GEOMETRY
PART I.—PURE GEOMETRY.
GEOMETRY has been divided since the time of Euclid into an “elementary” and a “higher” part. The contents and limits of the former have been fixed by Euclid’s Elements. The latter included at the time of the Greek mathematicians principally the properties of the conic sections and of a few other curves. The methods used in both were essentially the same. These began to be replaced during the 17 th century by more powerful methods, invented by Roberval, Pascal, Desargues, and others. But the impetus which higher geometry received in their works was soon arrested, in consequence of the discoveries of Descartes,—the new calculus to which these gave rise absorbing the attention of mathematicians almost exclusively, until Monge, at the end of the 18th century, re-established “pure” as distinguished from Descartes’s “coordinate” (or analytical) geometry. Since then the purely geometrical methods have been continuously extended, especially by Poncelet, Steiner, Von Staudt, and Cremona, and in England by Hirst and Henry Smith, to mention only a few of the leading names.
Whilst higher geometry thus made most rapid progress, the elementary part remained almost unaltered. It has been taught up to the present day on the basis of Euclid’s Elements, the latter being either used directly as a textbook (in England), or being replaced (in most parts of the Continent) by text-books which are essentially Euclid’s Elements rewritten, with a few additions about the mensuration of the circle, cone, cylinder, and sphere. Only within a very recent period have attempts been made to change the character of the elementary part by introducing some of the modern methods.
We shall give in this article—first, a survey of elementary geometry as contained in Euclid’s Elements, and then, in form of an independent treatise, an introduction to higher geometry, based on modern methods. In the former part we shall suppose that a copy of Euclid’s Elements is in the hands of the reader, so that we may dispense, as a rule, with giving proofs or drawing figures. We thus shall give only the contents of his propositions grouped together in such a way as to show their connexion, and often expressed in words which differ from the verbal translation in order to make their meaning clear. It will make little difference which of the many English editions of Euclid’s Elements the reader takes. Of these we may mention Simeon’s, Potts’s, and Todliunter’s.
Section I.— Elementary or Euclidian Geometry. The Axioms.
§ 1. The object of geometry is to investigate the properties of space. The first step must consist in establishing those fundamental properties from which all others follow by processes of deductive reasoning. They are laid down in the Axioms, and these ought to form such a system that nothing need be added to them in order fully to characterize space, and that nothing may be left out without making the system incomplete. They must, in fact, completely “define” space. Several such systems are conceivable. Euclid has given one, others have been put forward in recent times by Riemann (Abhandl. der königl. Gesellsch. zu Göttingen, vol. xiii.), by Helmholz (Göttinger Nachrichten, June 1868), and by Grassmann (Ausdehnungslehre von 1844). How many axioms the system ought to contain» and which system is the simplest, may be said to be still an open question. We shall consider only Euclid s system.
§ 2. The axioms are obtained from inspection of space and of solids in space,—hence from experience. The same source gives us the notions of the geometrical entities to which the axioms relate, viz., solids, surfaces, lines or curves, and points. A solid is directly given by experience; we have only to abstract all material from it in order to gain the notion of a geometrical solid. This has shape, size, position, and may be moved. Its boundary or boundaries are called surfaces. They separate one part of space from another, and are said to have no thickness. Their boundaries are curves or lines, and these have length only. Their boundaries, again, are points, which have no magnitude but only position. We thus come in three steps from solids to points which have no magnitude; in each step we lose one extension. Hence we say a solid has three dimensions, a surface two, a line one, and a point none. Space itself, of which a solid forms only a part, is also said to be of three dimensions. The same thing is intended to be expressed by saying that a solid has length, breadth, and thickness, a surface length and breadth, a line length only, and a point no extension whatsoever.
Euclid gives the essence of these statements as definitions :— [table]
If we allow motion in geometry,—and it seems impossible to avoid it,—we may generate these entities by moving a point, a line, or a surface, thus :— [table]
And we may then assume that the lines, surfaces, and solids, as defined before, can all be generated in this manner. From this generation of the entities it follows again that the boundaries—the first and last position of the moving element—of a line are points, and so on; and thus wo come back to the considerations with which we started.
Euclid points this out in his definitions,—Def. 3, I., Def. 6, I., and Def. 2, XI. He does not, however, show the connexion which these definitions have with those mentioned before. When points and lines have been defined, a statement like Def. 3, I., “The extremities of a line are points,” is a proposition which either has to be proved, and then it is a theorem, or which has to be taken for granted, in which case it is an axiom. And so with Def. 6, I., and Def. 2, XI.
§ 3. Euclid’s definitions mentioned above are attempts to describe, in a few words, notions which we have obtained by inspection of and abstraction from solids. A few more notions have to be added to these, principally those of the simplest line—the straight line, and of the simplest surface —the flat surface or plane. These notions we possess, but to define them accurately is difficult. Euclid’s Definition 4, I., “A straight line is that which lies evenly between its extreme points,” must be meaningless to any one who has not the notion of straightness in his mind. Neither does it state a property of the straight line which can be used in any further investigation. Such a property is given in Axiom 10, I. It is really this axiom, together with Postulates 2 and 3, which characterizes the straight line.
Whilst for the straight line the verbal definition and axiom are kept apart, Euclid mixes them up in the case of the plane. Here the Definition 7, L, includes an axiom. It defines a plane as à surface which has the property that every straight line which joins any two points in it lies altogether in the surface. But if we take a straight line and a point in such a surface, and draw all straight lines which join the latter to all points in the first line, the surface will be fully determined. This construction is therefore sufficient as a definition. That every other straight line which joins any two points in this surface lies altogether in it is a further property, and to assume it gives another axiom.
Thus a number of Euclid’s axioms are hidden among his first definitions. A still greater confusion exists in the present editions of Euclid between the postulates and axioms so-called, but this is due to later editors and not to Euclid himself. The latter had the last three axioms put together with the postulates (alτημaτfi), so that these were meant to include all assumptions relating to space. The remaining assumptions which relate to magnitudes in general, viz., the first eight “axioms” in modern editions, were called “common notions” (κoιvai twoιαι). Of the latter a few may be said to be definitions. Thus the eighth might be taken as a definition of “equal,” and the seventh of halves. If we wish to collect the axioms used in Euclid’s Elements, we have therefore to take the three postulates, the last three axioms as generally given, a few axioms hidden in the definitions, and an axiom used by Euclid in the proof of Prop. 4 and on a few other occasions, viz., that figures may be moved in space without change of shape or size.
We shall not enter into the investigation how far the assumptions which would be included in such a list are sufficient, and how far they are necessary. It may be sufficient here to state that from the beginning of a geometrical science to the present century attempts without end have been made to prove the last of Euclid’s axioms, that only at the beginning of the present century the futility of this attempt was shown, and that only within the last twenty years the true nature of the connexion between the axioms has become known through the researches of Riemann and Helmholz, although Grassmann had published already, in 1844, his classical but long-neglected A usdehnungsleh re.
§ 4. The assumptions actually made by Euclid may be stated as follows :—
1. Straight lines exist which have the property that any one of them may be produced both ways without limit, that through any two points in space such a line may be drawn, and that any two of them coincide throughout their indefinite extensions as soon as two points in the one coincide with two points in the other. (This gives the contents of Def. 4, part of Def. 35, the first two Postulates, and Axiom 10.)
2. Plane surfaces or planes exist having the property laid down in Def. 7, that every straight line joining any two points in such a surface lies altogether in it.
3. Right angles, as defined in Def. 10, are possible, and all right angles are equal; that is to say, wherever in space we take a plane, and wherever in that plane we construct a right angle, all angles thus constructed will be equal, so that any one of them may be made to coincide with any other. (Axiom 11.)
4. The 12th Axiom of Euclid. This we shall not state now, but only introduce it when we cannot proceed any further without it.
5. Figures may be . freely moved in space without change of shape or size. This is assumed by Euclid, but not stated as an axiom.
6. In any plane a circle may be described, having any point in that plane as centre, and its distance from any other point in that plane as radius. (Postulate 3.)
The definitions which have not been mentioned are all “nominal definitions,” that is to say, they fix a name for a thing described. Many of them overdetermine a figure. (Compare notes to definitions in Simson’s or Todhunter’s edition.)
§ 5. Euclid’s Elements are contained in thirteen books. Of these the first four and the sixth are devoted to “plane geometry,” as the investigation of figures in a plane is generally called. Tlιe õth book contains the theory of proportion which is used in Book VI. The 7th, 8th, and 9th books are purely arithmetical, whilst the 10th contains a most ingenious treatment of geometrical irrational quantities. These four books will be excluded from our survey. The remaining three books relate to figures in space, or, as it is generally called, to “solid geometry.” The 7th, 8th, 9th, 10th, l3th, and part of the 11th and 12th books are now generally omitted from the school editions of the Elements. In the first four and in the 6th book it is to be understood that all figures are drawn in a plane.
Book I. of Euclid’s “Elements.”
§ 6. According to the third postulate it is possible to draw in any plane a circle which has its centre at any given point, and its radius equal to the distance of this point from any other point given in the plane. This makes it possible (Prop. 1) to construct on a given line AB an equilateral triangle, by drawing first a circle with A as centre and AB as radius, and then a circle with B as centre and BA as radius. The point where these circles intersect—that they intersect Euclid quietly assumes—is the vertex of the required triangle. Euclid does not suppose, however, that a circle may be drawn which has its radius equal to the distance between any two points unless one of the points be the centre. This implies also that we are not supposed to be able to make any straight line equal to any other straight line, or to carry a distance about in space. Euclid therefore next solves the problem: It is required along a given straight line from a point in it to set off a distance equal to the length of another straight line given anywhere in the plane. This is done in two steps. It is shown in Prop. 2 how a straight line may be drawn from a given point equal in length to another given straight line not drawn from that point. And then the problem itself is solved in Prop. 3, by drawing first through the given - point some straight line of the required length, and then about the same point as centre a circle having this length as radius. This circle will cut off from the given straight line a length equal to the required one. Now-a-days, instead of going through this long process, we take a pair of compasses and set off the given length by its aid. This assumes that we may move a length about without changing it. But Euclid has not assumed it, and this proceeding would be fully justified by his desire not to take for granted more than was necessary, if he were not obliged at his very next step actually to make this assumption, though without stating it.
§ 7. We now come (in Prop. 4) to the first theorem. It is the fundamental theorem of Euclid’s whole system, there being only a very few propositions (like Props. 13, 14,15, I.), except those in the 5th book and the first half of the 11th, which do not depend upon it. It is stated very accurately, though somewhat clumsily, as follows :—
If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angles contained by those sides equal to one another, they shall also have their bases or third sides equal; and the two triangles shall be equal; and their other angles shall be equal, each to each, namely, those to which the equal sides are opposite.
That is to say, the triangles are “identically” equal, and one may be considered as a copy of the other. The proof is very simple. The first triangle is taken up and placed on the second, so that the parts of the triangles which are known to be equal fall upon each other. It is then easily seen that also the remaining parts of one coincide with those of the other, and that they are therefore equal. This process of applying one figure to another Euclid scarcely uses again, though many proofs would be simplified by doing so. The process introduces motion into geometry, and includes, as already stated, the axiom that figures may be moved without change of shape or size.
If the last proposition be applied to an isosceles triangle, which has two sides equal, we obtain the theorem (Prop. 5), if two sides of a triangle are equal, then the angles opposite these sides are equal.
Euclid’s proof is somewhat complicated, and a stumbling-block to many schoolboys. The proof becomes much simpler if we consider the isosceles triangle ABC (AB=AC) twice over, once as a triangle BAG, and once as a triangle CAB; and now remember that AB, AC in the first are equal respectively to AC, AB in the second, and the angles included by these sides are equal. Hence the triangles are equal, and the angles in the one are equal to those in the other, viz., those which are opposite equal sides, i.e., angle ABC in the first equals angle ACB in the second, as they are opposite the equal sides AC, and AB in the two triangles.
There follows the converse theorem (Prop. 6). If two angles in a triangle are equal, then the sides opposite them are equal,— i.e., the triangle is isosceles. The proof given consists in what is called a reductio ad absurdum, a kind of proof often used by Euclid, and principally in proving the converse of a previous theorem. It assumes that the theorem to be proved is wrong, and then shows that this assumption leads to an absurdity, i.e., to a conclusion which is in contradiction to a proposition proved before— that therefore the assumption made cannot be true, and hence that the theorem is true. It is often stated that Euclid invented this kind of proof, but the method is most likely much older.
§ 8. It is next proved that two triangles which have the three sides of the· one equal respectively to those of the other are identically equal, hence that the angles of the one are equal respectively to those of the other, those being equal which are opposite equal sides. This is Prop. 8, Prop. 7 containing only a first step towards its proof.
These theorems allow now of the solution of a number of problems, viz.:—
To bisect a given angle (Prop. 9).
To bisect a given finite straight line (Prop. 10).
To draw a straight line perpendicularly to a given straight line through a given point in it (Prop. 11), and also through a given point not in it (Prop. 12).
The solutions all depend upon properties of isosceles triangles.
§ 9. The next three theorems relate to angles only, and might have been proved before Prop. 4, or even at the very beginning. The first (Prop. 13) says, The angles which one straight line makes with another straight line on one side of it either are two right angles or are together equal to two right angles. This theorem would have been unnecessary if Euclid had admitted the notion of an angle such that its two limits are in the same straight line, and had besides defined the sum of two angles.
Its converse (Prop. 14) is of great use, inasmuch as it enables us in many cases to prove that two straight lines drawn from the same point are one the continuation of the other. So also is
Prop. 15. If two straight lines cut one another, the vertical or opposite angles shall be equal.
§ 10. Euclid returns now to properties of triangles. Of great importance for the next steps (though afterwards superseded by a more complete theorem) is
Prop. 16. If one side of a triangle be produced, the exterior angle shall be greater than either of the interior opposite angles.
Prop. 17, Any two angles of a triangle are together less than two right angles, is an immediate consequence of it. By the aid of these two, the following fundamental properties of triangles are easily proved :—
Prop. 18. The greater side of every triangle has the greater angle opposite to it;
Its converse, Prop. 19. The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it;
Prop. 20. Any two sides of a triangle are together greater than the third side;
And also Prop. 21. If from the ends of the side of a triangle there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle.
§ 11. Having solved two problems (Props. 22,23), he returns to two triangles which have two sides of the one equal respectively to two sides of the other. It is known (Prop. 4) that if the included angles are equal then the third sides are equal; and conversely (Prop. 8), if the third sides are equal, then the angles included by the first sides are equal. From this it follows that if the included angles are not equal, the third sides are not equal, and conversely, that if the third sides are not equal, the included angles are not equal. Euclid now completes this knowledge by proving, that “if the included angles are not equal, then the third side in that triangle is the greater which contains the greater angle;" and conversely, that “if the third sides are unequal, that triangle contains the grecder angle which contains the greater side" These are Prop. 24 and Prop. 25.
§ 12. The next theorem (Prop. 26) says thati/ two triangles have one side and two angles of the one equal respectively to one side and two angles of the other, viz., in both triangles either the angles adjacent to the equal side, or one angle adjacent and one angle opposite it, then the two triangles are identically equal.
This theorem belongs to a group with Prop. 4 and Prop. 8. Its first case might have been given immediately after Prop. 4, but the second case requires Prop. 16 for its proof.
§ 13. We come now to the investigation of parallel straight lines, i.e., of straight lines which lie in the same plane, and cannot be made to meet however far they be produced either way. The investigation, which starts from Prop. 16, will become clearer if a few names be explained which are not all used by Euclid. If two straight lines be cut by a third, the latter is now generally called a “transversal” of the figure. It forms at the two points where it cuts the given lines four angles with each. Those of the angles which lie between the given lines are called interior angles, and of these, again, any two which lie on opposite sides of the transversal but one at each of the two points are called “alternate angles.”
We may now state Prop. 16 thus :— If two straight lines which meet are cut by a transversal, their alternate angles are unequal. For the lines will form a triangle, and one of the alternate angles will be an exterior angle to the triangle, the other interior and opposite to it.
From this follows at once the theorem contained in Prop. 27. If two straight lines which are cut by a transversal make alternate angles equal, the lines cannot meet, however far they be produced, hence they are parallel. This proves the existence of parallel lines.
Prop. 28 states the same fact in different forms. If a straight Une, falling on two other straight lines, make the exterior angle equal to the interior and opposite angle on the same side of the line, or make the interior angles on the same side together equal to two right angles, the two straight lines shall be parallel to one another.
Hence we know that, “if two straight lines which are cut by a transversal meet, their alternate angles are not equal”; and hence that, “if alternate angles are equal, then the lines are parallel.”
The question now arises, Are the propositions converse to these true or not? That is to say, ‘ ‘ If alternate angles are unequal, do the lines meet?” And “if the lines are parallel, are alternate angles necessarily equal ?”
The answer to either of these two questions implies the answer to the other. But it has been found impossible to prove that the negation or the affirmation of either is true.
The difficulty which thus arises is overcome by Euclid assuming that the first question has to be answered in the affirmative. This gives his last axiom (12), which we quote in his own words.
Axiom 12.— If a straight line meet two straight lines, so as to make the two Ulterior angles on the same side of it taken together less than two right angles, these straight lines, being continually produced, shall at length meet on that side on which are the angles which are less than two right angles.
The answer to the second of the above questions follows from this, and gives the theorem Prop. 29. If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another, and the exterior . angle equal to the interior and opposite angle on the same side, and also the two interior angles on the same side together equal to two right angles.
§ 14. With this a new part of elementary geometry begins. The earlier propositions are independent of this axiom, and would be true even if a wrong assumption had been made in it. They all relate to figures in a plane. But a plane is only one among an infinite number of conceivable surfaces. We may draw figures on any one of them and study their properties. We may, for instance, take a sphere instead of the plane, and obtain “spherical” in the place of “plane” geometry. If on one of these surfaces lines and figures could be drawn, answering to all the definitions of our plane figures, and if the axioms with the exception of the last all hold, then all propositions up to the 28th will be true for these figures. This is the case in spherical geometry if we substitute “shortest line” or “great circle” for “straight line,” “small circle” for “circle,” and if, besides, we limit all figures to a part of the sphere which is less than a hemisphere, so that two points on it cannot be opposite ends of a diameter, and therefore determine always one and only one great circle.
For spherical triangles, therefore, all the important propositions 4, 8, 26; 5 and 6; and 18, 19, and 20 will hold good.
This remark will be sufficient to show the impossibility of proving Euclid’s last axiom, which would mean proving that this axiom is a consequence of the others, and hence that the theory of parallels would hold on a spherical surface, where the other axioms do hold, whilst parallels do not even exist.
It follows that the axiom in question states an inherent difference between the plane and other surfaces, and that the plane is only fully characterized when this axiom is added to the other assumptions.
§ 15. The introduction of the new axiom and of parallel lines leads to a new class of propositions.
After proving (Prop. 30) that “two lines which are each parallel to a third are parallel to each other ,” we obtain the new properties of triangles contained in Prop. 32. Of these the second part is the most important, viz., the theorem, The three interior angles of every triangle are together equal to two right angles.
As easy deductions not given by Euclid but added by Simson follow the propositions about the angles in polygons, they are given in English editions as corollaries to Prop. 32.
These theorems do not hold for spherical figures. The sum of the interior angles of a spherical triangle is always greater than two right angles, and increases with the area.
§ 16. The theory of parallels as such may be said to be finished with Props. 33 and 34, which state properties of the parallelogram, i.e., of a quadrilateral formed by two pairs of parallels. They are—
Prop. 33. The straight lines which join the extremities of two equal and parallel straight lines toιoards the same parts are themselves equal and parallel ; and
Prop. 34. The opposite sides and angles of a parallelogram are equal to one another, and the diameter {diagonal) bisects the parallelogram, that is, divides it into two equal parts.
§ 17. The rest of the first book relates to areas of figures.
The theory is made to depend upon the theorems—
Prop. 35. Parallelograms on the same base and between the same parallels are equal to one another ; and
Prop. 36. Parallelograms on equal bases, and between the same parallels, are equal to. one another.
As each parallelogram is bisected by a diagonal, the last theorems hold also if the -word parallelogram be replaced by “triangle,” as is done in Props. 37 and 38.
It is to be remarked that Euclid proves these propositions only in the case when the parallelograms or triangles have their bases in the same straight line.
The theorems converse to the last form the contents of the next three propositions, viz. :—
Theorem (Props. 40 and 41).— Equal triangles, on the same or on equal bases, in the same straight line, and on the same side of it, are between the same parallels.
That the two cases here stated are given by Euclid in two separate propositions proved separately is characteristic of his method.
§ 18. To compare areas of other figures, Euclid shows first, in Prop. 42, how to draw a parallelogram which is equal in area to a given triangle, and has one of its angles equal to a given angle. If the given angle is right, then the problem is solved to draw a “rectangle” equal in area to a given triangle.
Next this parallelogram is transformed into another parallelogram, which has one of its sides equal to a given straight line, whilst its angles remain unaltered. This may be done by aid of the theorem in
Prop. 43. The complements of the parallelograms which are about the diameter of any parallelogram are equal to one another.
Thus the problem (Prop. 44) is solved to construct a parallelogram on a given line, which is equal in area to a given triangle, and which has one angle equal to a given angle (generally a right angle).
As every polygon can be divided into a number of triangles, we can now construct a parallelogram having a given angle, say a right angle, and being equal in area to a given polygon. For each of the triangles into which the polygon has been divided, a parallelogram may be constructed, having one side equal to a given straight line, and one angle equal to a given angle. If these parallelograms be placed side by side, they may be added together to form a single parallelogram, having still one side of the given length. This is done in Prop. 45.
Herewith a means is found to compare areas of different polygons. We need only construct two rectangles equal in area to the given polygons, and having each one side of given length. By comparing the unequal sides we are enabled to judge whether the areas are equal, or which is the greater. Euclid does not state this consequence, but the problem is taken up again at the end of the second book, where it is shown how to construct a square equal in area to a given polygon.
§ 19. The first book concludes with one of the most important theorems in the whole of geometry, and one which has been celebrated since the earliest times. It is stated, but on doubtful authority, that Pythagoras discovered it, and it has been called by his name. If we call that side in a right-angled triangle λvhich is opposite the right angle the hypotenuse, we may state it as follows :—
Theorem of Pythagoras (Prop. 47).— In every right-angled triangle the square on the hypotenuse is equal to the sum of the squares of the other sides.
And conversely—
Prop. 48. If the square described on one of the sides of a triangle be equal to the squares described on the other sides, then the angle contained by these two sides is a right angle.
On this theorem (Prop. 47) almost all geometrical measurement depends, which cannot be directly obtained.
Book II.
§ 20. The propositions in the second book are very different in character from those in the first; they all relate to areas of rectangles and squares. Their true significance is best seen by stating them in an algebraic form. This is often done by expressing the lengths of lines by aid of numbers, which tell how many times a chosen unit is contained in the lines. If there is a unit to be found which is contained an exact number of times in each side of a rectangle, it is easily seen, and generally shown in the teaching of arithmetic, that the rectangle contains a number of unit squares equal to the product of the numbers which measure the sides, a unit square being the square on the unit line. If, however, no such unit can be found, this process requires that connexion between lines and numbers which is only established by aid of ratios of lines, and which is therefore at this stage altogether inadmissible. But there exists another way of connecting these propositions with algebra, based on modem notions which seem destined greatly to change and to simplify mathematics. We shall introduce here as much of it as is required for our present purpose.
At the beginning of the second book we find a definition according to which “a rectangle is said tobe ‘ contained* by the two sides which contain one of its right angles”; in the text this phraseology is extended by speaking of rectangles contained by any two straight lines, meaning the rectangle which has two adjacent sides equal to the two straight lines.
We shall denote a finite straight line by a single small letter, a, b, c, . . . . x, and the area of the rectangle contained by two lines a and b by ab, and this we shall call the product of the two lines a and b. It will be understood that this definition has nothing to do with the definition of a product of numbers.
We define as follows :—
The sum of two straight lines a and b means a straight line c which may be divided in two parts equal respectively to a and b. This sum is denoted by a- s rb.
The difference of two lines a and b (in symbols, a - δ) means a line c which when added to b gives a ; that is,
α- b=c if b+c≈a.
The product of two lines a and b (in symbols, ab) means the area of the rectangle contained by the lines a and b. For aa, which means the square on the line a, we write α 2 .
§ 21. The first ten of the fourteen propositions of the second book may then be written in the form of formulae as follows :— [table]
It will be seen that 5 and 6, and also 9 and 10, are identical. In Euclid’s statement they do not look the same, the figures being arranged differently.
If the letters a, b, c, . . . denoted numbers', it follows from algebra that each of these formulae is true. But this does not prove them in our case, where the letters denote lines, and their products areas without any reference to numbers. To prove them we have to discover the laws which rule the operations introduced, viz., addition and multiplication of segments. This we shall do now; and we shall find that these laws are the same with those which hold in algebraical addition and multiplication.
§ 22. In a sum of numbers we may change the order in which the numbers are added, and we may also add the numbers together in groups, and then add these groups. But this also holds for the sum of segments and for the sum of rectangles, as a little consideration shows. That the sum of rectangles has always a meaning follows from the Props. 43-45 in the first book. These laws about addition are reducible to the two— [table]
or, when expressed for rectangles, [table]
The brackets mean that the terms in the bracket have been added together before they are added to another term. The more general cases for more terms may be deduced from the above.
For the product of two numbers we have the law that it remains unaltered if the factors be interchanged. This also holds for our geometrical product. For if ab denotes the area of the rectangle which has a as base and b as altitude, then ba will denote the area of the rectangle which has b as base and a as altitude. But in a rectangle we may take either of the two lines which contain it as base, and then the other will be the altitude. This gives
αδ∙=δα .... (5).
In order further to multiply a sum by a number, we have in algebra the rule:—Multiply each term of the sum, and add the products thus obtained. That this holds for our geometrical products is shown by Euclid in his first proposition of the second book, where he proves that the area of a rectangle whose base is the sum of a number of segments is equal to the sum of rectangles which have these segments separately as bases. In symbols this gives, in the simplest case, and
α(⅛ + c)- ab+ac ) (b+d)a=Ixι + ca ) * ' ' ' ’’
To these laws which have been investigated by Sir William Hamilton and by Hermann Grassmann, the former has given special names. He calls the laws expressed in
(1) and (3) the commutative law for addition;
(5) „ ,, multiplication;
(2) and (4) the associative laws for addition;
(6) the distributive law.
§ 23. Having proved that these six laws hold, we can at once prove every one of the above propositions in their algebraical form.
The first is proved geometrically, it being one of the fundamental laws. The next two propositions are only special cases of the first. Of the others we shall prove one, viz., the fourth:— (α÷δ) 2 = (α + δ) (α + δ) = (α + δ)α + (a j rb)b by (6).
But (a + b)a=aa + ba by (6), = aa + a7> by (5); and (a + b)b= ab + bb by (6).
Therefore . (a + b) i =aa + ab + (ab + bb) ì = aa + (db + ab) + bb > by (4).
= aa + 2ab + bb )
This gives the theorem in question.
In the same manner every one of the first ten propositions is proved.
It will be seen that the operations performed are exactly the same as if the letters denoted numbers.
Props. 5 and 6 may also be written thus—
(α+δ) (α-δ) = α s -δ s .
Prop. 7, which is an easy consequence of Prop. 4, may be transformed. If we denote by c the line a + b, so that c=a + b, a≈c-b, we get c 2 + (c-δ) 2 = 2c(c-δ) + δ 2 ≈2c 2 -2bc + b 2 .
Subtracting c 2 from both sides, and writing a for c, we get (a-b) 2 ≈a 2 -2ab + V t .
