-^' t — while his object ive figure (lying in the horizontal plane) is to us the horizontal projection^ of 2" the problems which he solves by this method the derivation of ^ from (_^) appears. The author now extends his methods to the deter mination of the perspectives of polygons and gives four ad ditional methods for finding the perspective of an arbitrary point in space. (He had previously dealt only with points lying in the horizontal plane). Thai he includes the case where the center of projection is at a very great distance, so that the lines of projection can be considered parallel, and also where the two planes are not at right angles, an im portant special case being where they are parallel. With a discussion of the subject of shadows and a simpli fication of the problems which he stated at the beginning of the book, the "essai de Perspective" comes to a close. Al though his methods seem highly specialized and the work as a whole rather lax from a mathematical stand—point, this was nevertheless a beginning of the conscious use of a method which, although it had already been made possible by the dis coveries of geometry, was not at that time appreciated at its fu ll worth. The work of Brook Taylor (1685-1731), entitled "Hew Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 prlnoiplea of linear perapeotiTe ... etc.", published in 1749, was no less inçortant than that of s'Gravesande, in spite of its oomparatively small amount of matter. Taylor begins his book with an eacplanation of his basic ideas and his rather novel nomenclature. The center of projection he calls "point of sight". He constructs a plane throngh this point of sight parallel to the "picture- plane" and calls it the directing plane. Its intersections with arbitrary planes are called, respectively, directing lines and directing points. The "seat" of a point or line is its orthogonal projection on the picture—plane. The pro jection of an element at infinity is a "vanishing element. Taylor observed that the vanishing— and directing—lines of a plane have the same vanishing—point. It should be bom in mind that the vanishing-line of any plane is the projec tion of the lihe at infinity in that plane upon the picture- plane, with the point of sight as center of projection. In other words, the vanishing-line of a plane is the intersec tion of a plane drawn parallel to it through the point of sight with the piotnre—plane. Similarly, the vanishing-point of a line is the point in which a line drawn parallel to it through the point of sight cuts the picture-plane. Parallel lines which are perpendicular to the picture-plane have their (common) vanishing point such that it is the seat (orthogonal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 6 projeotion) of the point of sight. This point is called the "center of the picture? The following are problems which Taylor solved by using his method of perspective: I, The projections of three col- linear points A, B, C are given, and also the vanishing— c point of the line ; to determine the ratio AC;Bf. II. The projections of three collinear points A, B, C and the ratio ^:BC are given; to find the vanishing-point of the line. III. Given the projection of a triangle, the vanishing—line of its plane, the center of the picture, and the distance {the word "distance" is taken to mean the distance from the center of the picture to the point of sight); to find the nature of the triangle (size, shape, position). IT. One knows the projection and the nature of a triangle and the vanishing-line of its plane; to find the distance and the center of the picture. T. The projection of a known trape— zoid is also known; to find the vanishing-line of its plane, the center of the picture and the distance. VI. Given the projection of a rectangular parallelepiped; to find the cen^- ter of the picture, the distance, and the nature of the space figure. The remarkable thing about Taylor»s work on perspective (5 does w*e that not only it contain, due to the elegance of its conception and working-out, a great deal of new and valuable naterial in only eighty pages of printed matter, but alsg^a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 large portion of it was in mach the same form in which we find it today in the best works on perspective. There remains one more great figure in eighteenth- < 5 century perspective theory. It «a* Johann Heinrich Lam bert. His work "Freye Perspective* was published first in 1759 and later, with inmortant additions, in 1774. In dis cussing this work we w ill consider one by one the eight chapters into which it is divided. In the first chapter Lambert makes the assumptions (a) that light is propagated reetilinearly, and (b) if the table is placed vertically then vertical lines w ill have projections upon it which w ill be vertical lines. The line in which the table intersects the horizontal plane is called the fundamental or ground lin e. The distance OS. (fig .*3) of the eye 0, from the horizontal plane is called the "level of the eye"; the foot P of the perpendicular dropped to the table from g, is the "point of the eye"; the length of the segment OP is called the "distance"; finally, the horizontal line through the point of the eye and lying in the table is called the "horizontal line". How let be the fourth ver tex of the rectangle POSQ,. jC an arbitrary point of the horizontal plane^and c its p r o j e c t io n , low we draw through £ in the horizontal c plane some arbitrary auxili- F Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 aiy lin© meeting the ground—line in M, and then we draw the line whioh is parallel to CM and passes thrcagh 0, meeting TO the table in the pointy on the horizontal line; p^w ill be the projection of where ^ is the point at infinity on CM* All lines parallel to CM w ill have as projections lines which go throngh p, a point which depends only on the angle oCwhioh these original lines make with the groimd line, or the angleP = whioh they make with a line perpendicular to the ground line. lo determine one of these lines it is necessary to give the point in which it cuts the line S_3 (fig. If ). Then we notice that ^ %CM POp = p> and therefore Pp = OP tan/3 . now if we know /3 , we can lay off from_P, along the horizontal line, a segment equal to OP tan/^ . , We thus obtain two points p and 4- a. The projections of all lines of the horizontal plane making an angle of/3 with the ground—line mast go through and all making an a n g le , through p. Further, if one sets OP equal to PQ., then the triangles PQp and POp are congruent and so z. pop = / PQjp “ /3 • If one describes on the table a circle Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 with center and radina Of and draws through, q. the lines 3^ing in the table msOcing the angles of 1°, 2°, ... etc, with Of, then a "scale” of points w ill be determined on the horizontal line. These are correspondingly denoted by 1°, - f 2 ,... etc. How two lines making angles of/3 and ^ , res— ' peotively, with the ground—lin e, make an angle oifi' with each other, and the projections of two lines of the horizontal plane making an angle with each other go through two points of the above-mentioned scale whose de noting numbers have the difference if' • Working with these ideas Lambert adopts the following conventions; I. Two lines of the table which go through the same point of the horizontal line w ill be called "parallel" because th^ are projections of parallel lines. II. A line of the table at right-g^les to the horizontal line is called "perpendicular" because it is the projection of a vertical line. III. Every angle on the table w ill be assigned the scale-number equal to the magnitude of the corresponding angle of the horizontal plane. 17. Finally, every segment on the table is denoted by the length of the segment in the horizontal plane whose projection it is. The following problems w ill serve to indicate just- how the above conceptions were used: I Through a given point to pass a line parallel to a given line. II. Through a point given on a line to draw another line making a given angle Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 With the first* If the sides of a figiire and their posi tions are given, and also the angles of the figure, to con struct its perspective* IT. To construct the perspective of a circle of which one knows a chord corresponding to a g iv e n a r c . In regard to measurements Lambert states that if A^B 'c 'l/ls a quadrangle on the table, and if A^C'^and B ^ are parallel, while à! and 0*^1)^intersect on the horizontal line, ABOD w ill be a parallelogram and therefore AB » Op, ^ » Bp, ^ = BO. Therefore if one knows a scale on a line parallel to the ground line, it is easy to measure a seg ment parallel to it. The analogous question is the case of the arbitrary segment, however, is more d ifficu lt; Lambert gave its solution with the help of the following problem: "G-iven on the table an angle whose aide. S Q^ia horizontal. To determine the point ^''so that ^ = 1^*” The second chapter of the work is entitled "On the proper position of the ^e and its distance from the table." This is a collection of notes and advice for artists. In the fifth chapter Lambert modifies the definitions and proofs which he set up in the first chapter so that th^ w ill apply to the case where the table is not vertical. Among the problems which he solves is one in which it is re quired to find the line perpendicular to a given plane and passing through a given point. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 In the seventh chapter the eye Is supposed at infinity and thus the projecting rays are parallel. The eighth (and last) chapter deals with the "Recipro cal problems of perspective". Here it is required to find the conditions under which a given perspective projec ted, that is, to find the position of the eye and the table. In this connection Lambert gives a rich store of problems and examples. Fifteen years after the appearance of the "Freye Per spective" Lambert published a new edition of the work in which a second volume was added. This contained inq)ortant notes and appendices. The most important contains a group of significant problems, which we shall enumerate briefly. I. Given four non-collinear points of which each lies outside the triangle determined by the remaining three, to determine more points of the clroumference of some ellipse passing through them, the construction to be accomplished by means of lines only. II. Given a parallelogram; to draw a line through a given point parallel to a given line, using only lines in the construction. III. A circle and its center are given; to erect a perpendicular to a given line through a given point, using only lines in the construction. IT. A segment is bisected; to divide it into an arbitrary number of equal parts, using only lines. T. Two lines are given which intersect in an inaccessible point; without ex— Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 tending these lines, to draw a line through a give^ point whioh shall pass through the point of intersection of the two given lines. Use only lines in the construction. (Lambert's solution of this problem is still used. It de pends upon the harmonic relation of a complete quadrilat eral.) 71. Two lines are each divided into two equal parts; to draw a line parallel to a given line. 7X1. To prove the Pythagorean proposition by perspective methods, using only lines. (Given, of course, the horizontal plane and the table perpendicular to it.) 7III. A circle and its center are given; to bisect an arbitrary arc of the circle. H . Two seg ments AC and ^ intersect in the point JB so that AE « EC, ED » 2BB. To construct a parallelogram, using only lines. I-2II. Determination of parallels with other systems of data. 1 1 7 . I f EBP = ZABE, to construct the perpendicular to ^ through B, using only lines. In order to give an idea of Lambert's procedure, we give his solution of (7) above: P B f ! / \ '' ■< Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 Let ^ and BD be the lines which oononraf in an In accessible point, and B the given point. Draw two lines 45, GB through E and then draw ^ and throughJÇ (inter section point of 13 and draw the line KCD and "then CE and Gl); if F is the intersection point of the latter two lines, then ^ is the reg^uired line." From these last few problems we see that Lambert is approaching a non-metric geometiy, with the conception of perspeotivity as fundamental. However, upon examination, it w ill be seen that although in each case the construction itself is entirely Hpnrmetric, there are metric elements involved in the problems, and, therefore, necessarily in the I data. Problems ^ and 7, however, are completely non-metric in n a tu r e . Although Lambert’s work in non-metric perspective geom etry was very small in amount, this explicit emphasis on non-metric constructions and proofs was nevertheless very important, for though many non-metric propositions had been proved in non-metric ways before that time there was as yet, apparently, no thought of developing a completely non metric geometry, or even of experimenting with such a geom e t r y . Lambert, then, by restricting the solutions of some of his problems to those in which only lines were used (linear) made the first move in the development of non-metric geom Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 etry* This geometry was at first limited to the plane, and so was called "Geometrie des Lineals" (in French, "geometrie de la reg^e"). In snmmlng up these excursions into the field of per spective geometry we w ill say that although perspective theory in the eighteenth century was primarily a technology for artists, yet it kept the idea and the word "perspective" before the mathematicians of that time and directed their attention to the problems it involved, making than familiar in both a theoretical and intuitive way with the relations between a figure and its "perspective"# Though projective geometry as an organized subject bears only a distant resemblance to perspective geometry, in reality the two branches of geometry have many things in com mon,, So although we cannot say that projective geometry grew out of perspective geometiy, nevertheless perspective geometry was an important part of that matrix of theories and propositions in which we must look for the origini^ of projective geometry. However, it seems quite likely that the individual the orems then known were the most important element in devel oping the new theoiy. A very few of these theorems are fun damental in projective geometry, and these had with but few exceptions been proved before 1700. However, - it oooma quite lik e ly -that-tho individual Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 In the year 1800 the following were, briefly, the prin cipal materials which lay at hand for the construction of the new geometry: (1) Theorems. Cross ratio {in pole and polar theory), Apollonius Ptolemy’s theorem (py. ^?) « Invariance of anharmonio (cross) ratio under projection, Pappus (p. 10 ). Harmonic property of complete quadrilateral, Pappus (Ppld^ll). Involution of four, five, and six points, Desargues ( P3P. ' Theorems on poles and polars, Desargues (p/îiy^* Desargues» theorem (p.14 )• Pascal's theorem (pp. 14^7 ). Theorems of Hewton and Maolaurin on transversals of geometric curves (p^MÿW. (2 ) Methods and theories. Projection and section, Desargues (p. ). Theory of transversals; Ptolemy, Pappus, Desargues Conception of infinity; Kepler, Desargues (pmf:y3). Perspective; Desargues, s ’Gravesande and his prede cessors (p.3^ ). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 îîon-metrio geometry; Pappus, lambert (pJ/<^W). Center of mean harmonics, Lhelaurin . Descriptive geometry, Konge (p.;t3)* We have now completed a survey of the development of geometry up to the end of the eighteenth century, with special regard to the hearing which this development had upon the origin of projective geometry. It remains to trace the development of projective geom etry proper from the first part of the nineteenth century to the present time. This will he the subject of the next two chapters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 Chapter III Jean Tiotor Ponoelet was born at Metz in the year 1788, He studied at the Ecole Polytechnique from 1808 to ISlO^wkevt he came under the Influence of Monge. As a lieutenant of engineers he served under Hapoleon from 1812 until he was taken prisoner by the Russians in the spring of 1813. In prison he commenced his great work, the "Traite' des prop— rië'tës projectives des figures". This was first published in 1822, and a second edition appeared in 1865, with the ad dition of a second volume. We shall discuss the edition of 1865, remembering that most.of the material which was added to the original work had been published in other forms prior to 1 83 0 . At the beginning of the work Ponoelet makes the state ment that in what follows the word "projection" shall^unless otherwise specified^have the same meaning as "perspective". A conic is taken to be a plane section of a cone of circular base. The first theorem is one on the invariance under pro jection of a metric relation among line segments. Hue to its fundamental importance in the work, we shall give both the theorem and its proof. "In an equation of two teims, if both terms are rational • Zo in form, and if the same letters occur the same number of times on each side of the equation, the letters being taken Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 in pairs and interpreted as representing the end—points of segments whose magnitudes they are interpreted as represent ing in pairs, and if these points are a ll on the same straight line, then the relation is "invariant nnder projection." In the figure area SAB _ SA*SB a rea SA* B' S3? •SB'T'’ so a r ea SAB area SA^B* ~ “SA*SS" SA^ 'SB' ^ in ^ B S A » Let SA be a, SB be b, etc. Then area SAB = ^œa*b. ♦ If p is the perpendicular from 8 to _AB, th en area SAB = ^P'AB, so J ^p»AB = ^ « a » b , and AB . P For another segment GB of the figure, calling o, d, m , and p^ the new values corresponding to it, we have CD m/.o*d How since the segments are collinear, p = p'= etc., and for any relation of the above form 3 » Jb, _o, e t c . , w i l l obviously divide out in the substitution for CD, etc. So we obtain a relation among the sines of the angles at _S. Evidently, then, for any other transversal, if À, 3^, e t # . . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 are the points on etc,, then we have the same rela tion among its segments ^ (/ 3p/ eto., as we had among the segments 12, CD, eto. This relation, therefore, is projec tive. %.E.D. Now hy this theorem a relation of the form AC*22 = klD'BC is protective. In the form = k it is called the anharmonic or cross ratio. The special case = 1 is called the harmonic ratio. The points A, 2, _0, 2 in this case are called harmonics points and the lines SA, ^2, 82 are called a harmonic pencil. It follows that any trans versal of a harmonic pencil (any projection of harmonic fu A t 'h ÎK points) harmonic set of points. The following relation among the segments determined on the / sides of a triangle by any circle is, it is easy to verify, projec t iv e * AP*AP'' •BQ'BQ' 'GR'CR' f i t A « 2P*2P *CQ*CQ "AR'AR . Therefore this relation holds in /6 any projection of the figure, and since the projection of a circle is a conic and the projection of a triangle still a triangle, thai for any conic intersecting a triangle in six points the above relation holds. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 Suppose now that A goes to infinity# sinoe the ratios ^ and g-gy beoome equal to one. How if the three sides of a triangle are tangent to any c o n ic a t 2 , S f aiiA R, th en AP'BQ'CR = BP*CQ*AR. Suppose BÇ drawn parallel to A PR. Then ^ and therefore BQ, - OQ, so that Q is on the line AO throTjgh the vertex A_ and the mid—po in t 0 o f T h erefo re: F‘ % -17 Parallel chords of conic sections have their mid-points and the points of intersection of the tangents drawn through their respective extremities on a single line called a diam e t e r . Drawing a new tangent B'C'parallel to BC, we can consider BQ,, B'Q as forming a circumscribed triangle BB^K, where K is at infinity. Therefore PB"Q'B'= PB'»QB. A ls o , and Then * That is, the diameter QQ^ is divided har monically by the mid-point of each chord ^ and the vertex A of the circumscribed angle. Bow, (fig.'J ) if MN and PR are two parallel chords of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 / any eouic section and AB the / diameter tlseotlng each of them. . ^ /k then proposition (1) will become ./ QMS' “ whence /H §3“^ « p*OA*OB, p b e in g a c o n sta n t -^^7 varying only with the direction of f For the case of the parabola, OB and BQ both beoome Inflntely large, so ÔM® = p*OA (p = These relatione are very convenient for certain purposes as w ill be seen. Given two conicsj they w ill be determined by QB^ = p»OA*OB 0^M* ^ = p#0'Af "O'B* How If we assume the conics to be sim ilar, and the diameters ^ and A'B' to be homologous diameters, and the chords CM and o'M^ to be likew ise homologous, then OA r OB , AB _ OM PT" 0^ IT mr* so that E = 1. Therefore p = In this way we obtain p' a criterion for the sim ilarity of conic sections. In the case of the parabola, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 Ô W - ô & . ■’ Therefore all parabolas are similar. From the above considérât ions_, if a line mn is drawn in the plane of a conic, (1) the mid—point 0^ of the chord W which the curve intercepts upon it is at the intersection of this line with the diameter ^ conjugate to its direction; (2) the point O' in which the tangents at M and intersect is on this same diameter and is the harmonic conjugate of jO with respect to ^ and B; that is, o'A , OA^ O'B OB* Bow if we consider a line m'n'^ drawn exterior to the curve, and if _o' is the point in which it meets the diameter of the curve conjugate to its direction, then this diameter ^ w ill contain the mid—points of the chords of the curve parallel to m'nl If two tangents O'M, o'B are drawn to the curve, the mid—point 0 of M w ill lie on AB so that 00'AB constitute a harmonic set. Here Ponoelet introduces the idea of imaginaries. Inas much as he was the first to make use of this idea in pure geometry, it w ill be interesting to g_uote a passage. ”Weshall designate by "imaginary" every element which, although real in a given figure, becomes entirely impossible or inconstructible in some correlative figure. (Correlative- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 ”that which is considered to have come from the first hy the progressive and continuous movement of certain parts without violating the primitive laws of the system.”)-; the word "ideal" shall designate the mode of existence of an element which on the contrary remains real in the transformation of the figure, but ceases to depend in a real manner on other elements which define it graphically, because these elements have become imaginary." With these definitions in mind, m^n' can be called the ideal secant of the conic, (m'n' is a real line with imaginary points of intersection with the curve.) How let us consider the relation = p.O'A'O'B. It defines the points and on m'n'. The diataae» M'W will be the ideal cord on the ideal secant m'n'. If we construct the totality of ideal cords M'E' satis fying the above relation we shall evidently obtain a second conic, which we shall call supplementary to the f i r s t . There is obviously one such for every direction of the diameter AB. The analytical counterpart of the fact that the points and_E^ are imaginary is that in W^'2 _ p.O'A'O'B, either O'A or O'B will be a negative quantity, if both OA Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 and OB are considered positive. So in solving the egnation we obtain imaginaiy solutions, showing that in the problem as stated, M and E are imaginary. Of two supplementary conics either can, of course, be considered real and the other is then imaginary. If we consider two conics in the same plane, we know that if they have a real common cord: (l) The diameters conjugate to its direction in each of the curves w ill meet in the mid—point of the cord in question; (E) If this point is 0, and AB and A^B 'the diameters, and p and p ' the con stants corresponding to these diameters, respectively . / / ( p'OA'OB = p 'OA *0B , since 0M= OM . These two conditions are necessary and sufficient to determine the cords which two coplanar conics have in com mon. The cords which the supplementaries of two such conics have in common we shall call ideal common cords of the conic sections, in accordance with the above discussion of ideal cords. It is then proved that two conic sections actually do have ideal cords meeting the above conditions. In this cwfc proof Ponoelet makes clear-as* use of the principle of continuity, whose importance- in geometry he was the first to s ta te - It is shown that real and ideal common cords as defined by the above relations are equivalent to the intersection— Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 lines of the planes determining these oonios by their inter sections with a cone. ’Then the line of intersection of two such planes cuts the cone, itis a real common secant, and when this line is exterior to the cone, it is an ideal com mon secant. X.*'* It is shown that if _S is the apex of the c one 0 m2 = P'OA'OB = p 'OS^, where p is the constant of any section of the cone by a plane parallel to the plane OSM for the diameter determined by its plane and the plane If this conic is a circle, p = 1 and OM = OS. leaving this subject, Boncelet turns his attention to series of circles, now called pencils of circles. The basic theorem is: "If from any point P of the ideal common secant of two or more circles one draws tangents PT, PT^... to the circles, these segments w ill be equal." Bow if we consider a series of circles (C), (o'), (c''), ...in a plane all having the same real or ideal common secant mn, from any point P of this secant we shall draw tangents PT, Plf..., which will all be equal. The points of tangency T, T^ ,..., will then lie on a circle with radius PT and center at P, which will evidently cut all the circles (0), (c/),..., orthogonally. The line CC^ w ill be the com mon secant of this orthogonal series (P), (P^),... Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 The following theorena are then proved: "All the mid—points of oords of contaot corresponding to any point in the plane of a series of circles having a common secant, are on a new oiroumference orthogonal to a ll the first." "Two circles intersecting orthogonally in a plane are reciprocally siich th at the diameters of one are cut harmoni cally by the circumference of the other-" In a series of circles having an ideal common secant the "limit points" are the two points on the axis C£' which the centers of the circles approach as their radii become smaller. These are synmetrically placed with respect to the common secant. "If a series of circles having a common secant has two limit—points, these two points divide harmonically all the diameters which correspond to them in the different circles of the series." (This is in consequence of the fact that in such a series all the circles of the orthogonal series pass through the two limit points.) "All the cords of contact, or polars, which correspond to some given point and a series of circles having a common secant will intersect in a unique point of the plane of the circles." Let the f ir s t of these points be called A and the seoeed Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 ond y ; the polars of will also pass throng A. A and A are called "reciprocal points" with respect to such a series of circles. The reciprocal points of all points on the common secant are likewise located on the common secant. The following theorem concludes the subject of series of circles; "All the points which are the reciprocals of those of an arbitrary straight line with respect to a series of circles lie on a conic section passing through the two limit points (real or imaginary) of the series." The proof of this theorem for the case where the limit— points are real is very easy indeed, but its extension to the case where the circles have a real common secant and the limit points are imaginary is quite another matter, and Ponce- let extends his theorem without proof to this instance by vir tue of the principle of continuity. He does, however, insert a note in which the proof is given in f u ll. This is given as an example of Ponce let *s conception and use of the prin ciple of continuity, which he admitted into his geometry with out proof. Here Ponoelet returns to the consideration of conic sec tions in general. Since in two hyperbolas which are similar and similarly oriented the aasymptotes are parallel, the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 ourves have two common points at in finity on these assymp— totes, or, in other words, they have a common secant at in finity. This idea is then extended to ellipses which are similar and similarly oriented; their supplementary hyper bolas will also be similar and similarly oriented, and will thus have a real common secant at infinity, which will be the ideal common secant at infinity of the two ellipses. If the curves are further supposed concentric, they will have a real secant of contant at infinity if hyperbolas and an ideal secant of contact at infinity if ellipses, or rather double imaginary contact at infinity. If parabolas are similarly oriented th ^ touch in a real point whose tangent is at infinity. Circles are evidently similar and similarly oriented; then all circles in a plane have an ideal common secant at infinity, or two imaginary common points at infinity. These points have since been called the circular points at infinit; or simply the circular points of the plane. They originated with Ponoelet. Sinoe two circles always have another common secant, real or ideal, at a finite distance, except when they are concentric, when this secant unites with the original com mon secant at in fin ity , we can consider any two circles as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 having four points in oommon. The secant at infinity oommon to several circles or ellipses is necessarily indeterminate in direction. Fow in general when a straight line is moved continuously to an infinite distance it becomes indeterminate in direction, so we can consider all the points at infinity in a plane as distributed on a unique line, itself situated at infinity on this plane. This idea, original with Ponoelet, is one which is in volved in nearly all modem geometry. By nse of it and the conception of common secants at Infinity he solves the fol lowing problems: Given a conic section (C) and any line M in its plane, to find a center and a plane of projection such that the line M is projected to infinity and the conic is projected into a circle. To project the above conic so that M goes to infinity and the conic goes into (1) an hype rbola sim ilar to a given hyperbola, (2} an ellipse similar to a given ellipse. These constructions are accomplished by use of the relation of the conic section to the cone stated above, namely, OM =• p*OA*OB = p *03 . For the f ir s t case the locus of the center of projection is found to be a sphere Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 of radius gM with center at 0, For the next two it is an an nular surface. The plane upon which the projection is made is always supposed p arallel to the plane OIB containing the center of projection S and the secant M. The following theorems are then proved: "When one projects a conic section and a straight line in its plane so that on the new plans the line is represented hy the line at infinity, the pole of this line has for projection on the new plane the center of the conic which is the projectioi of the original conic section." "If one projects some conic section lying on an arbitrary plane, the pole of each line in the plane of the conic will re main in projection the pole of the line corresponding to the original line." "Any conic and an arbitrary point in its plane can be projected into a circle having the projection of this point as ce n ter." "Any two conics in a plane can be projected into two circlf (In one perspectivity). "Two or more conic sections in a plane having a secant in common can in general be regarded as the projection of an equal number of circles for which the line in question is the line at infinity in their plane." "When one projects two or more ooplanor conic sections ha?i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 a oommon secant onto some new plane so that this secant passes to infinity, these oonios beoome, in general, all similar and similarly oriented, having the projection of the original com mon secant for common secant at infinity.” ”The real or ideal common secants of two or more conic sections in a plane remain common secants in any projective transformation.” This last theorem is obvious for real oommon secants, sinoe relations of contiguity are projective. It is easily shown that the theorem is likewise true for ideal common secants. That is, relation of imaginary contiguity are likewise projective The following few theorems are given as results of the above considerations: plane figure containing two or more conics which have a common double contact can in general be regarded as the projec tion of another in which the conics are represented by circles all concentric and having an ideal common secant of contact at infinity, the projection of the cord of contact in the first fig u re. ” "Any two or more parabolas in the same plane have the line at infinity as common tangent ; so that if one projects them onto an arbitrary plane, there w ill result an equal number of conies having a common tangent, with different points of contact Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 if the axes of the parabolas are not parallel," "Two oonios in a plane can in general be projected onto a new plane so that eith er one of them becomes a circle concen tric with the projection of the other." Here Ponoelet turns his attention to the "principle of continuity" upon which he la id such great stress. This prin ciple, as Ponoelet conceived it, consisted in regarding as general and applicable to all oases principles which have been deduced for some limited range of cases. A good example of Ponoelet'3 use of this principle is the case of ideal cords of a conic, above. In analysis these ideal or imaginary cases will be represented by imaginary solutions. It was Ponoelet’s idea that just as, in general, we do not exclude imaginaries as meanin^ess in analysis, so th^ should not be excluded from pure geometry. Ponoelet admitted this principle without proof, as a sort of sweeping assumption which he maintained was necessary in some cases and very helpful in many others. Before him Monge had distinguished between relations which persisted in all general constructions of a figure and elements which in one general case are real and in another are imaginary. It was thi doctrine of Ponoelet s principle of continuity that the gen eral relation persisted regardless of whether it was with real ideal or imaginary elements that it dealt, and not only this, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 but that this general relation continued to have meaning. This discussion of the principle of continuity concludes the first section of the work. The second section begins with a treatment of the ^theoiy of transversals,” based on the following theorem: ”If all the sides AB, BÇ, CD, DE, EA of a plane polygon ABODE, or their extensions, are cut by any straight-line trans versal m* in the points m, n, p, q., r, respectively, the trans versal w ill determine two segments on each side such that the product of a l l of these segments .nihieh have no common extremi ties will eq.ual the product of all the others; that is, Am*Bn*Cp*Dq_*Er'= Ar*Bm*Cn*Dp*Eq.. (l) This is proved by showing if we project the figure so that the transversal goes to infinity, then Am = Ar, etc, (for pur poses of ratio) and since this relation (relation {1}) is pro jective, then it must hold in the original figure. This same relation can be shown to hold fo r any skew poly gon cut by a transversal plane. If we consider a new trans versal m r in the plane of the. polygon ABODE above, we have Am'^'Bn^'Cp'^ • Dq • E r'= Ar*^* Bm'^- Gz/ • D p'' Bq_"^ Eow if we represent the product (Am*Am') by (Am), etc., we w ill have, multiplying term by term, (Am)(Bn)(Cp)(DcL)(Er) = ( Ar ) (Bm) (Gn) (Dp ) (Eq) This same relation holds if we replace the two lines by any Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 oonio section. The proof rests upon the relation. -BQ-Bof .CR*CR^ = BP'Bî*' •CQ-C(i^ • AE*AR^ where ABC is any triangle and P, 0.» Q.^ » R, R ^ its in te r sections with any conic. This proposition is easily extended, to skew polygons out hy a surface of the second order. The following two theorems follow easily from these considerations "If a polygon, plane or skew, has a ll its sides tangent to a given ï*ee or surface of the second order, there are two segments on each side, measured from the point of contact to the vertices; that product of a ll these segments which have no oommon extremities is equal to the product of all the rest." "In every skew quadrilateral o Ircumscrit ed about a surfac of the second order the four points of contact lie in the same plane." The very important property of a complete quadrilateral of determining a harmonic set of points is then shown. "In every complete quadrilateral having its three diagon als, each of the diagonals is divided harmonically by the two others." If E and F are the Intersection points of the two pairs of opposite sides, let the figure be projected so that the diagonal EF goes to infinity. Then the figure is a parallel ogram, and the remaining, two diagonals bisect each other. Bn the point of each in which it meets the line EF is at infinit: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 So the diagonals are out in harmonic points. In fig . /1 \ p-*~ ------■ ..— * as can te verified projecting . 6 ^ the figure so that W goes to s/ infinity and noticing that the C above relation is projective. \ p _ . n $'ii If B is considered to be any point in the plane of the triangle ABO, then the lines throu^ JJ and the vertices of the triangle will meet the sides BG, CA, A3, opposite, respective ly, to these vertices, in the points F, G_and E, respectively, so that AE'BF'CG = BE*CF*AGr. This can be verified in the same way as the statement immediately before. It is easily seen 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 of continuity one of the sides of the above polygon can be considered as becomii^ zerb in magnitude while keeping a give direction, in which case the two segments determined on it will become identical and cancel from the equation. It is next shown by considerations which are essential!; the same as the above that the mid—poinds of the three diag onals of a complete quadrilateral lie in a s t r a l ^ t line. This important theorem is then proved: "If, in fig. /I , one joins the four points L, M, 1, P, two by two, by lines, the points of intersection ^and H_of the opposite sides of the quadrilateral LMP. will be, re spectively, on the two diagonals of the quadrilateral ABCD." The proof of this is interesting. Let there be a skew //'/ quadrilateral ^IBCD perspective with ÆBCD. How since rela tions of contiguity are projective, the lines corresponding to EM and PL must intersect in a point corresponding to / / / G-. i.e., theÿ must be coplanar. Therefore the points L MI! must be coplanar, and for this to be so, since Q and A^C ' and P'n' are coplanar by pans but by hypothesis not all coplanar, then A^ 0 / , i/m'' and P^H' must all meet in a single point. Similarly , MJL, , must all meet in a s ingl! point. Therefore the lines IM, ^ and intersect in the point I, and the lines IP, MET and BD intersect in the point H. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 How from this oonfiguration we can prove directly; "Any two triangles being so placed in a plane that their respective vertices, two by two, determine three lines con verging in a single point, the sides- opposite to the corres ponding vertices will intersect two by two in the same order in three points in a straight line, and conversely/^ The proof consists simply in referring to the figare, in which certain relations of contiguity have already been proved. In particular, in fig.if the triangles are considered to be ALP and CM, the point of convergence will be JC and the line will be BL5, where ]B, D and H are the three points. This will be recognized as Desargues' theorem. The theorem of Pappus: "In every hexagon inscribed in two lines in a plane the respective intersection points of pairs of opposite sides lie in a straight line" is similarly proved from the figure. If EM and ^ are the two lines and if the hexagon is EHPPMLE, then the line w ill be CIA. How considering a conic in which is inscribed a simple quadrilateral, we know that we can project the conic so that it becomes a circle and a given line in its plane goes to in finity. If this line is the line connecting the -two points of intersection of opposite sides of the quadrilateral, we shall obtain a rectangle inscribed in a circle. If the fig-ore is cut by & line ae, we know Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 ae*af _ aA»aB oe'of oD'oG On aocount of similar triangles. a S â ^ = ab . oD od’ cC ob * so ae-af _ ab»ad 06*of ob'Od This, it will be remanbered, is Desargues’ involution of six points. Being projective, it holds for any q^nadrilateral inscribed in any conic. Suppose now that we know five points A, B, G. D and f of a conic section. Having drawn the quadrilateral ABCD and, any transversal through f we can, by the above relation, find the remaining point of the transversal, in which it meets the curve. In this way we can construct the conic determined by any five points. In view of the fact that a complete quadrilateral deter mines an involution of six points on any transversal, the con struction can be accomplished non-metrically as follows: con struct the new quadrilateral A's'c ^ whose diagonal B^D^ pass es through f and whose pairs of opposite sides intersect with the sides of the first quadrilateral, respectively, in a and 0 , b and d; the second diagonal will meet the transversal in ^ 0 , the required point. How if we suppose a, b , and c fixed and move the inscribei Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 q.uadrilateral so that it remains inscrited and its sides continue to pass through these points (in the same order), the fourth will pivot about the fixed point d of the trans versal. This is a special case of an important theorem which Ponoelet states much la te r in the work. About a circle with an inscribed rectangle (the pro jection of a conic with an inscribed quadrilateral) draw the circumscribed parallelogram abed, whose sides are tan gent to the circle at the points A, B, C, D, respectively, the vertices of the inscribed rectangle. The opposite sides of this parallelogram will, of course, intersect at infinity. The diagonals ^ and ^ and bd, will all pass through P, the center of the circle, and the two latter will be parallel to the sides of the rectangle ABCP. Tangents at E and ? and__H, where these diagonals out the circle, w ill be, two by two, parallel to each other and to the sides of the inscribed rectangle. Further, all the lines through P_are cut by th is point and the circum ference of the circle or the opposite sides of either of the two quadrilaterals ABOI or abed in two equal parts. They can be regarded as cut harmonically by these points and the point at infinity. If we return to the original figure, all the lines which were parallel to each other in the case of the circle Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 intersect on the line which corresponds to the line at infinity in the plane of the circle; % will he the pole of this line. We can now state the following theorem; "If one insorihes any quadrilateral ABCD in a conic, and if one circumscribes another, abed. whose sides are tangent to the curve at the vertices of the first, then: "1. The four diagonals of these two quadrilaterals will intersect in a single point P. "2. The points L and M, 1 and m, in which the opposit sides of the inscribed and circumscribed quadrilaterals ii terseot, respectively, will be located on the polar ofJP, "3. The diagonals of the circumscribed quadrilateral intersect respectively in the points L and where the opposite sides of the inscribed quadrilateral intersect, two by two. "4. Each of these latter points is the pole of the line or diagonal passing throu^ the other and the point P. "5. Every line passing through is divided harmon ically by the points in which it meets the conic, the point P, and the point in which it meets the polar of P; similarly for the points and M with respect to the lines PM and PL of which they are the poles." This very important theorem is a good example of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 how tellingly Ponoelet used the idea of projection and sec tion, which he was the first to submit to a systematic treat ment. Prom th is theorem we can derive the following: "In every quadrilateral Inscribed in a conic, the points of intersection of the diagonals and the opposite sides are three points such that any one of them is the pole of the line through the other two.” "In BYery complete q.uadrilateral circumscribed to a con ic, each of the three diagonals is the polar of the point of intersection of the other two." We can farther state that if we inscribe a series of cords IB, i / s ' 3 in a conic section, a ll directed toward a single arbitrary point P; 1. All the points _C_, £, , . . . , which are the harmonic con jugates of P on the cords AB, A ^b/ , ...etc., with respect to the points A, _B» } '-'etc. all lie on a single line, the polar or cord of contact of P. 2. All the points of intersection L and M, 1 and ^..., of the new oords connecting the extremities of the different pairs of original cords, two by two, will also lie on the polar in q.uestion. 3. All the points T, I , in vdiich the pairs of tan gents drawn through the aid-points of each cord ,..., also lie on this polar. 4. Reciprocally, if tangents to the curve are drawn from Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 the points T, __T of an arbitrary line in its plane, the oords of oontact corresponding to them will all pass throtg^h a -unique point _P, the pole of the line in q-uestion. We may outline the remainder of the discussion of polea and polars by the simple statement of theorems. The princi pal ones are: "If a certain point is on a straight line in the plane of a conic section, its polar will pass through the pole of this same line." This is the basic theorem of Poncelet’s theory of poles. It is a simple restatement of some of the above properties. Following this is Lambert’s solution of the problem to draw a line through a given point and through the inaccessible point of intersection of -two given lines. It is solved by considering the two given lines as forming a degenerate conic The polar of any point in the plane of this conic will pass through the intersection point of these two lines. The con struction is identically that of Lambert, already given. A special case is that in which a line is to be drawn through a given point parallel to a given line, given a parallelogram in the plane. Pascal’s theorem, "In every hexagon inscribed in a conic the points of in tersection of pairs of opposite sides are all three on the same straight line," is proved by projecting the figure so Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 that the conic becomes a circle a ni the line through two of the intersection points of opposite sides goes to infinity, and showing that the third such point does also. From this it is shown th at "Throngh five arbitrary points in a plane one can draw only one conic section.” B rianüon’a theorem, "In every hexagon circumscribed to a conic section the diagonals which join, two by two, the opposite vertices all cross in a single point,” is proved from the same figure as its dual, Pascal’s theotem. From this it is deduced, in a manner analogous to the demonstration from Pascal’s theorem that only one conic can be drawn th ro u ^ five arbitrary points, that only one conic can be drawn tangent to five arbitrary lines. From these two theorems the following two are drawn. "If the sides of a movable triangle FIE lying in the plane of two lines ICP, Egg, are restricted so that they pivot about three fixed points ^ E, 1 as poles, and if at the same time the two vertices I and_E are required to describe the lines DOI and BCE as directrices, the third vertex F will describe a conic section." This conic will pass through A, B, C, D, S. .The theorem is the immediate corollary of the construction of a conic through five points by use of Pascal’s theorem. It is a special case of an important theorem stated Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 in a later part of the woik. ?he aoaoad—tbe-eroi&i The second is simply the plane dual of the above, and is obtained in a similar way. In this way we have a non-metric method for cod ] atructing either point or line conics* After a - few purely metric considerations which, althou^ interesting, are rather specialized, tie following proposition on poles and polars are stated: If two polygons in the plane of a conic section are suchj that the vertices of one are respectively the poles of the sides of the other, reciprocally the vertices of the other are the poles of the sides of the first. If the three points of intersection of opposite sides of a hexagon of a pair of hexagons of the above nature are in a straight line, the diagonals joining the opposite ver tices of the other and which are the polars of these three points ooncurr in a single point, the pole of the line in question, and conversely. If one of the above hexagons is insoribable to a conic section, the other is c iroumscribable to such a curve, and conversely. By virtue of the construction of conics by quadrilateral (see above), these theorems are extended to any polygon. Fur ther, from the same considerations; If a point or pole, taken in the plane of a conic sectio moves on another cmic section, its polar will envelope a thir Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 in its motion, and conversely. This is extended in two ways to the case of curves of any degree. One is by considering the curves as limits of circumscribed and inscribed polygons of an infinite number of sides, and the other is by virtue of the "principle of continuity," observing that if a point is dis placed an infinitessimal distance on one of the curves, its polar, by hypothesis tangent to the other curve, will tend to turn about the point of contact of this tangent, which is evi dently the pole of the element or the tangent corresponding to the point we have considered on the first curve. These sys tems of points and tangents can be called reciprocal polars. Remembering that the tangents of one curve correspond to the points of another, and points correspond to lines, the de gree of a curve being determined by the number of points in which it meets a straight line transversal, "The degree of the reciprocal polar of a given curve is, at the most, egual to the number expressing how many tangents can be drawn to the latter curve from a given point." The chapter is concluded with this far-reaching remark: "One can...say, in general, that there exists no descriptive relation of a given figure in a plane that does not have its reciprocal in another figure..." By descriptive Poncelet means non-metric. By reciprocal, he means polar reciprocal, where points correspond to lines and lines to points. This contains the germ of the whole theory of duality. Poncelet Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 extended this oonsiderahly somewhat later, as we shall see. He hecame involved in a dispute with Gergonne as to priority, hut we shall defer this matter until afoaowlaat later. The third and last chapter of the second section may, in view of its simplicity and relative unimportance, he omitted from this discussion. It is entitled "Of the center of symr- metry in general, and of that of two circles in particular, - Of circles which intersect or