Tn Euclid’s Elements this form of the theorem does not appear, all propositions being so stated that the notion of subtraction does not enter into them.
§ 24. The remaining two theorems (Props. 12 and 13) connect the square on one side of a triangle with the sum of the squares on the other sides, in case that the angle between the latter is acute or obtuse. They are important theorems in trigonometry, where it is possible to include them in a single theorem.
§ 25. There are in the second-book two problems, Props. 11 and 14.
If written in the above symbolic language, the former requires to find a line x such that a(ax)-x 2 . Prop. 11 contains, therefore, the solution of a quadratic equation, which we may write x 2 + ax = a 2 . The solution is required later on in the construction of a regular decagon.
More important is the problem in the last proposition (Prop. 14). It requires the construction of a square equal in area to a given rectangle, hence a solution of the equation x 2 = ab.
In Book I., 42-45, it has been shown how a rectangle may be constructed equal in area to a given figure bounded by straight lines. By aid of the new proposition we may therefore now determine a line such that the square on that line is equal in area to any given rectilinear figure, or we can square any such figure.
As of two squares that is the greater which has the greater side, it follows that now the comparison of two areas has been reduced to the comparison of two lines.
The problem of reducing other areas to squares is frequently met with among Greek mathematicians. We need only mention the problem of squaring the circle.
In the present day the comparison of areas is performed in a simpler way by reducing all areas to rectangles having a common base. Their altitudes give then a measure of their areas.
The construction of a rectangle having the base u, and being equal in area to a given rectangle, depends upon Prop. 43, L This therefore gives a solution of the equation ab = ux, where x denotes the unknown altitude.
Book III.
§ 26. The third book of the Elements relates exclusively to properties of the circle. A circle and its circumference have been defined in Book I., Def. 15. We restate it here in slightly different words :—
Definition.— The circumference of a circle is a plane curve such that all points in it have the same distance from a fixed point in the plane. This point is called the “centre” of the circle.
Of the new definitions, of which eleven are given at the beginning of the third book, a few only require special mention. The first, which says that circles with equal radii are equal, is in part a theorem, but easily proved by applying the one circle to the other. Or it may be considered proved by aid of Prop. 24, equal circles not being used till after this theorem.
In the second definition is explained what is meant by a line which “touches” a circle. Such a line is now generally called a tangent to the circle. The introduction of this name allows us to state many of Euclid’s propositions in a much shorter form.
For the same reason we shall call a straight line joining two points on the circumference of a circle a “chord.”
Definitions 4 and 5 may be replaced with a slight generalization by the following :—
Definition.— By the distance of a point from a line is meant the length of the perpendicular drawn from the point to the line.
§ 27. From the definition of a circle it follows that every circle has a centre. Prop. 1 requires to find it when the circle is given, i.e., when its circumference is drawn.
To solve this problem a chord is drawn (that is, any two points in the circumference are joined), and through the point where this is bisected a perpendicular to it is erected. Euclid then proves, first, that no point off this perpendicular can be the centre, hence that the centre must lie in this line; and, secondly, that of the points on the perpendicular one only can be the centre, viz., the one which bisects the part of the perpendicular bounded by the circle. In the second part Euclid silently assumes that the perpendicular there used does cut the circumference in two, and only in two points. The proof therefore is incomplete. The proof of the first part, however, is exact. By drawing two non-parallel chords, and the perpendiculars which bisect them, the centre will be found as the point where these perpendiculars intersect.
§ 28. In Prop. 2 it is proved that a chord of a circle lies altogether within the circle.
What we have called the first part of Euclid’s solution of Prop. 1 may be stated as a theorem :—
Theorem.— Every straight line which bisects a chord, and is at right angles to it, passes through the centre of the circle.
The converse to this gives Prop. 3, which may be stated thus :—
If a straight line through the centre of a circle bisect a chord, then it is perpendicular to the chord, and if it be perpendicular to the chord it bisects it.
An easy consequence of this is the following theorem, which is essentially the same as Prop. 4 :—
Theorem (Prop. 4).— Two chords of a circle, of which neither passes through the centre, cannot bisect each other.
These last three theorems are fundamental for the theory of the circle. It is to be remarked that Euclid never proves that a straight line cannot have more than two points in common with a circumference.
§ 29. The next two propositions (5 and 6) might be replaced by a single and a simpler theorem, viz. :—
Theorem.— Two circles which have a common centre, and whose circumferences have one point in common, coincide.
Or, more in agreement with Euclid’s form:—
Theorem.— Two different circles, whose circumferences have a point in common, cannot have the same centre.
That Euclid treats of two cases is characteristic of Greek mathematics.
The next two propositions (7 and 8) again belong together. They may be combined thus :—
Theorem.— If from a point in a plane of a circle, which is not the centre, straight lines be drawn to the different points of the circumference, then of all these lines one is the shortest, and one the longest, and these lie both in that straight line which joins the given point to the centre. Of all the remaining lines each is equal to one and only one other, and these equal lines lie on opposite sides of the shortest or longest, and make equal angles with them.
Euclid distinguishes the two cases where the given point lies within or without the circle, omitting the case where it lies in the circumference.
From the last proposition it follows that if from a point more than two equal straight lines can be drawn to the circumference, this point must be the centre. This is Prop. 9.
As a consequence of this we get Theorem.— If the circumferences of the two circles have three points in common they coincide.
For in this case the two circles have a common centre, because from the centre of the one three equal lines can be drawn to points on the circumference of the other. But two circles which have a common centre, and whose circumferences have a point in common, coincide. (Compare above statement of Props. 5 and 6. ) This theorem may also be stated thus :—
Through three points only one circumference may be drawn ; or, Three points determine a circle.
Euclid does not give the theorem in this form. He proves, however, that the two circles cannot cut another in more than tιvo points (Prop. 10), and that two circles cannot touch one another in more points than one (Prop. 13).
§ 30. Propositions 11 and 12 assert that if two circles touch, then the point of contact lies on the line joining their centres. This gives two propositions, because the circles may touch either internally or externally.
§ 31. Propositions 14 and 15 relate to the length of chords. The first says; that equal chords are equidistant from the centre,and that chords which are equidistant from the centre are equal;
Whilst Prop. 15 compares unequal chords, viz., Of all chords the diameter is the greatest, and of other chords that is the greater which is nearer to the centre ; and conversely, the greater chord is nearer to the centre.
§ 32. In Prop. 16 the tangent to a circle is for the first time introduced. The proposition is meant to show that the straight line at the end point of the diameter, and at right angles to it is a tangent. The proposition itself does not state this. It runs thus :—
Prop. 16. The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle; and no straight line can be drawn from the extremity, between that straight line and the circumference, so as not to cut the circle. Corollary.— The straight line at right angles to a diameter drawn through the end point of it touches the circle.
The statement of the proposition and its whole treatment show the difficulties which the tangents presented to Euclid.
Prop. 17 solves the problem through a given point, cither in the circumference or without it, to draw a tangent to a given circle.
Closely connected with Prop. 16 are Props. 18 and 19, which state (Prop. 18), that the line joining the centre of a circle to the point of contact of a tangent is perpendicular to the tangent ; and conversely (Prop. 19), that the straight line through the point of contact of, and perpendicular to, a tangent to a circle passes through the centre of the circle.
§ 33. The rest of the book relates to angles connected with a circle, viz., angles which have the vertex either at the centre or on the circumference, and which are called respectively angles at the centre and angles at the circumference. Between these two kinds of angles exists the important relation expressed as follows :—
Prop. 20. The angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same arc.
This is of great importance for its consequences, of which the two following are the principal :—
Prop. 21. The angles in. the same segment of a circle are equal to one another·,
And Prop. 22. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.
Further consequences are :—
Prop. 23. On the same straight line, and on the same side oj it, there cannot be two similar segments of circles, not coinciding with me another;
And Prop. 24. Similar segments of circles on equal straight lines are equal to one another.
The problem Prop. 25, A segment of a circle being given to describe the circle of which it is a segment, may be solved much more easily by aid of the construction described in relation to Prop. 1, III., in §27.
§ 34. There follow four theorems connecting the angles at the centre, the arcs into which they divide the circumference, and the chords subtending these arcs. They are expressed for angles, arcs, and chords in equal circles, but they hold also for angles, arcs, and chords in the same circle.
The theorems are :—
Prop. 26. In equal circles equal angles stand on equal arcs, whether they be at the centres or circumferences;
Prop. 27 (converse to Prop. 26). In equal circles the angles which stand on equal arcs are equal to one another, whether they be at the centres or the circumferences·,
Prop. 28. In equal circles equal straight lines (equal chords) cut off equal arcs, the greater equal to the greater, and the less equal to the less;
Prop. 29 (converse to Prop. 28). In equal circles equal arcs are subtended by equal straight lines.
§ 35. Other important consequences of Props. 20-22 are :—
Prop. 31. In a circle the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is less than a right amgle; and the angle in a segment less than a semicircle is greater than a right angle;
Prop. 32. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles which this Une makes wiih the line touching the circle shall be equal to the angles which are in the alternate segments of the circle.
§ 36. Propositions 30, 33, 34, contain problems which are solved by aid of the propositions preceding them :—
Prop. 30. To bisect a given arc, that is, to divide it into two equal parts;
Prop. 33. On a given straight line to describe a segment of a circle containing an angle equal to a given rectilineal angle;
Prop. 34. From a given circle to cut off a segment containing an angle equal to a given rectilineal angle.
§ 37. If we draw chords through a point A within a circle, they will each be divided by A into two segments. Between these segments the law holds that the rectangle contained by them has the same area on whatever chord through A the segments are taken. The value of this rectangle changes, of course, with the position of A.
A similar theorem holds if the point A be taken without the circle. On every straight line through A, which cuts the circle in two points B and C, we have two segments AB and AC, and the rectangles contained by them are again equal to one another, and equal to the square on a tangent drawn from A to the circle.
The first of these theorems gives Prop. 35, and the second Prop. 36, with its corollary, whilst Prop. 37, the last of Book III., gives the converse to Prop. 36.' The first two theorems may be combined in one :—
Theorem.— If through a point A inthe plane of acircle astraight line be drawn cutting the circle in B and C, then the rectangle ABAC has a constant value so long as the point A be fixed; and if from A a tangent AD can be drawn to the circle, touching at D, then the above rectangle equals the square bn AD.
Prop. 37 may be stated thus :—
Theorem.— If from a point A without a circle a line be drawn cutting the circle in B and C, and another line to a point D on the circle, and if AB.AC=AD i , then the line AD touches the circle at D.
It is not difficult to prove also the converse to the general proposition as above stated. It may be expressed as follows :—
Iffour points ABCD be taken on the circumference of a circle, and if the lines AB, CD, produced if necessary, meet at E, then
EA . EB = EC . ED; and conversely, if this relation holds then the four points lie on a circle, that is, the circle drawn through three of them passes through the fourth.
That a circle may always be drawn through three points, provided that they do not lie in a straight line, is proved only later on in Book IV.
Book IV.
§ 38. The fourth book contains only problems, all relating to the construction of triangles and polygons inscribed in and circumscribed about circles, and of circles inscribed in or circumscribed about triangles and polygons. They are nearly all given for their own sake, and not for future use in the construction of figures, as are most of those in the former books. In seven definitions at the beginning of the book it is explained what is understood by figures inscribed in or described about other figures, with special reference to the case where one figure is a circle. Instead, however, of saying that one figure is described about another, it is now generally said that the one figure is circumscribed about the other. We may then state the definitions 3 or 4 thus :—
Definition.— A polygon is said to be inscribed in a circle, and the circle is said to be circumscribed about the polygon, if the vertices of the polygon lie in the circumference of the circle.
And definitions 5 and 6 thus :—
Definition.— A polygon is said to be circumscribed about a circle, and a circle is said to be inscribed in a polygon, if the sides of the polygon are tangents to the circle.
§ 39. The first problem is merely constructive. It requires to draw in a given circle a chord equal to a given straight line, which is not greater than the diameter of the circle. The problem is not a determinate one, inasmuch as the chord may be drawn from any point in the circumference. This may be said of almost all problems in this book, especially of the next two. They are :—
Prop. 2. In a given circle to inscribe a triangle equiangular to a given triangle;
Prop. 3. About a given circle to circumscribe a triangle equiangular to a given triangle.
§ 40. Of somewhat greater interest are the next problems, where the triangles are given and the circles to be found.
Prop. 4. To inscribe a circle in a given triangle.
The result is that the problem has always a solution, viz., the centre of the circle is the point where the bisectors of two of the nterior angles of the triangle meet. The solution shows, though Euclid does not state this, that the problem has but one solution; and also, Theorem.— The three bisectors of the interior angles of any triangle meet in a point, and this is the centre of the circle inscribed in the triangle.
The solutions of most of the other problems contain also theorems. Of these we shall state those which are of special interest; Euclid does not state any one of them.
§ 41. Prop. 5. To circumscribe a circle about a given triangle.
The one solution which always exists contains the following :— Theorem.— The three straight lines which bisect the sides of a triangle at right angles meet in a point, and this point is the centre of the circle circu mscribed about the triangle.
Euelid adds in a corollary the following property :—
The centre, of the circle circumscribed about a triangle lies within, on a side of, or without the triangle, according as the triangle is acute-angled, right-angled, or obtuse-angled.
§ 42. Whilst it is always possible to draw a circle which is inscribed in or circumscribed about a given triangle, this is not the case with quadrilaterals or polygons of more sides. Of those for which this is possible the regular polygons are the most interesting. In each of them a circle may be inscribed, and another may be circumscribed about it.
Euclid does not use the word regular, but he describes the polygons in question as equiangular and equilateral. We shall use the name regular polygon. The regular triangle is equilateral, the regular quadrilateral is the square.
Euclid considers the regular polygons of 4, 5, 6, and 15 sides. For each of the first three he solves the problems—(1) to inscribe such a polygon in a given circle; (2) to circumscribe it about a given circle; (3) to inscribe a circle in, and (4) to circumscribe a circle about, such a polygon.
For the regular triangle the problems are not repeated, because more general problems have been solved.
Props. 6, 7, 8, and 9 solve these problems for the square.
The general problem of inscribing in a given circle a regular polygon of n sides depends upon the problem of dividing the circumference of a circle into n equal parts, or what comes to the same thing, of drawing from the centre of the circle n radii such that the angles between consecutive radii are equal, that is, to divide the space about the centre into n equal angles. Thus, if it is required to inscribe a square in a circle, we have to draw four lines from the centre, making the four angles equal. This is done by drawing two diameters at right angles to one another. The ends of these diameters are the vertices of the required square. If, on the other hand, tangents be drawn at these ends, we obtain a square circumscribed about the circle.
§ 43. To construct a. regular pentagon, we find it convenient first to construct a regular decagon. This requires to divide the space about the centre into ten equal angles. Each will be τ ⅛th of a right angle, or ⅜th of two right angles. If we suppose the decagon constructed, and if we join the centre to the end of one side, we get an isoceles triangle, where the angle at the centre equals ⅜th of two right angles; hence each of the angles at the base will be ⅜ths of two right angles, as all three angles together equal two right angles. Thus we have to construct an isoceles triangle, having the angle at the vertex equal to half an angle at the base. This is solved in Prop. 10, by aid of the problem in Prop. 11 of the second book. If we make the sides of this triangle equal to the radius of the given circle, then the base will be the side of the regular decagon inscribed in the circle. This side being known the decagon can be constructed, and if the vertices are joined alternately, leaving out half their number, we obtain the regular pentagon.
Euclid does not proceed thus. He wants the pentagon before the decagon. This, however, does not change the real nature of his solution, nor does his solution become simpler by not mentioning the decagon.
Once the regular pentagon is inscribed, it is easy to circumscribe another by drawing tangents at the vertices of the inscribed pentagon. This is shown in Prop. 12.
Prop. 13 and 14 teach how a circle may be inscribed in or circumscribed about any given regular pentagon.
§ 44. The regular hexagon is more easily constructed, as shown in Prop. 15. The result is that the side of the regular hexagon inscribed in a circle is equal to the radius of the circle.
For this polygon the other three problems mentioned are not solved.
§ 45. The book closes with Prop. 16. To inscribe a regular quindecagon in a given circle. That this may be done is easily seen. If we inscribe a regular pentagon and a regular hexagon in the circle, having one vertex in common, then the arc from the common vertex to the next vertex of the pentagon is ’ th of the circumference, and to the next vertex of the hexagon is ⅜th of the circumference. The difference between these arcs is, therefore, ⅜. - ½=^⅛th °f the circumference. The latter may, therefore, be divided into thirty, and hence also in fifteen equal parts, and the regular quindecagon be described.
§ 46. We conclude with a few theorems about regular polygons which are not given by Euclid.
Theorem.— The straight lines peιpendicular to and bisecting the sides of any regular polygon meet in a point. The straight lines bisecting the angles in the regular polygon meet in the same point. This point is the centre of the circles circumscribed about and inscribed in the regular polygon. The proof, which is easy, is left to the reader.
We can bisect any given arc (Prop. 30, III.). Hence we can divide a circumference into 2n equal parts as soon as it has been divided into n equal parts, or as soon as a regular polygon of n sides has been constructed. Hence—
Theorem.— If a regular polygon ofn sides has been constructed, then a regular polygon of 2n sides, of 4n, of 8n sides, Ac., may also be constructed. Euclid shows how to construet regular polygons of 3, 4, 5, and 15 sides. It follows that we can construct regular polygons of [table]
The construction of any new régulai· polygon not included in one of these series will give rise to a new series. Till the beginning of this century nothing was added to the knowledge of regular polygons as given by Euclid. Then Gauss, in his celebrated Arithmetic, proved that every regular polgon of 2 n +1 sides may be constructed if this number 2»> +1 be prime, and that no others can be constructed by elementary methods. This shows that regular polygons of 7, 9, 13 sides cannot thus be constructed, but that a regular polygon of 17 sides is possible; for 17≈2 4 + 1. The next polygon is one of 257 sides. The construction becomes already rather complicated for 17 sides.
Book V.
§ 47. The fifth book of the Elements is not exclusively geometrical. It contains the theory of ratios and proportion of quantities in general. The treatment, as here given, is admirable, and in every respect superior to the algebraical method by which Euclid’s theory is now generally replaced. It has, however, the reputation of being too difficult for schools, and is therefore very seldom read. We shall try to make the subject clear, and to show why the usual algebraical treatment of proportion is not really sound. We begin by quoting those definitions at the beginning of Book V. which are most important. These definitions have given rise to much discussion.
Tlιe only definitions which are essential for the fifth book are Defs. 1, 2, 4, 5, 6, and 7. Of the remainder 3, 8, and 9 are more than useless, and probably not Euclid’s, but additions of later editors, of whom Theon of Alexandria was the most prominent. Defs. 10 and 11 belong rather to the sixth book, whilst all the others are merely nominal. The really important ones are 4, 5, 6, and 7.
§ 48. To define a magnitude is not attempted by Euclid. The first two definitions state what is meant by a “part,” that is, a submultiple or measure, and by a “multiple” of a given magnitude. The meaning of Def. 4 is that two given quantities can have a ratio to one another only in case that they are comparable as to their magnitude, that is, if they are of the same kind.
Def. 3, which is probably due to Theon, professes to define a ratio, but is as meaningless as it is uncalled for, for all that is wanted is given in Defs. 5 and 7.
In Def. 5 it is explained what is meant by saying that two magnitudes have the same ratio to one another as two other magnitudes, and in Def. 7 what we have to understand by a greater or a less ratio. The 6th definition is only nominal, explaining the meaning of the word proportional.
Euclid represents magnitudes by lines, and often denotes them either by single letters or, like lines, by two letters. We shall use only single letters for the purpose. If a and b denote two magnitudes of the same kind, their ratio will be denoted by a: b ; if c and d are two other magnitudes of the same kind, but possibly of a different kind from a and b, then if c and d have the same ratio to one another as a and b, this will be expressed by writing— a: b:: c: d.
Further, if m is a (whole) number, ma shall denote the multiple of a which is obtained by taking it m times.
§ 49. The whole theory of ratios is based on Def. 5.
Def. 5. The first of four magnitudes is said to have the same ratio to the second that the third has to the fourth when, any equimultiples whatever of the first and the third being taken, and any equimultiples whatever of the second and the fourth, if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth; and if the multiple of the first is equal to that of the second, the multiple of the third is also equal to that of the fourth √ and if the multiple of the first is greater than that of the second, the multiple of the third is also greater than that of the fourth.
It will be well to show at once in an example how this definition can be used, by proving the first part of the first proposition in the sixth book. Triangles of the same altitude arc to one another as their bases, or if a and b are the bases, and a and β the areas, of two triangles which have the same altitude, then a: b:: a: β.
To prove this, we have, according to Definition 5, to show— [table]
That this is true is in oui’ case easily seen. We may suppose that the triangles have a common vertex, and their bases in the same line. "We set off the base a along the line containing the bases m times; we then join the different parts of division to the vertex, and get m triangles all equal to a. The triangle on ma as base equals, therefore, ma. If we proceed in the same manner with the base b, setting it off n times, we find that the area of the triangle on the base nb equals nβ, the vertex of all triangles being the same. But if two triangles have the same altitude, then their areas are equal if the bases are equal; hence ma≈nβ if ma=nb, and if their bases are unequal, then that has the greater area which is on the greater base; in other words, ma is greater than, equal to, or less than nβ, according as ma is greater than, equal to, or less than nb, which was to be proved.
§ 50. It will be seen that even in this example it does not become evident what a ratio really is. It is still an open question whether ratios are magnitudes which we can compare. We do not know whether the ratio of two lines is a magnitude of the same kind as the ratio of two areas. Though we might say that Def. 5 defines equal ratios, still we do not know whether they are equal in the sense of the axiom, that two things which are equal to a third are equal to one another. That this is the case requires a proof, and until this proof is given we shall use the :: instead of the sign =, which, however, we shall afterwards introduce.
As soon as it has been established that all ratios are like magnitudes, it becomes easy to show that, in some cases at least, they are numbers. This step was never made by Greek mathematicians. They distinguished always most carefully between continuous magnitudes and the discrete series of numbers. In modern times it has become the custom to ignore this difference.
If, in determining the ratio of two lines, a common measure can be found, which is contained m times in the first, and n times in the second, then the ratio of the two lines equals the ratio of the two numbers m: n. This is shown by Euclid in Prop. 5, X. But the ratio of two numbers is, as a rule, a fraction, and the Greeks did not, as we do, consider fractions as numbers. Far less had they any notion of introducing irrational numbers, which are neither whole nor fractional, as we are obliged to do if we wish to say that all ratios are numbers. The incommensurable numbers which are thus introduced as ratios of incommensurable quantities are nowadays as familiar to us as fractions; but a proof is generally omitted that we may apply to them the rules which have been established for rational numbers only. Euclid’s treatment of ratios avoids this difficulty. His definitions holds for commensurable as well as for incommensurable quantities. Even the notion of incommensurable quantities is avoided in Book V. But he proves that the more elementary rules of algebra hold for ratios. We shall state all his propositions in that algebraical form to which we are now accustomed. This may, of course, be done without changing the character of Euclid’s method.
§ 51. Using the notation explained above we express the first propositions as follows :—
Prop. 1. If a-ma ’ , b≈mb', c=md, then a + b + c = m(a' + b ’ + d).
Prop. 2. If a = mb, and c≈mã, e=nb, and f≈nd, then a + e is the same multiple of b as c +∕ is of d, viz. :— α + e=(γn + n)δ, and c+∕=(wι+n)tZ.
Prop. 3. If α = mδ, c≈md, then is na the same multiple of b that nc is of d, viz., na≈nmb, nc=nmd.
Prop. 4. If a: b:: c: d, then ma ∙.rib :: me: nd.
Prop. 5. If α = mδ, and c = md then a-c≈m(b-d).
Prop. 6. If a≈mb,c=md, then are a - nb and c - nd either equal to, or equimultiples of, b and d, viz., a-rib=(m- risb and c- nd≈(m-n)d, where m-n may be unity.
All these propositions relate to equimultiples. Now follow propositions about ratios which are compared as to their magnitude.
§ 52. Prop. 7. If α = δ, then a: c:: b: c and c: a:: c: b.
The proof is simply this. As α = δ we know that ma≈mb; therefore if ma>nc, then mb>nc, if ma - nc, then mb = nc, if ma<fnc, then mb<fnc, therefore the first proportion holds by Definition 5.
Prop. 8. If dfi>b, then a: cfi>b ∙.c, and c:aQc.b.
The proof depends on Definition 7.
Prop. 9 (converse to Prop. 7). If
<z: c:: δ: c, or if c: a:: c : δ, then α=≈δ.
Prop. 10 (converse to Prop. 8). If a: cfi>b : c, then α>δ and if c: a<fc : δ, then a<fi,
Prop. 11. If a : δ:: c: d, and a : δ:: e then c: d:: e :fi
In words, if two ratios are equal to a third, they are equal to one another. After these propositions have been proved, we have a right to consider a ratio as a magnitude, for only now can we consider a ratio as something for which the axiom about magnitudes holds: things which are equal to a third are equal to one another.
We shall indicate this by writing in future the sign = instead of:: . The remaining propositions, which explain themselves, may then be stated as follows:—
§53. Prop. 12. If a: b≈c: d≈e :fi then a+c+e ∙.b+d+f≈a :b.
Prop. 13. If a: b≈c: d and c : <ζ>c: f, then a : δ≥c ∙.f.
Prop. 14. If a: b≈c: d, and α>c, then b~f>d.
Prop. 15. Magnitudes have the same ratio to one another that their equimultiples have—
ma: mb≈a : δ.
Prop. 16. If a, b, c, d are magnitudes of the same kind, and if a: b≈c: d, then a:c≈byd.
Prop. 17. If a + b :b≈c + d :d, then a : δ = c: d.
Prop. 18 (converse to 17). If a : δ=c: d then a + b:b≈c+d:d.
Prop. 19. If a, b, c, d are quantities of the same kind, and if a :b≈c :d, then a-c : δ -d≈a :b.
§ 54. Prop. 20. If there be three magnitudes, and other three, which have the same ratio, taken two and two, then if the first be greater than the third, the fourth shall be greater than the sixth; and if equal, equal; and if less, less.
If we understand by a :b :e: d :c: ... =a' :b' :d :d' :d: . . . that the ratio of any two consecutive magnitudes on the first side equals that of the corresponding magnitudes on the second side, we may write this theorem in symbols, thus :—
If a, b, c be quantities of one, and d, e,f magnitudes of the sama or any other kind, such that a : δ: c≈d :e :f and if afi>c, then <ζ> f, but if ∙ a=c, then d= f and if α<c, then d∙<ff.
Prop. 21. If a: b=e :/and δ :c≈d: e, or if a : δ: c= -: -: 1
f e d i and if afi>c, then tζ> f, but if a = c, then d≈ f, and if α<c, then d<Qf.
By aid of these two propositions the following two are proved.
§ 55. Prop. 22. If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in order, the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last.
We may state it more generally, thus :—
If a: b: c: d: e: . . . ≈a' :b': cl :d' :d: . . , , then not only have two consecutive, but any two magnitudes on the first side, the same ratio as the corresponding magnitudes on the other. For instance—
a∙.c≈a ’ :d; b: e≈b':el, &c.
Prop. 23 we state only in symbols, viz. :—
If a:b:c:d:e:...≈^,:\.\.\,st..., a b d d d then a:c≈c':a', ft :e≈e': b', and so on.
Prop. 24 comes to this: If a: b≈c: d and e: b≈f∙. d, then a + c : δ≡c+∕ :d.
Some of the proportions which are considered in the above propositions have special names. These we have omitted, as being of no use, since algebra has enabled us to bring the different operations contained in the propositions under a common point of view.
§ 56. The last proposition in the fifth book is of a different character.
Prop. 25. If four magnitudes of the same kind be proportional, the greatest and least of them together shall be greater than the other two together. In symbols—
If a, b, c, d be magnitudes of the same kind, and if a: b≈c :d, and if a is the greatest, hence d the least, then a + dfi>b + c.
§ 57. We return once again to the question, What is a ratio? We have seen that we may treat ratios as magnitudes, and that all ratios are magnitudes of the same kind, for we may compare any two as to their magnitude. It will presently be shown that ratios of lines may be considered as quotients of lines, so that a ratio appears as answer to the question, How often is one line contained in another? But the answer to this question is given by a number, at least in some cases, and in all cases if we admit incommensurable numbers. Considered from this point of view, we may say the fifth book of the Elements shows that some of the simpler algebraical operations hold for incommensurable numbers. In the ordinary algebraical treatment of numbers this proof is altogether omitted, or given by a process of limits which does not seem to be natural to the subject.
r Book VI.
§ 58. The sixth book contains the theory of similar figures. After a few definitions explaining terms, the first proposition gives the first application of the theory of proportion.
Prop. 1. Triangles and parallelograms of the same altitude are to one another as their bases.
The proof has already been considered in § 49.
From this follows easily the important theorem
Prop. 2. If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those sides produced, proportionally; and if the sides or the sides produced be cut proportionally, the straight line which joins the points of section, shall be parallel to the remaining side of the triangle.
§ 59. The next proposition, together with one added by Simson as Prop. A, may be expressed more conveniently if we introduce a modern phraseology, viz., if in a line AB we assume a point C between A and B, we shall say that C divides AB internally in the ratio AC: CB; but if C be taken in the line AB produced, we shall say that AB is divided externally in the ratio AC: CB.
The two propositions then come to this:
Theorem (Prop. 3).— The bisector of an angle in a triangle divides the opposite side internally in a ratio equal to the ratio of the two sides including that angle; and conversely, if a line through the vertex of a triangle divide the base internally in the ratio of the two other sides, then that line bisects the angle at the vertex. Theorem (Simson’s Prop. A).— The line which bisects an exterior angle of a triangle divides the opposite side externally in the ratio of the other sides; and conversely, if a line through the vertex of a triangle divide the base externally in the ratio of the sides, then it bisects an exterior angle at the vertex of the triangle.
If we combine both we have—
Theorem.— The two lines which bisect the interior and exterior angles at one vertex of a triangle divide the opposite side internally and externally in the same ratio, viz., in the ratio of the other two sides.
§ 60. The next four propositions contain the theory of similar triangles, of which four cases are considered. They may be stated together.
Theorem.— Two triangles are similar,—
1. (Prop. 4). If the triangles are equiangular;
2. (Prop. 5). If the sides of the one are proportional to those of the other;
3. (Prop. 6). If two sides in one are proportional to two sides in the other, and if the angles contained by these sides are equal;
4. (Prop. 7). If two sides in one are proportional to two sides in the other, if the angles opposite homologous sides are equal, and if the angles opposite the other homologous sides are both acute, both right, or both obtuse; homologous sides being in each case those which are opposite equal angles.
An important application of these theorems is at once made to a right-angled triangle, viz. :—
Prop. 8. In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another. Corollary.— From this it is manifest that the perpendicular drawn from the right angle of a right-angled triangle to the base is a mean proportional between the segments of the base, and also that each of the sides is a mean proportional between the base and the segment of the base adjacent to that side.
§ 61. There follow four propositions containing problems, viz., in language slightly different from Euclid’s :—
Prop. 9. To divide a straight line into a given number of equal parts.
Prop. 10. To divide a straight line in a given ratio.
Prop. 11. To find a third proportional to two given straight lines.
Prop. 12. To find a fourth proportional to three given straight lines.
Prop. 13. To find a mean proportional between two given straight lines.
The last three may be written as equations with one unknown quantity,—viz., if we call the given straight lines a, b, c, and the required line x, we have to find, a line x so that
Prop. 11. a: b = b: x;
Prop. 12. a: b ≈ c: x;
Prop. 13. a: x = x: b.
We shall see presently how these may be written without the signs of ratios.
§ 62. Euclid considers next proportions connected with parallelograms and triangles which are equal in area.
Prop. 14. Equal parallelograms which have one angle of the one equal to one angle of the other have their sides about the equal angles reciprocally proportional; and parallelograms which have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another.
Prop. 15. Equal triangles which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional; and triangles which have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another.
The latter proposition is really the same as the former, for if, as in the accompanying diagram, in the figure belonging to the former the two equal parallelograms AB and BC be bisected by the lines DF and EG, and if EF be drawn, we get the figure belonging to the latter.
It is worth noticing that the lines FE and DG are parallel. We may state therefore the theorem—
Theorem.— If two triangles are equal in area, and have one angle in the one vertically opposite to one angle in the other, then the two straight lines which join the remaining two vertices of the one to those of the other triangle are parallel.
§ 63. A most important theorem is
Prop. 16. If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means; and if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines are proportionals.
In symbols, if a, b, c, d are the four lines, and if a: b≈c: d, then ad ≈ be; and conversely, if ad≈bc, then a:b≈c:d, where ad and be denote (as in § 20), the areas of the rectangles contained by a and d and by b and c respectively.
This allows us to transform every proportion between four lines into an equation between two products.
It shows further that the operation of forming a product of two lines, and the operation of forming their ratio are each the inverse of the other.
If we now define a quotient ~ of two lines as the number which multiplied into b gives a, so that ↑ b = a, b we see that from the equality of two quotients a c_ b ~d follows, if we multiply both sides by bd, ⅛b.d =^d.b, o a ad ≈ cb.
But from this it follows according to the last theorem that a: b = c: d.
Hence we conclude that the quotient st and the ratio a: b are different forms of the same magnitude, only with this important difference that the quotient would have a meaning only if a and b have a common measure, until we introduce incommensurable numbers, while the ratio a: b has always a meaning, and thus gives rise to the introduction of incommensurable numbers.
Thus it is really the theory of ratios in the fifth book which enables us to extend the geometrical calculus given before in connexion with Book II. It will also be seen that if we write the ratios in Book V. as quotients, or rather as fractions, then most of the theorems state properties of quotients or of fractions.
§ 64. Prop. 17 contains only a special case of 16. After the problem, Prop. 18. On a given straight line to describe a rectilineal figure similar and similarly situated to a given rectilineal figure, there follows another fundamental theorem:
Pro;». 19. Similar triangles are to one another in the duplicate ratio of their homologous sides. In other words, the areas of similar triangles are to one another as the squares on homologous sides. This is generalized in
Prop. 20. Similar polygons may be divided into the same number of similar triangles, having the same ratio to one another that the polygons have; and the polygons are to one another in the duplicate ratio of their homologous sides.
§ 65. Prop. 21. Rectilineal figures which are similar to the same rectilineal figure are also similar to each other, is an immediate consequence of the definition of similar figures. As similar figures may be said to be equal in “shape” but not in “size,” we may state it also thus:
“Figures which are equal in shape to a third are equal in shape to each other.”
Prop. 22. If four straight lines be pιoportionals, the similar rectilineal figures similarly described on them shall also be proportionals; and if the similar rectilineal figures similarly described on four straight lines be proportionals, those straight lines shall be proportionals.
This is essentially the same as the following :—
If a :b ≈ c :d, then a 2 : b 2 = c i : d-.
§ 66. Noav follows a proposition which has been much discussed with regard to Euclid’s exact meaning in saying that a ratio is compounded of two other ιatios, viz.:
Prop. 23. Parallelograms which are equiangular to one another, hare to one another the ratio which is compounded of the ratios of their sides.
The proof of the proposition makes its meaning clear. In symbols the ratio a: c is compounded of the two ratios a: b and b : c, and if a: b = a!: b' , b :c = b" ∙.c", then a: c is compounded of a ': b' and ∙a ’ : b" .
If we consider the ratios as numbers, we may say that the one satio is the product of those of which it is compounded, or in -symbols, a ab a' b" . p a a! .b b" c b c b c b b c c
The theorem in Prop. 23 is the foundation of all mensuration of areas. From it we see at once that two rectangles have the ratio of their areas compounded of the ratios of their sides.
If A is the area of a rectangle contained by a and b, and B that of a rectangle contained by c and d, so that A = ab, B = cd, then A: B = ab : cd, and this is, the theorem says, compounded of the ratios a: c and b: d. In forms of quotients, a b ab c d cd'
This shows how to multiply quotients in our geometrical calculus.
Further, Two triangles have the ratios of their areas compounded of the ratios of their bases and their altitude. For a triangle is equal in area to half a parallelogram which has the same base and the same altitude.
To bring these theorems to the form in which they are usually given, we assume a straight line u as our unit of length (generallv an inch, a foot, a mile, &c.), and determine the number α which expresses how often u is contained in a fine a, so that a denotes the ratio a: u whether commensurable or not, and that a = a,u. We call this number α the numerical value of a. If ⅛ the same manner 0 be the numerical value of a line b we have a: b = a: β ; in words: The ratio of two lines (and of two like quantities in general) is equal to that of their numerical values.
This is easily proved by observing that a = au, b = βu, therefore a: b — au : βu, and this may without difficulty be shown to equal a: β.
If now a, b be base and altitude of one, a', b' those of another parallelogram, a, β and a, β' their numerical values respectively, and A, A' their areas, then
A _ a b _a β aβ å! a! ' b' a' ’ β' a!β''
In words: The areas of two parallelograms are to each other as the products of the numerical values of their bases and altitudes.
If especially the second parallelogram is the unit square, i.e., a square on the unit of length, then α' = β' = 1, A' = tι 2 , and we have
A
- r . =aβ . or A = aβ. u 2 .
A
This gives the theorem: The number of unit squares contained in a parallelogram equals the product of the numerical values of base and altitude, and similarly the number of unit squares contained in a triangle equals half the product of the numerical values of base and altitude.
This is often stated by saying that the area of a parallelogram is equal to the product of the base and the altitude, meaning by this product the product of the numerical values, and not the product as defined above in § 20.
§ 68. Propositions 24 and 26 relate to parallelograms about diagonals, such as are considered in Book I., 43. They are—
Prop. 24. Parallelograms about the diameter of any parallelogram are similar to the whole parallelogram and to one another; and its converse (Prop. 26), If two similar parallelograms have a common angle, and be similarly situated, they are about the same diameter.
Between these is inserted a problem.
Prop. 25. To describe a rectilineal figure which shall be similar to one given rectilinear figure, and equal to another given rectilineal figure.
§ 69. Prop. 27 contains a theorem relating to the theory of maxima and minima. We may state it thus:
Prop 27 If a parallelogram be divided into two by a straight line cutting the base, and if on half the base another parallelogram be constructed similar to one of those parts, then this third parallelogram is greater than the other part.
Of far greater interest than this general theorem is a special case of it, where the parallelograms are changed into rectangles, and where one of the parts into which the parallelogram is divided is made a square; for then the theorem changes into one which is easily recognized to be identical with the following:—
Theorem.— Of all rectangles ιvhich have the same perimeter the square has the greatest area.
This may also be stated thus :—
Theorem.— Of all rectangles which have the same area the square has the least perimeter.
§ 70. The next three propositions contain problems which may be said to be solutions of quadratic equations. The first two are, like the last, involved in somewhat obscure language. We transcribe them as follows: —
Problem.— To describe on a given base a parallelogram, and to divide it either internally (Prop. 28) or externally (Prop. 29) from a point on the base into two parallelograms, of which the one has a given size (is equal in area to a given figure), whilst the other has a given shape (is similar to a given parai leiogi am).
If we express this again in symbols, calling the given base a, the one part x, and the altitude y, we have to determine x and y in the first case from the equations (a-x)y=k 2 , x _p y~st k 2 being the given size of the first, and p and q the base and altitude of the parallelogram which determine the shape of the second of the required parallelograms.
If we substitute the value of y, we get (a - x)x=^- , or, ax -x 2 =b 2 , where a and b∙ are known quantities, taking b 2 =sh⅛-. a
The second case (Prop. 29) gives rise, in the same manner, to the quadratic ax + ae 2 = i> 2 .
The next problem—
Prop. 30. To cut a given straight line in extreme and mean ratio, leads to the equation ax+x 2 ≈a 2 .
This is, therefore, only a special case of the last, and is, besides, an old acquaintance, being essentially the same problem as that proposed in II. 11.
Prop. 80 may therefore be solved in two ways, either by aid of Prop. 29 or by aid of II. 11. Euclid gives both solutions.
§71. Prop. 31 (Theorem). In any right-angled, triangle, any rectilineal figure described on the side subtending the right angle is equal to the similar and similarly-described figures on the sides containing the right angle,— is a pretty generalization of the theorem of Pythagoras (I. 47).
Leaving out the next proposition, which is of little interest, we come to the last in this book.
Prop. 83. In equal circles angles, xchether at the centres or the circumferences, have the same ratio which the arcs on which they stand have to one another; so also have the sectors.
Of this, the part relating to angles at the centre is of special importance; it enables us to measure angles by arcs.
With this closes that part of the Elements which is devoted to the study of figures in a plane.
Book XI.
§ 72. In this book figures are considered which are not confined to a plane, viz., first relations between lines and planes in space, and afterwards properties of solids.
Of new definitions we mention those which relate to the perpendicularity and the inclination of lines and planes.
Def. 3. A straight line is perpendicular, or at right angles, to a plane when it makes right angles with every straight line meeting it in that plane.
The definition of perpendicular planes (Def. 4) offers no difficulty. Euclid defines the inclination of lines to planes and of planes to planes (Defs. 5 and 6) by aid of plane angles, included by straight lines, with which we have been made familiar in the first books.
The other important definitions are those of parallel planes, which never meet (Def. 8), and of solid angles formed by three or more planes meeting in a point (Def. 9).
To these we add the definition of a line parallel to a plane as a line which does not meet the plane.
§ 73. Before we investigate the contents of Book XL, it will be well to recapitulate shortly what we know of planes and lines from the definitions and axioms of the first book. There a plane has been defined as a surface which has the property that every straight line which joins two points in it lies altogether in it. This is equivalent to saying.that a straight line which has two points in a plane lias all points in the pla-e. Hence, a straight line which does not lie in the plane cannot have more than one point in common with the plane. This is virtually the same as Euclid’s Prop. 1, viz.: —
Prop. 1. One part of a straight line cannot be in a plane and another part without it.
It also follows, as was pointed out in § 3, in discussing the definitions of Book I., that a plane is determined already by one straight line and a point without it, viz., if all lines be drawn through the point, and cutting the line, they will form a plane.
This may be stated thus :—
A plane is determined—
1st, By a straight line and a point which does not lie on it;
2d, By three points which do not lie in a straight line ; for if two of these points be joined by a straight line we have case 1;
3d, By two intersecting straight lines; for the point of intersection aud two other points, one in each line, give case 2;
4th, By two parallel lines (Def. 35, 1.).
The third case of this theorem is Euclid’s
Prop. 2. Two straight lines which cut one another are τn one plane, and three straight lines which meet one another arc in one plane.
And the fourth is Euclid’s
Prop. 7. If two straight lines be pa.rallcl, the straight line drawn from any point in one to any point in the other is in the same plane with the parallels. From the definition of a plane further follows *
Prop. 3. If two planes cut one another, their common section is a straight line.
§ 74. Whilst these propositions are virtually contained in the definition of a plane, the next gives us a new and fundamental property of space, showing at the same time that it is possible to have a straight line perpendicular to a plane, according to Def. 3. It states—
Theorem (Prop. 4).— If a straight line is perpendicular to two straight lines in a plane which it meets, then it is perpendicular to all lines in the plane which it meets, and hence it is perpendicular to the plane.
Def. 3 may be stated thus: If a straight line is perpendicular to a plane, then it is perpendicular to every line in the plane which it meets. The converse to this would be Theorem.— All straight lines which meet a given straight line in the same point, and arc perpendicular to it, lie in a plane which is perpendicular to that line.
This Euclid states thus:
Prop. 5. If three straight lines med all at one point, and a straight line stands at right angles to each of them at that point, the three straight lines shall be in one and the same plane.
§75. There follow theorems relating to the theory of parallel lilies in space, viz. :—
Theorem (Prop. 6).— Any two lines which are perpendicular to the same plane are parallel to each other ; aud conversely Theorem (Prop. 8).— If of two parallel straight lines one is perpendicular to a plane, the other is so also.
Further, the important theorems
Prop. 9. Two straight lines which arc each of them parallel to the same straight line, and not in the same plane with it, are parallel to one another; where the words, “and not in the same plane with it,” may be left out, for they exclude the case of three parallels in a plane, which has been proved before; and Theorem (Prop. 10).— If two angles in different planes have the two limits of the one parallel to. those of the other, then the angles are equal. That their planes are parallel is shown later on in Prop. 15.
This theorem is not necessarily true, for the angles in question may be supplementary; but then the one angle will be equal to that which is adjacent and supplementary to the other, and this latter angle will also have its limits parallel to those of the first.
From this theorem it follows that if we take any two straight lines in space which do not meet, and if we draw through any point P in space two lines parallel to them, then the angle included by these lines will always be the same, whatever the position of the point P may be. This angle has in modern times been called the angle between the given lines :—
Definition.— By the angles between two not intersecting lines we understand the angles which two intersecting lines include that are parallel respectively to the two given lines.
§ 76. It is now possible to solve the following two problems :—
Problem.— To draw a straight line perpendicular to a givenplane from a given point which lies
1. Not in the plane (Prop. 11).
2. In the plane (Prop. 12).
The second case is easily reduced to the first—viz., if by aid of the first we have drawn any perpendicular to the plane from some point without it, we need only draw through the given point in the plane a line parallel to it, in order to have the required perpendicular given. The solution of the first part is of interest in itself. It depends upon a construction which may be expressed as a theorem.
Theorem.— If from a point A without a plane a perpendicular AB be drawn to the plane, and if from the foot B of this perpendicular another perpendicular BC be drawn to any straight line in the plane, then the straight line joining A to the foot C of this second perpendicular will also be perpendicular to the line in the plane.
The theory of perpendiculars to a plane is concluded by the theorem—
Prop. 13. Through any point in space, whether in or without a plane, only one straight line can be drawn perpendicular to the plane.
§ 77. The next four propositions treat of parallel planes. It is shown that planes which have a common perpendicular arc parallel (Prop. 14); that two planes are parallel if tιco intersecting straight lines in the one are parallel respectively to two straight lines in the other plane (Prop. 15); that parallel planes arc cut by any plane in parallel straight lines (Prop. 16); and lastly, that any two straight lines are cut proportionally by a scries of parallel planes (Prop. 17).
This theory is made more complete by adding the following theorems, which are easy deductions from the last :— Two parallel planes have common perpendiculars (converse to 14); and Two planes which are parallel to a third plane arc parallel to each other.
It will be noted that Prop. 15 at once allows of the solution of the problem: “Through a given point to draw a plane parallel to a given plane.” And it is also easily proved that this problem allows always of one, and only of one, solution.
§ 78. We come now to planes which are perpendicular to one another. Two theorems relate to them.
Prop. 18. If a straight line be at right angles to a plane, every plane vehich passes through it shall be at right angles to that plane.
Prop. 19. If two planes which cut one another be each of them perpendicular to a third plane, their common section shall be perpendicular to the same plane.
§ 79. If three planes pass through a common point, and if they bound each other, a solid angle of three faces, or a trihedral angle, is formed, and similarly by more planes a solid angle of more faces, or a polyhedral angle. These have many properties which are quite analogous to those of triangles and polygons in a plane. Euclid states some, viz :—
Prop. 20. If a solid angle be contained by three plane angles, any two of them are together greater than the third.
But the next—
Prop. 21. Every solid angle is contained by plane angles, which are together lese than four right angles —has no analogous theorem in the plane.
We may mention, however, that the theorems about triangles contained in the propositions of Book I., which do not depend upon the theory of parallels (that is all up to Prop. 27), have their corresponding theorems about trihedral angles. The latter are formed, if for “side of a triangle” we write “plane angle” or “face” of trihedral angle, and for “angle of triangle” we substitute “angle between two faces” where the planes containing the solid angle are called its faces. We get, for instance, from I. 4, the theorem, If two trihedral angles have the angles of tιuo faces in the one equal to the angles of two faces in the other, and have likewise the angles included by these faces equal, then the angles in the remaining faces are equal, and the angles between the other faces are equal each to each, viz., those which are opposite equal faces. The solid angles themselves are not necessarily equal, for they may be only symmetrical like the right hand and the left.
The connexion indicated between triangles and trihedral angles will also be recognized in
Prop. 22. If everytwo of three plane angles be greater than the third, and if the straight lines which contain them be all equal, a triangle may be made of the straight lines that join the extremities of those equal straight lines.
And Prop. 23. solves the problem, To construct a trihedral angle having the angles of its faces equal to three given plane angles, any two of them being greater than the third. It is, of course, analogous to the problem of constructing a triangle having its sides of given length.
Two other theorems of this kind are added by Simson in his edition of Euclid’s Elements.
§ 80. These are the principal properties of lines and planes in space, but. before we go on to their applications it will be well to define the word distance. In geometry, distance means always “shortest distance”; viz., the distance of a point from a straight line, or from a plane, is the length of the perpendicular from the point to the line or plane. The distance between two nonintersecting lines is the length of their common perpendicular, there being but one. The distance between two parallel lines or between two parallel planes is the length of the common perpendicular between the lines or the planes.
§ 81. Parallelepipeds. — The rest of the book is devoted to the study of the parallelepiped. In Prop. 24 the possibility of such a solid is proved, viz. :—
Prop. 24. If a solid be contained by six planes two and two of which are parallel, the opposite planes are similar and equal parallelograms.
Euclid calls this solid henceforth a parallelepiped, though he never defines the word. Either face of it may be taken as base, and its distance from the opposite face as altitude.
Prop 25. If a solid parallelepiped be cut by a plane parallel to two of its opposite planes, it divides the whole into two solids, the base of one of τohich shall be to the base of the other as the one solid is to the other.
This theorem corresponds to the theorem (VI. 1) that parallelograms between the same parallels are to one another as their bases. A simi]ar analogy is to be observed among a number of the remaining propositions.
§ 82. After solving a few problems we come to
Prop. 28. If a solid parallelepiped be cut by a plane passing through the diagonals of two cf the opposite planes, it shall be cut in two equal parts.
In the proof of this, as of several other propositions, Euclid neglects the difference between solids which are symmetrical like the right hand and the left.
Prop. 31. Solid parallelepipeds, which arc upon equal bases, and of the same altitude, are equal to one another.
Props. 29 and 30 contain special cases of this theorem leading up to the proof of the general theorem.
As consequences of this fundamental theorem we get
Prop. 32. Solid parallelepipeds, which have the same altitude, are to one another as their bases ; and Prop. 33, Similar solid parallelepipeds are to one another inthe triplicateratio of'their homologous sides.
If we consider, as in § 67, the ratios of lines as numbers, we may also say—
The ratio of the volumes of similar parallelepipeds is equal to the ratio of the third powers of homologous sides. t Parallelepipeds which are not similar but equal are compared by aid of the theorem
Prop. 34. The bases and altitudes of eq’d,al solid parallelepipeds are reciprocally proportional; and if the bases and altitudes be reciprocally proportional, the solid parallelepipeds are equal
§ 83. Of the following propositions the 37th and 40th are of special interest.
Prop. 37. If four straight lines be proportionals, the similar solid parallelepipeds, similarly described from them, shall also be proportionals; and if the similar parallelepipeds similarly described from four straight lines be proportionals, the straight lines shall be proportionals.
In symbols it says—
If a :b≈.c :d, then α 3 : δ 3 =c 3 : √ 3 .
Prop. 40 teaches how to compare the volumes of triangular prisms with those of parallelepipeds, by proving that a triangular prism is equal in volume to a parallelepiped, which has its altitude and its base equal to the altitude ancl the base of the triangular prism.
§ 84. From these propositions follow all results relating to the mensuration of volumes. We shall state these as we did in the case of areas. The starting-point is the “rectangular” parallelepiped, which has every edge perpendicular to the planes it meets, and which takes the place of the rectangle in the plane. If this has all its edges equal we obtain the “cube.”
If we take a certain line u as unit length, then the square on u is the unit of area, and the cube on t¢ the unit of volume, that is to say, if we wish to measure a volume we have to determine how many unit cubes it contains.
A rectangular parallelepiped has, as a rule, the three edges unequal, which meet at a point. Every other edge is equal to one of them. If a, b, c be the three edges meetin" at a point, then we may take the rectangle, contained by two of them, say by b and c, as base and the third as altitude. Let V be its volume, V' that of another rectangular parallelepiped which has the edges a’, b, c, hence the same base as the first. It follows then easily, from Prop. 25 or 32, that V: ^N' = a : a' ; or in words, Rectangular parallelepipeds on equal bases are proportional to their altitudes.
If we have two rectangular parallelepipeds, of which the first has the volume V and the edges a, bfc, and the second,the volume V' and the edges a', b', cl, we may compare them by aid of two new ones which have respectively the edges a', b, c, and a', b', c, and the volumes V 1 and V 2 . We then have
V: V 1 =α: a' ; V 1 : V 2 = δ: b', V 2 : V'=c: c'.
Compounding these, we have
V:V'=(a:a')(6:6')(c:c'), or L=*Λ.f..
V' al V e
Hence, as a special case, making V' equal to the unit cube V on u we get
V _ a be o
U w u u^~ a ' where a, β, y are the numerical values of a, b, c ; that is, The number of unit cubes in a rectangular parallelepiped is equal to tire product of the numerical values of its three edges. This is generally expressed by saying the volume of a rectangular parallelepiped is measured by the product of its sides, or by the product of its base into its altitude, which in this case is the same.
Prop. 31 allows us to extend this to any parallelepipeds, and Props. 28 or 40, to triangular prisms.
Theorem.— The volume of any parallelepiped, or of any triangular prism, is measured by the product of base and altitude.
The consideration that any polygonal prism may be divided into a number of triangular prisms, which have the same altitude and the sum of their bases equal to the base of the polygonal prism, shows further that the same holds for any prism whatever.
Book XII.
§ 85. In the last part of Book XL we have learnt how to compare the volumes of parallelepipeds and of prisms. In order to determine the volume of any solid bounded by plane faces we must determine the volume of pyramids, for every such solid may be decomposed into a number of pyramids.
As every pyramid may again be decomposed into triangular pyramids, it becomes only necessary to determine their volume. This is done by the Theorem.— Every triangular pyramid is equal in volume to one third of a triangular prism having the same base and the same altitude as the pyramid.
This is an immediate consequence of Euclid’s
Prop. 7. Every prism having a triangular base may be divided t ∣ do three pyramids that have triangular bases, and arc equal to one another.
The proof of this theorem is difficult, because the three triangular pyramids into which the prism is divided are by no means equal in shape, and cannot be made to coincide. It has first to be proved that two triangular pyramids have equal volumes, if they have equal bases and equal altitudes. This Euclid does in the following manner. He first shows (Prop. 3) that a triangular pyramid may be divided into four parts, of which two are equal triangular pyramids similar to the whole pyramid, whilst the other two are equal triangular prisms, and further, that these two prisms together are greater than the two pyramids, hence more than half the given pyramid. He next shows (Prop. 4) that if two triangular pyra∙
nιids are given, having equal bases and equal altitudes, and if each be divided as above, then the two triangular prisms in the one are equal to those in the other, and each of the remaining pyramids in the one has its base and altitude equal to the base and altitude of the remaining pyramids in the other. Hence to these pyramids the same process is again applicable. We are thus enabled to cut out of the two given pyramids equal parts, each greater than half the original pyramid. Of the remainder we can again cut out equal parts greater than half these remainders, and so on as far as we like. This process may be continued till the last remainder is smaller than any assignable quantity, however small. It follows, so we should conclude at present, that the two volumes must be equal, for they cannot differ by any assignable quantity.