touch in a plane, - Of similar and similarly oriented conics, in general." It is almost en tirely a discussion of the centers and axes of symmetry of cii cles, homologous cords, etc., and is not necessary to an under standing of the next section, although related to it. In section III of the work, however, the f ir s t chapter Is quite important. It is the first systematic treatment of the idea of homology. The terms center and axis of homology were invented by Poncelet, A few notions are taken from the chap ter, which we omitted from consideration, so we will give then here, If two figures have a center of symmetry they are similar and sim ilarly oriented, and thus have th e ir homologous lines parallel. Pairs of homologous lines thus intersect on the line at infinity, while pairs of homologous points are collinear with the center of symmetry. In the case of two circles, there correspond to each poic / / A of one circle two points, A and E , of the other, with resr Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 ^to either one of the centers of syianetry. It is obvions that / only one, _A ^ is homologous with ^ by sim ilarity. We shall say that is directly and E 'inversely homologous with k with respect to the given center of symmetry. We shall extend this terminology- to apply to two cords, two arcs, two tangents, e tc., which have as extremities or points of contact points of one or the other kind. Two circles can be regarded in two different ways as per-' apective or projective with one another, with respect to each of the two centers of symmetry, according as the points, etc., which one considers are d irectly or inversely homologous. The word projection in this sense is quite restricted, since the homologous lines either intersect on a given line or are parallel. It_S is a center of symmetry of two circles, and if we draw two arbitrary transversals SA, SB, meeting the two circles, the f i r s t in A. and jE, a ' and the second in B. and D. I f B and D , and then if we connect the four points in each c irc le, two by two, by cords, 1. The oords ^ and a '^B ', DB and D ^e/, etc., which are directly homologous, will intersect on the secant at infinity common to the two circles. f ^ { r 2. The cords ^ and D E , ^ and A B , e tc ., which are inversely homologous, will intersect on the radical axis of the two circles. If we draw the tangents to the two circles at the points Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 A and 2, A and E { where one of the transversals SA in te r sects them, then, 1. The tangents AP and a '^P^, 2P and E^P^, which are directly homologous, w ill in tersect on the secant at in fin i ty common to the two circles. S. The tangents ^ and B ^P*", ^ and A^P % which are in versely homologo-ua, will intersect on the radical axis of the two circles. If from any point of either common secant of two circles one draws tangents to hoth circles, and joins these four points of oontact, two by two, the lines thus determined will intersect on one or the other center of symmetry of the cir cles in question. These theorems on the circle are extended to conics in general, by means of simple projection. The conics, of course are supposed to be similar and similarly oriented. The pro jection is one in which the center of projection is at inf initj and the plane containing the conics is any plane. From these considerations we see that any two conies in a plane can be considered, in two different vja.ys, as projec tions of one another with respect to each of the points of intersection of their common tangents taken in particular for center of projection. Further, any two conics in a plane can be considered as the projection of two plane sections of a con whose vertex is represented by one of the intersection points Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 of common tangents. Moreover, any two conics in a plane can be considered as the projections of two conics in some other plane which are sim ilar and sim ilarly oriented. The projec tion of the center of symmetry we shall call the center of homology, and the projection of the axis of symmetry we shall call the axis of homology. These terms were coined by Ponce le t. The theory of homology, of course, applies much more generally than merely to conics. Any two figures in a plane which have between them a projective correspondence can be considered as homological in at least one way. The purely descriptive (non-metric) properties of homologous figures do not depend on those of similar figures, however, except in that they are always the same as in similar figures. The significant characteristic of homologous figures is that the lines connecting points which are homologous in a projective transformation all pass through a given point, the center of homology, and the. points of intersection of such homologous lines are a ll on some uniq^ue line, the axis of homology. All homological figures are projective with each other, and therefore all relation^ metric or non-metric, which are true for one, are true for the other, if these relations are pro jective . A number of fam iliar problems on constructing homologous figures are given next. Given a center and axis of homology. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 and a conic section, to construct the homologous curve, when either one point or one tangent of the required curve is given. Under the same circumstances, if we do not know the axis of homology, to construct the homologous curve i f three points, two points and one tangent, etc., of the required cur| are given. Similarly, we can solve these la tte r problems if we are given the axis instead of the center of homology. These are all solved by means of the ruler only, using the fundamental property of an homology. The idea of homology is made use of to solve many d if ferent kinds of problems, among which is the problem to con struct a conic passing through n points and tangent to 5-n lines, n not being greater than 5. By use of an auxiliary circle, the center, axes, asymp totes, etc., can be constructed as well as the conic itself. By means of a given circle, in fact, a l l the problems of the second degree can be solved by use of the ruler alone. This principle, which was used so exhaustively bySteiner some time later, seems to have been discovered byPoncelet. Chapters II and III of this section can for our purposes be neglected. Chapter II is on the complete of common secant; and tangents of two conics in a plane, etc. Chapter III is the theory of double contacts of conic sections. These chap ters are applications, sometimes rather complicated, of the general theories we have outlined above. A host of interest! Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 properties are exhibited, but there is nothing of great signi- fioanoe. The following problems on common secants might be stated, however; Given two conics in a plane^ describe another having the same real or ideal common secants with each of these curves and passing through a given point or tangent to a given line. The reader w ill lik ely see at once how the machineiy al ready developed will apply in this case. The first chapter of the fourth section, dealing with constant or variable angles in the plane of conic sections, fall in the same class as the above two chapters. The next chapter, however, on polygons which are inscribed or c ir cumscribed to other polygons or to conic sections. The most important theorem is the following: If aU the sides of any polygon in a plane are subject ed to the restriction of pivoting about as many fixed points, while all the vertices except one described given lines, taken as directrices, the free vertex will describe, in the movement, a conic section passing through the two fixed points or poles upon its adjacent sides. ^ /A / " ^ This theorem was proved as follows: / /__ —»J / ___ f - — ' / In fig.20 abed is an arbi- / trary polygon, p, pt p^ are the arbitary points taken ' on the sides da, ab, be, and cd, A," # / 9/ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 respectively, about which they pivot. BC, and CD are the arbitrary lines through a, b , and c, respectively, along which they move. P is the fixed intersection point of pp'and p^^ The line Pb determines two points, x and y, on the lines da and od, respectively, which move in the general movement of the figure. It can be shown that the trian te dxy is such that two of its vertices, x and y, move along the two fixed lines Ex and Oy, while its sides pivot about the fixed points p, p , P. The th ird vertex, d, of such a t r i a n t e described a conic section, so that we have the above theorem proved for the case of the quadrilateral. By taking four consecutive vertices of any such polygon, not adjacent to the free vertex, we can determine a polygon of one less side whose free vertex will have the same motion. In this way we can finally reduce the polygon to a quadrilateral, and thus the theorem is proved. It follows immediately from this that if we consider the points p, p,^etc. as the vertices of an inscribed poly gon and AB, BG, etc., asthe sides of a circumscribed poly gon, any polygon moving so th at it remains continually c ir cumscribed to another polygon of the same sort, and, a single vertex excepted, inscribed to a third such polygon, this single point will describe a unique curve of the second order- We know by Pascal's theorem that if a hexagon is in scribed in a conic, its opposite sides intersect by pairs in three points of a straight line. Prom this and the above Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 theorem i t is not cLiffieult to prove that any polygon of an even number of aides inscribed in a conic moving so that all its sides but one pivot on points on a given straight line, the last side will also pivot about a point on this line. In the remainder of the chapter the subject is developed in this wqy applying to polygons at the same time inscribed and circum scribed to some conic whose vertices trace other conics having the same common tangents with the first, and so forth. The reader can imagine from what has been given above how this subject is treated. The third chapter of this section is an extension of the same theories to the case where the directri ces are of any order, etc. The principal theorem of this chap ter is that if one replaces the straight lines of the theorem of the last chapter by curves of the orders m, n, p, the free vertex will describe a curve whose order will at the most be 2 mnpq.... and will reduce to simply mnpq.... when all the fixed points are on a straight line. This depends principally on the idea that "two geometric curves in a plane can never meet in a greater number of points than the product of those which express*: the degree of these curves." After an exten sion of this subj ect, the subject matter of the second chap te r is further examined, revealing additional particular properties and relations. The entire la tte r part of the book, in fact, seems to be a development from the fundamental ideas of a host of interesting and in large measure hitherto unknown Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 properties. Many of these are very curious and interesting, bn it is with the fundamental ideas that we wish to deal here. A supplement on the projective properties of figures in space is an extension of many of the ideas which were developed only in the plane to the general three-dimensioned form. It is shown that all the points at infinity in space can he considered to belong to a unique plane, necessarily indeterminate in position The theorem on the Mobius tetrahedra is given. The extension o the theory of poles and polars to space is given, as well as of the theory of homology. As we said at the beginning of this discussion, the seconl volume was not published until the edition of 1865 was brought out. It contains two articles which are of interest to us, the second in particular- They a r e the theory of centers of mean harmonics, and a much extended treatment of the theory of reciprocal polars. The theory of center of mean harmonics depend^ on the 1 _ 1 1 1 ^ statement of the harmonic relatio n in the form ^ ~ 2 ’^pa’*’pb '’ where the relation in the ordinary form is = 1 . This is PB'QA due to Maolaurin. This relation is, of course, projective in the original form, and consequently in the above form. A special projection of th is is where = PB, th at is, where JQ, is the mean distance between PA and EB. If we have a large number of equal segments on a line, there is still a center of mean distances- P. The analogy to this in the general case is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 where P is the center of mean harmonics - that is, where we have a number of points AB, A'^B^ , etc., such that, for each of the pairs of segments QA, QB^ 9 :^ > 9 ? » ^ will be the mean harmonic. In the case of the center of mean distances, Q will be the point at infinity. Of course, the factor i 2 is only for the case of two points, A and B. For three points it will be i, etc. Thus — = —(A + i + + • • • • J PQ m PA PB PC ^ to m terms. This can be put in the foim iQ " 5 Ï * ^ * ...... “ i û " ^ 30 M + là + ££ + •••= 0 AP BP CD (If any points A, B, etc., are on the side of Q. opposite from the others, they will, of course, in this, as in all the above relations, be prefixed with a minus sign.) This relation can of course be extended to pencils of lines,where sines of angles replace the abovesegmenta Suppose now that it is required toconstruct the point jg, such that one has n 1 1 1 ..., pq pq pb pc m being the number of points a, b ,..., whence and pQ, = 5pq PQ. PCL ^ Let us suppose that one has between the points p, q, a, b , Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 c.,,, on a line the relation m+m t + m f f + .... m m_ / / / .... pq pa pb pc in which m, m are numerical coefficients Indepaendent of the position of the points a, b, c,..., q., with regard to p. Subjectlng this to the above rule of signs, SIM ^ sLë£ + sIm + + ... = 0 ap bp op dp pa pb pc pq A being some new origin. The author develops this theory in several ways, and shows how it can be used to find the center of mean harmonics of a system of points or lines located heterogeneously in a plane or in space. For our purposes this is quite sufficient space to devote to this subject. The usefulness of the theory will be realized upon recalling liaolaurin's theorems on transversals of curves of any degree. Poncelet makes use of these properties! in his Analysis of Transversals, which forms the third section of the second volume. The second section, as we have said before, is a discus sion of reciprocal polars. It was probably this for which Poncelet was most famous, at least during the first half of the nineteenth century. The discussion in the first volume lays the foundation for the theory, so that we shall outline only the additional developments that Porcelet gives here. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 Mere statement of theorems will for the most part be sufficient. It is shown that the reciprocal polar of a curve of the mth degree is of a degree no higher than the m(ra - 1 )th , whence a ll conics have other conics as reciprocal polars. Points of inflexion, double points, and singular points are taken up, and it is shown that their reciprocal polars are also special points in the new curves. A rather detailed extension to reciprocal polars in space., is then given, and to curves of any degree. Special attention is given to reciprocal surfaces of the second degree. The most interesting new properties demonstrated in this discussion are the metric relations between a figure and its reciprocal polar with respect to a circle. If there is any projective relation amoig the segments of a given figure, the sines of the angles corresponding to these segments in its re ciprocal polar will be in the same relation. In other words, reciprocal polars are projective figures with the elements corresponding reciprocally also corresponding projectively. This theory is immediately extended to apply to reciprocal figures in space. Here we will end the enumeration of Poncelet’s methods and theorems, and, in closing, attempt to give a critical summary of the work. The most notable feature of the entire "Traite" is the o o v j ,'rfie k T IK ? consist projective new-point, lending methodical unity to a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 great mass of material. The notion of the invarlanoe of the harmonic ratio under projection is the single fundamental kqy to the entire work, althou^ the conception of the invariance of relations of contiguity {non-metric relations) is also used considerably. The idea of continuity is depended upon for generality in some cases. The theory of homology is given new, elegant treatment, and, what has never been done before, extended to three dimensions. The theoiy of reciprocal polars is the basis for complete duality, in the plane and in space, between non-metric theories, and, as we have seen, if the aux iliary line or surface is a circle or sphere, for duality in projective metric relations. This latter kind of duality caused a considerable stir among the contemporary mathemati cians. On the subject of duality there was considerable argu ment between Poncelet and Gergonne, principally as to priority. Gergonne, in 1826, made the following statement; "In plane geometry to every theorem there necessarily cor responds another which is derived from the first by simply ex changing the two words point and line; while in solid geometry one must exchange the words point and plane to pass from one theorem to its correlative." The first edition of the Traite des Propriétés Projectives des Figures appeared in 1822, in which was contained the f irs t discussion of reciprocal polars outlined above. The second treatment, which appears in the second volume of the edition Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 of 1865 as seotlon II was read., before the Académie Français in 1824, We may conclnde, then, that although the statement of G-ergonne was more explicit, Poncelet»s treatment was not only earlier but more complete, for it is very doubtful that G-ergonne intended to include, or was in a position to include, metric relations in his statement. We must not forget the importance of Poncelet’s use of elements at infinity, for he was the first to give a clear- out treatment of this important question. It was he who showed the existence of the line and plane at infinity and the circular points at infinity. The metric introduction of imagineries in connection with the principle of continuity was an important step, paving the way for Von Staudt’s non-metric introduction of imaginaries some years later- In conclusion we are justified in saying that Poncelet was the founder of projective geometry. Hot only did he de velop its central idea, but he outlined its principal sub ject matter and the subsidiary ideas of duality, homology, etc., by which this material is handled. He created at a single stroke a new branch'of mathematics, far f romain i ts final form, to be sure, and far-from/figorousin a certain sense, but con taining possibilities of development and generalization which he never imagined. In 1847, twenty-three years after the appearance of the J’Traite des Propriétés Pro jectives des Figures” of Poncelet, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. .89 a l i t t l e work entitled "Geometrle der Lage", by K. G-- C. Yon Staudt, wag published. It was guite different in spirit from Poncelet's work, being, in the first place, entirely confined to non-metric properties. It showed considerably greater attention to the axiomatic foundation, which was much simpler than Poncelet»s. A serious attempt was made to define what was meant by the terms involved, although the notion of undefined elements aid relations as defining others was not at all clear, if present, In the firs t section the following remarks are notable: Every surface has two sides, every line on a surface has two sides, and every point on a line has two sides. A point is indivisible. If a point moves, it describes a line ; if a line moves, it describes a surface; if a surface moves, it describes a bodily space. Bodily spaces are bounded and divided by surfaces, surfaces by lines, and lines by points. A line passes through every point on it, and a sur face through all the points on lines on it. Bodily spaces, surfaces, lines, points, and aggregations of any of these are geometric forms. A figure is usually a completely bounded surface. The periphery of a figure is.the line which enclos es i t , or the aggregate of lines which bounds i t . Every point of a closed line (returning line) can be considered as its beginning and end. If a point describes a closed line to which the three points A, B, and 0 belong, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 i t must moTe e ith e r in the sense ABC (wMoh is the same as BCA or Gab ) or in the opposite sense, which can be denoted by BAG or ACB or GBA. If a line is not closed, and if three points are taken on it, the two outer ones are separated by the middle one. If on a closed line we take any four points, two pairs are separated by each other. One says that a surface and a line which go through one and the same point intersect or touch at this point, according as the adjacent parts of the line lie on the opposite or the same sides of the surface. If a line is drawn connecting two points, it intersects any surface an even or odd number of times according as the points lie on the same or opposite sides of the surface. Similarly, two lines on a surface touch or intersect in a given common point according as the adjacent parts of either lie on the same or opposite sides of the other- Two surfaces passing through one and the same line inteiv sect or touch each other in this line, etc., and two surfaces are tangent in a point if, etc.. A line which is fixed by two points so that it cannot change its position is called a straight line. A line is called curved when no part of it is straight. A straight- line form is in general the aggregate of all points (elements) on i t . A surface which can be determined by the motion of a straight line is called a ruled surface. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 A straight line considered as an element is called a ray, which is divided by any point on it into two h alf-ray s, of which one is called the opposite of the other. A point is projected from a point A by the ray ^ going through both points. A is called the center of projection. A bundle of rays is the aggregate of all rays that are conceived through a single point (center). A bundle of half rays is the aggregate of all half-rays originating at a single point (center). A simple ”argle-space” is a part of a bundle of half-rays such that every half-ray has the same "center" with it and goes through some point in it is entirely within it. A surface which can be thou^t of as filled with half-rays all having a common center is called a simple angle - or conic-surface. Every half-ray of a simple angle-surface can be considered as the beginning and end of it, so four elements of such a surface separate each other by pairs (only two pairs). In a bundle of half-rays which is divided into two parts by an angle-surface, if two half-rays not lying on this sur face are connected by another simple angle-surface, these surfaces will intersect an, even or odd number of times accord ing as the half-rays lie on the same or opposite sides of the first surface. Two simple closed angle-surfaces having a common center intersect an even number of times. Two simple angle spaces or angle surfaces are called Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 oomplementary spaces or surfaces to each, other if every half ray in one of the forms has its complement in the other. The boundaries of complementary spaces are complementary surfaces to each other. The complete angle-space and complete §ngle surface are obtained by simply substituting rays for half-rays. It has properties exactly analogous to the ones outlined above for the angle—space and angle-surface. A complete closed angle-surface is considered to be of even order if a ray describing it finally comes to its ori ginal position; it is considered to be of odd order if the ray comes to its original position except that the half-rays are interchanged. Two complete closed angle-surfaces with a common center Intersect an even number of times if both are of even order, or one is of even and the other of odd order. If both are of odd order, they intersect an odd number of times. In the second section the plane is introduced and pencils of lines and planes are defined. A plane is an angle-surface of uneven (first) order, in which any point can be considered as center. If a s t r a i ^ t line passes through two points in a plane, it lies entirely in the plane. If a plane and a line outside it have a point in common, they intersect in this point, which is called the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 trace of the line on the plane or of the plane on the line. An infinity of planes are oonceirable throngh two points or through a lin e. Through three points not on the same line, or through a line and a point not on it, there can be only one plane. A plane connecting a line a and a point P is said to project a from the point P or to project P from the axis a. An angle-surface of even or odd order is out in an even or odd number of rays by a plane through its center. A simple closed conic surface is cut an even nunher of times by a plane going through its center. The totality of all rays having a common point of inter section and lying in a single plane is called a pencil of rays. The totality of all planes through a single line (axis) is called a pencil of planes. The above statement about order of points on a line ap plies directly to the order of elements of a pencil. The straight-line form, the pencil of rays (or half-rays) and the pencils of planes are called one-dimensional funda mental forms or fundamental forms of the first degree. The point in the element in the straight-line form, the line in the pencil of rays, and the plane in the pencil of plane s. A straight-line form A3CP consisting of four points is projected from a point lying outside the line by a pencil Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 of lines aboâ, of which the foxm A3Cp is a section. The form B^C is projected from the point hy the pencil bade, etc...In th is way, when one of two forms which are correlated to each other undergoes a permutation, the other, for the forms to remain in this relation, must undergo a corresponding permu tation. A plane in which there are infinitely many straight-line forms and pencils of rays is called a planar system or planar field. The planar field and the bundle of rays are the funda mental forms of the second degree. In the planar fie ld a pencil of rays is the aggregate of all rays through a single point, and in the bundle of rays it is the aggregate of all rays on a single plane. The fundamental form of the third degree is the space system or unbounded space, in which in fin ite ly many funda mental forms of the f i r s t and second degree exist. A special word, "^chein", is used for the set of lines through a given point outside a figure and all the points of the figure, if the figure is on a surface or line. The fig ure is considered to be a section of this '’|chein" (that is, if it is a plane or straight-line figure.) After this peliminary discussion, Ton Stoudt takes up the question of parallelity. He uses the Euclidean defini tion of parallelity, namely that two lines in the same plane Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 having no point in common are p arallel. A few more axioms are stated, qnite informally, however, so that one scarcely knows whether they are intended to he assumptions or merely remarks. Two lines lying in a single plane most either inter sect or he parallel. Throngh any point outside a line there is a line parallel to it. If three lines are in a single plane; a) they intersect in three points, h) they all intersect in a single point. c ) two of the lines are parallel and are cut hy the other, d) thoy are a ll p arallel. If two lines are parallel, any third must he parallel either to hath or neither. Several more assumptions of this nature are given, which for the most part can he readily anticipated. The idea of the direction of a line is introduced, and the position of a plane as determined hy two such directions. A line is said to pro ject in its direction all the points lying on it, and a plane sim ilarly projects a ll its points in its position, A plane system is thus projected in a given direction hy the bundle oi p arallel rays in th is direction. Every point of the plane corresponds, as in the case of the general bundle of rays, to a ray of the bundle, every line corresponds to a plane, every .angle corresponds to a dihedral angle, and every line to a cylindrical surface. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 The next seotion is on n-pointa, sides and polyhedrons. It begins with a discussion of plane polygons and the pyramids whose bases th ^ are. It is shown that if every vertex of a polyhedron can be connected with every other by an edge or a broken line made up of edges, and if its surface is divided in two parts by every closed line made up of edges and not going more than once through any vertex, then if E is the number- of vertices, F the number of planes, and C the number of edges, E"* F»=K^ 2. Further, since at least three edges meet in a vertex, and each edge connects only two vertices, 3E is not greater than Since every surface is bounded by at least three edges, about only two surfaces intersect in an edge, 3F is not greater than 2K. Since neither of the two numbers 3E, 3F, whose sum is 5K +-G. is greater than 2K, so neither is less than K + whence each is greater than E, so that either more than five edges can meet in each vertex, or each surface can be bounded by more than five edges. If m edges meet in every vertex, then in E = 2E, E:F - 2 :K = 2 ;m - 2 ;m. If every surface is bounded by n edges, nF = 2E and F:E - E:K = 2:n - 2:n. Elements at infinity are introduced to replace the "direction" mentioned above. Two parallel lines have a point at infinity in common, etc. Upon examination it will Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 be aeen that if, in the above, the wards point or line at Infinity are replaced for the words direction and position, and the slight required adjustment made in the language, the statement will remain essentially unchanged. ÿ e t * t . Thes-e are a—few statements in this section that do not directly involve elements at infinity but,^are fundamental in the remainder of the woi3c. For instance, two fundamental forms are said to be correlated if each element of one is as sociated with some element of the other, which is called the element corresponding to i t . If of any number of forms the first is correlated to the second, the second to the third, etc., they are all correlated. The simplest case of this is a strai^t-line form and a pencil of planes whose axis does not intersect the first foim. If the line-form is con sidered as a section of the pencil of planes by the line S, the two forms will be correlated. If a pencil of rays is a section of a pencil of planes, and a line-form are section of the pencil of rays, the line-form is also a section of the pencil of planes. In two pencils of rays which are sections of a single pencil of planes each two lines which lie in the same plane of the pencil of planes correspond. In two pencils of rays which are "soheins" of the same straight-line form, each two rays which project the same point of the form correspond to each oth er. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 If we eut a tundle of rays whose seotion P is a planar field, by a plane of the bundle parallel to P, the line at in finity in P will correspond to this plane. A real line will correspond to any other plane or the bundle. If one bas two skew lines of which he has taken one as axis, from which to project, and the other as projection line, upon which to project, the projection of a point is understood to be the intersection point of the projection line with the plane that projects the point from the given axis. The projec tion of a form upon a plane is simply a section by the plane of the "^chein" of the form with center at the point from which one wishes to project, etc. Two parallel half-rays meet the plane at infinity in a single point P and are onthe same or opposite sides ac cording as they are directed in the same or opposite directions. Therefore, in the first case the plane at infinity touches the broken line APB, and in the second case cuts it. This rather long list of elementary propositions, almost all of them assumptions or definitions, is only partial, but it is believed the selection given is, by Ton Staudt’s logical standards, adequate. The principle of duality is here introduced in axiomatic form. The statement is th at every proposition in which no d istinction is made between real and ideal elements, w ill have a correlative in which the words point and plane (straig h t-lin e Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 fom and pencil of planes» etc. ) have been interchanged, and which is likewise true. The axioms here stated dually are ) A line ^ is determined ) A line AB is determined by two points A and J , through by two planes A and B, whose which i t goes. line of intersection it is. ) A line a and a point B ) A line a and a plane B not on it determine a plane not going through it determine AB» which connects the line a point a^» in which the line with the point. and plane intersect. ) A plane ABC is deter )A point ABC is determined mined by three points A,:^, C_, by three planes A,B,C, which do which do not lie on the same not go through the same line. line, aM goes through them. The three planes intersect in this point. } Two lines which have a ) Two lines lying in the same common point lie in the same plane have a point in common. plane. The following simple theorems are proved from these; If A,B,C,p are four If A,B,C,B are four planes, points, and the lines and the lines CD intersect, CD intersect, then the four then the four planes and all the points and all lines con lines determined by them by pair si necting them by pairs lie go through the same point. in the same plane. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 If of any mimber of linos each two intersect but not all intersect in a single point, lie in the same plane, all all lie in the same plane. intersect in the same point. A form which consists of elements A, B, Ç, D, is usually de noted by ABCD. A form which consists of elements ^ ^ , SB w ill be denoted by S(ABCB). If A, B, 0, D are points, _S must be either a planes, S must be either a point or a line. plane or a line. If two forms are correlated and an element of one form coincides with the corresponding element of the other (is iden tical with it), this element is said to be self-corresponding. In two planar fields which In two bundles of rays are sections of the same bmndle which project the same planar of rays, any two elements of the fie ld , any two elements of the fields which are the traces of bundles which project one and one and the same element of the the same element of the planar bundle correspond to each other- field correspond to each other. The line of intersection of the The ray common to the two bun- two fields is self corresponding, dies is self corresponding, and the same is true of every and the same is true of every point on it. plane through it. Two pencils of rays which Two pencils of rays which are sections of the same pencil project the same line-form of planes have either a self either have a self corresponding corresponding line or project ray or are sections of the same the same line-form, according pencil of planes, according as as they are concentric or not they do or do not lie in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 oonoentric. same plane. It is noted that in a similar way the point and line are reciprocal in the plane. The next section deals with polygons again, this time in a different way. A plane n-point is supposed to be the figure formed by points lying in the plane (vertices) and the n sides which con nect each two successive points. The n points are thus con sidered in a particular order, so that the first follows the last again, and no three consecutive vertices are supposed to be in a straight line. The ^ elements of the plane n-point AiAgAg.-.An are the origins at the point Ai, the line AiAg, the point ^ 2 , the line A2 A3 , etc. The (ii-fl)th element is called opposite from the f ir s t, the (n-* 2 )th opposite from the second, etc., so that if n^ is even, a vertex is opposite to a vertex and a side to a side, but if n is odd, sides are opposite to vertices and vertices to sides. Every lin e connecting 2 non-consecutive vertices of a plane n-point is called a diagonal. A complete plane n-point A complete plane n-side coi consists of n points (vertices) sists of n lines (sides) lying lying in the same plane, no in the same plane, no three of three of which lie in a which meet in the same point, straight line, ani the n(n-l) and the n(n-l) points (vertices 2 2 lines (sides) which connect ' which they determine, two by tw Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 the n points, two hy two. (n-1 (There are n-1 vertices on each sides meet in each vertex.) line.) A complete n-point in A consiste n -flat consasbs space consists of n points of _n planes (surfaces) of which (vertices of which no four no four go through the same lie in the same plane, the point, the lines (edges) in which lines (edges) connecting each each two surfaces intersect and two vertices and the planes the points (vertices) in which (surfaces) connecting each three each three vertices intersect. vertices, n —1 edges intersect n - 1 edges lie in each surface, in each vertex, and n-g sur n-3 vertices lie in each edge, faces intersect in each edge so these are n(n- 1 ) edges and 2 so there are n(n-l) edges and n(n-l)(n— 2 ) vertices. 2 2*3 n(n—1 ) (n r -2 ) surfaces. 2*3 If in two reciprocal statements in plane or space geometry an n-point is considered as its own reciprocal, it is obvious that an n-point which is a Is j an n-side or n-flat is meant . In the above statements this is the case only if we ignore the diagonals. Two plane n_-points are considered correlated if each vertex of one corresponds to a vertex of the other, and therefore each side of one corresponds to a side of the other. If two correlated triangles lie in two planes whose line of intersection coincides with none of the six sides, but each two corresponding (homologous) sides intersect (lie in the same Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 plane), then the two triangles are sections of the same three-e&j If two correlated complete plane g_nadrangles lie in two planes whose line of intersection goes throng none of the eight vertices, and if five aides of one intersect the five homologous sides of the other, the two quadrangles are sections of the same complete four-edge, so that the remaining two homologous sides must intersect. This follows from the above, if we notice that three-edges which have two edges in common are concentric. Two n-points or n-sides lying in a single plane and which are correlated are called perspective if are projections of the same third plane ^-point or n-side E (from two points M, M]_), and the lines connecting each pair of homologous vertices all pass through a single point (the trace of the line M, , while the points in which each pair of homologous sides intersect all lie on a single line (the trace of the plane ¥). Wow if two correlated triangles ^C, lie in the same plane but have no elements in common, and the three points in which the pairs of homologous sides intersect lie in a single line u which is not a side of either triangle, the triangles are perspective, so the three lines connecting the three pairs of homologous vertices intersect in a single point, for if we project ABC onto any plane going through u, one obtains a third triangle AgBgCg of which, as we know, triangle IxB^Ci is also a projection. Similarly, if two correlated complete quadrangles lie in a plane, and a line u which goes through none of their vertices Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0 4 is cut by fire sides of one in the same points in which it is cut by the homologous sides of the other, the quadrangles are perspective, and therefore the two remaining homologous sides will cut the time in one and the same point. This is proved from the preceding theorem on g^uadrangles just as the la st theorem was proved from the preceding one on triangles. The plane duals of both of these theorems are given. In concluding th is section the theorem on Ifobins’ te tra - hedra is given. It is not thought necessary to write it out here. In the next section Ton Staudt defines the harmonic form on the line in terms of the complete quadrangle. The points A, C are separated harmonically by the points B, D, on a straight line, if A, C or B, are the points of inter section of pairs of opposite sides of the quadrangle and the other points the ones in which the diagonals cut the line. These points are also said to be harmonically separated. A pencil of lines or plane projecting (whose section is) a har monic form on a line is shown to be, under a dual definition, a harmonic pencil. It is further shown that if ABCB is a har monic form, then also BCBA, CBAB, DABC, DCBA. CBAD, BADC. ADCB are harmonic forms, and that if three elements of a harmonic form are known, and it is known which must be harmonically separated from the fourth, the fourth is determined. Therefore if ^ABOD. and abed are two dissim ilar harmonic forms, and the elements A, B_, C are correlated to the elements b, c and lie Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 in them, respectively, the first form is a section of the latter. The relation of projectivity (not the process of projection) is defined as the relation existing between two correlated one dimensional fundamental forms when every harmonic form in one corresponds to a harmonic form in the other. It is denoted-by Two projective one-dimensional primitive forms can also be perspective. The following are projective and also perspec tive; oi. } A straight-line form ani pencil of rays or a pencil of planes, when one is a section of the other; ^ ) Two straight-line forms, when they are sections of the same pencil of rays; Y ) Two pencils of rays, if they are sections of the same pencil of planes or if they project the same straight-line form, or both; %) Two pencils of planes which project the same pencil of lin e s . If one projects, in a plane, a line-form u onto a line from two points S^, Sg, one obtains two projective iine-forms ui, Tig, on the same line V and which have one or two points self-corresponding, according as the lines n, v, and inter sect in one or three points. If two projective one-dimensional primitive forms have three elements self-corresponding, all their elements are so. If a straight-line form and a pencil of rays or planes are projective, and three elements of one lie.in the corresponding Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 three elements of the other, the one form is a section of the other. Further, if two projective line-forms u anii u-, which r * lie in the same plane but not on the same line have their inter sect ion-polnt self correspond.ing, they are sections of the sane pencil of lines, and dually. Also, if in two such Iine-forms the lines connecting three pairs of homologous points all meet in a point, they are sections of the same pencil of rays. When two projective line -forms u, U]_ do not lie in the same plane, the lines (Y) connecting every pair of homologous points constitute a closed ruled surface R, which also contains another group of lines U. Each line of one group cuts each line of the other, while no two lines in the same group inter sect. Each point lying on a Each plane going through line of one group lies also on a line of one group also goes a line of the other. through a line of the other. Such a surface is called a regulus. Two projective pen cils of planes u, u^ whose axes do not lie In the same plane likewise determine a regulus, for each pencil determines a line- form on the other, and these line forms are projective and con nected by the intersection lines of homologous planes. v/e can also take three lines as determining such a surface, if the_ lines are skew, for each line through the three lines deter mines three planes with these lines and likewise three points. If these elements are considered to be homologous, we have three Iine-forms and three pencils of planes, all projective. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 If ABCD^j-alsod, then BCDi^boda, etc. Also, if ABGI^boil and ACDB^aode, then ABGDE^bode and ABBE^abde, etc. All harmonic forms are projective. I t-is q.uite easy, from what has preceded, to show that these things are so. Further, it Is shown that if MUA3p^l®AiBx, then IWAAx^IIEBBx, and also ABOI^GBAD, AOBBjçOABD, BBA^BBCA, etc. lext we take up projective relations among fundamental forms of the second order. Two such forms are called collinear or reciprocal according as they are so correlated that each two unlike elements P, g of one form, of which P_ lies in g, correz ond to two unlike elements of the other, of which the first, Pg, likewise lies in the second, or passes through the second. In other words, in a collinear correlation lines correspond to lines, and in a reciprocal corre lation lines correspond to points, etc. If the elements A, B of one form correspond to the elements Ag, B^ of the other, and if the first two determine a third element the second two must determine an element AgBg corresponding to the element AB. Two forms which are reciprocal to a third are collinear with each other. Two systems which are either collinear or reciprocal are said to be projective, because in each ease each harmonic form in one system corresponds to a harmonic form in the other, and thus each two homologous one-dimensional forms are projective. If two projective systems, either collinear or reciprocal, have Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0 8 three elenents of one and the same one-dimensional form self- corresponding, then they have all their elements self-oorrespond- ing. If two collinear forms of the second order have fonr simi lar elements self-corresponding, no three of which belong to the same one-dimensional form, they have a 11 th eirelements self-corresponding. (This is meant to apply in its widest sense - i.e., for instance, if the elements are planes in one system and lines in the second, the second will be a section of the first). Two collinear systems can, farther, be perspective. The following are perspective with each other: a) k plane system and a bundle of rays, if the first is a section of the last. b) Two plane systems in d iffere n t planes, if they are sections of the samebundle of rays. c) Two coplanar collinear systems if they have a line-form and a pencil of rays (a ll elements of these forms) self-corres ponding. d) Two non-conoentrie bundles of rays, if they project the same plane system. From these considerations it follows that in two perspective systems each two homologous one-dimensional forms are perspective with each other. Two collinear systems in a plane which have a straig h t-lin e form self-corresponding also have a pencil of rays self-corres ponding, for if one projects the system on another plane through Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0 9 the line of the line-form, one has a third system of which the second is also a projection, so the two f ir s t systems are per spective . Here there is a lengthy discusaioh of the conditions for putting systems in projective relation, collinearly or reci procally, which is just as well omitted from our discussion. How if wc-'tate-G the idea of a pencil in a wider sense as being any fixed sequence of lines in a plane. Thus, in two reciprocal plane systems a pencil of rays w ill correspond to a line (not necessarily straight. It can be considered as a fixedd sequence of points.) This conception, of course, can be extend ed to surfaces. If, in one of two reciprocal plane systems, one pivots two rays about two fixed points 1 and B so that their intersection- point describes a line S, the line of the reciprocal system which corresponds to this point whose traces move on two fixed lines a and b, describes the pencil of rays S, which corres ponds to the line S. This likewise can easily be extended to surfaces. In this manner Von Staudt finds a number of relations in the reciprocal systems, such as rectilinearity (in curves), tan- gency, and a great many other reciprocal properties, both in the plane and in space. For instance, to every curve in one system there corresponds one in the other which is composed of tangents to a curve which in turn corresponds to the curve in the first system composed of tangents to the first curve. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 0 A closed line is said to be of even or odd order according as it is ont by a plane containing no part of it an even or odd nnmber of times* It follows that a closed plane line is of even or odd order according as it is ont an even or odd nnmber of times by a straight line in the plane containing no part of the curve. Let the line _S be cut by the plane H in m points and by the plane Y_in n points. By hypothesis the line is closed, and in every point in which it cnts one of the two planes, it goes from one of the two dihedral angles W, I T into the other, so mj_n is an even number, and the line is out either an even or an odd num ber of times by each plane. If three lines AMB, AHB. A ^ are bounded by the same two points, then the lines which are formed by putting these together by twos will either all three be of even order, or one will be of even order while the other two are of odd order, for if a plane is passed either through A or B, it will cut AMB in m points, ABB in n points, and ARB in r points, so the three curves formed by taking these two by two w ill be out in m+-n, n^r, n»r points, respectively, and the sum of these, 2 m+Bn4gr, must be an even number, etc. If one substitutes, in a line of even or odd order which is made up of segments, the complement of any segment for the segment its e lf, one obtains a line of odd or even order. This comes difectly from the above considerations. Of course when the order of a line is spoken of, it is supposed to be a closed Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I l l line. Any two lines lying in the same plane intersect an even nnmber of times i f they are both of even order, or one is even and the other of odd order; they intersect an odd number of times if th ^ are both of odd order. Two closed forms intersect in a form of odd order, if they are both of odd order; in any other case they intersect in a form of even order- Three closed forms intersect in an odd num ber of elements if they are all of odd order, but otherwise they intersect in an even number of elements. If two closed forms are projective, they are, either both of even or both of odd order. A line of even or odd order is projected from a point out side it by an angle-snrfaoe of even or odd order. It is also projected from an axis outside it by a pencil of planes of even or odd order. A ruled surface of even or odd order is project ed from a point not on it by a pencil of planes of even or odd order which contains the point and all the lines of the surface. The next two sections, on plane figures and bodies and their projections may be omitted. Section 15, dealing with "returning elements" is somewhat more in^ortant however- The point_P is thonght to move, describing a straight line f I P, or the line P, describing a plane P, remaining drawn throu^ the point__P, or the plane 'turns about the line P,^ or all of these motions are conceived. ÎJow if the point P has moved and has, in the position Bj changed the sense of its motion on the line 2* then the point B is called a "returning point" of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 2 lin e described by the point P, or i f the lin e is cnrved {the line P^has also turned about the point P), it is called a vertex. If the line P turns, and in the positionchanges the sense of its turning, the line B (a called a returning-line of the luled surface described by the line P'f" If we denote the case in which an element _B_is not a 'Return ing element" by — , and the case in which it is by-4 , then for any point ^ of a curve ^ and its ray B'^of the pencil of rays belonging to the curve, we have one of the four cases — —, — +, + —, + +. In the first case, the point B is an ordinary point of the curve, in the second an ordinary turning point, in the third an ordinary vertex, and in the fourth, in which the point B is a returning element of the curve 2 and also the ray B, is / a returning element of the pencil S^, it is both a vertex and turning-point. The curve S will be touched in the first case and out in the third case by the ray B_ in the point B^, but any other ray through ^ will in the first, ease cut the curve, and in the second case touch it. In the second case the curve is cut, and in the fourth touched, by this line through the point B, In the f ir s t case the curve is called convex on the side in / which it is touched by the line _B at the point B^ It is called convex on the other side. If the curve S in one of two reciprocal plane systems cor responds to a pencil of rays S,'^ of the other, (and so the pencil corresponds also to the curve) and if we replace the point B and / / ray B by xy, then in the reciprocal we will replace the line _Bj Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 3 and point by yx, so the number of turning points in one of the two curves is the same as the number of returning points of the other. We have a similar case for collinear systems. If xy represents a combination of the signs —, +, then we will understand ay to be — or +, according as it contains an odd or even number of — signs. If xy is again substituted for the point B of a plane curve / / S and the line ^ of the pencil of rays ^ belonging to the curve S, then; (1) If a point describes the curve 3 the line connecting it with a fixed point 4 describes a pencil of rays which pro jects it. In accordance with the above we substitute _x or xy or y_for the ray of the pencil which projects the point according as ^ is outside the straight line B, or is a point of this line different from the point B, or is the point B it self. This idea and this terminology are extended to all of the different curves, surfaces and pencils. (2) Every closed form which is contained in a one-dimen sional primitive form has an even number of returning elements. A closed plane curve_S which therefore determines its /■ closed pencil of rays ^ , has an even or odd number of turning points, according as returning points, according it is of even or odd order. as the pencil of rays S is of even or odd order- If the curve has ^ turning points and is cut in n. points by a lin e u going through any one of these m points and belongii^ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 4 to the pencil of rays S, then, by (1) above, the section of this pencil by the line u contains returning points, whence, by ( 2 ), the numbers m and n are either both even or both odd. If xy is substituted for the point JB of a plane curve and the ray B'^belongii^ to it and to the pencil of rays determined by the curve, in the manner of the above, and if we represent xy by y^ (xy = y^), then one can interpret the combination xy^, in which — means intersection and + means targency, so that the curve will be touched or cut by one of the lines going through / 3 , The f i r s t sign has to do with lines different from 2* and the other with the line B, If one wishes, conversely, to obtain the relation xy from xy-,, he needs only to notice that y = xy^. That is, if xy^ = -----, so that all the lines through the point B intersect the curve in this point, then xy = — + and the point B_ is an ordinary turning point. This theojy is extended further, but this is sufficient to show the whole idea. The involution is the next idea to be developed. An involu tion is defined as a system consisting of two projective forms in vhich each two homologous elements correspond doubly, that is, an. element P of one form corresponds to an element P, of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 gaicL to be in involution if any projections or sections of them are in involution. If in two projective one—dimensional forms some two elements A, Ai correspond doubly, the forms are in involution, for if B is a third element in one of the forms and 2 % is the correspond ing one in the other, then since %A%AB%B (as we have al ready shown) then the element B^ of the first form must corres pond to the element B of the latter. Further, ABAittAiB^^A, so the forms must have no elements or two elements self—correspond -1 ing, according as the elements A, A^ are separated by the elementj B, B]_ or not. A one-dimensional form contains either no elements which coincide with the elements coordinate with them, or two such, each of which is coordinate with itself, according as two co ordinate elements are or are not separated by two other such elements. In the latter case the self-corresponding elements are M, B, by which every two coordinate elements P, so that MIPPi 7 ^MNP]_P, are harmonically separated. An involution is determined by two pairs of elements, or a double element and one pair of coordinate elemenb s , or two double elements. This is easily shown, considering all that has already been said. The usual graphic construction of an in volution, using a complete quadrilateral, is given : The idea of involution is now applied to the theories of collinear and reciprocal systems already developed. If two collinear systems are, besides being projective, also in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 6 involutLon, the entire aystem la called, an InTOlute system. If the two systems are reciprocal and in inrolution, the entire system is called, a polar ^stem or simply a polarity. The in volute system is treated at considerable length, but there is nothing that is very fa r removed from what we have already seen. The polarity is of much greater interest to us. The dis cussion is begun by stating a criterio n that two reciprocal sys tems be in involution. It is that if two such systems lie in the same plane, and if the vertices of some triangle in it cor respond to the sides opposite them, then the ^sterns are in in volution. This is of course merely the determination of an in volution by a pair of elements of a projeotivity whiah cor respond doubly, for if the points A, C. of one system corres pond to the lines AC, AB of the other, then the lines ^ , AC, AB of the first system will correspond to the points A, ^ C of the latter. In a plane polar system each point is called the pole of the line coordinate with it, and the line is called the polar of this point. Obviously, from the above, the polar of a ll points which belong to the same line intersect in the pole of this line. If a point lies on its polar, every other point of line lies outside its polar. Every line form P which does not go through its poleis in involution with its coordinate pencil of rays, as can easily be shown from v&iat has already been proved. Every plane five—point determines a polar system in which Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 7 each Tertez is the pole of the side opposite. This short dlsoTossion of polar systems seta the stage fo r the introduction of conic sections in the next section. The definition is that the curve containing all the points of a polar system which are on th e ir polars is a curve of the second order. Also all curves projective with this are likewise of the second order. A curve of the second order is projected from a point lying outside its plane by a conic surface of the second order- The usual non-metric relations between poles and polars are developed, including Pa^cai^s and Brlanehor^ theorems. It is shown that a curve of the second order is determined by five points. (that if the sides of an n-point pivot on fixed points and the vertices, save one, describe straight lines, the remaining vertex will describe a line of the second or der) is proved. If two curves of the second order are put in projection, then one can take A^, B%, C]_, the homologous elements to A, B and G_, at w ill. The reader will readily see that now that the frame-work for Ton Staudt>s geometry has been set up, one can prove a great many well known theorems quite simply by its aid. The more im portant of the theorems which Ton Staudt proves w ill be given here, without any proof, or with the proof merely indicated. If a line n is a common targent to two curves of the second Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 8 order and one considers every two tangents to the curves which intersect this line in the same point to correspond, then the curves are protective. If the curves touch this line in the same point, then they are perspective, so that the rays connecting pairs of homologous points all intersect in a single point. Two curves of the second order in two planes whose line of intersection is tangent to both curves at a single point are sections of a single conic surface of the seeoM order. If the intersection line of the two planes cuts the curves in the same two points, there are two such conic surfaces through both curves. The following extension of Pascal’s theorem is given; If of the 2n+l points in which the pairs of opposite sides of a (4n+2) point intersect, respectively, 2n lie in a single straight line, the remaIning point does also. If the elements of a curve of the second order are paired in an involution so that each two coordinate points of the curve determine a line passing through a single point common to all these lines, and all the points in vhich coordinate tan gents of the curve intersect by pairs will lie in a single line, then every involution of three pairs of points of the curve is projected from any point of the curve by an involution of three pairs of rays, and similarly every involution of three pairs of tangents is out by another tangent in an involution of three pairs of points. If the vertices of an n-point lie in a given curve of the second order and n - 1 sides of the n-point go Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 9 through given points not on the carve, the last side either goes throogh a given point or is tangent to a given curve of the seconj order- Two curves of the second order have at the most four points in common, and their tangent pencils of rays have at the most four rays in common. If two curves of the second order have four common tangents, they have either four common points or none. If we are given five points A, C_, D, E of a curve of the second order and five points of another. A, 2* %» which has the three points A, B, and 0 common with the first, we can find out wh#ier the curves are tangent in one of their three common points, or if they intersect in a fourth. First we find the point Dg of the first curve such that E ( ABODg ) yç-'S'i ( ABGD-| ), and then the point _F which is projected from the point Dg by the line Dg^l• Then, since two pencils of rays projecting the same curve of the second degree are projec tiv e, F(ABOBg) ) and also F(ABCD^ ) ) . Then the point F of the first curve also lies on the second, since two projective non-perspective pencils determine a curve of the second order by the Intersection of their homologous rays. This point, then, if it is distinct from A, B and _C, is the fourth point of intersection of the two curves. If it coincides with one of these, the two curves touch in this point. There is a discussion of problems of the second degree and an extension of the idea of polarity to space, and also an extensj Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 2 0 of the idea of curves of the second order to space, where the procedure is much the same as that outlined ahove. These, with numerous interesting theorems, space does not permit us to dis cuss. The essential structure of the work, it is believed, has been sufficiently indicated as far as simple fact is concerned. It remains to show just what historical sigaificance it has, and how its fundamental ideas arose. It is necessary*to state first that Poneelet was not 7on Staudfc’s only great predecessor in projective geometry, but that Jacob Steiner had already published his work "Systematische Entwickel4ng.. -e tc .” in 1832. This we will take up rather b rief ly, after discussing Ton Staudt*s work. The reasons for thus violating the chronological order are two. The first is to bring Ton Staudt*s work in close jh^taposition with Poneelet’s. The reason for this will be seen shortly. The second is that although Steiner was the originator of much of the terminology and many of the methods of Ton Staudt*s book, in the form which Ton Staudt gave them they are an excellent introduction to the methods of Steiner and Reye. To return to the work itself, the single most striking feature is of course the complete avoidance of metric considera tions. The treatment of conic sections in non-metric geometry, of course, requires a non-metric definition of conics. Ton Staudt’s definition, it will be remembered, is that the self conjugate points of a polarity form a curve of the second order- This is shown to be equivalent to Steiner’s definition, given in 1832, stating that the points of intersection of homologous Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 2 1 raya of two projective non—perspective flat pencils is a curve of the second order. It is interesting to speculate upon Ton Staudt’s definition, which at f ir s t glance seems rather unnatural. The answer to our curiosity undoubtedly lies in Poncelet’s veiy extensive treatment of the subject of reciprocal polars. Bowhere, I belieVe, did Poncelet prove or state that in any system two reciprocal polars were actually projective, although he proved the existence of all non-metric projective relations among them, and also, in some cases, of projective metric relations. Ton Staudt, by showing how a figure composed of lines, for instance,, could be considered as projective with one composed of points, and by introducing the idea of the Involution as a projactivity in which elements correspond doubly, he was able to reproduce Poncelet’s recipro cal polar figures non-nætrically. By noticing that, tangents to the conic with respect to which one constructs reciprocal polars are the reciprocals of their points of tangency, and vice versa, the definition of the curve of the second degree arose very naturally. It gave rise to a very extended nonr-metrio treatment of involutions, and the conceptions of collineations and polari ties as units with which to work. The work is truly non-metric, for although the words element at infinity, parallel, imaginaiy, etc., are used, they are merely words for special cases whose restrictions we do not know or con sider as apart from the oases not so designated. These words merely form a convenient transformation to the metric geometry Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 S 2 where they are given content of th e ir own. The "G-eometrie der Lage” was far from being Ton Staudt ’ 3 only important woaSc. In 1857, ten years after its publication, he published his "Beitrage zur Geometrie der Lage." It contained the first example of the introduction of analytic methods into geometry on a strictly projective basis, and is much like the one which w ill be described in the discussion of Teblen’s work. This also dealt with the q^uestion of the interpretation of imaginary elements in geometry. Jakob Steiner (1796-1863), as we have already said, published his work on projective geometry in 1832. His principal contribu tions to the development of projective geometry were; the idea of different one-dimensional forms and two-dimensional forms as a basis from which to work ; the definition of a point conic as the points of intersection of homologous rays of projective non-per spective flat pencils in the same plane; the conception of this point conic and its duals, the line conic and cone of planes, and the dual of the second, the cone of lin es, as four one-dimen sional forms of the second degree; the theorem: "If A and_B are any two given points of a conic and P is a variable point of this conic, then A(P)t^ B(P)", known as Steiner’s theorem; the con sistent dualizing of assumptions and the 0 rems ; and, due of course to his definition of the conic, the development of an extended non-metric geometry, a thing which had not been done before then. His geometry is not exclusively projective, but large parts of it are completely independent of the metric considerations introduced. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 123 The following propositions are due to Steiner; "Given six points Pg, P 3 , P4 , P5 , Pg, on a conic. By taking these in all possible orders 60 different hexagons are ob tained. The Pascal lines of the three hexagons P 1P2P3P4P5P6 , P1P4 P3PGP5P2 , and P 1P6P3P2P5P4 are concurrent. The point in which they concur is called a Steiner^po_^t There are 20 Steiner points in a ll. If the vertices of a hexagon are permuted in the above way except that alternate vertices are not changed, we obtain five other simple hexagons. The Pascal lines of these pass through two Steiner points, called conjugate Steiner points. The 20 Steiner points constitute ten pairs of conjugates. The 20 Steiner points lie by fours on 15 lines called Steiner lin e s. Two triangles self—polar with respect to the same conic, or perspective and have six vertices on a second conic and six sides tangent to a third conic. In summary, Steiner developed a new nomenclature (pencils, bundles, etc.) and showed how, by his definition, conic sections could be treated non—metrically. There are less really importait achievements so far as the development of this particular subject is concerned. The next important figure is Theodor Reye. He published his "Geometrie der Lage" in 1866-67. It then appeared in two volumes only. The fourth edition of the first volume was published in 1899, while the third editions of the second and third volumes Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 8 4 appeared, in 1892. We sh all discuss the fourth edition of the first volume and the third editions of the other two. By way of introduction it may he said that although there was little that was new and also fundamental in these three volumes, they nevertheless formed a rather complete survey of the entire field, with a somewhat more ambitious program than any of the others as far as extent was concerned. The terminology of this work is somewhat different from that of Von Staudt’s and Steiner’s, and is rather closely that used today. The six fundamental forms are the row of points, pencil of lines, pencil of planes, planar field., bundle, space system. The principle of duality is stated at the beginning here, as in Von Staudt’s and S tein er’s works, and, as in th e irs, is kept sight of constantly all through the work, the dual proposi tions and proof being printed side by side. Four dual assumptions are stated, and a few simple theorems proved from them. This follows almost exactly the discussion in Von Staudt ’ 3 book. Ilext' a complete n-point is defined. Then Desargues’ theorem is proved, and the theorem on the intersection of the six th pair of sides of complete four-points in different planes. A harmonic set of points is then defined by the complete quadrangle. It is shown th a t if three points A, B, C_of a harmonic set are known, and also their order, the fourth point is determined. Also harmonic forms are always projected into harmonic forms. Then Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 metric considerations are introduced and it is stated that if of four harmonic lines two which are separated by the remaining two are perpendicular to each other they bisect the angles between the other two lines. If ABCD is a harmonic set, then ^ ~ A3). EC CD Two fundamental fbrms are said to be projective if they are so placed with respect to each other that each four harmonic elements correspond to four harmonic elements in the other- If two forms are projective with a third then they are projective with each other- Reye»s definition of projectivity is the same as Ton Staudt’s, but not like S teiner’s, which is; If the two forms A, B are so related that their elements are determined by the order in which thqy correspond pair-wise, they are projective. next, the fundamental theorem of projective geometry is stated, namely, that i f two simple projective forms have three elements A, 1, C self-corresponding, then all their elements are self-corresponding, and thus the forms are identical. This is apparently the first recognition of the importance of this theorem. Steiner's definition of the curve of the second order is used. Also, the above definition of projectivity is shown to be equivalent to Cremona’s definition of the projectivity as a sequence of perspectivities. The invariance of the harmonic ratio under projection is demonstrated by Poncelet»s method. The various problems on the construction of conics with certain elements given are solved by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 3 6 means of Pascal’s and Brlanehon's theorems, jnat as th ^ were in Ton Standt’s "Geometric der lage." It is shown that the tangents at fo\r harmonic points of a curve of the second, order are four harmonic tangents, and the pencil of tangents to a curve of the second order is cut by every two tangents in projective rows of points. ïïext the theory of poles and polars is developed entirely non- metrically by means of the cross ratio, which can be constructed by a complete quadrangle. The usual theorems are proved, the fol lowing being notable: If a point _P describes a row of points u, its polar p describes a pencil of lines U, which is projective with the row of points. That is, u(P)-^U(p}. Two points of the plane are defined as conjugate with regard to a curve of the second order if each lies on the polar of the other. If two points A,are conjugate with a third, C, then their conecting-line is the polar of If in a plane we are given a curve of the second order and a lin e T and a point U not on T, and i f we determine on each lin e through U the point which is conjugate with the intersection point of this line with_%, all these points lie on a curve of the second order going through T, the pole of 7, U, and the points of tangency of the two tangents which can be drawn to the given curve from U and T. If a curve of the second order is cut by two conjugate lines AG and Bl, the intersection points A, 3, C, D are four harmonic curve-points. The metric relations of the axis, center, and line at infinity, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 2 7 all with, regard to the theoiy of poles and polars, are stated. From these relations the analytic representations of these curves are derived. Here the regains is introduced as determined by two skew projective rows of points.Such a surface contains two groups of lines, each of one meeting every line of the other group but no line of its own group. Each group consists of all lines which three arbitrary lines of the other group. In one group of such a surface the other group is called the cross-group, and its lines I the cross-lines. A group is cut by two of its cross-lines in ' projective rows of points, and is projected from two of its oross^ lines by projective pencils of planes. These pencils of planes a: perspective with these rows of points. Four lines of a group are called harmonic lines i f they are cut by one, and therefore by a ll, of the cross-lines of the group in four harmonic points. A group is cut by every plane that has no line in common with it j in a curve of the second order. j Returning to conic sections, fpur harmonic points of a curve of the second order are projected from every fifth point of the curve by four harmonic lines. If two projective curves of the second order have four point self-corresponding, then they have all their points so, and are thus identical. Two similar projective forms u, u^ which do not have a ll the elements self-corresponding but are conjective are involutes or have the relation of involutions if in them each two homologous Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 8 0 elements correspond donlaly. Two dissim ilar forms constitute an imrolution if one is the involute of a section or projection of the other. If two projective rows of points of the second order are involutes, the conneoting-lines of their homologous points all go through a point U; the intersection points of their tangents, however, a l l lie on the polar u of U. The line u is called the axis of involution and the point _IJ is called the center of in volution of the row of points. Two similar projective and con jective elementary foriæ are an involution if in them some two elements correspond doubly to each other. An involution has two or no double elements and is called elliptic or hyperbolic according as two coordinate elements are separated by two others or not. In each double element two co ordinate elements coincide. To return to matric considerations, in a line involution of the first order there are two and in general only two conjugate lines which intersect at right angles; these are called the axes of the involution. A line involution of the first order is orthog onal if in it some two lines, ^andjb, not conjugate, make right- angles with their coordinate lines a^ and b^. Similarly for plane involutions. This discussion of involutions is quite naturally followed by problems of the second degree. This includes a ciiapter on con- focal conics. On the basis of what we have already said about this work and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 2 9 q^uoted from it, it w ill serve our p-urpose if for the most part we simply indicate the remainder of the work» Collineations and correlations of the second order are next. Collineations are dealt with much as Ton Staudt dealt with them. A correlation is simply a projectivity between the elements of a plane of lines and a plane of points. Then follow "Collinear and reciprocal plane curves” and "Collineations and correlations of space systems," "Surfaces of the second order," "Polar theory of surfaces of the second order," "Affinity, sim ilar!congruence and symmetry of plane fields. Affine conic sections." Two plane fields are said to be affine when their lines at infinity correspond. If ele ments at infinity are then given the usual metric significance, it will be seen that in an affinity parallelograms will correspond to parallelograms, etc. We have already seen sufficient of reciprocal systems and polarities so that they do not require discussion here. The idea of a complex, though, is new, as far as projective geometry is concerned, and it was Reye’s definition that intro duced i t into projective geometry. He discussed the complex as the system of lines linearly dependent on five independent lines or on the five sides of a skew pentagon. This was the fimt defi nition of this sort to be given to a complex. In connection with the linear complex Reye introduced the null system, which is the system in which there exists a correspondence between plains and the points on them through which all the lines of a linear complex which lie in those planes will pass. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 3 0 How that we have shown Reye’s methods, in which there was nothing particularly new or otherwise remarkable, and indicated his subject matter, we have attained an object as far as his work is concerned. There are several names besides these four that are associat ed with the development of projective geometry from its beginning in 1822 to the end of the century, a list of which would include Klein, Pluoker, Grassmann, Clebsoh, and others, but these men were not interested primarily in projective geometry. Toward the end of the century and at the beginning of the new there was Hilbert, who worked with the foundations of geometry, and Enriques, who published h is "Vorlesunge% über die Geometric der lage" in 1906. This work, thou^ small, contained the most beautiful parts of projective geometry as it then stood, presented in beautiful form. This work was the last of the old order. In the twentieth century projective geometry was to undergo a very important and fundamental change in organization. This will be the subject of the next chapter- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 3 1 CHâPTSR IT The f ir s t volume of "Projective Geometry" by Teblen and Young was published in October, 1910. It was the outgrowth of the extensive work which had been done at the University of Chicago on the foundation of Mathematics. This latter was largely a continuation of the fundamental work done by Hilbert and others in the la s t decade of the nineteenth century. ^ Aside from the vigor and clarity of the work of Teblen and Young, its moat remarkable aspect is the unity which is given by the simple extension of the original set of axioms in var ious directions. The most important single portion of the work, from the point of view of obtaining a clear understanding of the funda mental idea.s involved, is the introduction. We shall discuss this in some detail. In order to give an example of a mathematical science, and of the fundamental rules of mathematical logic a set of as sumptions is given, couched in the most general language. They are seven in number and foim a consistent, categorical set. "A set of assumptions is said to be consistent if a single concrete representation of the assumptions can be given"; that is, if we can find any one system corresponding to these as sumptions in one-to-one correspondence, and which we admit to be consistent, then the set of assumptions is consistent. "A set of assumptions is said to be categorical if there is es sentially only one system for which the assumptions are valid- i.e., consistent; that is, if all systems containing these assump^ tions, if the systems are consistent, correspond to the original Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 132 in one-to-one correspendenoe . This means that if an assumption is added it is either not independent of the others or not con- aistent with them. The latter statement fall-s-pat easily from our previous remarks. The first req.uires justification. If an assumption of a set is independent of the others then in .its absence its contrary may be assumed, and the set w ill remain consistent. This means that if we (above) added the contrary of the assumption we proposed adding, then either this assumption would be dependent on the others or, if independent, would form another system not isomorphic with the one in which its converse held. But this is contrary to the definition of categoricall- ness. From the above discussion of independence it will be seen that to prove one assumption independent of the other in a set a system will be set up where this assumption is replaced by its converse. If this system satisfies the test for consistency, is then the assumption in question was independent of the other. Otherwise, not. The assumption in question, however, must be consistent with the others of its system in the first place, for otherwise the test is not valid. From the set of assumptions mentioned above several theorems are proved. It is shown that the ommission of one of the axioms leaves the set non-categorical, and tl^ a categorical set is more restricted in its applications than a non-oategorical. The idea of homogeneous coordinates in solid analytic Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 geometry is adopted as the ”ooncrete example" mentioned above, used to prove the ocnsistenoy of terminology and assumptions. The assumption here is, of course, that the number system of algebra is consistent - i.e ., eannoVinvolve contradictions. These coordimtes are of the form (%%, xg, xg, X4). It is agreed that (x^, xg, Xg, X 4 ) shall represent the same point as (cxi, cxg, cxg, CX 4 ), where o .is an arb itrary constant. (0 = 0 }, These homogeneous coordinates are equivalent in ordinary coordinates to ( 1 , m, n, i ) , where d is the distance from the origin. Thus d — » cm x^+y*+ ■/x2+y2+22 ^ - wm 2« = * » » s = 2 ■/x 2 +y2+ 22 -/x 2 +y2+ 22 & where x, y, and z are ordinary coordinates. (x%, xg, xg, x^) are called the homogeneous coordinates of a point. The inter pretation for X 4 = 0 is that the point in question is on the plane at infinity, whose equation is, naturally, x^= 0 . A plane is defined as the set of all points (x^, Xg, xg, X4) satisfying the linear homogeneous equation: ax^ + bxg + cxg + dX4 = 0 A line is the set of all points (xq, Xg, Xg, x^) which satis fy the two distinct linear homogeneous equations aqxq + b^Zg + c^Zg + cLqZ^ = 0 agXi + bgxg + cgXg + dgX4 = 0 Since these must be distinct, the corresponding coefficients thronghout must not be proportional. From this definition we see Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 3 4 that the intersection of any ordinary plane and the plane ^ 4=0 is a line, called the line at Infinity. Also, from this, every line contains one point at infinity and parallel lines have their points at infinity in common. We have now defined an analytic space of three dimensions consisting of Points: All sets of four numbers zg, Zg, Z 4 ) ezcepfe the set ( 0 , 0 , 0 , 0 ), where (cz^, czg, ozg, cz^) is regarded as identical with (zq_, zg, Zg, Z4), provided c is not zero. Planes: All sets of points satisfying one linear homo geneous equation. lines; All sets of points satisfying two distinct linear homogeneous equations. The validity of this test for consistency lies in the fact that in analytic space there is a one—to—one correspondence between points and numbers, and we are assuming the number system of algebra to be consistent. It should be noticed that the above algebraic system may be regarded, if one desires, as entirely non-metric, the equa tions only serving to distinguish point from point, line from line, etc., and to show relations of contiguity (whether a point Is on a line, etc.). All this, of course, is preparatory to the actual work. Before we commence with this, there is one idea that if stated will serve greatly to clarify the plan of the work. Given a (non-oategorieal) set of assumptions. If we add FolU^ei by 13 6" Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 136 some assault ion which is independent of these and consistent with them, but which may or may not make the set a categorical set, it will restrict the application of the system of assump tions, for in the old system we could develop theorems special cases of which will lie in the new system, but possibly other cases, in contradiction to this new assumption, will be elimin ated in the new system. Any theorem true in the new set of assumptions is also true in the old. The plan of this treatise is to start with a general pro jective geometry, defined by two sets of assumptions denoted by A and S. These assumptions, in conjunction with tne new assump tion introduced later, define all the other"spaces" or "geom etries" that are considered. Thus all these "spaces" or "geometries" are subdivisions of the general one defined by the assumption common to them all- Some of these are also subdivisions of others, that is, some third assumption that is added is common to them both. The assumptions denoted by ^(Assumptions of alignment), are the following: If A and ^ are distinct points, there is at least one line on both A and B. A2. If A and B are distinct points, there is not more than one line on both A and A3. If A, B, 0 are points not all on the same line, and _____ ' ------' .w J- D and E (D=/E) are points such that B, 0, D are on a line and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 3 7 E are on a line, there ia a point F such that A, B, F are on a line and also D, JB, F are on a line. Theorem 1. Two distinct points are on one and only one line. (^, Ag) Theorem 3. If C andJD (CfD) are points on the line AB, A and B are points on the line CD. (A^, Ag) Theorem 3, Two d istin ct lines cannot be on more than one common point. (A^^, Ag) Definition. If P, Q, E are three points not on the same lin e, and 1 is a line joining Q, and R, the class Sg of all points such that every point of Sg is collinear with P and some point of 1 is called the plane determined by and 1 . Theorem 4. If ^ and B_are points on a plane'tT", then every point on the line AB is onTfT s Assumption of extension, BO. There are at least three points on eveiy line. This is called an assumption of extension because it insures a minimum extension of the class in question. An assumption, on the contrary, which insures a maximum extension of the class is called an assumption of closure. Theorem 5. Any two lines on the same plane TT~are on a common point. (A, EG). Theorem 6 .‘ The plane TTdetermined by a line 1 and point P is identical with the plane yT"determined by a line m and a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 138 point Q, provided m and Q, are onlT". EO) Theorem 7. Two distinct planes which are on two common points are on ail the points of the line IB, and on no other common points. (A, EO) Corollary. Two distinct planes cannot be on more than one common line. (A, EO) Définit ion. If P, Q,, R, T are f onr points not on the same plane, and if jris the plane containing Q,, E, and T, the class S3 of all points such that every point of Sg is collinear with P and some point of-rr is called the space of three dimensions, or the three-space determined by_P andTT» Theorem 8 . If A and B are distinct points on a three-space 3^, every point on the line ^ is on (A) Corollary 1. If Sg is a three-space determined by a point P and a planerfr , thenTT and any line on ^ but not 9 *^]Ta.re on one and only one common point. (A, EO) Corollary 2 . Every point on any plane determined by three non—collinear points on a three-space Sg is on Sg, (A) Collary 3. If a three—space Sg is determined by a point P and a plane If , thenTTand any plane on Sg distinct fromTr are on one and only on common lin e. (A, EO) Theorem 9. If a plane^and a line ^not onT^ are on the three-space Sg, then TT and a are on one and only one common point. (A, EO) Corollary 1. Any two distinct planes on a three-space are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 139 oa one and only eomraon lin e. (A, EO) Corollary 2. Conversely, if two planes are on a oommon line, there exists a three—spaoe on both. (A, EO) Corollary 3. Three planes on a three-space which are not on a common line are on one and only common point. (A, EO) Corollary 4. I f ^ , yare .three distinct planes on the same Sg hut not on the same line, and if a line i is on each of two planesy^ which are on the lines^3^and£cC respectively then it is on a plane ^ which is on the l i n e . (A, EO) Theorem 10. The three—space Sg determined by a plane IT and. a point P is identical with the three—space S,'", determined by a plane and a point p/, provided JT^ and^^ are on Sg. (A,:^) Corollary. There is one and only one three-space or four given points not on the same plane, or a plane and a line not on the plane, or two non-intersecting lines. (A, EO) Assumptions of extension, E; There exists at least one line. BE. All points are not on the same line. E3. All points are not on the same plane. g3 . If Sg is a three—space, every point is on Sg. The last may, in accordance with what was saidabove, be called an assunçtion of closure. The following corollaries of extension are derived from these: Corollary 1. At least three coplanar lines are on every po in t. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 4 0 Corollary 3. At least three distinct planes are on every line. Corollary 3. All planes are not on the same line. Corollary 4. All planes are not on the same point. Corollary 5. If Sg is a three-space, eveiy plane is on Sg, These assumptions and theorems can he arranged so as to form a dual set of statements. If we arrange all the assumptions thus fa r made in a column, we w ill find that we have already proved their space duals: (A1, AE). Theorem 9, Cor. 1. A3. Theorem 9, Cor. 4. Cor. 2. of extension. El. E 1. Cor. 3, of extension. Cor. 4, of extension. E3'. Cor. 5. of extension. Since the definitions can he worded in space dual language, the following theorem can he proved. Theorem 11. Any proposition deducihle from assumptions A and E concerning points, lin e s, and planes of a three-space remains valid if stated in the "on” terminology, when the words "point” and "plane" are interchanged. (A, E) The corresponding theorems for plane duality and point duality can also be stated. This is the first ease where the principle of duality has heen^igorously proved, for it is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 4 1 shown definitely that for any procedure from the set A, E, there is an exactly similar procedure from the exactly analogous set on the right, which at each step lead to results exactly analo gous in the canner which we call dual. Thus the principle of duality is proved once for all, and we need feel no hesitancy in changing from a theorem to any of its duals without a new proof. The theory of duality is traced to its origin in the assumptions. Hext we have projection, section, etc. Définit ion. A fj^^e is any set of points, lines, and planes in space- A plane figure is any set of points and lines on the same plane. A point figure is any set of planes and lines on the same point. Definition. Given a figure Definition. Given a figure F and a point P; every point and a plane7T ; every plane of F distinct from ? determines of F distinct fromU determines with P a line, and every line cfwith ^ a line, and eveiy line of F not on P determines with F not onTT-determines with P a plane; the set of these IT a point; the set of these lines and planes through lines and points onTT is P is called the projection called the section of F byIf . of F from _P^ The individualThe individual lines and points lines and planes of the pro- of the section are also called jectlon are also called the the traces of the respective projectors of the respective planes and lines of _F. points and lines of F. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14-2 Définitlon» G-iren a plane figure W and a line 1 in the plane of _F; the set of points in which the lines of F distinct from 1 meet l i s called the section of F by 1. The line 1 is called a transversal, and the points are called the traces of the re spective lines of 2 ' Definition. Two figures Fg are said to be in (1, 1) cor respondence or to correspond in a one-to-one reciprocal way if every element of F^ corresponds to a nniq^ue element of Fg in such a way that every element of Fg is the correspondent of a nnlgne element of F^. Two elements that are associated in this way are said to be corresponding or homologous elements. ■ Definition. If any two homo- Définit ion. If any two homo logous elements of two cor- logous elements of two cor responding figures have the responding figures have the same projector from a fixed same trace in a fixed planew, point 0 , such that all the snoh that all the traces of projectors are distinct, the either figure are distinct, figures are said to be per- the figures are said to be speotive from 0, The point 0 perspective fromi*). The plane is called the center of per- a> is called the pla^ ^f pen^ speotivity. speetivity. Définition. If any two homologous lines in two corresponding figures in the same plane have the same trace on a line 1 , such that all the traces of either figure are distinct, the figures are said to be perspective from 1. The line 1 is called the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 4 3 axis of perspeetivity. Definition» To project a figure in a plane from a point 0 ____ -— onto a plane distin ct from © 6 , is to form the section byo{'of the projection of the given figure from 0. To project & set of points of a line from a point _0 is to form the section by l"" of the projection of the set of points from 0 . (The definitions of the conçlete n-points, etc., are omitted here). Définit ion. A figure is called a configuration if it eon— sists of a finite number of points, lines,and planes, Mth the property that each point is on thesame number a^g of lines and also on the same number of planes; each line is on the same number ag^ of points and the same number agg of planes; and each plane is on the same number a^^ of points and the same number agg of lines. A configuration may be conveniently represented by a square matrix; 1 2 3 point line plane 1 point 1 ^ 1 1 & 12 ^13 : 3 line &21 ^ 2 2 ^23 : 3 plane agi ^32 &33 ^ The number a.|j thus gives the number of elements of the^ thof kind on every element of the ± thp kind. The numbers app, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 4 4 ®2 2 » &3 3 » give the total numbers of points, lines and planes, respectively. Desargues configuration; 5 4 6 2 1 0 3 3 3 1 0 is obtained by taking a section of a con^lete space five-point. The configuration in the plane has the symbol 1 0 3 3 1 0 From this configuration Desargues’ theorem is easily proved. As a corollary from this the theorem Ton Mobius tetrahedra is proved. The complete plane four-point is defined, and then the next assumption is introduced. Assumption H. The diagonal points of a complete g_ua dr angle are non—collinear. / / / / Theorem. If two complete g_ua dr angles P]_PgP 3 P^ and P ]_P gP 3 P 4 / / correspond - P% to P 1 , Pg to P g, etc., - in such a way that five of the pairs of homologous sides intersect in points of a line 1 , then the sixth pair of homologous sides will intersect in a point of 1. (A, E). The set of these six points on a line is called a quadrangular set. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 4 5 The nine primitive geometric forms are next defined. Per- speotivity between them is defined, and is Indicated by the symbol ^ . The relation of projectivity is defined as a se quence of perspectivities, and is denoted byÿr • A number of theorems on projectivity are proved, the main ones being: If jA, ^ ^ are three points of a line 1 and A ^ ^ O'' / / three points of a line 1 , then ^ can be projected into A , B into ^ ^ and G into _C^ ^by means of two centers of perspectivi- ty- (A, E) The projectivity iBGB -tt-BADO holds for any four distinct points A, B, G, D of a line. (A, E). If (P), (P'’), (P^), are pencils of points on three distinct concurrent lines _1, 3^ , respectively, such that (P) tt (P ) and (P'')7'^ (P'’'), then likewise (P) (P^^), and the three cen ters of perspectivity _S, , _Sare collinear- E) Gorrespondences are indicated by large letters A, B, The resultant/'" of two correspondences A and_B is indicated by That i s , iB = r~ . The correspondence making every element correspond to itself is indicated by 2^. Al = lA = A. Définit ion. A class _C_ of elements, which we denote by a, b, c,...,, is said to form a group with respect to an opera- tion on law of conhination , acting on pairs of elements of G , £>V Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 146 proTided the following postulatars are satisfied: Grl. For every pair of ( equal or distinct ) elements a, b of the result a o b of acting with the operation^ on the pair in the order given is a uniquely determined element of _G. G-2, The relation (aob) oc = ao(boc) holds for any three (equal or d istin ct) elements a,J>, a of G-3. There occurs in G- an element i, such that the relation aoi = a holds fo r every element a of G. / G4. For every element _a in G there exists an element a satisfying the relation aoa^ = i. These follow as theorems: The relations aoa^ = i and aoi = a imply respectively the relations a'oa = 1 and ioa = a. An element i of G is called an identity element, and an element a'' satisfying the relation aoa '= i is called an inverse element of a. There is only one identity element in G. For every element a of G there is only one inverse. A group which, further, satisfies the following postulate is said to be commutative. G5. The relation aob = boa is satisfied for every pair of elements a, b, in G. A set of correspondences forms a group provided the set contains the inverse of any correspondence in the set and pro vided the resultant of any two correspondences is in the set- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 47 D efinition. If a correspondence A transfornis eveiy element of a given figure F Into an element of the same figure, the fig ure _F Is said to he Invariant under A. Theorem. The inverse of any projectIvlty and the resultant of any two projeotlvltles are pro j eotlvlt les. Theorem. The set of all projeotlvltles leaving a given pencil of points Invariant form a group. Définit Ion. A projective transformation between the ele ments of two two-dimensional or two three-dimensional forms is any one-to-one reciprocal correspondence between the elements of the two forms, such that to every one—dimensional form of one corresponds a one-dlmenslonal form of the other. Definition. A collIneatlon is any (1, 1) correspondence between two two-dimensional or two three-dimensional forms In which to every element of one there corresponds an element of the same kind in the other form, and In which every one-dl mensl onal form of the other. A projective collineation is one In which th is correspondence Is projective. Unless other wise Indicated, the word collineation will denote a projective collineation. Définit ion. A perspective collineation In a plane is one leaving Invariant every point on a given line o and every line on a given point 0. The line o and point 0 are called the axis and center, respectively, of the collineation. If these are distinct, the collineation is called a planar homology; if not, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 148 a planar elation» Theorem. A perspective collineation' in a plane Is uniquely defined if the center, axis, and any two homologous points (not on the axis or center) are given, with the single restriction that the homologous points must he oollinear w ith (A, E) Theorem. Any complete q.uadran^e of a plane can be tran s formed into any complete g.uadrangle of the same or a different plane by a projective collineation whioh, if the quadrangles are in the same plane, is the'resultant of a finite number of perspeotive c ollineat 1 ons. Theorem. The section by a transversal of a g^uadrangular set of lines is a quadrangular set of points. (A, E) Corollary. A set of oollinear points which is projective with a quadrangular set is a quadrangular set. (A, E). Corollary. If a set of elements of a pc imitive one-dimen sional form is projective with a quadrangular set, it is itself a quadrangular set. Definition. A quadrargular set Q(1E3,124) is called a harmonic set and is denotedby H(12, 34). Theorem« The harmonic conjugate of an element with respect to two other elements of a one-dimensional primitive form is a unique element of the form. (A, E). Définit ion. A point P of a line is said to be harmonically related to three given distinct points A, of the line pro vided P is one of a sequence of points A, B, G, Hq, Eg, Eg,... Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 4 9 of the line, fin ite in numher, such that is the harmonic conjugate of one of the points A, B, C with respect to the other two, and such th at every other point % is harmonic with three of the set A, B^, G, Hi, Eg,.,., class of all points harmonically related to three distinct points A, B, C on a line is called the one—dimensional-Kjet of rationality defined by A, B, G; it is denoted by H(ABC). Lemma. If a projectivity leaves three distinct points of a line fixed, it leaves fixed every point of the linear net de fined by these points. Theorem. If A, B, C^, D are distinct points of a linear net of rationality, and A B ^ 0 ""are any three distinct points of another or the same linear net, then for any projeotivities giving ABCL'xrA^B'^G and ABCLt-A'^B'^G'"Di , we have I) ^ = Di . (A, S). This is the fundamental theorem of pro j ectivity for a net of ratio n ality on a lin e. The fhndamental theorem of projective geometry is: Theorem. If 1, 2, 3, 4 are any four elements of a one dimensional primitive form, and 1 '", e'", Z are any three elements of another or the same one-dimensional primitive form, then, for f t / f / / / any pro jeo tiv ities giving 1234;^ 1 2 3 4 and 1234 7 tl 2 3 4,, we have 4 / = 4% . In order to prove th is theorem it is necessary to make the assumption of projectivity : Assumption P. If a projectiv ity leaves one of each of three Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 distinct points of a line invariant, it leaves every point of the line invariant. This assumption plays the mme part in proving the funda mental theorem that the above lemma does in proving the theorem fo r nets of ratio n a lity . The plane and space duals of this assumption are immediate consequences of the assumption, so that the principle of duality is still valid. Any space in which is valid is called a properly projective sgase. Theorem. A net of rationality in space is a properly pro jective space. The theorem of Pappus is proved, making use of assumption ^ (necessarily). Its configuration, in accordance with the con vention already adopted, is 9 3 3 9 Theorem. A necessary and sufficient condition for the pro jectiv ity on a line MAS ;;ç-MA^B ^ ( II / S) is Q(MAB, EB^a'). (a, ^ P) Corollary. A necessary and sufficient condition for the / / projectivity on a line MMAB *A"MÆA 3 (li = M) is Q,(MAB, MB^a '). (A, E, P) Such a projectivity is called a parabolic projectivity. Définit ion. If a correspondence is repeated ^ times (written A^), it is said to be of period _n when n is the s m a lle s t Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 6 1 positive integer for whioti the relation = 1 is satisfied. Definition. If a projectivity in a one-dimensional fo im is of period two, it is called an involution. Any pair of homologous points of an involution is called a conjugate pair or pair of conjugates. Theorem. If for a single point _A of a line which is not a double point of a projectivity f f on the line we have the rela- tions TT (a) = Af andJT(-6^ ) = A, the projectivity is an involution (A, E, P) Theorem. A necessary and sufficient condition thË: three pairs of points A, ; B, C, he conjugate pairs of an involution is Q,(ABC, A^:^ C^} E, P) Theorem. If (A) and (B) are any Theorem. If (l) and (m) are two projective pencils of points ar^ two projective pencils of in the same plane on distinct lines in the same plane on lines li, Ig, there exists a line distinct points S%, Sg, there 1 such that if Ai,B]_ and Ag.Bg exists a point S such that if are any two pairs of homologous a^, h^ and ag, hg are any two points of the two pencils, the pairs of homologous lines of lines A^Bg and Ig®! i&tersect on the two pencils , the points 1(A, E, P) a^hg and agh-| are oollinear with 3. (A, E, P) Definition. The line is called Definiti on. The point S is ' the axis of homology of the two called the center of homology pencils of points. of the pencils of lines. In outlining the logical structure of a work of this type, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 5 2 many of the theorems and corollaries may be disregarded, as coming so d irectly from others that they may be introduced whenever necessary without discussion. The definitions, however, are very valuable as furnishing a key to the work, and we are quot ing most of them here. Where we have an eye to logical /rigor, a complete statement of the assumptions is indespensible. Conic sections are introduced with Steiner» s definition. Upon an investigation of the possibility of constructing a conic of which five points are given, it is found that the hex agon containing these and the required sixth point has the prop erty stated in Pascal’s theorem. The■degenerate ease where one side is a tangent is proved. (It is not proved as a degenerate case, however.) The polar system of a conic is developed from the défini—, tion of a polar of a point P Ils the line on which tangents to the conic at all the pairs of points oollinear with P will inter sect by pairs. Definition. The pairing of points and lines of a plane brought about by associating with every point its polar and with every line its pole with respect to a given conic in the plane is called a polar system. The algebra of points is an interesting development, being a refinement of Yon Staudt» s original algebra of throws. Définit ion. In any plane through a line 1, on which there are three distinct points Pg P]_, P^ let 1 and /p^be any two Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 5 3 lines throogh and let ^ be any line through meeting l^and in two points __A and k \ le t and P^. he any two points of 1 , and let the lines P^^A' and PyA^ meet and l^a in the points ^ a n d respectively. The point P^-^y in^the line ZY meets 1, is called the of the points P^j- and (P^+Py=P%+y). The operation of obtaining such a sum is called addition. Theorem. If P^ and Py are distin ct from P^ and P^Pg, ^PyPx+y) is a necessary and sufficient condition for the equality Px+Py=P^+yj. E ). This comes from the fact that AXA^Y is a quadrangle determining the above quadrangular set. In corollaries it is shown that P%:+Po=Po+2z=2%, and P^;+:^= P^+Px=^(P 2 ;’^PeJ , and that the operation of addition is one - valued for every pair of points except the pair P^., and associative fo r any three points fo r which the above expressions are defined; that is ; Px f (Py + Pz) = (Px + Py) + Pz. Also, the operation is commentâtive; i.g ., Px+Py = Py+Px. D efinition. In any plane through 1 le t 1^, 1^, 1-obe any three lines through Pq , P%, Eo, respectively, and le t J_i meet 1q and 1«* in points A and B respectively.' Let P^, Py be any two points of and let the lines PxA and PyB meet E* and 1 q in the points Z and Y respectively. The point Pxy in which the line ZY meets _1 is called the product of Px by Py (P^'Py=Pxy ). The orem. If P^ and Py are any two points of i distinct from Pq , P^, ^(P qPx PI. %»PyP%y) is necessary and sufficient Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 5 4 for the equality Px*% = P%y. In corollaries it is shown; Pl^Px = P%'Pi = Po-I’x = fx'Po = Po; P P% = Px'P : P ; (P%.Py).P2 = Px«(Py*P*)‘ In a theorem it is shown: P£*{Px+^y) = pf'P% + Pg'Py, e t c . . . . . All of these theorems are proved hy use of assumptions ^ and S. only. However, to prove that the operation of multiplica tion is commutative, i .e ., Px’^y = Py-P^; we must use assumption _P, for the proof rests upon the fact that in the projectivity ^PpPxPzyT^ ^^jCpoPxPyx ^yx- So we have Theorem. Assumption P is necessary and sufficient fo r the commutative law of m ultiplication. In a further discussion of this question, a number system exactly analogous to the ordinary number system is shown to exist, with operations exactly analogous to corresponding opera tions in the ordinary number system. A field of numbers in one- to-one correspondence with the field of points defined by the four rational operations in our purely geometrical system is introduced, and thus we have coordinates of points on a line. (Also of lines in a pencil, etc.) By considering two lines in a plane, we can obtain a system O-V of coordinates of points in a plane. (The lines have Pq , a»d 0, as intersection-points.) In both of these cases we have the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 5 5 objectionable elements at infinity. These elements are objec tionable beoanse, in order that the set of points (coordinates) on a line, for instance, form a group (of correspondences. See above) with respect to both addition and multiplication, must be exiuded from the set. (There is also no inverse operation with respect to multiplication for Pq ). There is a theorem which arises from the considerations above, but not directly so. It w ill have to be supposed true: Theorem. The set of all points on a line on which a scale has been established, and from which the point Pyg,is excluded, forms a field with respect to the operations of addition and mult^ication previously defined. (A, S, P) (A fie ld is a system of numbers forming commutative groups with respect to both addition and multiplication.) If homogeneous coordinates are introduced, (much as at the beginning of the work), these element at infinity can be elim inated, as far as uniqueness is concerned. Passing over a considerable number of necessary intermediate considerations, we have the Theorem. The necessary and sufficient condition that a point (xp, Xg, Xg) in a plane bé on a line (up, ug, Ug) is that uixp + ugXg + UgXg = 0. Theorem. The equation of the line joining the points (ap, ag, ag) and (bp, bg, bg) is %1 zg Xg ' ai ag ag = o bi bg bg Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 5 6 The condition that three points he oollinear is ai ag ag ^ 1 bg = 0 ®1 Og Any projectivity in a one-dimensional primitive form in the plane is given hy a relatio n of the form V _ o'- X +/3 A projectivity between two such forms in the plane is ob tained hy establishing a projective relation =«iA_t£ (-S_/3ar/o) erx +5 between the parameters^ , \ of the two forms. P ^position Kg. If any finite number of involutions are given in a space S satisfying assumptions A, E, p, there exists a space^ of which ^ is a subspace, such that all the given in volutions have double points in S Without proving this, for the time being, if the elements of / ^ are called proper and those of _S but not of ^ are called im proper, Theorem. A proper one-dimensional projectivity without proper double elements may always be regarded in an extended space as having two improper double elements. (H, E, P, Hq , Kg) Assumption ^ , i t w ill be remembered, is that the diagonal points of a complete q^uadrangle are non-collinear. By the in troduction of Kg it is shown that points exist satisfying aiiXi^+aagXg^+aggXg^+Ba^^gXiXg+SapsziXg +2 aggXgXg = 0 , the equation of a conic. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 5 7 It is shown that this expression always represents a conic. An expression of the form, a^xi + agxg is called a linear binary form. The eg_natlon 8% = &ixi + agxg = 0 defines a unique element of a one-dimensional form, whose homogeneous coordinates are (x%; xg) = (agai). The condition that two such elements shall coincide is that /a i & 2 1 A = (b^bg) = 0 Suppose the two elements subjected to any projective transformation"^ ; = bi^ =ocbq_ +3rbg bg^ =/3 b i = f bg But i f A = 0, then A^= 0 a i ag - 0 ^ 1 ^ 2 &1& 2 4P or bibg r s l A is called an next the polar system is developed in much the same way as Ton Staudt developed it. A projective correspondence be tween the elements of a plane of points and the elements of a plane of lines is called a correlation. A correlation in which the points and lines correspond doubly is a polari^. Two homologous elements of a polarity are called pn^ and polar. If two points are so situated that one is on the polar Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 5 8 of the other, thqjr are said to he conjugate. If a point is on its polar, it is self-oonjngate. Theorem. The self-conjngate points of a polarity are on a conio. Theorem. Eveiy polarity is the polar system of a conic, the conic determined by its self-conjngate points. Every pole and polar pair are pole and polar with respect to this conic. The regnlns, the l inear congmence, and the Ij^e^r .cpm- plex are next discussed. The .treatment of these figures de pends upon the Definition. If two lines are coplanar, the lines of the flat pencil containing them are said to be linearly dependent on them. If two lines are skew, only the lines themselves are linear dependent on themselves. If three lines are skew, a ll lines on a ll three are linearly dependent on them. If Ip, Ig Ijj are any number of lines and mp, mg, . . . , m^ such that mpls linearly dependent on two or three of 1 ,, Igi.'-la, and mg is linearly dependent on two or three of Ip, lg,...,lQ , mp, and so on, being linearly dependent on two or three of Ip, lg,...,ljj, %, mg,..., i%_p, then m^ is said to be linearly dependent on Ip, lg,,...l% . Thus a re^gulus is the set of a l l lines lin early dependent on three linearly independent lines, a co^ruefioe is the set of all lines linearly dependent on four linearly independent lines, and the complex, all lines dependent on five such lines. The Pluoker line coordinates are introduced. Two points Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 5 9 whose homogeneOTia coordinates are (x]_, Xg, Zg, x^), (y^, Yg, Yg, y^) determine a line 1. These coordinates determine six numbers. XiXg = 1 = 2 j = 1 = 4 = ^ 1 3 = ^ 1 4 = 71JZI ; 7 1 7 3 7 1 7 4 X4 X2 = 3 ^ 4 1 = 2 = 3 ^ 3 4 = ^ 4 2 “ ^ 2 3 " 7 3 7 4 1 7 7 4 7 2 ; 7 2 7 3 Pig, P^g, etc.,^called the Plncker coordinates of the line. It is shown that fonr independent coordinates determine a line. In exactly the same way, plane coordinates are fonnd. A chapter on "Ponndationa" is next, A recapitulation of a ll the assumptions is made, and new ones added in different ways. In the first place, assumptions A and E_ are common to all the sets. The space defined by assurait ions A and E alone is called the general projective space. All the other spaces are contained in this space. The space defined by A, E, P is a proper projective space. (o& definition above) Assumption E. If any harmonic sequence ex ists, not every one contains only a finite number of points. The space defined by A, E, H, is a non-modular projective sp ace. Assumption E. If any harmonic sequence exists, at least one contains only a finite number of points. The space defined by A, E, E,is a modular projective space. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 6 0 Asanmptlon S. SJ., For any three distinat oollinear points A, B, C there is a sense S(ABC). S^. For any three d istin ct oollinear points there is not more than one sense S(lBC). S3. S(ABC) = S(BCA) - AcB S(ABC) 9^ S{éSG) 85. If 3 (ABC) = 3 (4 / 3 ' C^) and S(A*b"^c/) = S{aV''c^), then S(ABC) = S(A^'bV''}. 36. If S(ABO) = S(BCG), then S(ABO) = S(ACO)- If OA and OB are d istin ct lin es, and S(OAA ) = 3(OAAg) and OAA^Ag^ OBB]_Bg, then S(OBB]_) = S(0BB2) The space defined hy 2, ^ is an ordered projeoti^ Assumption Q,. There is not more than one net of rationality on a lin e . The space defined hy A, 2, E, Q, is a rational modular projective space, and the'apace defined hy A, 2, E, Q, is a ratlonal^non-modular projective space. Definition. Two subsets, (A) and (B), of a net of ration ality R{EqE-j_;^) constitute a cut (A, B) with respect to the scale E q , Ep, Bj^^oif and only if they satisfy the following conditions; (1) Sveiy point of the net except E ^ is in (A) or (B); (2 ) with respect to the scale Eq , E-, , every point of (A) precedes every point of (B). If there is a point _0^ in (A) or Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 6 1 in (b ) snoh. that every point of (A) distinct from 0 precedes i t and every point of (B) d istin ct from 0 follows it, the cut la said to he closed and to have as its cut-point ; otherwise the out is said to he open. The class (A) is said to he the lower side and (B) the upperside of the cut. Définit ion. With respect to the scale Hq , E]_, an open cut precedes all the points of its upper side and is preceded by all the points of its lower side. A closed cut precedes all the points which its cut point precedes and is preceded hy all points hy which its cut—point is preceded. A cut (A, B) precedes a cut (C, D) if and only if there is a point _B preceding a point C. Définit ion. With respect to the scale H^, E^, a out (A^, Ag) is said to he between two cuts (B^, Bg) and {G^, Cg) in ease (Bp, Bg) precedes (A]_, Ag) and (Ap, Ag) precedes (Cl, Cg) or in case (Cp, Cg) precedes (A^, Ag) and (Ap, Ag) precedes (Bp, Bg). Assumption of continuity. If every net of ratio n ality contains an infinity of points, then on one line 1 in one net El^EgEp:^) there is associated with every open cut (A, B ) with respect to the scale Eq, Ep, a point P(A, B) which is on 1 and such that the following conditions are satisfied: (1) If two open cuts (A, 3) and (C, D) are d istin ct, tie points P(a , B) and P(C, D) are d istin c t; (2) If (Ap, Ag) and (Bp, Bg) are any two cuts and (Cp, Cg) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 6 2 any open cat "between two points ^ and B of R(HoHiI 4 t»), and i f is a projectivity such that T(lioAB) = %o^(Ai,Ag)^(Bl,Bg), then ^(^(Ci.Cg) Is a point associated with some cut (Di,Dg) be tween (A]_,Ag) and (B]_,Bg). Définit ion. The set of a ll points R(HoH]_H«,), together W.th all points associated with cats in R(HoHiB^), with respect to the scale Ho,Hi,Rae, is called the ^haj:^ G(EoHiRbo). The points of R(HoH3_B^) are called ra tio n a l, and any other point of the chain is called Irrational with respect to R{EoE%Bg^). A point associated with a cut which follows Eg is called positive. and one associated with a cut which precedes Eg is called negative. Assumption R. On at least one lin e, if there is one there is not more than one chain. It is proved as a theorem that the principle of duality is valid for all theorems deducible from A, E, H, C, R. The space defined by these assumptions is called the real_projec- It is proved that these assumptions are consistent if the real number system exists, and in a manner equivalent to that above, at the beginning of the present chapter, it is shown that these constitute a categorical set. It is also demon strated that all are independent. If we substitute for R Assumption R. On some lin e 1, not a ll points belong to the same chain. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 3 1 6 $ and Aas-gmptlon I . Thi^-ugh a point P of any chain C of the line 1, and any point J on 1 but not in C, there is not more than one chain of 1 which is on no other point than P in common with G. It is proved as'a theorem that "the geometric number system in any space satisfying Assumptions A, S, H, C, R, I, is isomorphic with the complex number system of analysis." In this we obtain, by purely projective means, not only the net of ratio n ality and the chain, but a complete, continuous geometric number system, isomoiphie either with the real or complex number systems of analysis. In summary, the distinguishing features of this work are, first, the postulational method, together with complete^ care ful definition, insuring perfect logical rigour, and second, the brilliant reproduction of the number system by projective methods. It is vastly superior to any of the works preceding it, from any possible viewpoint. Its unity is greater, its range and volume of subject matter covers the entire field, and it is completely satisfying from the purely logical view point. 3ven its elegance of treatment is greater. A good example of this is the proof the principle of duality as a theorem. In the works of Steiner, Yon Staudt,and Reye, as well as of many other mathematicians, it was the custom to print a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 6 ^ theorem and its dnaL side by side. This was done away with in a single simple stroke hy Yehlen and Young. This superiority was made possible by essentially only one thing — the postulational nethod employed. This method centers the attention continually upon the foundations, which exhibits clearly the similarities and differences of the different branches, and whether or not they are compatible. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 165 CHAPTER Y 1 In reviewing the history of this sabjsot it is sur prising with what small ana erratic steps even the great est of minds go forward over territory which, once trav ersed, is seen to be c[aite firm. In geometry, as we have seen, there has been no single innovation which was far removed from what went before. Appearances are often to the contrary, bat this is due to the fact that in many instances the innovation consisted in seizing upon what was quite inconspicuous and incidental be fore and putting it to its full use. It is also due to the fact that a new idea which does not seem to amount to very imich in itself becomes, upon general application, the means of revealing scores of new relations, and even of massive underlying generalities. A very small advance may thus alter the entire aspect of the subject. With these things in mind, let us make a brief review of the entire subject to show specifically how the develop ment is a gradual, connected one, even though it is not smooth and continuous, and how every important idea has a considerable historical background. In the original conception of conic sections as plane sections of right circular cones lies, beyond doubt, the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 166 historical origin of the projective method.. This concep tion was due to Plato, and was thus quite early in the prog ress of Greek geometry, from which projective geometry is now sharply differentiated. Considering the point of view from which the Greeks worked, it would have been remarkable if they had discovered these curves in any other way, and almost as remarkable if they had never discovered them, or even if they had never la id emphasis on them. The theory of conics was developed rather thoroughly by the Greeks who came after Plato, among whom Euclid was impor tant in this respect. If finally reached its culmination, as far as the Greeks were concerned, with the work of Apol lonius, who, by the force of his tremendous genius, evolved the ideas of poles and polars, cross ratio, and many others. It is unfortunate that we do not know more about the development of the theory of conics among the Greeks. It is likely that we would find a more or less gradual transition from the first ideas of the Platonian school to the quite highly developed theory of Apollonius. There is some disa greement upon this subject, however. Much later Pappus, in his "Mathematical Collections",^ proved the invariance of the anharmonio ratio under projec tion and the harmonic property of the complete quadrilateral. These two theorems, v/hich were of tremendous importance in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 167 the development of projective geometry were, it appears, proved as aids to the reading of the works of Euclid and others. The proofs were perfectly straightforward Euclid ean geometry. There is no reason to believe that Pappus un derstood the role which these theorems were to play much l a te r . We are probably indebted to astronomy for the mathemat ical concept of infinity, for Kepler was first of all an astronomer, and as such he may well have noticed that ellip tical orbits which were very gareJk resembled parabolas in the visible portions at the vertices of the ellipse. To Desargues we owe the explicit formulation of the con ception of infinity. More important still, Desargues was the first to use projection in the proof of a theorem. This use was not explicit, of course, but it was actual. In Desargues work we see the work of Apollonius and Pappus combined to form what closely resembles the projective geometry of the time of Poncelet. The theory of perspective grew out of the requirements of artists, notably pairters and stone-cutters. It led to the work of Taylor and Lambert, and finally assisted in directing attention toward the method of projection, and, through Lambert, to non-metric properties and methods. When one reflects on the nature of perspective geome try, and takes into account that some of the greatest math- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 6 8 ematiciana had lent their aid to its development, and fur ther that some of the important theorems of pure geometry which were discovered by the European geometers partook of this nature, it is not remarkable that Poncelet began a thoroughgoing investigation of the protective properties of figures, and considering the rich store of materials which lay ready at hand, and his own ability, it is far from sur prising that he met with such fortunate results* Again, in Poncelet*s own work, the foundations for the theory of reciprocal polars had been laid by Pesargues, the foundations of the theory of transversals by Mclaurin, and even the theory of homology and the famous theorem on the variable polygon whose free vertex describes a conic were original only in their generality. Poncelet»s one supreme con tribution was the systematic use of the method of projection and sectio n , a method which in i t s e l f was not at a l l new. ?rom Poncelet *3 theorem on the variable polygon Steiner drew his definition of a conic in terms of projective flat pencils. From the theory of reciprocal polars Von Staudt drew his definition of a conic and built up his theories of collinearities, polarities, etc., and it was due to this that he conceived a non-metric geometrical algebra. So it was for all the other new ideas. All arose quite naturally and very gradually. There is not one which does not have its own particular history. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 169 In tracing the history of the idea which led to the re organization presented by Teblen and Yonng it is necessary to go outside the field of projective geometry proper. All of the great geometers of history had a working familiarity with logic, but it was not until very recently that there were clear-cut ideas about logic, that is, that the nature of the logical structure of geometry was under stood. Perhaps the first step in this direction was lobachevjiky*s work "The Theory of Parallels", first published in 1829. This was the f i r s t instance of an independence proof. Xobachevjiky showed that if in Euclid’s axioms the axiom of parallelity is assumed to be false, the resulting set of axioms is consis te n t. This work led the way to a mass of speculations and con fusion, proceeding mainly from the fact that the essentially formal nature of logic was nowehre understood, and the founda tions of mathematics were obscured by intuitive entanglements. Those whose names are associated with the investigations of non-Euclidean space are notably Riemann, Helmholtz, Cayley, Klein and Poincare. An excellent account of the work of these men is contained in "The Foundations of G-eometry", by Bertrand Russell, published in 1897. In the w inter of 1898—1899 David H ilb ert delivered a series of lectures at the University of Gottingen, on the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 7 0 foundation of geometry. These were published in an English translation in 1902. Here there appeared a new conception of logical rigor, with formal definitions, undefined elements and relations, and, in short, the whole basis for the new organization of geometry. From this we see that there was a continuous change on the frontier of mathematical thou^t, resulting in the new conceptions of the foundations of geometry. It is curious, in looking over the development of this subject, to notice how a very simple idea W ll remaingover looked for years. The idea of duality is a simple and oon- seq.uently striking example of this. As we have seen, the principle follows immediately from the fact that the logical foundation, and therefore the whole logical framework, can be duplicated by another in which we simply interchange point and plane, or point and line, etc., and that this logical framework is consistent with the first because it can be de duced from it, since its foundations can be deduced from the original logical structure. This very simple idea is thus a direct consequence of the logical foundations of geometry. This could have been demonstrated with ease to any of those who, like Poncelet, Steiner, etc., evidently had such a hazy idea of duality. They considered it necessary to the rigor of their work that they prove or indicate the proof not only Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 7 1 of their original theorems but of the duals of these. They had no idea that the principle of dm lity would not hold, but, not having proved it, it would have been undue logical laxneas to regard it as being true, at least without making it one of the assumptions upon which they built. This al ternative, of course, was not desirable, although it might have revealed to them just how the'principle of duality was necessarily true if their other assumptions were true. For nearly a century the workers in projective geom etry proved the dual theorems side by side without realizing that they were enabled to do this because the logical struc ture with which they were working was designed from the very foundations so that if certain elements were inter changed throughout there would come into being a second structure, as consistent as the first, and consistent with it, exactly similar to it except for this change —that is, if the logical structure is considered to comprise both of these subsidiary ones, it is symmetrical with respect to certain elem ents. Thus, at a considerable material cost in the publica tion of their works, mathematicians accepted the principle of duality as actually true, but would not rest any logical weight upon it, since they were not satisfied that it was necessarily true. Surely it was a very simple step to show that it was necessarily true, but the step was not taken Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 7 2 fo r more than seventy—fiv e y ears. We could enumerate many other examples similar to this. If there is any conclusion to he drawn from these phenomena, it is that our intellectual progress is, considering the simplicity of the steps we take, quite slow and uncertain, and its direction is largely a matter of accident, depending, for example, upon which of two possible new ideas w ill be discovered first, which in turn depends on a number of things, such as^the proclivities and attainments of some very able man. As surely as we advance in one direction, there will be some other direction which w ill be neglected longer than would otherwise be the case. Further, it is not always the simplest steps that are taken first. The Greeks are an ex cellent example in point. It was not because of inferiority that the Greeks did not develop analytic geometry or algebra, and it was not because their span of civilization was too short. It was not because analytic geometry is not as simple as classical geometry, for the Greeks rose to tremendous heights in their own particular field. It was simply that they happened to be so occupied with one method that the other did not occur to them. So it has been in protective geometry, with the exception that as we have become wiser we are more likely to conduct a systematic search for new ideas and methods, and to search Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 7 3 for the underlyijog principles that give unity to our appar ently heterogeneous mass of knowledge. II There is an aspect of this subject which has consider able philosophical interest, namely the logical one. A great deal has been learned about the logical foundations of mathematics since the confused discussions of the last cen tu ry . A remarkable thing about the logical structure is its essential simplicity. We have undefined elements and rela tions, and assumptions involving these. The assumptions, if consistent, determine an entire logical structure which can be built up from them. True, we may determine a system whose possibilities^ it would be very difficult to exhaust, but, nevertheless, the system is determined by the assumptions, much as two points determine a line, or three points a plane. The most common error of the 19th century was to pro ceed with the assistance of intuitive concepts. Intuitive concepts associated with the undefined elements and relations are in themselves harmless, but they gave rise to illogicali ties, in that definitions were partly formal and partly in tuitive, and the assumptions were often not worded uniformly, so that, in strict logical procedure, they could never be con fined to form theorems. The deductive logical structure, to which all deductive Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 7 4 prooesses rmiat conform. If they are to be rigorous, emerged from all this confusion as a form. The function of deduction is to say if A is true, then B follows. It does not matter whether A is true or not. The statement holds in any case. Further, if we have set up a deductive system in terms of certain symhols, and if we substitute other symbols (words, etc.) for these, the structure itself is not vitiated, provide^ we substitute new symbols corresponding to old ones. This must surely be obvious# A simple ezample would be a geometric system and its dual system. Another would be a geometry and its translation into some other language. In general, if we have two logical systems which are identical formally, except for a difference of words, sym bols, etc., and if these words and symbols carry special in tuitive interpretations of their own, then by an excharge of nomenclature between the two systems, we w ill effectively exchange intuitive interpretations, thus demonstrating that any change of interpretation of the symbols of a -logical system has no effect upon the logical structure Itself, This structure, if not contentless, is at any rate adaptable to any class of things (words, symbols, etc., accompanied or not accompanied by intuitionà) interpretations) which sat isfies the assumptions upon which the structure is built. The net result, then, is that if we develop a logical system (a geometry, for example) all of the propositions, corollaries, etc., which we have proved in this system will also be true if the propositions are stated with new elements, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 7 5 relations, etc., so that the new propositions are analogous to the old in a definite way, provided that the assumptions which are analogous in this way to the old assumptions are also true, (that is, acceptable). Whether we call these two systems identical, hut with different applications, or whether we call them different hut analogous (in one to one correspondence), is a question which calls for attention. Tehlen and Young commit them selves d e fin ite ly . "We understand the term a mathematical science tp. m e^.any set of propositions arranged according to a sequence of logical deduction ••• such a science is purely abstract. If any concrete system of things may he regarded as satisfying the fundamental assumptions, this system is a concrete applica tion or representation of the abstract science." Keyser also commits himself repeatedly in favor of the first concep tion. In other words, mathematicians (at least some math ematicians) are inclined to favor the abstract view, in which that which is common to two analogous logical structures is considered to be the actual structure. In contrast to this many philosophers favor the second conception, maintaining that we think in images and that therefore an abstract logical system is simply a fiction. An analogy which seems fair in this connection is one with physical science. Is a physical law concrete or abstract? The answer is, unhesitatingly, abstract. True, the law de- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 176 pends for its existence upon certain behavior of concrete things, or, from another view-point, manifests itself in such behavior and such behavior only. The law is essen tially a pattern, which is surely an abstract thing. In the same way, a logical structure can be regarded as a pattern. Such a pattern is common to two concrete ex amples of Itself in the same way that a physical law is com mon to two concrete manifestations of itself. Turning to the psychological aspect of the problem, is thought contentless? If abstract symbols, although content- less, are actually content, then thought is not contentless. We can say that thought, whether contentless or not, is es sentially patternistic, for it follows a pattern regardless of the content or symbols used. Whether thought this pattern or merely follows it is the question under considér ât ion. Ratiocination is possible in terms of the most abstract symbols, as can easily be verified. Ihthematically speaking, we then have an abstract logical■structure applied to a set of symbols, and ordinarily we consider this application as abstract if the symbols are abstract. From this point of view 7 /e can say that thought is abstract - i.e., contentless, and th ere seems to be no p ra c tic a l objection to taking th is point of view. A psychological investigation of deductive Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 177 logic would be an interesting and very fundamental study, if it bore fruit, especially in this connection. It is interesting to notice here that deductive {and inductive) mental processes are by no means p ecu liar to man. lower animals use deductive methods quite constantly. These mental operations are of course confined to simple oases,^are probably for the most part, if not quite, uncon scious. Insofar as an animal is not acting purely instinct ively and mechanically, it classifie^, and recognizes in stantly that an entity belonging to a given class has the properties of that class. To go to the mo re formal aspect of deductive logic, there are certain important things involved at least in operational logic. The most important are the assumptions, which have already been given considerable attention. The fundamental unit of deductive logic is the syllogism. This can be put into several forms, all of which are self-evident. The usual form is the following; If A implies B and B implies C then A implies C. Another, which is even more obvious, is this; If A belongs to a class B and class 3 belongs to class C then A belongs to class 0. In the first case the undefined relation is "implies". In the second it is "belongs to". In the first A, B and C are undefined symbols, and in the second. A, class B, class G. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 178 How If we consider a restriction to be the assignment of any element or elements to a given class (and it is easily seen that this is in the nature of a restriction) then a definition is merely the aggregate of restrictions upon any element or elements, denoted by some convenient word or sym bol. .Hot all things with which we deal can be defined, of course, for the'f'e will always be at least one class not comr- pletely contained in any other, and upon which, therefore, no restriction can be made# An assnmption or a theorem is a statement that elements belonging to c e rta in one or more classes belong to c e rta in other one or more classes. A theorem is ^merely a summation of syllogisms, by the statement of which we prove the theorem A syllogism, of course, involves certain rel&tlons which are already known. Therefore, to prove a theorem, we must assume certain relations, that is, make certain assumptions. Assum ptions^ like theorems, state that elements belonging to cer tain one or more classes belong to some other one or more c la sse s. To illustrate these statements, we can prove a theorem, making the re q u is ite assum ptions. Theorem. I f A, B; G, B belong to tv/o c lasse s AB, CD, respectively, and if the classes AB, CD belong to the same claas Z, the elements A, B, C, D belong to the same cla ss Y, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 7 9 Assumption. If A, B, C or D belongs to a class AB or CD which belongs to another c la ss Y, th en A, B, 0, or D be longs to the class Y. As sumption. If two classes AB, CD belong to the class Z they belong to the class Y. Here we have a theorem based on entirely adeq.uate as sumptions. The undefined relation Is "belongs to". The un defined (unrestricted) classes or elements are A, B, C, D, classes AB, CD, Y, Z. (A, B, C and D may be considered as classes). If we substitute the relation "Is on" for "be longs to" and Interpret A, B, C and D to be points and classes cUïS AB, CD to be lines, while class Y Is a point and^Z la a plane, we w ill have a theorem in geometry. The net result of these considerations Is that In any logical system, In which, of course, we have theorems, assump tions are necessary to the proof of the theorems, and unde- Co V*. « fined elements and rootrlotiens are necessary to the state ment of the assumptions. Definitions are merely conveniences, tying together many restrictions under one name or symbol. As we multiply assumptions, In general we restrict the appli cation of the mathematical science, In spite of the fact that we are enabled to prove more theorem s. That i s , we say "more and more about less and less." To pay somewhat more attention to definitions, in spite dlwuyy Of their utter logical insignificance they have qitwyo been Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 8 0 very important as directing forces» The existence of any sort of geometrical entity more complicated than mere point, line and plane, for instance, depends upon definitions. It would be almost impossible to operate in geometry if the geometrical forms were wiped out and the often very long series of restrictions defining them were substituted for them. Thus definitions, although, from the logical viewpoint, mere conveniences, are really the life of geometry. Deductive logic has been accused of sterility, since, it is maintained, we obtain nothing from the conclusions which was not contained in the assumptions. In a sense this is true. The immediate answer to this accusation, however, is to direct attention to modern projective geometry, which is a brilliant example of the fertility of pure deductive logic. Why is it that logic is not sterile? The answer is undoubtedly that we cannot possibly real ize, by merely glancing at a set of assumptions, or even by close examination of them, all of their implications. It is necessary to resort to detailed ratiocination to discover them in any number at all. V/hen it is realized that any two propo sitions dealing with the same elonents and relations can be combined to form a theorem it is realized what a dizzy logical network can be erected upon the foundation of only a few simple assumptions. In any geometry we deal with only a small part of this network, and only the meagre outlines of this small part. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 8 1 Further, a few assanctions can lead not only to many results hut to valuable ones. The best example of this in projective geometry is Veblen and Young's book, in which, proceeding in a rigorous way from simple assumptions, they build up the whole field of projective geometry, much of which was previously thought to belong only to %uite different forma of geometry, but all of which was conceded to be very valuable and interesting. In concluding we w ill discuss one more aspect of deduc tive logic, quite as striking as the above considerations, i f somewhat le s s c o n tro v e rsial. To put if bluntly, deductive logic is infallible. At least, it has hever yet been known to fail. If we accept certain assumptions we find ourselves compelled to accept any conclusion drawn from them in accordance with the 'laws of de ductive logic. This is an Inflexible law of human reason, and, do in as far as animals^^,reason, of animal reason. There have been many failures, in history, to arrive at the correct conclusion from valid assumptions, but invariably the rules of deductive logic have in some way been violated. A typical procedure is to tacitly and unconsciously make some additional assumption which seems to be obviously true, but . which, as a matter of fact, is not. Cf course, in many cases, the original assumptions were not all true, and perhaps not all consistent. Slips in logical operation are also made, of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1S 2 cotirse, but they are relatively unimportant, because they do not lead to the confusion and Inconsistency that arises from vague and slippery assuo^tiona. Aside from the understanding of the logical structure in general and in its application to mathematics, the most Important recent advance in geometry^ was the recognition of the necessity of focussing care fill at tention upon the axiomatic foundations of any logical system. In summary, deductive logic must be regarded as essentialls formal, patternistic, as opposed to intuitional. In a sense it may be the latter — this is a subject for further specula tion — but in any ordinary sense, and for the purpose of easy, clear-minded^ logical operation, any logical system is a pattern or form complying -to certain universal laws. It is in this way that it is, or should be, dealt with in mathe matics. 'The only field in which any other view of deductive lo g ic should receive any^serious co n sid eratio n seems to be, as was mentioned above, psychology. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I0 !T 1 With the advice of the department, specific referen ces were omitted as being impractical. It is hoped that this note will in part discharge the function of such references. In the chapter on the Greeks, Ohasles* "Aperçu" was used as a guide, with material taken from Gow's "Histo 1 7 of Greek Mathematics" and Cantor's "Geschichte". The part on Apollonius follows closely (with important om issions) Chasles* account (which is g.uot ed at great length by Gow and agrees perfectly with Cantor's later and therefore more rmml work). The theorems attributed to Apollonius, Pappus, etc., were taken almost without excep tion from Chasles. There were no figures in the original, however. In the second chapter Chasles was likewise followed. The portions on Pesargues, Pascal, and Maclaurin were par ticularly useful. The part on the theory of perspective in the nineteenth century and on the development of descriptive geometry, with a discussion of Monge, followed the treat ment of Gino Loria in the fourth volume of Cantor's history. Such alterations as were made were on the authority principally of Chasles. In the remainder of the work there were obviously no references used. The books in question were dealt with Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. d irectly . In the fourth chapter the statements headed Definition, etc., are all quoted verbatim from Veblen and Young * 8 book. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3IB1IGGRA2EY Cantor, Iloritz, Yorlesungen iiber G-esohlchte der I'athematik. 4 Y d s. Vol. I, second edition, Leipzig, 1894. Yol. II, second edition, Leipzig, 1900. Yol. Ill, second edition, Leipzig, 1901. Yol. lY, , Berlin, 1924. This work, the three firs t.volumes of which were written by Cantor alone, is the standard work on the history of math ematics. It was an excellent check throughout for statements found elsewhere. The part on Greek geometry and the articles by Gino Loria on Perspective and Descriptive Geometry were particularly valuable. Chasles, Aperçu Historique ... des Méthodes en Géométrie. Paris, 1875. This is a classic v;ork on the history of geometry. It is written from the standpoint of what was at the time of- its writing "modern geometry", thus being specially suited to the present purpose. The part on the Greeks, Apollonius in par ticular, is excellent, and the work proved unite valuable in regard to Desargues and Monge, Cremona, Luigi, Elements of Projective Geometry. Oxford, 1893. The edition used was the second edition of the English tra n s la tio n by Charles Leudesdorf, Though prim arily a te x t book, it provides a simple, readable unification of the main ideas previous to 1860. The historical notes, especially in the preface, were particularly helpful in a directing cap- ac ity . Enriques, Federigo, Yorlesungen uber Projektive Geometrie. Leipzig, 1903. The German e d itio n is by Hermann F leisch er. This work gives a beautiful presentation of projective geometry, for the most part from Yon Standt's view-point, although there is a slight metric admixture. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. G-ow, James, A Short History of Greek Mathematics. Cambridge, 1884. This hook is concerned mainly with the city of Alex andria, where most of the Greek work in Mathematics was done. It proved particularly valuable in regard to the development of the theory of conies sections among the Greeks. Eenrioi, O.M.F., in Encyclopaedia Brittaniea, eleventh edition, Cambridge, 1910, articles Projection and Geometry, Projective. These articles were valuable for the historical refer ences made, and as in d icatio n s of th e conservative view of projective geometry at that time. Other articles in the Encyclopaedia Brittaniea which proved valuable were under the headings: Alexandria — Apollonius — Euclid — Geometry (descriptive, line, non-Euclidean, axioms) — Lambert — Legendre — Logic — Monge — Pappus — Pascal — Perspective — Poncelet - Steiner, and many others. Hilbert, David, The Foundations of Geometry. Chicago, 1902. This book, a tra n s la tio n by E, J . Townsend of the lec tures delivered by Hilbert in the winter of 1898—99 at the University of GOttingen, shows clearly the starting point of the im portant modem work in geometry. Though le s s simple and clear than th e la te r work, i t shows the beginnings of an un derstanding of the nature of deductive logic. Lobaohevski, Nicholas, The Theory of Parallels. Chicago, 1914. This is a translation by Halsted of the original which was publsihed at Berlin, 1840, the original work having been done previous to 1838* This is the first example of a non- Euclidean geometry. Ponoelet, Jean Victor, Traite'des Propriétés Projectives des Figures. 2 Vols. Paris, 1865. The first edition of the first volume of this work appeared in 1822. The second volume is composed of memoirs almost all written previous to 1830. This was the starting point of projective geometry. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Poudra, M., Oeufres Desargues. 2 Yolg. Paris, 1864, This collection of the works of the "Father of Euro pean Geometry", previously thought lost, was very valuable, providing material not obtainable elsewhere. It is very curious, being written in the original old French. Reye, Theodor, Die Geometrie der Ldge, Vol. I, fourth edition, Leipzig, 1899. Vol. II, third edition, Leipzig, 1892, Vol. Ill, third edition, Leipzig, 1892. The first two volumes were first published in the yeary 1866 and 67. This work is more a te x t than a tr e a ti s e . Rusell, Bertrand A. W., The Foundations of Geometry. Cambridge, 1897. This work, combining a philosophical and a critical- historical treatment of the subject, gives an excellent picture of the philosophical struggles of the nineteenth century over the new geometries, particularly elliptic geonv- etry. It shows clearly the nineteenth-century theories of geometry, and is a brilliant attempt to outline the founda tions of geometry, with unsatisfactory results, in the light of the present understanding of the subject. Von Standt, G.K.C., Geometrie der Ldge. Burnberg, 1847. This book is the result of work which was begun in 1832. As mentioned in the text, it is a brilliant attack upon the problem of developing a non-metric geometry. It is probably the second most important work in projective geometry in the nineteenth centuiy. Jacob Steiner's Gesammette Werke, Z, Weierstrass. 2 Vols. Berlin, 1880-81. The particular part of this edition of Steiner’s works which was valuable is the "Systematische Eutwiokelunÿ der Abliangigkeit geometrischer Gesta%ten von einander", ® Vol. I, pp. 229—460. This work, first published in 1832, was Steiner's principal claim to fame in projective geometry. It is one of the three great works in projective geometry in the last century. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Jacob Steiner*a Torlesungen uber Synthetisohe Geometrie. H. Schrô'ter. 2 Tels., Leipzig, 1876. This edition of Steiner's connected work in geometry, first published in 1866, gives the work which Steiner did after the publication of his Shfttwickelung, together with the subject-matter of the Entwiokelung. It comprises congruences, nets, nets of conics, etc. , Teblen, Oswald and Young, John Wesley, Projective Geometry. 2 Vols. Vol. I, Boston, 1910, Vol. I I , Boston, 1918 (By Oswald Veblen only). The appearance of this work was undoubtedly one of the most inportant events in the history of geometry. Suffi cient has been said of it in the text, but it remains to be mentioned that the logical theory of the work is for the most part developed in the first volume and the first part of the second. The remainder is devoted principally to applications. Young, John Wesley, Projective Geometry, Chicago, 1930. This, the fourth of the Garus Mathematical Monographs, is a rather elementary presentation of the subject, with only slight interspersions of metrical considerations. Its principal value was in the historical notes, which, though few, were different in character from those found elsewhere. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. VITA Born at Chautauqua, Hew York, July 10, 1910. Attended Roosevelt school, Missoula, and grade school at Chautauqua, Hew York* Attended Chautauqua County High School, 1923-24, and Missoula County High School, 1924—26. Matriculated in University of Montana, 1926. B. A., University of Montana, 1929, in French. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.