To Greek mathematicians this conclusion offers far greater difficulties. They prove elaborately, by a reductio ad absurdum, that the volumes cannot be unequal. This proof must be read in the Elements. A further discussion of this method of exhaustion, as it is called, would belong to a treatise on the history of geometry. We refer readers to Hankel, Geschichte der Mathematik (p. 115 sq). We must, however, state that we have in the above not proved Euclid’s Prop. 5, but only a special case of it. Euclid does not suppose that the bases of the two pyramids to be compared are equal, and hence he proves that the volumes are as the bases. The reasoning of the proof becomes clearer in the special case, from which the general one may be easily deduced.
§ 86. Prop. 6 extends the result to pyramids with polygonal bases. From these results follow again the rules at present given for the mensuration of solids, viz., a pyramid is the third part of a triangular prism having the same base and the same altitude. But a triangular prism is equal in volume to a parallelepiped which has the same base and altitude. Hence if B is the base and h the altitude, we have
Volume of prism = Bä, Volume of pyramid = -∣ Bä, statements which have to be taken in the sense that B means the number of square units in the base, h the number of units of length in the altitude, or that B and h denote the numerical values of base and altitude.
§ 87. A method similar to that used in proving Prop. 5 leads to the following results relating to solids bounded by simple curved surfaces :—
Prop. 10. Every cone is the third part of a cylinder which has the same base, and is of an equal altitude with it.
Prop. 11. Cones and cylinders of the same altitude are to one an other as their bases.
Prop. 12. Similar cones and cylinders have to O7ic aτιother the triplicate ratio of that which the diameters of their bases have.
Prop. 13. If a cylinder be cut by a plane parallel to its opposite planes or bases, it divides the cylinder into two cylinders, one of which is to the other as the axis of the first to the axis of the other ; which may also be stated thus:—
Cyli7tdaers on the same base are proportional to their altitudes.
Prop. 14. Cones and cylinders upon equal bases are to one another as their altitudes.
Prop. 15. The bases and altitudes of equal cones and cylinders are rec iprocally proportional, and if the bases and altitudes be reciprocally proportional, the coτιes and cylinders are eqτzal to oτιe another.
These theorems again lead to formulae in mensuration, if we compare a cylinder with a prism having its base and altitude equal to the base and altitude of the cylinder. This may be done by the method of exhaustion. We get, then, the result that their bases are equal, and have, if B denotes the numerical value of the base, and ä that of the altitude,
Volume of cylinder = Bä.
Volume of cone = ¾∙ Bä. ð
§ 88. The remaining propositions relate to circles and spheres. Of the sphere only one property is proved, viz. :—
Prop. 18. Spheres have to one another the triplicate ratio of that which their diameters have. The mensuration of the sphere, like that of the circle, the cylinder, and the cone, had not been settled in the time of Euclid. It was done by Archimedes.
Book XIII.
§ 89. The 13th and last book of Euclid’s Elements is devoted to the regular solids. It is shown that there are five of them, viz. :—
1. The regular tetrahedron, with 4 triangular faces and 4 vertices;
2. The cube, with 8 vertices and 6 square faces;
3. The octohedron, with 6 vertices and 8 triangular faces;
4. The dodecahedron, with 12 pentagonal faces, 3 at each of the 20 vertices;
5. The icosahedron, with 20 triangular faces, 5 at each of the 12 vertices.
It is shown how to inscribe these solids in a given sphere, and how to determine the lengths of their edges.
These results are—if r denotes the radius of the circumscribed sphere, and a the side of the regular solid—
For tetrahedron a- — - ∣ r-, ó
,, octahedron α 2 ≈ 2r 2 ,
4
,, hexahedron or cube a- = -r i , û
,, icosahedron α 2 = 2^1 - r 2 ,
,, dodecahedron α 2 = 2^1 - ^Q^r i .
§ 90. The 13th book, and therefore the Elements, conclude with the scholium, “that no other regular solid exists besides the five ones enumerated.”
The proof is very simple. Each face is a regular polygon, hence the angles of the faces at any vertex must be .angles in equal regular polygons, must be togetheχ∙ less than four right angles (XL 21), and must be three or more in number. Each angle in a regular triangle equals two-thirds of one right angle. Hence it is possible to form a solid angle with three, four, or five regular triangles or faces. These give the solid angles of the tetrahedron, the octohedron, and the icosahedron. The angle in a square (the regular quadrilateral) equals one right angle. Hence three will form a solid angle, that of the cube, and four will not. The angle in the regular pentagon equals -ð-of a right angle. Hence three of them equal y (t,e., less than 4) right angles, and form the solid angle of the dodecahedron. Three regular polygons of six or more sides cannot form a solid angle. Therefore no other regular solids are possible.
Section II.— Higher or Projective Geometry.
It is difficult, at the outset, to characterize Projective Geometry as compared with Euclidian. But a sew examples will at least indicate the difference between the two.
In Euclid’s Elements almost all propositions refer to the magnitude of lines, angles, areas, or volumes, and therefore to measurement. The statement that an angle is right, or that two straight lines are parallel, refers to measurement. On the other hand, the fact that a straight line does or does not cut a circle is independent of measurement, it being dependent only upon the mutual “position” of the line and the circle. This difference becomes clearer if we project any figure from one plane to another. By this the length of lines, the magnitude of angles and areas, is altered, so that the projection, or shadow, of a square on a plane will not be a square; it will, however, be some quadrilateral. Again, the projection of a circle will not be a circle, but some other curve more or less resembling a circle. But one property may be stated at once,—no straight line can cut the projection of a circle in more than two points, because no straight line can cut a circle in more than two points. There are, then, some properties of figures which do not alter by projection, whilst others do. To the latter belong nearly all properties relating to measurement, at least in the form in which they are generally given. The others are said to be projective properties, and their investigation forms the subject of Projective Geometry.
Different as are the kinds of properties investigated in the old and the new sciences, the methods followed differ in a still greater degree. In Euclid each proposition stands by itself; its connexion with others is never indicated; the leading ideas contained in its proof are not stated; genoral principles do not exist. In the modern methods, on the other hand, the greatest importance is attached to the leading thoughts which pervade the whole; and general principles, which bring whole groups of theorems under one aspect, are given rather than separate propositions. The whole tendency is towards generalization. A straight line is considered as given in its entirety, extending both ways to infinity, while Euclid is very careful never to admit anything but finite quantities. The treatment of the infinite is in fact another fundamental difference between the two methods. Euclid avoids it. In modern geometry it is systematically introduced, for only thus is generality obtained.
Of the different modern methods of geometry, we shall treat principally of the methods of projection and correspondence which have proved to be the most powerful. These have become independent of Euclidian Geometry, especially through the Geometrie der Lage of V. Staudt, and the Ausdehnungslehre of Grassmann.
For the sake of brevity we shall presuppose a knowledge of Euclid’s Elements, although we shall use only a few of his propositions.
§ 1. We consider space as filled with points, lines, and planes, and these we call the elements out of which our figures are to be formed, calling any combination of these elements a “figure.”
By a line we mean a straight line in its entirety, extending both ways to infinity; and by a plane, a plane surface, extending in all directions to infinity. We suppose
That through any two points in space one and only one line may be drawn (Euch I., Def. 4, Ax. 10, Post. 2);
That through any three points which are not in a line, one and only one plane may be placed (compare p. 386, § 73, above);
That the intersection of two planes is a line (Encl. XI. 13);
That a line which has two points in common with a plane lies in the plane (Euch I., Def. 7), hence that the intersection of a line and a plane is a single point; and
That three planes which do not meet in a line have one single point in common.
These results may be stated differently in the following form;— I. A plane is determined— A point is determined—
1. By three points which do 1. By three planes which do not lie in a line; not pass through a line;
2. By two intersecting lines; 2. By two intersecting lines;
3. By a line and a point which 3. By a line and a plane which does not lie in it. does not pass through it.
II. A line is determined—
1. By two points; 2. By two planes.
The reader will observe that not only are planes determined by points, but also points by planes·; that therefore we have a right to consider the planes as elements, like points; and also that in any one of the above statements we may interchange the words point and plane, and we obtain again a correct statement, provided that these statements themselves are true. As they stand, we ought, in several cases, to add “if they are not parallel,” or some such words, parallel lines and planes being evidently left altogether out of consideration. To correct this we have to reconsider the theory of parallels.
§ 2. Let us take in a plane a line p (fig. 1), a point S not in this line, and a line q drawn through S. Then this line q will meet the line p in a point A. If we turn the line q about S towards (f , its point of intersection with p will move along p towards B, passing, on continued turning, to a greater and greater distance, until it is moved out of our reach. If we turn q still farther, its continuation will meet p, but now at the other side of A. The point of intersection has disappeared to the right and reappeared to the left. There is one intermediate position where q is parallel to p— that is where it does not cut p. In every other position it cuts p in some finite point. If, on the other hand, we move the point A to an infinite distance in p, then the line q which passes through A will be a line which does not cut p at any finite point. Thus we are led to say: Every line through S which joins it to any point at an infinite distance in p is parallel to p. But by Euclid’s 12th axiom there is but one line parallel to p through S. The difficulty in which we are thus involved is due to the fact that we try to reason about infinity as if we, with our finite capabilities, could comprehend the infinite. To overcome this difficulty, we may say that all points at infinity in a line appear to us as one, and may be replaced by a single “ideal” point, just as all points in a fixed star—which is not at an infinite, only at a great distance—cannot be distinguished by us and to beings on the earth count as one. We may therefore now give the following definitions and axiom:—
Definition. —Lines which meet at infinity are called parallel.
Axiom. — All points at an infinite distance in a line may be considered as one single point. 4 Definition.— This ideal point is called the point at infinity in the line.
The axiom is equivalent to Euclid’s Axiom 12, for it follows from either that through any point only one line may be drawn parallel to a given line.
This point at infinity in a line is reached whether we move a point in the one or in the opposite direction of a line to infinity. A line thus appears closed by this point, and we speak as if we could move a point along the line from one position A to another B in two ways, either through the point at infinity or through finite points only.
It must never be forgotten that this point at infinity is ideal, that the results based on this assumption are true for that finite region of space which is within our reach, and that beyond this region they may or may not be true, —we do not know.
The advantage of this view of parallels will become apparent at every step as we go on.
§ 3. Having thus arrived at the notion of replacing all points at infinity in a line by one ideal point, there is no difficulty in replacing all points at infinity in a plane by one ideal line.
To make this clear, let us suppose that a line p, which cuts two fixed lines a and b in the points A and B, moves parallel to itself to a greater and greater distance. It will at last cut both a and b at their points at infinity, so that a line which joins the two points at infinity in two intersecting lines lies altogether at infinity. Every other line in the plane will meet it therefore at infinity, and thus it contains all points at infinity in the plane.
AU points at infinity in a plane lie in a line, which is called the line at infinity in the plane.
It follows that parallel planes must be considered as planes having a common line at infinity, for any other plane cuts them in parallel lines (Euch XI. 16), which have a point at infinity in common.
If we next take two intersecting planes, then the point at infinity in their line of intersection lies in both planes, so that their lines at infinity meet. Hence every line at infinity meets every other line at infinity, and they are therefore all in one plane.
All points at infinity in space may be considered as lying in one ideal plane, which is called the plane at infinity.
§ 4. We have now the following definitions :—
Parallel lines are lines which meet at infinity;
Parallel planes are planes which meet at infinity;
A line is parallel to a plane if it meets it at infinity.
Theorems like this—Lines (or planes) which are parallel to a third are parallel to each other—follow at once.
This view of parallels leads therefore to no contradiction of Euclid’s Elements.
As immediate consequences we get the propositions :—
Every line meets a plane in one point, or it lies in it;
Every plane meets every other plane in a line;
Any two lines in the same plane meet.
§ 5. We have called points, lines, and planes the elements of geometrical figures. We also say that an element of one kind contains one of the other if it lies in it or passes through it.
All the elements of one kind which are contained in one or two elements of a different kind form aggregates which have to be enumerated. They are the following :—
I. Of one dimension.
1. The row, or range, of pointe formed by all points in a line, which is called its base.
2. The flat pencil formed by all the lines through a point in a plane. Its base is the point in the plane.
3. The axial pencil formed by all planes through a lino which is called its base or axis.
II. Of two dimensions.
1. The field of points and lines—that is, a plane with all its points and all its lines.
2. The pencil of lines and planes— that is, a point in space with all lines and all planes through it.
III. Of three dimensions.
The space of points—that is, all points in space.
The space of planes—that is, all planes in space⅛ IV. Of four dimensions.
The space of lines, or all lines in space.
§ 6. The word dimension in the above needs explanation.
If in a plane we take a row p and a pencil with centre Q, then through every point in p one line in the pencil will pass, and every ray in Q will cut p in one point, so that we are entitled to say a row contains as many points as a fiat pencil lines, and, we may add, as an axial pencil planes, because an axial pencil is cut by a plane in a flat pencil.
The number of elements in the row, in the flat pencil, and in the axial pencil is, of course, infinite and indefinite too, but the same in all. For a moment we shall treat it as being a definite number which we denote by a. Then a plane contains α 8 points and as many lines. To see this, take a flat pencil in a plane. It contains a lines, and each line contains a points, whilst each point in the plane lies on one of these lines. Similarly, in a plane each line cuts a fixed line in a point. But this line is cut at each point by a lines and contains α points; hence there are α 2 lines in a plane.
A pencil in space contains as many lines as a plane contains points and as many planes as a plane contains lines, for any plane cuts the pencil in a field of points and lines. Hence a pencil contains α 2 lines and α 2 planes. The field and the pencil are of two dimensions.
To count the number of points in space we observe that each point lies on some line in a pencil. But the pencil contains α 2 lines, and each line α points; hence space contains α 3 points. Each plane cuts any fixed plane in a line. But a plane contains α 2 lines, and through each pass a planes; therefore space contains α 8 planes.
Hence space contains as many planes as points, but it contains an infinite number of times more lines than points or planes. To count them, notice that every line cuts a fixed plane in one point. But α 2 lines pass through each point, and there are α 2 points in the plane. Hence there are α 4 lines in space. The space of points and planes is of three dimensions, but the space of lines is of four dimensions.
A field of points or lines contains an infinite number of rows and flat pencils; a pencil contains an infinite number of flat pencils and of axial pencils; space contains a triple infinite number of pencils and of fields, α 4 rows and axial pencils, and α 5 flat pencils— or, in other words, each point is a centre of α 2 flat pencils.
§ 7. The above enumeration allows a classification of figures.
Figures in a row consist of groups of points only, and figures in the flat or axial pencil consist of groups of lines or planes. In the plane we may draw polygons; and in the pencil or in the point, solid angles, and so on.
We may also distinguish the different measurements. We have— In the row, lengtli of segment;
In the flat pencil, angles;
In the axial pencil, dihedral angles between two planes;
I a the plane, areas;
In the pencil, solid angles;
In the space of points or planes, volumes.
Segments of a Line.
§ 8. Any two points A and B in space determine on the line through them a finite part, which may be considered as being described by a point moving from A to B. This we shall denote by AB, and distinguish it from BA, which is supposed as being described by a point moving from B to A, and hence in a direction or in a “sense” opposite to AB. Such a finite line, which has a definite sense, we shall call a “segment,” so that AB and BA denote different segments, which are said to be equal in length but of opposite sense. The one sense is often called positive and the other negative.
In introducing the word “sense” for direction in a line, we have the word direction reserved for direction of the line itself, so that different lines have different directions, unless they be parallel, whilst in each line we have a positive and negative sense.
We may also say, with Professor Clifford, that AB denotes the “step ’’ of going from A to B.
§ 9. If we now have three points A, B, C in a line (fig. 2), the step AB will bring us from A to B, and the step BC from B to C. Hence both steps are equivalent to the one step AC. This is expressed by saying that AC is the “sum” of AB and BC; in symbols—
AB + BC = AC, where account is to be taken of the sense.
This equation is true whatever be the position of the three points on the line. As a special case we have
AB + BA=0 (1), and similarly
AB + BC + CA=0 (2), which again is tine for any three points in a line.
We further write
AB=-BA, where — denotes negative sense.
We can then, just as in algebra, change subtraction of segments into addition by changing the sense, so that AB—CB is the same as AB + (-CB) or AB + BC. A figure will at once show the truth of this. The sense is, in fact, in every respect equivalent to the “sign” of a number iu algebra.
§ 10. Of the many formulae which exist between points in a line we shall have to use only one more, which connects the segments between any four points A, B, C. D in a line. We have—
BC = BD+DC, CA = CD +DA, AB≈AD+DB; or multiplying these by AD, BD, CD respectively, we get—
BC . AD = BD. AD + DC. AD = BD. AD-CD. AD
CA . BD = CD . BD + DA. BD = CD . BD - AD. BD
AB. CD = AD . CD + DB . CD = AD. CD - BD. CD.
It will be seen that the sum of the right hand sides vanishes, hence that
BC. AD + CA. BD + AB. CD - 0 . . . (3)
for any four points on a line.
§ 11. If C is any point in the line AB, then we say that C AC
divides the segment AB in the ratio ð— , account being taken of the sense of the two segments AC and CB. If C lies between A and C the ratio is positive, as AC and CB have the same sense. But if C lies without the segment AB, i.e., if C divides AB externally, then the ratio is negative. To see how the value of this ratio changes with C, . _
we will move C ft † ¾! B Ç
along the whole _. „
line (fig. 3), whilst 1 S- 0 ∙
A and B remain fixed. If C lies at the point A, then AC = 0, hence the ratio AC: CB vanishes. As C moves towards B, AC increases and CB decreases, so that our ratio increases. At the middle point Μ of AB it assumes the value +1, and then increases till it reaches an infinitely large value, when C arrives at B. On passing beyond B the ratio becomes negative. If C is at P we have AC=AP=AB + BP, hence
AC AB BP AB 1
CB~PB + PB BP -1 *
In the last expression the ratio AB: BP is positive, has its greatest value ∞ when C coincides with B, and vanishes when BC becomes infinite. Hence, as C moves from B to the right to the point at infinity, the ratio AC: CB varies from —∞ to —1.
If on the other hand C is to the left of A, say at Q, we have AC = AQ = AB + BQ = AB-QB, hence —
Here AB <QB, hence the ratio AB: QB is positive and always less than one, so that the whole is negative and <1. H C is at the point at infinity it is —1, and then increases as C moves to the right, till for C at A we get the ratio =0. Hence—
“As C moves along the line from an infinite distance to the left to an infinite distance at the right, the ratio always increases; it starts with the value —1, reaches 0 at A, +1 at Μ, ∞ at B, changes now sign to — ∞ , and increases till at an infinite distance it reaches again the value —1. It assumes therefore all possible values from — ∞ to + oo, and each value only once, so that not only does every position of C determine a definite value of the ratio AC: CB, but also, conversely, to every positive or negative value of this ratio belongs one single point in the line AB.
Projection and Cross-Ratios.
§ 12. If we join a point A to a point S, then the point where the line SA cuts a fixed plane τr is called the projection of A on the plane π from S as centre of projection. If we have two planes τr and π' and a point S, we may project every point A in π to the other plane. If A' is the projection of A, then A is also the projection of A', so that the relations are reciprocal. To every figure in π we get as its projection a corresponding figure in π'.
It will be our business to find such properties of figures as remain true for the projection, and which are called projective properties. For this purpose it will be sufficient to consider at first only constructions in one plane.
Let us suppose we have given in a plane two lines p and sk and a centre S (fig. 4); we may then project the points in p from S to p'. Let A', B'... be the projections of A, B..., the point at infinity in p which we shall denote by J will be projected into a finite point Γ inp', viz., into the point where the parallel to p through S cuts sk. Similarly one point J in p will be projected into the point Γ at infinity in V. This point J is of course the point where the parallel to sk through S cuts p. We thus see that every point in 79 is projected into a single point in p ’ .
A glance at fig. 5 shows that a segment AB will be projected into a segment A'B' which is not equal to it, at least not as a rule; and also that the ratio AC: CB is not equal to the ratio A'C': C'B' formed by the projections. These ratios will become equal only if p and p' are parallel, for in this case the triangle SAB is similar to the triangle SA'B'. Between three points in a line and their projections there exists therefore in general no relation. But between four points a relation does exist.
§ 13. Let A, B, C, D be four points in p, A!, B', C', D' their projections in p', then the ratio of the two ratios AC: CB and AD: DB into which C and D divide the segment AB is equal to the corresponding expression between A', B', C', D'. In symbols we have—
AC . AD = AO' AT'
CB ’ DB C'B' : D'B' ’
. This is easily proved by aid of similar triangles.
Through the points A and B on p draw parallels to p', which cut the projecting rays in C 2 , D 2 , B 2 and A 1 , C 1 , D 1 , as indicated in fig. 6. Tlιe two triangles ACC 2 and BCC 1 will be similar, as will also be the triangles ADD 2 and BDD 1 .
We have therefore—
AC AC 2 AD AD.,
CB C 1 B ’ DB~D 1 B , where account is taken of the sense.
Hence—
AC . AD = AC a AD 2 _ AC 2 - C 1 B .
CB ' DB C 1 B : D 1 B AD, : D,B ’
but
∆a=4'C' antl c 1 b~cw .
AD 2 A'D' D 1 B~D'B' ’
so that the above expression becomes
A'C' C'B' A'D' : D'B' ’ A'∩' A'∏'
which is equal to; —— as required.
C13 Do
This result is of fundamental importance.
The expression formed has been called by Chasles the anharmonic ratio of the four points A, B, C, D. Instead of this Professor Clifford has proposed the shorter and more expressive, name of “crossratio.’' We shall adopt the latter. We have then the Fundamental Theorem.— The cross-ratio of four points in a tine is equal to the cross-ratio of their projections on any other line which lies in the same plane with it.
§ 14. Before we draw conclusions from this result, we must investigate the meaning of a cross-ratio somewhat more fully.
If four points A, B, C, D are given, and we wish to form their cross-ratio, we have first to divide them into two groups of two, the points in each gιoup being taken in a definite order. Thus, let A, B be the first, C, D the second pair, A and C being the first points in each pair. The cross-ratio is then the ratio AC: CB divided by AD: DB. This will be denoted by (ABCD), so that (ABCD)≈^:^.
This is easily remembered. In order to write it out, make first the two lines for the fractions, and put above and below these
A A
the letters A and B in their places, thus, —g : ~jj î and then fill up, crosswise, the first by C and the other by D.
§ 15. If we take the points in a different order, the value of the cross-ratio will change. We can do this in twenty-four different ways by forming all permutations of the letters. But of these twenty-four cross-ratios groups of four are equal, so that there are really only six different ones.
We have the following rules :—
I. If in a cross-ratio the two groups be interchanged, its value remains unaltered.
(ABCD) = (CDAB).
II. If in a cross-ratio the two points belonging to one of the two groups be interchanged, the cross-ratio changes into its reciprocal.
( abcd )-(λ⅛C)∙
III. If in a cross-ratio the two middle letters be interchanged, the cross-ratio κ changes into its complement 1 - κ.
(ABCD) = 1- (ACBD).
The first two are easily proved by writing out their values. The third is proved by aid of the formula (3), § 10,
BC. AD + CA. BD + AB. CD = 0.
If we divide this by CB . AD we get
, CA. BD , AB . CD ~ 1 + CB. AD + CB .AD~° ’ or,
AC AD AB AD .
CB : DB + BC : DC - ’
that is,
(ABCD) + (ACBD) = 1, which was 11 be proved.
IV. From II. it follows at once that if we interchange the elements in each pair, the cross-ratio remains unaltered; and thus we see that
(ABCD) = (CDAB) by I.
= (BADC) = (DCBA) by II.
§ 16. By aid of these rules we get the following results :— (ABCD) ≈ (BADC) = (CDAB) - (DCBA) = κ. ( ABDC) = ( B ACD) - (CDBA) = (DCAB)=J.
(ACBD) = (BDAC) = (CADB) = (DBCA) = 1 - κ .
( ACDB) = ( BDCA) = (CABD) = (DB AC) =
(ADBC) = (BCAD) ≈ (CBDA) = (DACB) = — - 1 .
(ADCB) = (BCDA) = (CBAD) = (DABC) = ^−.
In the theorem that the cross-ratio of four points equals that of 'the projections, the points have, of course, to be taken in the sañie order.
§ 17. If one of the points of which a cross-ratio is formed is the point at infinity in the line, the cross-ratio changes into a simple ratio. It is convenient to let the point at infinity occupy the last place in the symbolic expression for the cross-ratio. Thus if 1 is a point at infinity, we have
(ABCI)=-~, because
AI:IB--1.
Every common ratio of three points in a line may thus De expressed as a cross-ratio, by adding the point at infinity to the group of points.
§ 18. If the points have special positions, the cross-ratios κ may have such a value that, of the six different ones, two and two become equal. If the first two shall be equal, we get κ = ^> °r κ 2 =l, κ=±l.
AC AD
If we take κ= +1, we have (ABCD) = 1, or ðB = DB > that ιs > the points C and D coincide, provided that A and B are different. This is too special a case to be of much interest.
If, however, we take κ=-l, so that (ABCD)=-1, we have θβ= - ∙θβ . Hence C and D divide AB internally and externally in the same ratio.
The four points are in this case said to be harmonic points, and C and D are said to be harmonic conjugates with regard to A and B.
But we have also (CDAB) = -1, so that A and B are harmonic conjugates with regard to C and D.
The principal property of harmonic points, upon which almost all applications depend, is this, that their cross-ratio remains unaltered if we interchange the two points belonging to one pair, viz.:
( ABCD) - (ABDC) - (BACD).
For four harmonic points the six cross-ratios which are generally different become two and two equal:
κ= -1, l-κ = 2, jτy = i∙
l- 1 ,+= i ,⅛
Hence if we get four points whose cross-ratio is 2 or ⅜, then they are harmonic, but not arranged so that conjugates are paired. If this is the case the cross-ratio = —1.
§ 19. If we equate any two of the above six values of the crossratios, we get either κ = l, 0, ∞, or κ= -1, 2, ⅛, or else κ becomes a root of the equation κ 2 - κ +1 = 0, that is, an imaginary cube root of - 1. In this case the six values become three and three equal, so that only two different values remain. This case, though important in the theory of cubic curves, is for our purposes of no interest, whilst harmonic points are all-important.
§ 20. From the definition of harmonic points, and by aid of §11, the following properties are easily deduced.
If C and D are harmonic conjugates with regard to A and B, then one of them lies in, the other without AB; it is impossible to move from A to B without passing either through C or through D; the one blocks the finite way, the other the way through infinity. This is expressed by saying A and B are “separated” by C and D.
For every position of C there will be one and only one point D which is its harmonic conjugate with regard to any point pair A, B.
If A and B are different points, and if C coincides with A or B, D does. But if A and B coincide, one of the points C or D, lying between them, coincides with them, and the other may be anywhere in the line. It follows that, “if of four harmonic conjugates two coincide, then a third coincides with them, and the fourth may be any j>oint in the line."
If C is the middle point between A and B, then D is the point at infinity for AC:CB=+1, hence AD: DB must be equal to -1. The harmonic conjugate of the point at infinity in a line with regard to two points A, B is the middle point of AB.
This important property gives a first example how metric properties are connected with projective ones.
§ 21. Complete Quadrilateral.— A figure formed by four lines in a plane is called a complete quadrilateral or, shorter, a four-side (fig. 7). The figure has six vertices, that is, points where the sides meet, and three diagonals AB, EF, GH, which join opposite vertices.
Similarly a figure formed by four points in a plane is called a four-point. It has six sides, which join the points, or vertices, and three diagonal points, where the sides meet.
The three diagonals of a four-side cut each other harmonically.
If we project the points E, F, K, D from H to AB, we get (EFKD) = (ABCD), and if we project them from G, we get
(EFKD)≈(BACD); so that
(ABCD) = (BACD), that is, the cross-ratio (ABCD) is equal to its reciprocal, hence = -1, as the four points are all different. This gives the important theorem :—
Theorem. — In every four-side any diagonal is cut harmonically by the other two.
This allows the solution of the problem :—
Problem.— To construct the harmonic conjugate D to a point C with regard to two given points A and B.
Solution.— Through A draw any two lines, and through C one cutting the former two in G and H. Join these points to B, cutting the former two lines in E and F. The point D where EF cuts AB will be the harmonic conjugate required.
This remarkable construction requires nothing but the drawing of lines, and is therefore independent of measurement. It follows, also, that all four-sides which have two vertices at A. and B, and, one diagonal passing through C, will each have the third diagonal passing through D.
§ 22. The theory of cross-ratios may be extended from points in a row to lines in a flat pencil, and to planes in an axial pencil. We have seen (§ 13) that if the lines which join four points A, B, C, D to any point S be cut by any other line in A', B', C, D', then (ABCD) = (A'B'C'D'). In other words, four lines in a flat pencil are cut by every other line in four points whose cross-ratio is constant.
Definition.— By the cross-ratio of four rays in a flat pencil is meant the cross-ratio of the four points in which the rays are cut by any line. If a, b, c, d be the lines, then this cross-ratio is denoted, by (abed). Definition.— By the cross-ratio of four planes in an axial pencil is understood the cross-ratio of the four points in which any line cuts the planes, or, what is the same thing, the cross-ratio of the four rays in which any plane cuts the four planes.
In order that this definition may have a meaning, it has to be proved that all lines cut the pencil in points which have the same cross-ratio. This is seen at once for two intersecting lines, as their plane cuts the axial pencil in a flat pencil, which is itself cut by the two lines. The cross-ratio of the four points on one line is therefore equal to that on the other, and equal to that of the four rays in the flat pencil.
If two non-intersecting lines p and q cut the four planes in A, B, 0, D and A', B', C', D', draw a line r to meet both p and q r and let this line cut the planes in A",B",C",D". Then (ABCD) = (A'B'C'D ’), for each is equal to (A"B"C"D").
§ 23. We may now also extend the notion of harmonic elements, viz. :—
Definition.— Four rays in a flat pencil and four planes in air axial pencil are said to be harmonic if their cross-ratio equals - 1, that is, if they are cut by a line in four harmonic points.
Harmonic pencils are constructed by aid of the theorem in § 21, which may now be stated thus :—
In a four-side two sides are harmonic conjugates with regard to the diagonal through their intersection and the line from this point to the point where the other diagonals meet. Or thus: In a four-point the lines joining one diagonal point to the other two are harmonic conjugates with regard to the sides passing through the first.
If we understand by a “median line” of a triangle a line which joins a vertex to the middle point of the opposite side, and by a “median line” of a parallelogram a line joining middle points of opposite sides, we get as special cases of the last theorem :—
The diagonals and median lines of a parallelogram form an harmonic pencil ; and At a vertex of any triangle, the two sides, the median line, and the line y>arallel to the base form an harmonic pencil.
Taking the parallelogram a rectangle, or the triangle isosceles, we get :—
Theorem.— Any two lines and the bisections of their angles form an harmoic pencil. Or :— In an harmonic pencil, if two conjugate rays are perpendicular, then the other two arc equally inclined to them ; and, conversely, if one ray bisects the angle between conjugate rays, it is perpendicular to its conjugate.
This connects perpendicularity and bisection of angles with projective properties.
§ 24. We add a few theorems and problems which are easily proved or solved by aid of harmonics.
An harmonic pencil is cut by a line parallel to one of its rays in three equidistant points.
Through a given point to draw a line such that the segment determined on it by a given angle is bisected at that point.
Having given two parallel lines, to bisect on either any given segment without using a pair of compasses.
Having given in a line a segment and its middle point, to draw through any given point in the plane a line parallel to the given line.
To draw a line which joins a given point to the intersection oí two given lines which meet off the drawing paper (by aid of § 21).
Correspondence.
§ 25. Two rows, p and p', which are one the projection of the other (as in fig. 5), stand in a definite relation to each other, characterized by the following properties.
1. To each point in either corresponds one point in the other; that is, those points are said to correspond which are projections of one another.
2. The cross-ratio of any four points in one equals that of the corresponding points in the other.
3. The lines joining corresponding points all pass through the same point.
If we suppose corresponding points marked, and the rows brought into any other position, then the lines joining corresponding points will no longer meet in a common point, and hence the third of the above properties will not hold any longer; but we have still a correspondence between the points in the two rows possessing the first two properties. Such a correspondence has been called a one-one correspondence, whilst the two rows between which such correspondence has been established are said to be projective or homologoιιs. Two rows which are each the projection of the other are therefore projective. We shall presently see, also, that any two projective rows may always be placed in such a position that one appears as the projection of the other. If they are in such a position the rows are said to be in perspective position, or simply to be perspective.
§ 26. The notion of a one-one correspondence between rows may be extended to flat and axial pencils, viz., a flat pencil will be said to be projective to a flat pencil if to each ray in the first corresponds one ray in the second, and if the cross-ratio of four rays in one equals that of the corresponding rays in the second.
Similarly an axial pencil may be projective to an axial pencil. But a flat pencil may also be projective to an axial pencil, or either pencil may be projective to a row. The definition is the same in each case: there is a one-one correspondence between the elements, and four elements have the same cross-ratio as the corresponding ones.
§ 27. There is also in each case a special position which is called perspective, viz. :—
1. Two projective rows are perspective if they lie in the same plane, and if the one row is a projection of the other.
2. Two projective flat pencils are perspective—(a) if they lie in the same plane, and have a row as a common section; (β) if they lie in the same pencil (in space), and are both sections of the same axial pencil; (7) if they are in space and have a row as common section, or are both sections of the same axial pencil, one of the conditions involving the other.
3. Two projective axial pencils, if their axes meet, and if they have a flat pencil as a common section.
4. A row and a projective flat pencil, if the row is a section of the pencil, each point lying in its corresponding line.
5. A row and a projective axial pencil, if the row is a section of the pencil, each point lying in its corresponding line.
6. A flat and a projective axial pencil, if the former is a section of the other, each ray lying in its corresponding plane.
That in each case the correspondence established by the position indicated is such as has been called projective follows at once from the definition. It is not so evident that the perspective position may always be obtained. We shall show in § 30 this for the first three cases. First, however, we shall give a few theorems which relate to the general correspondence, not to the perspective position.
§28. Theorem.— Two rows or pencils, flat or axial, which are projective to a third are projective to each other, as follows at once from the definitions.
§ 29. Fundamental Theorem.— If two rows, or two pencils, either flat or axial, or a row and a pencil, shall be projective, we may assume to any three elements in the one the three corresponding elements in the other, and then the correspondence is τιniquely determined. Proof.— If in two projective rows we assume that the points A, B, C in the first correspond to the given points A', B', C' in the second, then to any fourth point D in the first will correspond a point D' in the second, so that
(ABCD) = (A'B'C'D').
But there is only one point, D', which makes the cross-ratio (A'B'C'D') equal to the given number (ABCD).
The same reasoning holds in the other cases.
§ 30. Theorem.— If two rows are perspective, then the lines joining corresponding points all meet in a point, the centre of projection; and the point in which the two bases of the rows intersect as a point in the first row coincides with its corresponding point in the second.
This follows from the definition. The converse also holds, viz. :— Theorem.— If two projective rows have such a position that one point in the one coincides with its corresponding point in the other, then they are perspective, that is, the lines joining corresponding points all pass through a common point, and form a flat pencil. Proof.— Let A, B, C, D . . . be points in the one, and A', B', C', D' . . . the corresponding points in the other row, and let A be made to coincide with its corresponding point A'. Let S be the point where the lines BB' and CC' meet, and let us join S to the point D in the first row. This line will cut the second row in a point D", so that A,B,C,D are projected from S into the points A,B',C',D". The cross-ratio (ABCD) is therefore equal to (AB'C'D"), and by hypothesis it is equal to (A ’ B'C , D'). Hence (A'B'C ’ D") = (A'B'C'D'), that is, D" is the same point as D'.
§ 31. Theorem. — If two projected flat pencils in the same plane are perspective, then the intersections of corresponding lines form a row, and the line joining the two centres as a line in the first pencil corresponds to the same line as a line in the second. And conversely, Theorem.— If two projective pencils in the same plane, but with different centres, have one line in the one coincident with its corresponding line in the other, then the two pencils are perspective, that is, the intersection of corresponding lines lie in a line.
The proof is the same as in § 30.
§ 32. Theorem. —If two projective flat pencils in the same point (pencil in space), but not in the same plane, are perspective, then the planes joining corresponding rays all pass through a line (they form an axial pencil), and the line common to the two pencils (in which their planes intersect) corresponds to itself. And conversely, Theorem.— If two flat pencils which have a common centre, but do not lie in a common plane, are placed so that one ray in the one coincides with its corresponding ray in the other, then they are perspective, that is, the planes joining corresponding lines all pass through a line. .
§ 33. Theorem.— If two projective axial pencils are perspective, then the intersection of corresponding planes lie in a plane, and the plane common to the two pencils (in which the two axes lie) corresponds to itself. And conversely, Theorem. —If two projective axial pencils are placed in such a position that a plane in the one coincides with its corresponding plane, then the two pencils are perspective, that is, corresponding planes meet in lines which lie in a plane.
The proof again is the same as in § 30.
§ 34. These theorems relating to perspective position become illusory if the projective rows of pencils have a common base. We then have:—
Theorem.— In two projective rows on the same line—and also in two projective and concentric flat pencils in the same plane, or in two projective axial pencils with a common axis—every element in the one coincides with its corresponding element in the other as soon as three elements in the one coincide with their corresponding elements in the other.
Proof (in case of two rows).—Between four elements A, B, C, D and their corresponding elements A', B', C', D' exists the relation (ABCD) = (A'B'C'D'). If now A', B', C' coincide respectively with A, B, C, we get (ABCD) = (ABCD'); hence D and D' coincide.
The last theorem may also be stated thus :—
Theorem.— In two projective rows or pencils, which have a common base but are not identical, not more than two elements in the one can coincide with their corresponding elements in the other.
Thus two projective rows on the same line cannot have more than two pairs of coincident points unless every point coincides with its corresponding point.
It is easy to construct two projective rows on the same line, which have two pairs of corresponding points coincident. Let the points A, B, C as points belonging to the one row correspond to A, B, and C' as points in the second. Than A and B coincide with their corresponding points, but C does not. It is, however, not necessary that two such rows have twice a point coincident with its corresponding point; it is possible that this happens only once or not at all. Of this we shall see plenty of examples as we go on.
§ 35. If two projective rows or pencils are in perspective position, we know at once which element in one corresponds to any given element in the other. If p and q (fig. 8) are two projective rows, so that K corresponds to itself, and if we know that to A and B in p correspond A' and B' in q, then the point S, where AA' meets BB', is the centre of projection, and hence, in order to find the point C z corresponding to C, we have only to join C to S; the point C', where this line cuts q, is the point required.
If two flat pencils, S 1 and S 2 , in a plane are perspective (fig. 9), we need only to know two pairs, a, a ’ and b, b ’ , of corresponding rays in order to find the axis s of projection. This being known, a ray c' in S 2 , corresponding to a given ray c in S 1 , is found by joining S 2 to the point where c cuts the axis s.
Asimilar construction holds in the other casesof perspective figures. On this depends the solution of the following general problem.
§ 36. Problem.— Three pairs of corresponding elements in two projective rows or pencils being given, to determine for any element in one the corresponding element in the other.
We solve this in the two cases of two projective rows and of two projective flat pencils in a plane.
Problem I.—Let A, B. C lie three Problem II,—Let α, ò, c be three points in a row s, A',B', C' the corre-rays in a pencil S, α', b', c' the corresponding points in a projective row sponding rays in a projective pencil both being in a plane; it is reS', both being in the same plane; it quired to find for any point D in s is required to find for any ray d in S the corresponding point D' in s'. the corresponding ray d' in S'.
The solution is made to depend on the construction of an auxiliary row or pencil which is perspective to both the given ones. This is found as follows :—
Solution· of Problem I.—On the line joining two corresponding points, say AA' (fig. 10), take any two points, S and S', as centres of auxiliary pencils. Join the intersection B 1 of SB and S'B' to the intersection C 1 of SC and S'C ’ by the line s 1 . Then a row on s 1 will be perspective to s with S as centre of projection, and to s' with S' as centre. To find now the point D' on s' corresponding to a point D on s we have only to determine the point D 1 , where the line SD cuts s 1 , and to draw S'D 1 ; the point where this line cuts s' will be the required point D'.
Proof.— The rows s and s' are both perspective to the row s 1 , hence they are projective to one another. To A, B, C, D on s correspond A 1 , B 1 , C j , D 1 ons 1 , and to these correspond A', B', C', D' on s'; so that D and D' are corresponding points as required.
Solution of Problem II.—Through the intersection A of two corresponding rays a and α' (fig. 11), take two lines, s and s', as bases of auxiliary rows. Let S 1 be the point where the line b l , which joins B and B', cuts the line c 1 , which joins C aud C'. Then a pencil S 1 will be perspective to S with s as axis of projection. To find the ray d' in S' corresponding to a given ray d in S, cut d by sat D; project this point from S 1 to D' on s' and join D' to S'. This will be the required ray.
Proof.— That the pencil S 1 is perspective to S and also to S' follows from construction. To the lines α 1 , b 1 , c 1 , d 1 in S 1 correspond the lines a, b, c, d in S and the lines α', b', d, d' in S', so* that d and d' are corresponding rays.
In the first solution the two centres, S, S', are any two points on a line joining any two corresponding points, so that the solution of the problem allows of a great many different constructions. But whatever construction be used, the point D', corresponding to D, must be always the same, according to the theorem in § 29. This gives rise to a number of theorems, into which, however, we shall not enter. The same remarks hold for the second problem.
§ 37. As a further application of the theorems about perspective rows and pencils we shall prove the following important theorem.
Theorem.— If ABC and A'B'C' (fig. 12) be two triangles, such that the lines AA', BB', CC' meet in a point S, then the intersections of BC and B'C', of CA and C'A', and of AB and A'B' will lie in a line.
Proof. — Let a, b, c denote the lines AA', BB', CC , , which meet at S. Then these may be taken as bases of projective rows, so that A, A', S on a correspond to B, B', S on b, and to C, C', S on c. As the point S is common to all, any two of these rows will be perspective.
If S, be the centre of projection of rows b and c, S 2 ,, ,, ,, c and a,
S 3 ,, ,, ,, a and b, and if the line S 1 S 3 cuts a in A 1 , and b in B 1 , and c in C 1 , then Λ 1 , B 1 will be corresponding points in a and b, both corresponding to C 1 in c. But a and b are perspective, therefore the line A 1 B 1 , that is S 1 S 2 , joining corresponding points must pass through the centre of projection S 3 of a and b. In other words, S 1 , S 2 , S 3 lie in a line. This is Desargues’s celebrated theorem if we state it thus :—
Theorem of Desargues. —If each of two triangles has one vertex on each of three concurrent lines, then the intersections of corresponding sides lie in a line, those sides being called corresponding which are opposite to vertices on the same line.
The converse theorem holds also, viz.:—
Theorem.— If the sides of one triangle meet those of another in three points which lie in a line, then the vertices lie on three lines which meet in a point.
The proof is almost the same as before.
§ 38. Metrical relations between projective rows.— Every row contains one point which is distinguished from all others, viz., the point at infinity. In two projective rows, to the point I at infinity in one corresponds a point I' in the o"ther, and to the point J' at infinity in the second corresponds a point J in the first. The points I' and J are in general finite. If now A and B are any two points in the one, A', B' the corresponding points in the other row, then
(ABJI) = (A'B'JT).
Or.
AJ Al A'J' AT.
JB : IB J'B' : ΓB' ’ but, by § 17,
Al _ A'J' _
IB “J'B' = ’
therefore the last equation changes into
AJ AT _ JB ∙ ΓB' = 1, or into
AJ. AT = BJ. BT, that is to say—
Theorem.— The product of the distances of any two corresponding points in two projective rows from the points which correspond to the points at infinity in the other is constant, viz., AJ . AT — k. Steiner has called this number k the Power of the correspondence.
§ 39. Similar Bows.— If the points at infinity in two projective rows correspond so that Γ and J are at infinity, this result loses its meaning. But if A, B, C be any three points in one A', B', C' the corresponding ones on the other row, we have
(ABCI) = (A'B'CT), which reduces to
AC AO' AC _ BC CB “C'B' ’ ° r A'C' B'C' ’
that is, corresponding segments are proportional. Conversely, if corresponding segments are proportional, then to the point at infinity in one corresponds the point at infinity in the other. If we call such rows similar, we may state the result thus—
Theorem.— Two projective rows are similar if to the point at infinity in one corresponds the point at infinity in the other, and conversely, if two rows are similar then they are projective, and the points at infinity are corresponding points.
From this the well-known propositions follow :—
Two lines are cut proportionally (in similar rows) by a series of parallels. The rows arc perspective, with centre of projection at infinity.
If two similar rows are placed parallel, then the lines joining homologous points pass through a common point.
§ 40. Theorem.— If two flat pencils be projective, then there exists in either one single pair of lines at right angles to one another, such that the corresponding lines in the other pencil are again at right angles.
To prove this, we place the pencils in perspective position (fig. 13) by making one ray coincident with its corresponding ray. Corresponding rays meet then on a line p. And now we draw the circle which has its centre 0 on p, and which passes through tho centres S and S' of the two pencils. This circle cuts p in two points H and K. The two pairs of rays, It, k and h', k' , joining these points to S and S' will be pairs of corresponding rays at right angles. The construction gives in general but one circle, but if the line p is the perpendicular bisector of SS', there exists an infinite number, and to every right angle in the one pencil corresponds a right angle in the other. Principle of Duality.
§ 41. It has been stated in § 1 that not only points, but also planes and lines, are taken as elements out of which figures are built up. We shall now see that the construction of one figure which possesses certain properties gives rise in many cases to the construction of another figure, by replacing, according to definite rules, elements of one kind by those of another. The new figure thus obtained will then possess properties which may be stated as soon as those of the original figure are known.
We obtain thus a principle, known as the principle of duality or of reciprocity, which enables us to construct to any figure not containing any measurement in its construction a reciprocal figure, as it is called, and to deduce from any theorem a reciprocal theorem, for which no further proof is needed.
It is convenient to print reciprocal propositions on opposite sides of a page broken into two columns, and this plan will occasionally be adopted.
We begin by repeating in this form a few of our former statements :—
Two points determine a line. Two planes determine a line.
Three points which are not in a Three planes which do not pass line determine a plane. through a line determine a point.
A line and a point without it deA line and a plane not through it termine a plane. determine a point.
Two lines in a plane determine a Two lines through a point deterpoint. mine a plane.
These propositions show that it will be possible, when any figure is given, to construct a second figure by taking planes instead of points, and points instead of planes, but lines where we had lines.
For instance, if in the first figure we take a plane and three points in it, we have to take in the second figure a point and three planes through it. Tlιe three points in the first, together with the three lines joining them two and two, form a triangle; the three planes in the second and their three lines of intersection form a trihedral angle. A triangle and a trihedral angle are therefore reciprocal figures.
Similarly, to any figure in a plane consisting of points and lines will correspond a figure consisting of planes and lines passing through a point S, and hence belonging to the pencil which has S as centre.
The figure reciprocal to four points in space which do not lie in a plane will consist of four planes which do not meet in a point. In this case each figure forms a tetrahedron.
§ 42. As other examples we have the following :—
To a row is reciprocal an axial pencil,
„ a flat pencil „ a flat pencil,
,, a field of points and lines „ a pencil of planes and lines,
„ the space of points „ the space of planes.
For the row consists of a line and all the points in it, reciprocal to it therefore will be a line with all planes through it, that is, an axial pencil; and so for the other cases.
This correspondence of reciprocity breaks down, however, if we take figures which contain measurement in their construction. For instance, there is no figure reciprocal to two planes at right angles, because there is no segment in a row which has a magnitude as definite as a right angle.
We add a few examples of reciprocal propositions which are easily proved.
Theorem.— If A, B, C, Dare any four Theorem.— If a, β, γ, ð are four Jn i ι 1 r∏ ,1 sp , c xJ and if the lines AB planes in space, and if the lines aβ Slp⅛"o⅛ 1 ⅛ u i¾ 1 ≤ “<! χ> ≡⅛ “∞ well as AD and BC, meet. he m a P 0mt (pencil), hence also aβ aud γδ, as well as aδ and βγ, meet.
Theorem.— If of any number of lines everyone meets every other, whilst all do not lie in a point then all lie in a lie in a plane, then all lie in a Plane. point (pencil).
§ 43. Reciprocal figures as explained lie both in space of three dimensions. If the one is confined to a plane (is formed of elements which lie in a plane), then the reciprocal figure is confined to a pencil (is formed of elements which pass through a point).
But there is also a more special principle of duality, according to which figures are reciprocal which lie both in a plane or both in a pencil. In the plane we take points and lines as reciprocal elements, for they have this fundamental property in common, that two elements of one kind determine one of the other. In the pencil, on the other hand, lines and planes have to be taken as reciprocal, and here it holds again that two lines or planes determine one plane or line.
Thus, to one plane figure we can construct one reciprocal figure in the plane, and to each one reciprocal figure in a pencil. We mention a few of these. At first we explain a few names :—
A figure consisting of n points in A figure consisting of n lines in a a plane will be called an n-point. plane will be called an n-side.
A figure consisting of n planes in a A figure consisting of n lines in a pencil will be called an n-flat. pencil will be called an n-edge.
It will be understood that an n-side is different from a polygon of n sides. The latter has sides of finite length and n vertices, the former has sides all of infinite extension, and every point where two of the sides meet will be a vertex; A similar difference exists between a solid angle and an n-edge or an n-flat. We notice particularly—
A four-point has six sides, of which A four-side has six vertices, of two and two are opposite, and three which two and two are opposite, diagonal points, which are intersec-and three diagonals, which join tions of opposite sides. opposite vertices.
A four-flat has six edges, of which A four-edge has six faces, of which two and two are opposite, aud three two and two are opposite, and three diagonal planes, which pass through diagonal edges, which are intersecopposite edges. tions of opposite faces.
A four-side is usually called a complete quadrilateral. The above notation, however, seems better adapted for the statement of reciprocal propositions.
§44.
If a point moves in a plane it de-If a line moves in a plane it en-scribes a plane curve. velopes a plane curve (fig. 14).
If a plane moves in a pencil it en-If a line moves in a pencil it developes a cone. scribes a cone.
A curve thus appears as generated either by points, and then we call it a “locus,” or by lines, and then we call it an “envelope.” In the same manner a cone, which means here a surface, appears either as the locus of lines passing through a fixed point, the “vertex” of the cone, or as the envelope of planes passing through the same point.
To a surface as locus of points corresponds, in the same manner, a surface as envelope of planes; and to a curve in space as locus of points corresponds a developable surface as envelope of planes. Of the latter we shall not say any more at present.
It will be seen from the above that we may, by aid of the principle of duality, construct for every figure a reciprocal figure, and that to any property of the one a reciprocal property of the other will exist, as long as we consider only properties which depend upon nothing but the positions and intersections of the different elements and not upon measurement.
For suc∙lι propositions it will therefore be unnecessary to prove more than one of two reciprocal theorems.
Curves and Cones of Second Order or Second Class.
§ 45. If we have two projective pencils in a plane, corresponding rays will meet, and their point of intersection will constitute some locus which we have to investigate. Reciprocally, if two projective rows in a plane are given, then the lines which join corresponding points wul envelope some curve. We prove first :—
Theorem.— If two projective flat Theorem.—It two projective rows pencils lie in a plane, but are neither lie in a plane, but are neither per-perspective nor concentric, then the spective nor on a common base, then locus of intersections of correspond-the envelope of lines joining coιτe-ing rays is a curve of the second spending points is a curve of the order, that is, no line contains more second class, that is, through no point than two points of the locus. pass more than two of the enveloping lines.
Proof.— We draw any line t. This Proof— ^We take any point T and cuts each of the pencils in a row, so join it to all points in each row. that we have on t two rows, and these This gives two concentric pencils, are projective because the pencils which are projective because the are projective. If corresponding rows are projective. If a line joinrays of the two pencils meet on the ing corresponding points in the two line t, their intersection will be a rows passes through T, it will be a point in the one row which coincides line in the one pencil which coincides with its corresponding point in the with its corresponding line in the other. But two projective rows on other. But two projective con-the same base cannot have more centric flat pencils in the same plane than two points of one coincident cannot have more than two lines of with their corresponding points in one coincident with their correspond-the other (§ 34). ing line in the other (§ 34).
It will be seen that the proofs are reciprocal, so that the one may be copied from the other by simply interchanging the words point and line, locus and envelope, row and pencil, and so on. We shall therefore in future prove seldom more than one of two reciprocal theorems, and often state one theorem only, the reader being recommended to go through the reciprocal proof by himself, and to supply the reciprocal theorems when not given.
§ 46. We state the theorems in the pencil reciprocal to the last, without proving them :—
Theorem.— If two projective flat Theorem — If two projective axial pencils are concentric, but are neither pencils lie in the same pencil (their perspective nor co-planor, then the axes meet in a point), but are neither envelope of the planes joining cor-perspective nor co-axial, then the responding rays is a cone of the locus of lines joining corresponding second class; that is, no line through planes is a cone of the second order; the common centre contains more that is, no plane in the pencil con-
than two of the enveloping planes. tains more than two of these lines.
§ 47. Of theorems about cones of second order and cones of second class we shall state only very few. We point out, however, the following connexion between the curves and cones under consideration :—
The lines which join any point in Every plane section of a cone of space to the points on a curve of the the second order is a curve of the second order form a cone of the second order, second order.
The planes which join any point in Every plane section of a cone of space to the lines enveloping a curve the second class is a curve of the of the second class envelope them-second class, selves a cone of the second class.
By its aid, or by the principle of duality, it will be easy to obtain theorems about them from the theorems about the curves.
We prove the first. A curve of the second order is generated by two projective pencils. These pencils, when joined to the point in space, give rise to two projective axial pencils, which generate the cone in question as locus of the lines where corresponding planes meet.
§ 48.
Theorem.— The curve of second Theorem.— The envelope of second order which is generated by two proclass which is generated by two projective flat pencils passes through jective rows contains the bases of the centres of the two peucils. these rows as enveloping lines or tangents.
Proof. —If S and S' are the two Proof.— If s and s' are the two pencils, then to the ray SS' or p' in rows, then to the point ss' or P' as a the pencil S' corresponds in the pen-point in s' corresponds in s a point cil S a ray p, which is different from P, which is not coincident with P ’, p', for the pencils are not perspec-for the rows are not perspective. But five. But p and p' meet at S, so that P and P' are joined by s, so that s is
S is a point on the curve, and simi-one of the enveloping lines, and larly S'. similarly s'.
It follows that every line in one of the two pencils cuts the curve in two points, viz., once at the centre S of the pencil, and once where it cuts its corresponding ray in the other pencil. These two points, however, coincide, if the line is cut by its corresponding line at S itself. The line p in S, which corresponds to the line SS' in S', is therefore the only line through S which has but one point in common with the curve, or which cuts the curve in two coincident points. Such a line is called a tangent to the curve, touching the latter at the point S, which is called the point of contact.
In the same manner we get in the reciprocal investigation the result that through every point in one of the rows, say in s, two tangents may be drawn to the curve, the one being s, the other the line joining the point to its corresponding point in sl. There is, however, one point P in s for which these two lines coincide. Such a point in one of the tangents is called the “point of contact” of the tangent. We thus get—
Theorem.— To the line joining the Theorem — To the point of inter centres of the projective pencils as a section of the bases of two projective line in one pencil corresponds in the rows as a point in one row corre-other the tangent at its centre. sponds in the other the point of contact of its base.
§ 49. Two projective pencils are determined if three pairs of corresponding lines are given. Hence if α 1 , b 1 , c 1 are three lines in a pencil S 1 , and α 2 , ⅛ 2 , c t the corresponding lines in a projective pencil S 2 , the correspondence and therefore the curve of the second order generated by the points of intersection of corresponding rays is determined. Of this curve we know the two centres S 1 and S 2 , and the three points a 1 a i , b 1 b 2 , c j c 2 , hence five points in all. This and the reciprocal considerations enable us to solve the following two problems :—
Problem.— To construct a curve of Problem. —To construct a curve the second order, of which five of the second class, of which five points S 1 , S 2 , A, B, C are given. tangents w 1 , w 2 , a, b, c are given.
In order to solve the left-hand problem, we take two of the given points, say S 1 and S 2 , as centres of pencils. These we make projective by taking the rays α 1 , b 1 , c 1 , which join S 1 to A, B, C respectively, as corresponding to the rays α 2 , b i , c 2 , which join S 2 to A, B, C respectively, so that three rays meet their corresponding rays at the given points A, B, C. This determines the correspondence of the pencils which will generate a curve of the second order passing through A, B, C and through the centres S 1 and S 2 , hence through the five given points. To find more points on the curve we have to construct for any ray in S 1 the corresponding ray in S 2 . This has been done in § 36. But we repeat the con struction in order to deduce further properties from it. We also solve the right-hand problem. Here we select two, viz., u 1 , u 3 of the five given lines, u 1 , u 3 , a, b, c, as bases of two rows, and the points A 1 , B,, C 1 where a, b, c cut u 1 as corresponding to the points A 2 , B 2 , C 2 where a, b, c cut u 2 .
We get then the following solutions of the two problems :—
Solution.—Through the point A Solution.— In the line a take any draw any two lines, u 1 and u 2 (flg. 15), two points S 1 and S 2 as centres of the first «i to cut the pencil S, in a pencils (fig. 16), the first S 1 (A 1 B∣C 1 )
row AB 1 C 1 , the other u 2 to cut the to project the row m 1 , the other pencil S 2 in a row AB j jC 2 . These two S 2 (A 2 B 2 C 2 ) to project the row u t . rows will be perspective”, as the point These two pencils will be perspective, A corresponds to itself, and the centre the line S 1 A l being the same as the of projection will be the point S, corresponding line S 2 A 2 , and the axis where the lines B 1 B 2 and C 1 C 2 meet. of projection will be the line «, which
To find now for any ray d 1 iñ S 1 its joins the intersection B of S∣B 1 and corresponding ray d 2 in S 2 , we deterS 2 B 2 to the intersection C of S 1 C∣ and mine the point D 1 where d, cuts w 1 , S 2 C 2 . To find now for any point D 1 project this point from S to D 2 on u 2 iñ w 1 the corresponding point D 2 in and join S 2 to D 2 . This will be the tz 2 , we draw S 1 D 1 and project the required ray d 2 which cuts d l at some point D where this line cuts u from S 2 point D on the curve. to m 2 . This will give the required point D 2 , and the line d joining D 1 to D 2 will be a new tangent to the curve.
§ 50. These constructions prove, when rightly interpreted, very important properties of the curves in question.
If in fig. 15 we draw in the pencil Si the ray L· which passes through the auxiliary centre S, it will be found that the corresponding ray k 3 cuts it on u 3 . Hence—
Theorem.— In the above construe- Theorem.— In the above construc tion the bases of the auxiliary rows tion (fig. 16) the tangents to the curve w . and u 2 cut the curve where they from the centres of the auxiliary pencut the rays S 2 S and S.S respec-cils S 1 and S 2 are the lines which pass tively. through w 2 w and u 1 u respectively.
As A is any given point on the curve, and u 1 any line through it, we have solved the problems :—
Problem.— To find the second point Problem.— To find the second tan in which any line through a known gent which can be drawn from any point on the curve cuts the curve. point in a given tangent to the curve.
If we determine in S 1 (fig. 15) the ray corresponding to the ray S 2 S 1 in S a , we get the tangent at S 1 . Similarly we can determine the point of contact of the tangents u 1 or m 2 in fig. 16.
§ 51. If five points are given, of which not three are in a line, then we can, as has just been shown, always draw a curve of the second order through them, we select two of the points as centres of Erojective pencils, and then one such curve is determined. It will e presently shown that we get always the same curve if two other points are taken as centres of pencils, that therefore five points deter- mine one curve of the second order, and reciprocally, that five tangents determine one curve of the second class. Six points taken at random will therefore not lie on a curve of the second order. In order that this may be the case a certain condition has to be satisfied, and this condition is easily obtained from the construction in § 49, fig. 15. If we consider the conic determined by the five points A, S 1 , S 2 , K, L, then the point D will be on the curve if, and only if, the points on D 1 , S, D 2 be in a line.
Tliis may be stated differently if we take AKS 1 DS 2 L (figs. 15 and 17) as a hexagon inscribed in the conic, then AK and DS 2 will be opposite sides, so will be KS 1 and S 2 L, as well as S 1 D and LA. The first two meet in D 2 , the others in S and D 1 respectively. We may therefore state the required condition, together with the reciprocal one, as follows :—
Pascal’s Theorem—It a hexagon Brianchon's Theorem.— If a hexagon be inscribed in a curve of the second be circumscribed about a curve of order, then the intersections of oppo-the second class, then the lines join site sides are three points in a line. ing opposite vertices are three lines meeting in a point.
These celebrated theorems, which are known by the names of their discoverers, are perhaps the most fruitful in the whole theory of conics. Before we go over to their applications we have to show that we obtain the same curve if we take, instead of S 1 , S 2 , any two other points on the curve as centres of projective pencils.
§ 52. We know that the curve depends only upon the correspondence between the pencils S 1 and S 2 , and not upon the special construction used for finding new points on the curve. The point A (fig. 15 or fig. 17), through which the two auxiliary rows u 1 , u, 2 were drawn, may therefore be changed to any other point on the curve. Let us now suppose the curve drawn, and keep the points S 1 , S 2 , K, L, and D, and hence also the point S fixed, whilst we move A along the curve. Then the line AL will describe a pencil about L as centre, and the point D 1 a row on S 1 D perspective to the pencil L. At the same time AK describes a pencil about K and D 2 a row perspective to it on S 2 D. But by Pascal’s Theorem D 1 and D ä will always lie in a line with S, so that the rows described by D 1 and D 2 are perspective. It follows that the pencils K and L will themselves be projective, corresponding rays meeting on the curve. This proves that we get the same curve whatever pair of the five given points we take as centres of projective pencils. Hence—
Theorem.— Only one curve of the Theorem.— Only one curve of the second order can be drawn which second class can be drawn which passes through five given points. touches five given lines.
We have seen that if on a curve of the second order two points coincide at A, the line joining them becomes the tangent at A. If, therefore, a point on the curve and its tangent are given, this will be equivalent to having given two points on the curve. Similarly, if on the curve of second class a tangent and its point of contact are given, this will be equivalent to two given tangents.
We may therefore extend the last theorem :—
Theorem.— Only one curve of the Theorem.— Only one curve of the second order can be drawn, of which second class can be drawn, of which four points and the tangent at one four tangents and the point of con-of them, or three points and the tact at one of them, or three tangents tangents at two of them, are given. and the points of contact at two of them, are given.
§ 53. At the same time it has been proved :—
Theorem.— -If all points on a curve Theorem. —All tangents to a curve of the second order be joined to of second class are cut by any two of any two of them, then the two them in projective rows, those being pencils thus formed are projec-corresponding points which lie on tive, those rays being correspond-the same tangent. Hence— ïng which meet on the curve.
Hence—
The cross-ratio of four rays joining The cross-ratio of the four points a point S on a curve of second in which any tangent w is cut by order to four fixed points A, B, C, D four fixed tangents a, b, c, d is in in the curve is independent of the dependent of the position of w, and position of S, and is called the cross-is called the cross-ratio of the four ratio of the four points A, B, C, D. tangents a, b, c, d.
If this cross-ratio is = -1, the four If this cross-ratio equals -1, the points are said to be four harmonic four tangents are said to be four points. harmonic tangents.
We have seen that a curve of second order, as generated by projective pencils, has at the centre of each pencil one tangent; and further, that any point on the curve may be taken as centre of such pencil. Hence—
Theorem.— A curve of second order Theorem.— A curve of second class has at every point one tangent. has on every tangent a point of contact.
§ 54. We return to Pascal’s and Brianchon’s theorems and their applications, and shall, as before, state the results both for curves of the second order and curves of the second class, but prove them only for the former.
Pascal’s theorem may be used when five points are given to find more points on the curve, viz., it enables us to find the point where any line through one of the given points cuts the curve again. It is convenient, in making use of Pascal’s theorem, to number the points, to indicate the order in which they are to be taken in forming a hexagon, which, by the way, may be done in 60 different ways. It will be seen that 1 2 and (leaving out 3) 4 5 are opposite sides, so are 2 3 and (leaving out 4) 5 6, and also 3 4 and (leaving out 5) 6 1.
If the points 1 2 3 4 5 are given, and we want a 6th point on a line drawn through 1, we know all the sides of the hexagon with the exception of 5 6, and this is found by Pascal’s theorem.
If this line should happen to pass through 1, then 6 and 1 coincide, or the line 6 1 is the tangent at 1. And always if two consecutive vertices of the hexagon approach nearer and nearer, then the side joining them will ultimately become a tangent.
We may therefore consider a pentagon inscribed in a curve of second order and the tangent at one of its vertices as a hexagon, and thus get the theorem :—
Theorem. —Every pentagon in- Theorem.— Every pentagon cir-
scribed in a curve of second order cumscribed about a curve of the has the property that the intersec-second class has the property that tions of two pairs of non-consecutive the lines which join two pairs of sides lie in a line with the point non-consecutive vertices meet on where the fifth side cuts the tangent that line which joins the fifth at the opposite vertex. vertex to the point of contact of the opposite side.
This enables us also to solve the following problems.
Problem. —Given five points on a Problem.— Given five tangents to a curve of second order to construct curve of second class to construct the tangent at any one of them. the point of contact of any one of them.
If twice two adjacent vertices coincide, the hexagon becomes a quadrilateral, with tangents at two vertices. These we take to be opposite, and get the following theorems :—
Theorem. —If a quadrilateral be Theorem.— If a quadrilateral be inscribed in a curve of second order, circumscribed about a curve of the intersections of opposite sides, second class, the lines joining and also the intersections of the opposite vertices, and also the lines tangents at opposite vertices, lie in joining points of contact of opposite a line (fig. 18). sides, meet in a point
If we consider the hexagon made up of a triangle and the tangents at its vertices, we get—
Theorem. —If a triangle is inscribed Theorem.—If a triangle be circum-
in a curve of second order, the points scribed about a curve of second class, in which the sides are cut by the the lines which join the vertices to tangents at the opposite vertices the points of contact of the opposite meet in a point. sides meet in a point (fig. 19).
§ 55. Of these theorems, those about the quadrilateral give rise to a number of others. Four points A, B, C, 1) may in three different ways be formed into a quadrilateral, for we may take them in the order ABCD, or ACBD, or ACDB, so that either of the points B, C, D may be taken as the vertex opposite to A. Accordingly we may apply the theorem in three different ways.
Let A, B, C, D be four points on a curve of second order (fig. 20), and let us take them as forming a quadrilateral by taking the points in the order ABCD, so that A, C and also B, D are pairs of opposite vertices. Then P, Q will be the points where opposite sides meet, and E, F the intersections of tangents at opposite vertices. The four points P, Q, E, F lie therefore in a line. The quadrilateral ACBD gives us in the same way the four points Q, R, G, H in a line, and the quadrilateral ABDC a line containing the four points R, P, I, K. These three lines form a triangle PQR.
The relation between the points and lines in this figure may be expressed more clearly if we consider ABCD as a four-point inscribed in a conic, and the tangent at these points as a four-side circumscribed about it,—viz., it will be seen that P, Q, R are the diagonal points of the four-point ABCD, whilst the sides of the triangle PQR are the diagonals of the circumscribing four-side. Hence the theorem—
Theorem.— Any four-point on a curve of the second order and the four-side formed by the tangents at these points stand in this relation that the diagonal points of the four-point lie in the diagonals of the four-side. And conversely, If a four-point and a circumscribed four-side stand in the above relation, then a curve of the second order may be described which passes through the four points and touches there the four sides of these figures.
That the last part of the theorem is true follows from the fact that the four points A, B, C, D and the line a, as tangent at A, determine a curve of the second order, and the tangents to this curve at the other points B, C, D are given by the construction which leads to fig. 20.
The theorem reciprocal to the last is—
Theorem.— Any four-side circumscribed about a curve of second class and the four-point formed by the points of contact stand in this relation that the diagonals of the four-side pass through the diagonal points of the four-point And conversely, If a four-side and an inscribed four-point stand in the above relation, then a curve of the second class may be described which touches the sides of the four-side at the points of the four-points.
§ 56. The four-point aud the four-side in the two reciprocal theorems are alike. Hence if we have a four-point ABCD and a four-side abed related in the manner described, then not only may a curve of the second order be drawn, but also a curve of the second class, which both touch the lines a, b, c, d at the points A, B, C, D.
The curve of second order is already more than determined by the points A, B, C and the tangents a, b, c at A, B, and C. The point D may therefore be any point on this curve, and d any tangent to the curve. On the other hand the curve of the second class is more than determined by the three tangents a, b, c and their points of contact A, B, C, so that d is any tangent to this curve. It follows that every tangent to the curve of second order is a tangent of a curve of the second class having the same point of contact. In other words, the curve of second order is a curve of second class, and vice versa. Hence the important theorems :—
Theorem.— Every curve of second Theorem.— Every curve of second order is a curve of second class. class is a curve of second order.
The curves of second order and of second class having thus been proved to be identical shall henceforth be called by the common name of Conics.
For these curves hold, therefore, all properties which have been proved for curves of second order or of second class. We may therefore now state Pascal’s and Brianchon’s theorem thus—
Pascal's Theorem.— If a hexagon be inscribed in a conic, then the intersections of opposite sides lie in a line.
Brianchon ’ s Theorem.— If a hexagon be circumscribed about a conic, then the diagonals forming opposite centres meet in a point.
§ 57. If we suppose in fig. 20 that the point D together with the tangent d moves along the curve, whilst A, B, C and their tangents a, b, c remain fixed, then the ray DA will describe a pencil about A, the point Q a projective row on the fixed line BC, the point F the row b, and the ray EF a pencil about E. But EF passes always through Q. Hence the pencil described by AD is projective to the pencil described by EF, and therefore to the row described by F on b. ' At the same time the line BD describes a pencil about B projective to that, described by AD (§ 53). Therefore the pencil BD and the row F on b are projective. Hence—
Theorem.— If on a conic a point A be taken and the tangent a at this point then the cross-ratio of the four rays which join A to any four points on the curve is equal to the cross-ratio of the points in which the tangents at these points cut the tangent at A.
§ 58. There are theorems about cones of second order and second class in a pencil which are reciprocal to the above, according to § 43. We mention only a few of the more important ones.
Theorem.— The locus of intersections of corresponding planes in two projective axial pencils whose axes meet is a cone'of the second order.
Theorem.— The envelope of planes which join corresponding lines in two projective flat pencils, not in the same plane, is a cone of the second class.
Theorem.— Cones of second order and cones of second class are identical.
Theorem.— Every plane cuts a cone of the second order in a conic.
Theorem.— A cone of second order is uniquely determined by five of its edges or by five of its tangent planes, or by four edges and the tangent p>lanae at one of them, Ac., Ac. Theorem (Pascal’s).—If a solid angle of six faces be inscribed in a cone of the second order, then the intersections of opposite faces are three lines in a plane.
Theorem (Brianchon’s).—If a solid angle of six edges be circumscribed about a cone of the second order, then the planes through opposite edges meet in a line.
Each of the other theorems about conics may be stated for cones of the second order.
§ 59. We have not yet considered the shape of the conics. We know that any line in the plane of the conic, and hence that the line at infinity, either has no point in common with the curve, or one (counting for two coincident points), or two distinct points. If the line at infinity has no point on the curve the latter is altogether finite, and is called an Ellipse (fig. 20). If the line at infinity has only one point in common with the conic, the latter extends to infinity, and has the line at infinity a tangent. It is called a Parabola (fig. 21). If, lastly, the line at infinity cuts the curve in two points, it consists of two separate parts which each extend in two branches to the points at infinity were they meet. The curve is in this case called an Hyperbola (see fig. 19, 24, or 25). The tangents at the two points at infinity are finite because the line at infinity is not a tangent. They are called Asymptotes. The branches of the hyperbola approach these lines indefinitely as a point on the curves moves to infinity.
§ 60. That the circle belongs to the curves of the second order is seen at once if we state in a slightly different form the theorem that in a circle all angles at the circumference standing upon the same arc are equal. If two points S 1 , S 2 on a circle be joined to any other two points A and B on the circle, then the angle included by the rays S 1 A and S 1 B is equal to that between the rays S 2 A and S 2 B, so that as A moves along the circumference the rays S 1 A and S 2 A describe equal and therefore projective pencils. The circle can thus be generated by two projective pencils, and is a curve of the second order.
If we join a point in space to all points on a circle, we get a (circular) cone of the second order (§ 43). Every plane section of this cone is a conic. This conic will be an ellipse, a parabola, or an hyperbola, according as the line at infinity in the plane has no, one, or two points in common with the conic in which the plane at infinity cuts the cone. It follows that our curves of second order may be obtained as sections of a circular cone, and that they are identical with the “Conic Sections” of the Greek mathematicians.
§ 61. Any two tangents to a parabola are cut by all others in projective rows; but the line at infinity being one of the tangents, the points at infinity on the rows are corresponding points, and the rows therefore similar. Hence—
Theorem. — The tangents to a parabola cut each other proportionally. Pole and Polar.
§ 62. We return once again to the remarkable figure (fig. 20) which we obtained in § 55.
If a four-side be circumscribed about and a four-point inscribed in a conic, so that the vertices of the second are the points of contact of the sides of the first, then the triangle formed by the diagonals of the first is the same as that formed by the diagonal points of the other.
Such a triangle will be called a polar triangle of the conic, so that PQR in fig. 20 is a polar triangle. It has the property that on the side p opposite P meet the tangents at A and B, and also those at C and D. From the harmonic properties of four-points and four-sides it follows further that the points L, Μ, where it cuts the lines AB and CD, are harmonic conjugates with regard to AB and CD respectively.
If the point P is given, and we draw a line through it, cutting the conic in A and B, then the point Q harmonic conjugate to P with regard to AB, and the point H where the tangents at A and B meet, are determined. But they lie both on p, and therefore this line is determined. If we now draw a second line through P, cutting the conic in C and D, then the point Μ harmonic conjugate to P with regard to CD, and the point G where the tangents at C and D meet, must also lie on p. As the first line through P already determines p, the second may be any line through P. Now every two lines through P determine a four-point ABCD on the conic, and therefore a polar-triangle which has one vertex at P and its opposite side at p. This result, together with its reciprocal, give the theorem :—
Theorem.— All polar-triangles which have one vertex in common have also the opposite side in common. All polar-triangles which have one side in common have also the opposite vertex in common.
§ 63. To any point P in the plane of but not on a conic corresponds thus one line p as the side opposite to P in all polar triangles which have ope vertex at P, and reciprocally to every line p corresponds one point P as the vertex opposite to p in all triangles which have p as one side.
We call the line p the polar of P, and the point P the pole of the line p with regard to the conic.
If a point lies on the conic, we call the tangent at that point its polar; and reciprocally we call the point of contact the pole of tangent.
§ 64. From these definitions and former results follow—
Theorem-The polar of any point Theorem.— The pole of any line p hΛ Åî ≠ ιi' e ° 0nic ls a 1 l" e p ' which not a tangent to the conic is a point has the following properties :- p, which has the following pro perties :—
1. On every line through P which 1. Of all lines through a point on cuts the conic, it contains the har- p from which two tangents may be monic conjugate of P with regard to drawn to the conic, it contains the those points on the conic. line which is harmonic conjugate to p, with regard to the two tangents.
2. If tangents can be drawn from 2. If p cuts the conic, the tangents P, their points of contact lie on p. at the intersections meet at P.
3. Tangents drawn at the points 3. The point of contact of tangents where any line through P cuts the drawn from any point on p to the conic, meet on it; and conversely, conic lie in a line with P; and conversely, 4. If from any point on it tangents 4. Tangents drawn at points where be drawn, their points of contact any line through P cuts the conic will lie in a line with P. meet on p.
5. Any four-point on the conic 5. Any four-side circumscribed which has one diagonal point at P about a conic which has one diagonal has the other two lying on p. on p has the other two meeting at P.
The truth of 2 follows from 1. If T be a point where p cuts the conic, then one of the points where PT cuts the conic, and which are harmonic conjugates with regard to PT, coincides with T; hence the other does—that is, PT touches the curve at T.
That 4 is true follows thus: If we draw from a point H on the polar one tangent a to the conic, join its point of contact A to the pole P, determine the second point of intersection B of this line with the conic, and draw the tangent at B, it will pass through H, and will therefore be the second tangent which may be drawn from H to the curve.
§ 65. The second property of the polar or pole gives rise to the theorem—
Theorem.— From a point in the Theorem.— A line in the plane of a plane of a conic, two, one, or no tan-conic has two, one, or no points in gents may be drawn to the conic, common with the conic, according according as its polar has two, one, as two, one, or no tangents can be or no points in common with the drawn from its pole to the conic, curve.
Of any point in the plane of a conic we say that it was without, on, or within the curve according as two, one, or no tangents to the curve pass through it. The points on the conic separate those within the conic from those without. That this is true for a circle is known from elementary geometry. That it also holds for other conics follows from the fact that every conic may be considered as the projection of a circle, which will be proved later on.
The fifth property of pole and polar stated in § 64 shows how to find the polar of any point and the pole of any line by aid of the straight-edge only. Practically it is often convenient to draw three secants through the pole, and to determine only one of the diagonal points for two of the four-points formed by pairs of these lines and the conic (fig. 21).
These constructions also solve the problem :-—
Problem.— From a point without a conic, to draw the two tangents to the conic by aid of the straight-edge only.
For we need only draw the polar of the point in order to find the points of contact.
§ 66. The property of a polar-triangle may now be stated thus— Theorem.— In a polar-triangle each side is the polar of the opposite vertex, and each vertex is the pole of the opposite side.
If P is one vertex of a polar-triangle, then the other vertices, Q and R, lie on the polar p of P. One of these vertices we may chose arbitrarily. For if from any point Q on the polar a secant be drawn cutting the conic in A and D (fig. 22), and if the lines joining these points to P cut the conic again at B and C, then the line BC will pass through Q. Hence P and Q are two of the vertices on the polar-triangle which is determined by the four-point ABCD. The third vertex R lies also on the line p. It follows, therefore, also—
Theorem.— If Q is a point on the polar of P, then P is a point on the polar of Q; and reciprocally, If q is a line through the pole of p, then p is a line through the pole of q.
This is a very important theorem. It may also be stated thus—
Theorem.— If a point moves along a line describing a row, its polar turns about the pole of the line describing a pencil. This pencil is projective to the row, so that the cross-ratio of four poles in a row equals the cross-ratio of its four polars, which pass through the pole of the row.
To prove the last part, let us suppose that P, A, and B in fig. 22 remain fixed, whilst Q moves along the polar p of P. This will make CD turn about P and move R along p, whilst QD and RD describe projective pencils about A and B. Hence Q and R describe projective rows, and hence PR, which is the polar of Q, describes a pencil projective to either.
§ 67. Two points, of which one, and therefore each, lies on the polar of the other, are said to be conjugate with regard to the conic; and two lines, of which one, and therefore each, passes through the pole of the other, are said to be conjugate with regard to the conic. Hence all points conjugate to a point P lie on the polar of P; all lines conjugate to a line p pass through the pole of p.
If the line joining two conjugate poles cuts the conic, then the poles are harmonic conjugates with regard to the points of intersection; hence one lies within the other without the conic, and all points conjugate to a point within a conic lie without it.
Of a polar-triangle any two vertices are conjugate poles, any two sides conjugate lines. If, therefore, one side cuts a conic, then oue of the two vertices which lie on this side is within and the other without the conic. The vertex opposite this side lies also without, for it is the pole of a line which cuts the curve. In this case therefore one vertex lies within, the other two without. If, on the other hand, we begin with a side which does not cut the conic, then its pole lies within and the other vertices without. Hence—
Theorem.— Every polar triangle has one and only one vertex within the conic.
We add, without a proof, the theorem—
Theorem.— The four points in which a conic is cut by two conjugate polars are four harmonic points in the conic.
§ 68. If two conics intersect in four points (they cannot have more points in common, § 52), there exists one and only one four-point which is inscribed in both, and therefore one polar triangle common to both.
Theorem. — Two conics which intersect in four points have always one and only one common polar-triangle; and reciprocally,
Two conics which have four common tangents nave always one and only one common polar-triangle.
The proof that these polar triangles are identical in case of a conic which have four points and also four tangents in common is left to the reader.
Diameters and Axes of Conics.
§ 69. Diameters.— The theorems about the harmonic properties of poles and polars contain, as special cases, a number of important metrical properties of conics. These are obtained if either the pole or the polar is moved to infinity,—it being remembered that the harmonic conjugate to a point at infinity, with regard to two points A, B, is the middle point of the segment AB. The most important properties are stated in the following theorems :—
The middle points of parallel chords of a conic lie in a line—viz., on the polar to the point at infinity on the parallel chords.
This line is called a diameter. The polar of every point at infinity is a diameter. The tangents at the end points of a diameter are parallel, and arc parallel to the chords bisected by the diameter. All diameters pass through a common point, the pole of the line at infinity. All diameters of a parabola are parallel, the pole to the line at infinity being the point where the curve touches the line at infinity.
In case of the ellipse and hyperbola, the pole to the line at infinity is a finite point called the centre of the curve.
A centre of a conic bisects every chord through it. The centre of an ellipse is within the curve, for the line at infinity does not cut the ellipse.
The centre of an hyperbola is without the curve, because the line at infinity cuts the curve. Hence also From the centre of an hyperbola two tangents can be drawn to the curve which have their point of contact at infinity. These are called Asymptotes (§ 59).
To construct a diameter of a conic, draw two parallel chords and join their middle points.
To find the centre of a conic, draw two diameters; their intersection will be the centre.
§ 70. Conjugate Diameters.— A polar-triangle with one vertex at the centre will have the opposite side at infinity. The other two sides pass through the centre, and are called conjugate diameters, each being the polar of the point at infinity on the other.
Of two conjugate diameters each bisects the chords parallel to the other, and if one cuts the curve, the tangents at its ends are parallel to the other diameter.
Further—
Every parallelogram inscribed in a conic has its sides parallel to two conjugate diameters; and Every parallelogram circumscribed about a conic has as diagonals two conjugate diameters.
This will be seen by considering the parallelogram in the first case as an inscribed four-point, in the other as a circumscribed four-side, and determining in each case the corresponding polar-triangle. The first may also be enunciated thus—
The lines which join any point on an ellipse or an hyperbola to the ends of a diameter are parallel to two conjugate diameters.
§ 71. The Circle.— If every diameter is perpendicular to its conjugate the conic is a circle.
For the line which joins the ends of a diameter to any point on the curve include a right angle.
A conic which has more than one pair of conjugate diameters at right angles to each other is a circle.
Let AA' and BB' (fig. 23) be one pair of conjugate diameters at right angles to each other, CC' and DD' a second pair. If we draw through the end point A of one diameter a chord AP parallel to DD', and join P to A', then PA and PA' are, according to § 70, parallel to two conjugate diameters. But PA is parallel to DD', hence PA' is parallel to CC', and therefore PA and PA' are perpendicular. If we further draw the tangents to the conic at A and A', these will be perpendicular to AA', they being parallel to the conjugate diameter BB'. We know thus five points on the conic, viz., the points A and A' with their tangents, and the point P. Through these a circle may be drawn having AA' as diameter; and as through five points one conic only can be drawn, this circle must coincide with the given conic.
§ 72. Axes.— Conjugate diameters perpendicular to each other are called axes, and the points where they cut the curve vertices of the conic.
In a circle every diameter is an axis, every point on it is a vertex; and any two lines at right angles to each other may be taken as a pair of axes of any circle which has its centre at their intersection.
If we describe on a diameter AB of an ellipse or hyperbola a circle concentric to the conic, it will cut the latter in A and B (fig. 24). Each of the semicircles in which it is divided by AB will be partly within, partly without the curve, and must cut the latter therefore again in a point. The circle and the conic have thus four points A, B, C, D, and therefore one polar-triangle, in common (§ 68). Of this the centre is one vertex, for the line at infinity is the polar to this point, both with regard to the circle and the other conic. The other two sides are conjugate diameters of both, hence perpendicular to each other. This gives—
Theorem. — An ellipse as well as an hyperbola has one pair of axes.
This reasoning shows at the same time hoιυ to construct the axis of an ellipse or of an hyperbola. A parabola has one axis, if we define an axis as a diameter perpendicular to the chords which it bisects. It is easily constructed. The line which bisects any two parallel chords is a diameter. Chords perpendicular to it will be bisected by a parallel diameter, and this is the axis.
§ 73. The first part of the right hand theorem in § 64 may be stated thus: any tλvo conjugate lines through a point P without a conic are harmonic conjugates with regard to the two tangents that may be drawn from P to the conic.
If we take instead of P the centre C of an hyperbola, then the conjugate lines become conjugate diameters, and the tangents asymptotes. Hence—
Theorem.— Any two conjugate diameters of an hyperbola are harmonic conjugates with regard to the asymptotes.
As the axes are conjugate diameters at right angles to one another, it follows (§ 23)—
Theorem.— The axes of an hyperbola bisect the angles between the asymptotes.
Let 0 be the centre of the hyperbola (fig. 25), t any secant which cuts the hyperbola in C,D and the asymptotes in E,F, then the line OH which bisects the chord CD is a diameter conjugate to the diameter OK which is parallel to the secant t, so that OK and OM are harmonic with regard to the asymptotes. The point Μ therefore bisects EF. But by construction Μ bisects CD. It follows that DF = EC, and ED = CF; or Theorem. — On any secant of an hyperbola the segments between the curve and the asymptotes are equal.
If the chord is changed into a tangent, this gives The segment between the asymptotes on any tangent to an hyperbola is bisected by the y>oint of contact.
The first part allows a simple solution of the problem to find any number of points on an hyperbola, of which the asymptotes and one point are given. This is equivalent to three points and the tangents at two of them. This construction requires measurement.
§ 74. For the parabola, too, follow some metrical properties. A diameter PM (fig. 26) bisects every chord conjugate to it, and the pole P of such a chord BC lies on the diameter. Buta diameter cuts the parabola once at infinity. Hence—
Theorem.— The segment PM which joins the middle point Μ of a chord of a parabola to the pole P of the chord is bisected by the parabola at A.
§ 75. Two asymptotes and any two tangents to an hyperbola may be considered as a quadrilateral circumscribed about the hyperbola. But in such a quadrilateral the intersections of the diagonals and the points of contact of opposite sides lie in a line (§ 54). If therefore DEFG (fig. 27) is such a quadrilateral, then the diagonals DF and GE will meet on the line which joins the points of contact of the asymptotes, that is, on the line at infinity; hence they are parallel. From this the following theorem is a simplo deduction :—
Theorem.— All triangles formed by a tangent and the asymptotes of an hyperbola are equal in area.
If we draw at a point P (fig. 27) on an hyperbola a tangent, the part HK between the asymptotes is bisected at P. The parallelogram PQOQ' formed by the asymptotes and lines parallel to them through P will be half the triangle OHK, and will therefore be constant. If we now take the asymptotes OX and OY as oblique axes of coordinates, the lines OQ and QP will be the coordinates of P, and will satisfy the equation xy = const, =α 2 .
Theorem. — For the asymptotes as axes of coordinates the equation of the hyperbola is xy= const.
It is not difficult to get the equations to the ellipse and hyperbola referred to their axes as axes of coordinates. We are satisfied to have shown in one case that the curves of the second order as generated by projective pencils are the same as those which are in coordinate geometry defined by equations of the second degree.
involution.
§ 76. If we have two projective rows, ABC on u and A'B'C' on u', and place their bases on the same line, then each point in this line counts twice, once as a point in the row u and once as a point in the row u'. In fig. 28 we denote the points as points in the one row by letters above the line A, B, C . . ., and as points in the second row by A', B', C' .... below the line. Let now A and B' be the same point, then to A will correspond a point A' in the second, and to B' a point B in the first row. In general these points A' and B will be different. It may, however, happen that they coincide. Then the correspondence is a peculiar one, as the following theorem shows :—
Theorem. — If two projective rows lie on the same base, and if it happens that to one point in the base the same point corresponds, whether we consider the point as belonging to the first or to the second row, then the same will happen for every point in the base— that is to say, to every point in the line corresponds the same point in the first as in the second row.
Proof. In order to determine the correspondence, we may assume three pairs of corresponding points in two projective rows. Let then A', B , , C', in fig. 29, correspond to A, B, C, so that A and B', and also B and A', denote the same point. Let us further denote the point C ’ when considered as a point in the first row by D; then it is to be proved that the point D', which corresponds to D, is the same point as C. We kno∖v that the cross-ratio of four points is equal to that of the corresponding row. Hence
(ABCD) = (A'B'C'D')
but replacing the dashed letters by those undashed ones which denote the same points, the second cross-ratio equals (BADD'), which, according to § 15, iv., equals (ABD ’ D); so that the equation becomes
(ABCD) = ( ABD'D).
This requires that C and D' coincide.
§ 77. Two projective rows on the same base, which have the above property, that to every point, whether it be considered as a point in the one or in the other row, corresponds the same point, are said to be in involution, or to form an involution of points on the line.
We mention, but without proving it, that any two projective rows may be placed so as to form an involution.
An involution may be said to consist of a row of pairs of points, to every point A corresponding a point A', and to A' again the point A. These points are said to be conjugate.
From the definition, according to which an involution may be considered as made up of two projective rows, follow at once the following important properties :—
(1.) The cross-ratio of four points equals that of the four conjugate points.
(2.) If we call a point which coincides with its conjugate point a “focus” of the involution, we may say: An involution has either two foci, or one, or none, and is called respectively a hyperbolic, parabolic, or elliptic involution (§ 34).
(3.) In a hyperbolic involution any two conjugate points are harmonic conjugates with regard to the two foci.
For if A, A'be two conjugate points, F 1 , F 2 the two foci, then to the points F 1 , F 2 , A, A' in the one row correspond the points F 1 , F 2 , A', A in the other, each focus corresponding to itself. Hence (F 1 F 2 AA') = (F 1 F 2 A'A)—that is, we may interchange the two points AA' without altering the value of the cross-ratio, which is the characteristic property of harmonic conjugates (§ 18).
(4.) The point conjugate to the point at infinity is called the “centre” of the involution. Every involution has a centre, unless the point at infinity be a focus, in which case we may say that the centre is at infinity.
In a hyperbolic involution the centre is the middle point between the foci.
(5.) The product of the distances of two conjugate points A, A' from the centre 0 is constant :—
OA.OA'-c.
Proof.— Let A, A', and B, B' be two pairs of conjugate points, 0 the centre, I the point at infinity, then
(ABOI) = (A'B'IO), or
OA. OA'= OB. OB'.
In order to determine the distances of the foci from the centre, we write F for A and A' and get
ÕF 2 =c; OF=±√c.
Hence if c is positive OF is real, and has two values, equal and opposite. The involution is hyperbolic.
If c=0, OF = 0, and the two foci both coincide with the centre.
If c is negative, √c becomes imaginary, and there are no foci. Hence we may write—
In a hyperbolic involution, OA. OA'=⅛ 2 ,
In a parabolic involution, OA. OA' = 0, In an elliptic involution, OA. OA'= - ⅛ s .
From these expressions it follows that conjugate points A, A' in a hyperbolic involution lie on the same side of the centre, and in an elliptic involution on opposite sides of the centre, and that in a parabolic involution one coincides λvith the centre.
In the first case, for instance, OA. OA' is positive; hence OA and OA' have the same sign.
It also follows that two segments, AA' and BB', between pairs of conjugate points have the following positions:—in a hyperbolic involution they lie either one altogether within or altogether without each other; in a parabolic involution they have one point in common; and in an elliptic involution they overlap, each being partly within and partly without the other.
Proof.— We have OA. OA' = OB. OB'=I∙ 2 incase of a hyperbolic involution. Let A and B be the points in each pair which are nearer to the centre 0. If now A, A' and B, B' lie on the same side of ( ), and if B is nearer to 0 than A, so that OB<OA, then OB'≥OA'; hence B' lies further away from O than A', or the segment A A' lies within BB'. And so on for the other cases.
(6. ) An involution is determined—
(a) By two pairs of conjugate points. Hence also
(3) By one pair of conjugate points and the centre;
(γ) By the two foci;
(δ) By one focus aud one pair of conjugate points;
(e) By one focus and the centre.
(7.) The condition that A, B, C and A', B', C' may form an involution may be written in one of the forms—
(ABCC') = (A'B'C'C), or (ABCA') = (A'B'C'A), or (ABC'A')≈=(A'B'CA), for each expresses that in the two projective rows in which A, B, C and A', B', C' are conjugate points two conjugate elements may be interchanged.
(8.) Any three pairs, A, A', B, B', C, C', of conjugate points are connected by the relation—
BA' CB' AC' a'cbα'cb - 1, Proof.— We have by (7) (ABC'A') = (A'B'CA), which, when worked out, gives the above relation.
The latter is easily remembered by aid of the following rule of writing the first side. First write
B C A
CAB’
and then fill up the gaps in numerator and denominator by A', B', C' respectively.
§ 78. Theorem. — TV sides of any four-point arc cut by any line in six points in involution, opposite sides being cut in conjugate points.
Let A 1 B 1 C 1 D 1 (fig. 30) be the four-point. If its sides be cut by the line p in the points A, Λ', B, B', C, C', if further, C 1 D 1 cuts the line A.B 1 in C 2 , and if we project the row A 1 B 1 C 2 C to p once from D 1 and once from C 1 , we get
(A'B ’ σC) = (BAC'C).
Interchanging in the last cross-ratio the letters in each pair we get (A'B'C'C)=-(ABCC').
Hence by § 77 (7) the points are in involution.
The theorem may also be stated thus :—
Theorem.— The three points in which any line cuts the sides of a triangle and the projections, from .any point in the plane, of tho vertices of the triangle on to the same line are six points in involution.
Or again—
The projections from any point on to any line of the six vertices of a four-side are six points in involution, the projections of opposite vertices being conjugate points.
This property gives a simple means to construct, by aid of the straight edge only, in an involution of which two pairs of conjugate points are given, to any point its conjugate.
§ 79. The theory of involution may at once be extended from the row to the flat and. the axial pencil—viz., we say that there is an involution in a flat or in an axial pencil if any line cuts the pencil in an involution of points. An involution in a pencil consists of pairs of conjugate rays or planes; it has two, one, or no focal rays or ρ>lanes, but nothing corresponding to a centre.
An involution in a flat pencil contains always one, and in general only one, pair of conjugate rays which are perpendicular to one another. For in two projective flat pencils exist always two corresponding right angles (§ 40).
Each involution in an axial pencil contains in the same manner one pair of conjugate planes at right angles to one another.
As a rule, there exists but one pair of conjugate lines or planes at right angles to each other. But it is possible that there are more, and then there is an infinite number of such pairs. An involution in a flat pencil, in which every ray is perpendicular to its conjugate ray, is said to be circular. That such involution is possible is easily seen thus:—if in two concentric flat pencils each ray on one is made to correspond to that ray on the other which is perpendicular to it, then the two pencils are projective, for if we turn the one pencil through a right angle each ray in one coincides with its corresponding ray in the other. But these two projective pencils are in involution.
A circular involution has no focal rays, because no ray in a pencil coincides with the ray perpendicular to it.
§ 80. Theorem.— Every elliptical involution in a row may be considered as a section of a circular involution. Proof.— In an elliptical involution any two segments AA' and BB' lie partly within partly without each other (fig. 31). Hence two circles described on 'AA and BB' as diameters will intersect in two points E and E'. The line EE' cuts the base of the involution at a point O, which, from a well known proposition (Eucl.
III. 35), has the property that OA,OA' = OB.OB', for each is equal to OE. OE'. The point 0 is therefore the centre of tho involution. If we wish to construct to any point C the conjugate point O', we may draw the circle through CEE'. This will cut the base in the required point C' for OC.OC' = OA.OA'. But EC and EC' are at right angles. Hence the involution which is obtained by joining E or E' to the points in the given involution is circular. This may also be expressed thus :—
Every elliptical involution has the property that there are two definite points in the plane from which any two conjugate points are wen under a right angle.
At the same time the following problem has been solved :—
Problem. — To determine the centre and also the point corresponding to any given point in an elliptical involution of which two pairs of conjugate points are given.
§ 81. By the aid of § 53, the points on a conic may be made to correspond to those on a line, so that the row of points on the conic is projective to a row of points on a line. We may also have two projective rows on the same conic, an$ these will be in involution as soon as one point on the conic has the same point corresponding to it all the same to whatever row it belongs. An involution of points on a conic will have the property (as follows from its definition, and from § 53) that the lines which join conjugate points of the involution to any point on the conic are conjugate lines of an involution in a pencil, and that a fixed tangent is cut by the tangents at conjugate points on the conic in points which are again conjugate points of an involution on the fixed tangent. For such involution on a conic the following theorem holds :—
Theorem. — The lines which join corresponding points in an involution on a conic all pass through a fixed point ; and reciprocally, the points of intersection of conjugate lines in an involution among tangents to a conic lie on a line.
We prove the first part only. The involution is determined by two pairs of conjugate points, say by A, A' and B, B' (fig. 32). Let AA' and BB' meet in P. If we join the points in involution to any point on the conic, and the conjugate points to another point on the conic, we obtain two projective pencils. We take A and A ’ as centres of these pencils, so that the pencils A(A'BB') and A'(AB'B) are projective, and in perspective position, because AA' corresponds to A'A. Hence corresponding rays meet in a line, of which two points are found by joining AB' to A'B and AB to A'B'. It follows that the axis of perspective is the polar of the point P, where A A' and BB' meet. If we now wish to construct to any other point C on the conic the corresponding point C', we join C to A' and the point where this line cuts y> to A. The latter line cuts the conic again in C'. But we know from the theory of pole and polar that the line CO' passes through P.
Involution determined by a Conic on a Line.—Foci.
§ 82. The polars, with regard to a conic, of points in a row p form a pencil P projective to the row (§ 66). This pencil cuts the base of the row p in a projective row.
If A is a point in the given row, A' the point where the polar of A cuts p, then A and A' will be corresponding points. If we take A' a point in the first row, then the polar of A' will pass through A, so that A corresponds to A'—in other words, the rows are in involution. The conjugate points in this involution are conjugate points with regard to the conic. Conjugate points coincide only if the polar of a point A passes through’ A—that is, if A lies on the conic. Hence—
Theorem.— A conic determines on every line in its plane an involution, in which those points are conjugate which are also conjugate with regard to the conic. If the line cuts the conic the involution is hyperbolic, the points of intersection being the foci. If the line touches the conic the involution is parabolic, the two foci coinciding at the point of contact. If the line does not cut the conic the involution is elliptic, having no
If, on the other hand, we take a point P in the plain of a conic, we get to each line a through P one conjugate line which joins P to the pole of a. These pairs of conjugate lines through P form an involution in the pencil at P. The focal rays of this involution are the tangents drawn from P to the conic. This gives the theorem ’ reciprocal to the last, viz. :—
Theorem.— A conic determines in every pencil in its plane an involution, corresponding lines being conjugate lines with regard to the conic. If the point is without the conic the involution is hyperbolic, the tangents from the points being the focal rays. If the point lies on the conic the involution is parabolic, the tangent at the point counting for coincident focal rays. If the point is within the conic the involution is elliptic, having no focal rays.
It will further be seen that the involution determined by a conic on any line p is a section of the involution, which is determined by the conic at the pole P of p.
§ 83. Definition.— The centre of a pencil in which the conic determines a circular involution is called a “focus” of the conic.
In other words—
A focus is such a point that every line through it is perpendicular to its conjugate line.
The polar to a focus is called a directrix of the conic.
From the definition it follows that :—
Every focus lies on an axis, for the line joining a focus to the centre of the conic is a diameter to which the conjugate lines are perpendicular; and Every line joining two foci is an axis, for the perpendiculars to this line through the foci are conjugate to it. These conjugate lines pass through the pole of the line, the pole lies therefore at infinity, and the line is a diameter, hence by the last property an axis.
It follows that all foci lie on one axis, for no line joining a point in one axis to a point in the other can be an axis.
As the conic determines in the pencil which has its centre at a focus a circular involution, no tangents can be drawn from the focus to the conic. Hence each focus lies within a conic ; and a directrix does not cut the conic.
Further properties are found by the following considerations :— § 84. Through a point P one line p can be drawn, which is with regard to a given conic conjugate to a given line q, viz., that line which joins the point P to the pole cf the line q. If the line q is made to describe a pencil about a point Q, then the line p will describe a pencil about P. These two pencils will be projective, for the line p passes through the pole of q, and whilst q describes the pencil Q, its pole describes a projective row, and this row is perspective to the pencil P.
We now take the point P on an axis of the conic, draw any line p through it, and from the pole of p draw a perpendicular q to p. Let q cut the axis in Q. Then, in the pencils of conjugate lines, which have their centres at P and Q, the lines p and q are conjugate lines at right angles to one another. Besides, to the axis as a ray in either pencil will correspond in the other the perpendicular to the axis (§ 72). The conic generated by the intersection of corresponding lines in the two pencils is therefore the circle on PQ as diameter, so that every line in P is perpendicular to its corresponding line in Q.
To every point P on an axis of a conic corresponds thus a point Q, such that conjugate lines through P and Q are perpendicular.
We shall show that these pointpairs P, Q form an involution. To do this let us move P along the axis, and with it the line p, keeping the latter parallel to itself. Then P describes a row, p a perspective pencil (of parallels), and the pole of 72 a projective row. At the same time the line q describes a pencil of parallels perpendicular to p, and perspective to the row formed by the pole of p. The point Q, therefore, where q cuts the axis, describes a row projective to the row of points P. The two points P and Q describe thus two projective rows on the axis; and not only does P as a point in the first now correspond to Q, but also Q as a point in the first corresponds to P. The two rows therefore form an involution. The centre of this involution, it is easily seen, is the centre of the conic. A focus of this involution has the property that any two conjugate lines through it are perpendicular; hence, it is a focus to the conic.
Such involution exists on each axis. But only one of these can have foci, because all foci lie on the same axis. The involution on one of the axis is elliptic, and appears (§ 80) therefore as the section of two circular involutions in two pencils whose centres lie in the other axis. These centres are foci, hence the one axis contains two foci, the other axis none; or every central conic has two foci which lie on one axis equidistant from the centre.
The axis which contains the foci is called the principal axis ; in case of an hyperbola it is the axis which cuts the curve, because the foci lie within the conic.
In case of the parabola there is but one axis. The involution on this axis has its centre at infinity. One focus is therefore at infinity, the one focus only is finite. A parabola has only one focus.
§ 85. If through any point P (fig. 33) on a conic the tangent PT and the normal PN (t. e., the perpendicular to the tangent through the point of contact) be drawn, these will be conjugate lines with regard to the conic, and at right angles to each other.
They will therefore cut the principal axis in two points, which are conjugate in the involution considered in § 84; hence they are harmonic conjugates with regard to the foci. If therefore the two foci F 1 and F 2 be joined to P, these lines will be harmonic with regard to the tangent and normal. As the latter are perpendicular, they will bisect the angles between the other pair. Hence—
The line joining any point on a conic to the two foci are eqτuιlly inclined to the tangent and normal at that point.
In case of the parabola this becomes—
The line joining any point on a parabola to the focus and the diameter through the point, are equally inclined to the tangent and normal at that point.
From the definition of a focus it follows that—
The segment of a tangent between the directrix and the point of contact is seen from the focus belonging to the directrix under a right angle, because the lines joining the focus to the ends of this segment are conjugate with regard to the conic, and therefore perpendicular.
With equal ease the following theorem is proved :—
The two lines which join the points of contact of two tangents each to one focus, but not both to the same, are seen from the intersection of the tangents under equal angles.
§. 86. Other focal properties of a conic are obtained by the following considerations :—
Let F (fig. 34) be a focus to a conic, f the corresponding directrix, A and B the points of contact of two tangents meeting at T, and P the point where the line AB cuts the directrix. Then TF will be the polar of P (because polars of F and T meet at P). Hence TF
and PF are conjugate lines through a focus, and therefore perpendicular. They are further harmonic conjugates with regard to FA andFB (§§ 64 and 13), so that they bisect the angles formed by these lines. This by the way proves—
The segments between the point of intersection of two tangents to a conic and their points of contact are seen from a focus under equal angles.
If we next draw through A and B lines parallel to TF, then the points A 1 , B 1 where these cut the directrix will be harmonic conjugates with regard to P and the point where FT cuts the directrix. The lines FT and FP bisect therefore also the angles between FΛ 1 and FB 1 . From this it follows easily that the triangles FAA j and FBB 1 are equiangular, and therefore similar, so that
FA: AA 1 = FB: BB 1 .
The triangles AA 1 A a and BB 1 B 2 formed by drawing perpendiculars from A and B to the directrix are also similar, so that AA 1 : AA 2 = BB 1 : BB 2 ,
This, combined with the above proportion, gives
FA: AA 2 =FB: BB 2 .
Hence the theorem:—
The ratio of the distances of any point on a conic from a focus and the corresponding directrix is constant.
To determine this ratio we consider its value for a vertex on the principal axis. In an ellipse the focus lies between the two vertices on this axis, hence the focus is nearer to a vertex than to the corresponding directrix. Similarly in an hyperbola a vertex is nearer to the directrix than to the focus. In a parabola the vertex lies halfway between directrix and focus.
It follows in an ellipse the ratio between the distance of a point from the focus to that from the directrix is less than unity, in the parabola it equals unity, and in the hyperbola it is greater than unity.
It is here the same which focus we take, because the two foci lie symmetrical to the axis of the conic. If now P is any point on the conic having the distances r 1 and r 2 from the foci and the distances d 1 and d. 2 from the corresponding directrices, then d 1 ⅛ e ' where e is constant. Hence also r sh~≈e. d l ±d i
Ln the ellipse, which lies between the directrices, d 1 + rf 2 is constant, therefore also r 1 + r 2 , In the hyperbola on the other hand d t - d, 2 is constant, equal to the distance between the directrices, therefore in this case r i - r s is constant.
If we call the distances of a point on a conic from the focus its focal distances we have the theorem :—
In an ellipse the sum of the focal distances is constant; and in an hyperbola the difference of the focal distances is constant. This constant sum or difference equals in both cases the length of the principal axis. Pencil of Conics.
§ 87. Through four points A, B, C, D in a plane, of which no three lie in a line, an infinite number of conics may be drawn, viz., through these four points and any fifth one single conic. This system of conics is called a pencil of conics. Similarly all conics touching four fixed lines form a system such that any fifth tangent determines one and only one conic. We have here the theorems :—
Theorem.— The pairs of pointe in Theorem.— The pairs of tangents which any line is cut by a system of which can be drawn from a point to conics through four fixed points are a system of conics touching four in involution. fixed lines are in involution.
We prove the first theorem only. Let ABCD (fig. 35) be the four-point, then any line t will cut two opposite sides AC, BD in the points E,E', the pair AD, BC in points F,F', and any conic of the system in M,N, and we have
A(CDMN) = B(CDMN).
If we cut these pencils by t we get'
(EFMN) = (FΕ ’ MN)
or (EFMN) = (E ’ F'NM).
But this is, according to § 77 (7), the condition that M, N are corresponding points in the involution determined by the point pairs E, E', F, F' in which the line t cuts pairs of opposite sides of the four-point ABCD. This involution is independent of the particular conic chosen.
§ 88. There follow several important consequences:—
Theorem.— Through four points two, one, or no conic may be drawn which touch any given line, according as the involution determined by the given four-point on the line has real, coincident, or imaginary foci. Theorem.— Two, one, or no conics may be drawn which touch four given lines and pass through a given point, according as the involution determined by the given four-side at the point has read, coincident, or imaginary focal rays.
For the conic through four points which touches a given line has its point of contact at a focus of the involution determined by the four-point on the line.
As a special case we get, by taking the line at infinity: —
Theorem.— Through four points of which none is at infinity either two or no parabolas ma,y be drawn.
The problem of drawing a conic through four points and touching a given line is solved by determining the points of contact on the line, that is, by determining the foci of the involution in which the line cuts the sides of the four-point. The corresponding remark holds for the problem of drawing the conics which touch four lines and pass through a given point.
Ruled Quadric Surfaces.
§ 89. Formerly we have considered projective rows which lie in the same plane. In that case, lines joining corresponding points envelope a, conic. We shall now consider projective rows whose bases do not meet. In this case, corresponding points will be joined by lines which do not he in a plane, but on some surface, which like every surface generated by lines is called a ruled surface. This surface clearly contains the bases of the two rows.
If the points in either row be joined to the base of the other, we obtain two axial pencils which are also projective, those planes being corresponding which pass through corresponding points in the given rows. If A,A'be two corresponding points, a,a the planes in the axial pencils passing through them, then AA ’ will be the line of intersection of the corresponding planes a,cl, and also the line joining corresponding points in the rows.
If we cut the whole figure by a plane this will cut the axial pencils in two projective flat pencils, and the curve of the second order generated by these will be the curve in which the plane cuts the surface. Hence Theorem.— The locus of lines joining corresponding points in two príjective rows which do not lie in the same plane is a surface which contains the bases of the rows, and which can also be generated by the lines of intersection of corresponding planes in two projective axial pencils. This surface is cut by every plane in a curve of the second order, hence either in a conic or in a line-pair. No line which does not lie altogether on the surface can have more than two points in common with the surface, which is therefore said to be of the second order, or is called a ruled quadric surface.
That no line which does not lie on the surface can cut the surface in more than two points is seen at once if a plane be drawn through the line, for this will cut the surface in a conic. It follows also that A line which contains more than two points of the surface lies altogether on the surface.
§ 90. Through any point in space one line can always be drawn cutting two given lines which do not themselves meet.
If therefore three lines in space be given of which no two meet, then through every point in either one line may be drawn cutting the other two.
Theorem.— If a line moves so that it always cuts three given lines of which no two meet, then it generates a ruled quadric surface. Proof.— Let a, b, c be the given lines, andp, q, r .. . . lines cutting them in the points A, A', A". . .; B, B',B".. .; C, C', C". . . respectively; then the planes through a containing p, q, r, and the planes through b containing the same lines, may be taken as corresponding planes in two axial pencils which are projective, because both pencils cut the line c in the same row C, C', C".. .; the surface can therefore be generated by projective axial pencils.
Of the lines p, q, r . . . no two can meet, for otherwise the lines a, b, c which cut them would also lie in their plane. There is a single infinite number of them, for one passes through each point of a. These lines are said to form a set of lines on the surface.
If now three of the lines p, q, r be taken, then every line d cutting them will have, three points in common with the surface, and will therefore lie altogether on it. This gives rise to a second set of lines on the surface. From what has been said the theorem follows:—
Theorem.— A ruled quadric surface contains two sets of straight lines. Every line of one set cuts every line of the other, but no two lines of the same set meet. Any two lines of the same set may be taken as bases of two projective rows, or of two projective pencils which generate the surface. They are cut by the lines of the other set in two projective rows.
The plane at infinity like every other plane cuts the surface either in a conic proper or in a line-pair. In the first case the surface is called an Hyperboloid of one sheet, in the second an Hyperbolic Paraboloid.
The latter maybe generated by aline cutting three lines of which one lies at infinity, that is, cutting two lines and remaining parallel to a given plane.
Quadric Surfaces.
§ 91. The conics, the cones of the second order, and the ruled quadric surfaces complete the figures which can be generated by projective rows or flat and axial pencils, that is, by those aggregates of elements which are of one dimension (§§ 5, 6). We shall now consider the simpler figures which are generated by aggregates of two dimensions. The space at our disposal will not, however, allow us to do more than indicate a few of the results.
§ 92. We establish a correspondence between the lines and planes in pencils in space, or reciprocally between the points and lines in two or more planes, but consider principally pencils.
In two pencils we may either make planes correspond to planes and lines to lines, or else planes to lines and lines to planes. If hereby the condition be satisfied that to a flat, or axial, pencil corresponds in the first case a projective flat, or axial, pencil, and in the second a projective axial, or flat, pencil, the pencils are said to be projective in the first case and reciprocal in the second.
For instance, two pencils which join two points S 1 and S 2 to the different points and lines in a given plane π are projective (and in perspective position), if those lines and planes be taken as corresponding which meet the plane τr in the same point or in the same line. In this case every plane through both centres S 1 and S 2 of the two pencils will correspond to itself. If these pencils are brought into any other position they will be projective (but not perspective).
The correspondence between two projective pencils is uniquely determined, if to four rays (or planes) in the one the corresponding rays (or planes) in the other are given, provided that no three rays of either set lie in a plane. Proof.— Let a, b, c, d be four rays in the one, a', b', d, d' the corresponding rays in the other pencil. We shall show that we can find for every ray e in the first a single corresponding ray e' in the second. To the axial pencil a (b, c, d . . . ) formed by the planes which join a to b, c, d . .. , respectively corresponds the axial pencil a! (b', d, d'. .. ), and this correspondence is determined. Hence, the plane a'e' which corresponds to the plane ae is determined. Similarly the plane b'e' may be found and both together determine the ray e'.
Similarly the correspondence between two reciprocal pencils is determined if for four rays in the one the corresponding planes in the other are given.
§ 93. We may now combine—
1. Two reciprocal pencils.
Each ray cuts its corresponding plane in a point, the locus of these points is a quadric surface.
2. Two projective pencils.
Each plane cuts its corresponding plane in a line, but a ray as a rule does not cut its corresponding ray. The locus of points where a ray cuts its corresponding ray is a twisted cubic. The lines where a plane cuts its corresponding plane are secants
3. Three projective pencils.
The locus of intersection of corresponding planes is a cubic surface.
Of these we consider only the first two eases.
§ 94. If two pencils are reciprocal, then to a plane in either corresponds a line in the other, to a flat pencil an axial pencil, and so on. Every line cuts its corresponding plane in a point. If S 1 and S 2 be the centres of the two pencils, and P be a point where a line α 1 in the first cuts its corresponding plane α 2 , then the line b, 2 in the pencil S 2 which passes through P will meet its corresponding plane β i in P. For δ 2 is a line in the plane α 2 . The corresponding plane β 1 must therefore pass through the line α 1 , hence through P.
The points in which the lines in S l cut the planes corresponding to them in S 2 are therefore the same as the points in which the lines in S 2 cut the planes corresponding to them in S 1 .
The locus of these points is a surface which is cut by a plane in a conic or in a line-pair and by a line in not more than two points unless it lies altogether on the surface. The surface itself is therefore called a quadric surface, or a surface of the second order.
To prove this we consider any line p in space.
The flat pencil in Si which lies in the plane drawn through p and the corresponding axial pencil in S 2 determine onp two projective rows, and those points in these which coincide with their corresponding points lie on the surface. But there exist only two, or one, or no such points, unless every point coincides with its corresponding point In the latter ease the line lies altogether on the surface.
This proves also that a plane cuts the surface in a curve of the second order, as no line can have more than two points in common with it. To show that this is a curve of the same kind as those considered before, λve have to show that it can be generated by projective flat pencils. We prove first that this is true for any plane through the centre of one of the pencils, and afterwards that every point on the surface may be taken as the centre of such pencil. Let then α 1 be a plane through S x . To the flat pencil in S 1 which it contains corresponds in S 2 a projective axial pencil with axis a i , and this cuts α 1 in a second flat pencil. These two flat pencils in α 1 are projective, and, in general, neither concentric nor perspective. They generate therefore a conic. But if the line a, 2 passes through S 1 the pencils will have S 1 as common centre, and may therefore have two, or one, or no lines united with their corresponding lines. The section of the surface by the plane α 1 will be accordingly a line-pair or a single line, or else the plane α 1 will have only the point S 1 in common with the surface.
Every line l 1 through S l cuts the surface in two points, viz., first in S 1 and then at the point where it cuts its corresponding plane. If now the corresponding plane passes through S 1 , as in the case just considered, then the two points where l i cuts the surface coincide at S 1 , and the line is called a tangent to the surface with S 1 as point of contact Hence if Z 1 be a tangent, it lies in that plane τ 1 which corresponds to the line S 2 S 1 as a line in the pencil S 2 . The section of this plane has just been considered. It follows that —
All tangents to quadric surface at the centre of one of the reciprocal pencils lie in a plane which is called the tangent plane to the surface at that point as point of contact. To the line joining the centres of the two pencils as a line in one corresponds in the other the tangent plane at its centre. The tangent plane to a quadric surface cither cuts the surface in two lines, or it has only a single line, or else only a single point in common with the surface. In the first case the point of contact is said lobe hyperbolic, in the second parabolic, in the third elliptic.
§ 95. It remains to be proved that every point S on the surface may be taken as centre of one of the pencils which generate the surface. Let S be any point on the surface Φ' generated by the reciprocal pencils S 1 and S s . We have to establish a reciprocal correspondence between the pencils S and S 1 , so that the surface generated by them is identical with Φ. To do this we draw two planes α 1 and β v through S 1 , cutting the surface Φ in two conics which we also denote ∣>y α 1 and β i . These conics meet at S 1 , and at some other point T where the line of intersection of β 1 and β l cuts the surface.
In the pencil S we draw some plane σ which passes through T, but not through S 1 or S 2 . It will cut the two conics first at T, and therefore each at some other point which we call A and B respectively. These we join to S by lines a and b, and now establish the required correspondencebetweèn the pencils S 1 andS as follows:— To S 1 T shall correspond the plane <r, to the plane α l the line a, and to β 1 the line b, hence to the flat pencil in α 1 the axial pencil a. These pencils are made-projective by aid of the conic in a 1 .
In the same manner the flat pencil in β 1 is made projective to the axial pencil b by aid of the conic in β 1 , corresponding elements being those which meet on the conic. This determines the correspondence, for we know for more than four rays in S 1 the corresponding planes in S. The two pencils S and S 1 thus made reciprocal generate a quadric surface Φ', which passes through the point S and through the two conics α 1 and β 1 .
The two surfaces Φ and Φ' have therefore the points S and S j and the conics α l and β l in common. To show that they are identical, we draw a plane through S and S 2 , cutting each of the conics α 1 and β l in two points, which will always be possible. This plane cuts Φ and Φ' in two conics which have the point S and the points where it cuts α 1 and β l in common, that is five points in all. The conics therefore coincide.
This proves that all those points P on Φ' lie on Φ which have the property that the plane SS 2 P cuts the conics α 1 , β 1 in two points each. If the plane SS 2 P has not this property, then we dιaw a plane SS j P. This cuts each surface in a conic, and these conics have in common the points S, S j , one point on each of the conics α 1 , β l , and one point on one of the conics through S and S., which lie on both surfaces, hence five points. They are therefore coincident, and our theorem is proved.
§ 96. The following propositions follow :—
A quadric surface has at every point a tangent pilane. Every plane section of a quadric surface is a conic or a line-pair. Every line which has three points in common with a quadric surface lies on the surface. Every conic which has five points in common with a quadric surface lies on the surface. Through two conics zvhich lie in differentplanes, but have two points in common, and through one external point always one quadric surface may be drawn.
§ 97. Every plane which cuts a quadric surface in a line pair is a tangent plane. For every line in this plane through the centre of the line-pair (the point of intersection of the two lines) cuts the surface in two coincident points and is therefore a tangent to the surface, the centre of the line-pair being the point of contact . If a quadric surface contains a line, then every plane through this line cuts the surface in a line-pair (or in two coincident lines'). For this plane cannot cut the surface in a conic. Hence If a quadric surface contains one line p then it contains an infinite number of lines, and through every point Q on the surface, one line q can be drawn which cuts p. For the plane through the point Q and the linep cuts the surface in a line-pair which must pass through Q and of which p is one line.
No two such lines q on the surface can meet. For as both meet p their plane would contain p and therefore cut the surface in a triangle.
Every line which cuts three lines q will be on the surface; for it has three points in common with it.
Hence the quadric surfaces which contain lines are the same as the ruled quadric surfaces considered in §§ 89-93, but with one important exception. In the last investigation we have left out of consideration the possibility of a plane having only one line (two coincident lines) in common with a quadric surface.
§ 98. To investigate this case we suppose first that there is one point A on the surface through which two different lines a, b can be drawn, which lie altogether on the surface.
If P is any other point on the surface which lies neither on a nor b, then the plane through P and a will cut the surface in a second line a ’ wlιich passes through P and which cuts a. Similarly there is a line b ’ through P which cuts b. These two lines α' and b , may coincide, but then they must coincide with PA.
If this happens for one point P, it happens for every other point Q. For if two different lines could be drawn through Q, then by the same reasoning the line PQ would be altogether on the surface, hence two lines would be drawn through P against the assumption. From this follows :—
If there is one point on a quadric surface through which one, but only one, line can be drawn on the surface, then through every point one line can be drawn, and all these lines meet in a point. The surface is a cone of the second order. If through one point on a quadric surface, two, and onlytιco, lines can be drawn on the surface, then through every point two lines may be drawn, and the surface is a ruled quadric surface. If through one point on a quadric surface no line on the surface can be drawn, then the surface contains no lines.
Using the definitions at the end of § 95, we may also say :—
On a quadric surface the points are all hyperbolic, or all parabolic, or all elliptic.
As an example of a quadric surface with elliptical points, we mention the sphere which may be generated by two reciprocal pencils, where to each line in one corresponds the plane perpendicular to it in the other.
§ 99. Poles and Polar Planes.— The theory of poles and polars with regard to a conic is easily extended to quadric surfaces.
Let P be a point in space not on the surface, which we suppose not to be a cone. On every line through P which cuts the surface in two points we determine the harmonic conjugate Q of P with regard to the points of intersection. Through one of these lines we draw two planes a and β. The locus of the points Q in a is a line a, the polar of P with regard to the conic in which a cuts the surface. Similarly the locus of points Q in β is a line b. This cuts a, because the line of intersection of a and β contains but one point Q. The locus of all points Q therefore is a plane. This plane is called the polar plane of the point P, with regard to the quadric surface. If P lies on the surface zee take the tangent plane of P as its polar.
The following propositions hold :—
1. Every point has a polar plane, which is constructed by drawing the polars of the point with regard to the conics in which two planes through the point cut the surface.
2. If Q is a point in the polar of P, then P is a point in the polar of Q, because this is true with regard to the conic in which a plane through PQ cuts the surface.
3. Every plane is the pιolar plane of one point, zchich is called the Pole of the plane.
The pole to a plane is found by constructing the polar planes of three points in the plane. Their intersection will be the pole.
4. The points in which the polar plane of P cuts the surface are points of contact of tangents drawn from P to the surface, as is easily seen. Hence :—
5. The tangents drawn from a point P to a quadric surface form a cone of the second, order, for the polar plane of P cuts it in a conic.
6. If the pole describes a line a, its polar plane will turn about another line a', as follows from 2. These Unes a and a! are said to be conjugate with regard to the surface.
§ 100. The pole of the line at infinity is called the centre of the surface. If it lies at the infinity, the plane at infinity is a tangent plane, and the surface is called a paraboloid. The polar plane to any point at infinity passes through the centre, and is called a diametrical plane. A line through the centre is called a diameter. It is bisected at the centre. The Une conjugate to it lies at infinity. If a point moves along a diameter its polar plane turns about the conjugale line at infinity, that is, it moves parallel to itself, its centre moving on the first line. The middle points of parallel chords lie in a plane, viz., in the polar plane of the point at infinity through which the chords are drawn.
The centres of parallel sections lie in a diameter ιchich is a line conjugate to the line at infinity in which the planes meet. Twisted Cubic».
§ 101. If two pencils with centres S 1 and S 2 are made projective, then to a ray in one corresponds a ray in the other, to a plane a plane, to a flat or axial pencil a projective flat or axial pencil, and so on.
There is a double infinite number of lines in a pencil. We shall see that a single infinite number of lines in one pencil meets its corresponding ray, and that the points of intersection form a curve in space.
Of the double infinite number of planes in the pencils each will meet its corresponding plane. This gives a system of a double infinite number of lines in space. We know (§ 5) that there is a quadruple infinite number of lines in space. From among these we may select those which satisfy one or more given conditions. The systems of lines thus obtained was first systematically investigated and classified by Pliicker, in his Geometrie des Raumes. He uses the following names:—
A treble infinite number of lines, that is, all lines which satisfy one condition, are said to form a complex, of lines·, e.g., all lines cutting a given line, or all lines touching a surface.
A double infinite number of lines, that is, all lines which satisfy two conditions, or which are common to two complexes, are said to form a congruence of lines; c.g., all lines in a plane, or all lines cutting two curves, or all lines cutting a given curve twice.
A single, infinite number of lines, that is, all lines which satisfy three conditions, or which belong to three complexes, form a ruled surface; e.g., one set of lines on a ruled quadric surface, or developable surfaces which are formed by the tangents to a curve.
It follows that all lines in which corresponding planes in two projective pencils meet form a congruence. We shall see this congruence consists of all lines which cut a twisted cubic twice, or of all secants to a twisted cubic.
§ 102. Let l x be the line S 1 S 2 as a line in the pencil S 1 . To it corresponds a line l. 2 in S 2 . At each of the centres tιoo corresponding lines meet. The two axial pencils with l 1 and l, 2 as axes are projective, and, as their axes meet at S 2 , the intersections of corresponding planes form a cone of the second order (§ 58), with S 2 as centre. If π j and ιr 2 be corresponding planes, then their intersection will be a line p 3 which passes through S 2 . Corresponding to it in S 1 will be a linep 1 which lies in the plane π 1 , and which therefore meets p, 2 at some point P. Conversely, if p i be any line in S 2 which meets its corresponding line p 1 at a point P, then to the plane l< 2 p, i will correspond the plane l l p l , that is, the plane S l S 2 P. These planes intersect in p, 2 , so that p, 2 is a line on the quadric cone generated by the axial pencils l l and l, 2 . Hence :—
All lines in one pencil which meet their corresponding lines in the other form a cone of the second order which has it centre at the centre of the first pencil, and passes through the centre of the second.
From this follows that tho points in which corresponding rays meet lie on two cones of tho second order which have the ray joining their centres in common, and form therefore, together with the line S 1 S 2 or l i , the intersection of these cones. Any plane cuts each of the cones in a conic. These two conics have necessarily that point in common in which it cuts the line l l , and therefore besides either one or three other points. It follows that the curve is of the third order as a plane may cut it in three, but not in more than three, points. Hence :—
The locus of points in which corresponding lines on two projective pencils meet is a curve of the third order or a “twisted cubic” k, which passes through the centres of the pencils, and which appears as the intersection of two cones of the second order, which have one line in common. A line belonging to the congruence determined by the pencils is a secant of the cubic; it has two, or one, or no points in common with this cubic, and is called accordingly a secant proper, a tangent, or a secant improper of the cubic. A secant improper may be considered, to use the language of coordinate geometry, as a secant with imaginary points of intersection.
§ 103. Ifα 1 and α 2 be any two corresponding lines in the two pencils, then corresponding planes in the axial pencils having a 1 and α 2 as axes generate a ruled quadric surface. If P be any point on the cubic k, and if the corresponding rays in
S 1 and S 2 which meet at P, then to the plane α 1 7q in S 1 corresponds α 2 p, 2 in S 2 . These therefore meet in a line through P.
This may be stated thus :—
These secants of the cubic which cut a ray a 1 , drawn through the centre S 1 of one pencil, form a ruled quadric surface which passes through both centres, and which contains the twisted cubic k. Of such surfaces an infinite number exists. Every ray through S 1 or S 2 which is not a secant determines one of them.
If, however, the rays β 1 and a, 2 are secants meeting at A, then the ruled quadric surface becomes a cone of the second order, having A as centre. Or all lines of the congruence which pass through a point on the twisted cubic k form a cone of the second order. In other words, the projection of a twisted cubic from any point in the curve on to any plane is a conic.
If α 1 is not a secant, but made to pass through any point Q in space, the ruled quadric surface determined by α 1 will pass through Q. There will therefore bè one line of the congruence passing through Q, and only one. For if two such lines pass through Q, then the lines S 1 Q and S 2 Q will be corresponding lines; hence Q will be a point on the cubic k, and an infinite number of secants will pass through it. Hence :—
Through every point in space not on the twisted cubic one and only one secant to the cubic can be drawn.
§ 104. The fact that all the secants through a point on the cubic form a quadric cone shows that the centres of the projective pencils generating the cubic are not distinguished from any other points on the cubic. If we take any two points S, S' on the cubic, and draw the secants through each of them, we obtain two quadric cones, which have the line SS' in common, and which intersect besides along the cubic. If we make these two pencils having S and S' as centres projective by taking four rays on the one cone as corresponding to the four rays on the other which meet the first on the cubic, the correspondence is determined. These two pencils will generate a cubic, and the two cones of secants having S and S' as centres will be identical with the above cones, for each has five rays in common with one of the first, viz., the line SS' and the four lines determined for the correspondence; therefore these two cones intersect in the original cubic. This gives the theorem :—
On a hoisted cubic any two points may be taken as centres of projective pencils ιvhich generate the cubic, corresiìonding planes being those which meet on the same secant.
Of the two projective pencils at S and S' we may keep the first fixed, and move the centre of the other along the curve. The pencils will hereby remain projective, and a plane a in S will be cut by its corresponding plane α' always in the same secant a. Whilst S' moves along the curve the plane α' will turn about a, describing an axial pencil.
In this article we have given a purely geometrical theory of conics, cones of the second order, quadric surfaces, &c. In doing so we have followed, to a great extent, Reye’s Geometrie der Lage, and to this excellent work those readers are referred who wish for a more exhaustive treatment of the subject.
It will have been observed that scarcely any use has been made of algebra, and it would have been even possible to avoid this little, as is done by Reye. There are, however, other systems of geometry which start more or less from theorems known to the Greeks, "and using more or less algebra.
We cannot do more here than enumerate a few of the more prominent works on the subject, which, however, are almost all Continental. These are the following :—
Monge, Geometrie Descriptive·. Carnot, Géométrie de Position (1803), containing a theoι y of transversals; Fencelet's gιeat work. Traite des Propriétés Pro- jectiυes des Figures (1822); Möbius, Barycentritcher Calcul (1826); Steiner, Abhängigkeit Geometrischer Gestalten (1832), containing the first full discussion of the projective relations between rows, pencils, A∙c.; Von Staudt, Geometrie der Lage (1847) and Beiträge zur Geometrie der Lage (1856-60), in which a system of geometry is built up fιom the beginning without any reference to number, so that ultimately a number itself gets a geometrical definition, and in which imaginary elements are systematically introduced into pure geometry; Chasles, Aperçu Historique (1837), in which the author gives a brilliant account of the progress of modern geometrical methods, pointing out the advantages of the different purely geometrical methods as compared with the analytical ones, but without taking as much account of the German as of the Flench authors; Id., Rapport sur les Progrès de la Géométrie (1870), a continuation of the Aperçu ; Id., Traité de Géométrie Supérieure (1852); Cremona, Jntroduzione ad una Teoria Geometrica della Curve Piane (1862) and its continuation Preliminary di una Teoria Geometrica delle Superficie, which at present are most easily procurable in their German translations by Cnrtze. As more elementary books, we mention Steiner, Vorlesungen iιber Synthetische Geometrie, edited by Geiser and Schröder (1867); Cremona, Éléments de Géométrie Projective (1875), translated from the Italian by Dewulf; Townsend. Modern Geometry of the Point, Line, and Circle (1863). which contains a variety of modern methods, but, unfortunately, is confined to circles, without entering into conics. A great many of the propositions are, however, easily extended to conics. (O. ∏∙) PART II.—ANALYTICAL GEOMETRY.
This will be here treated as a method. The science is Geometry; and it would be possible, analytically, or by the method of coordinates, to develop the trutlιs of geometry in a systematic course. But it is proposed not in any way to attempt this, but simply to explain the method, giving such examples, interesting (it may be) in themselves, as are suitable for showing how the method is employed in the demonstration and solution of theorems or problems.
Geometry is one-, two-, or three-dimensional, or, what is the same thing, it is lineal, plane, or solid, according as the space dealt with is the line, the plane, or ordinary (three-dimensional) space. No more general view of the subject need here be taken :—but in a certain sense one-dimensional geometry does not exist, inasmuch as the geometrical constructions for points in a line can only be performed by travelling out of the line into other parts of a plane which contains it, and conformably to the usual practice Analytical Geometry will be treated under the two divisions, Plane and Solid.
It is proposed to consider Cartesian coordinates almost exclusively; for the proper development of the science homogeneous coordinates (three and four in plane and solid geometry respectively) are required; and it is moreover necessary to have the correlative line-and planecoordinates; and in solid geometry to have the six coordinates of the line. The most comprehensive English works are those of Dr Salmon, Th e Conics (5th edition, 1869), Higher Plane Curves (2d edition, 1873), and Geometry of Three Dimensions (3d edition, 1874); we have also on plane geometry Clebsch’s Vorlesungen über Geometrie, posthumous, edited by Dr F. Lindemann, Leipsic, 1875, not yet complete.
I. Plane Analytical Geometry (§§ 1-25).
1. It is assumed that the points, lines, and figures considered exist in one and the same plane, which plane, therefore, need not be in any way referred to. The position of a point is determined by means of its (Cartesian) coordinates; i.e., as explained under the article Curve, we take the two lines x'0x and y'Oy, called the axes of x and y respectively, intersecting in a point 0 called the origin, and determine the position of any υther point P by means of its coordinates x = OM (or NP), and y = MP (or ON).
The two axes are usually (as in fig. 1) at right angles to each other, and the lines PM, PN are then at right angles to the axes of x and y respectively. Assuming a scale at pleasure, the coordinates x, y of a point have numerical values.
It is necessary to attend to the signs: x has opposite signs according as the point is on one side or the other of the axis of y, and similarly y has opposite signs according as the point is on the one side or the other of the axis of x. Using the letters N, E, S, W as in a map, and considering the plane as divided into four quadrants by the axes, the signs are usually taken to be— [table]
A point is said to have the coordinates (a, b), and is referred to as the point (a, δ), when its coordinates are x = a, y≈b∙> the coordinates x, y of a variable point, or of a point which is for the time being regarded as variable, are said to be current coordinates.
2. It is sometimes convenient to use oblique coordinates; the only difference is that the axes are not at right angles to each other; the lines PM, PN are drawn parallel to the axes of y and x respectively, and the figure OMPN is thus a parallelogram. But in all that follows the Cartesian coordinates are taken to be rectangular; polar coordinates and other systems will be briefly referred to in the sequel
3. If the coordinates (x, y y ) of a point are not given, but only a relation between them f(x, y) = 0, then we have a curve. For, if we consider x as a real quantity varying continuously from - oo to + oo , then, for any given value of x, y has a value or values. If these are all imaginary, there is not any real point; but if one or more of them be real, we have a real point or points, which (as the assumed value of x varies continuously) varies or vary continuously therewith; and the locus of all these real points is a curve. The equation completely defines the curve; to trace the curve directly from the equation, nothing else being known, we obtain as above a series of points sufficiently near to each other, and draw the curve through them. For instance, let this be done in a simple case. Suppose y = 2x - I; it is quite easy to obtain and lay down a series of points as near to each other as we please, and the application of a ruler would show that these were in a line; that the curve is a line depends upon something more than the equation itself, viz., the theorem that every equation of the form y = ax + b represents a line; supposing this known, it will be at once understood how the process of tracing the curve may be abbreviated; we have x = 0, y = - 1, and x — ⅜, y = 0; the curve is thus the line passing through these two points. But in the foregoing example the notion of a line is taken to be a known one, and such notion of a line does in fact precede the consideration of any equation of a curve whatever, since the notion of the coordinates themselves rests upon that of a line. In other cases it may very well be that the equation is the definition of the curve; the points laid down, although (as finite in number) they do not actually determine the curve, determine it to any degree of accuracy; and the equation thus enables us to construct the curve.
A curve may be determined in another way; viz., the coordinates x, y may be given each of them as a function of the same variable parameter θ ; x, y=foθ), ≠(6) respectively. Here, giving to θ any number of values in succession, these equations determine the values of x. ιfo that is, the positions of a series of points on the curve. The ordinary form y = φ(x), where y is given explicitly as a function of x, is a particular case of each of the other two forms: we have fox, y),≈yφ(fo) f — θ5 aud x = Θ, y = φ(shfi
4. As remarked under Curve, it is a useful exercise to trace a considerable number of curves, first taking equations which are purely numerical, and then equations which contain literal constants (representing numbers); the equations most easily dealt with are those wherein one coordinate is given as an explicit function of the other, say y = φ(x) as above. A few examples are here given, with such explanations as seem proper.
(1.) y≈2x-l, as before; itis at once seen that this is a line; and taking it to be so, any two points, for instance, (0, -1) and (⅛, 0), determine the line.
(2.) y≈x e . The equation shows that x may be positive or negative, but that y is always positive, and has the same values for equal positive and negιtive values of x : the curve passes through the origin, and through the points (±1, 1). It is ah eady known that the curve lies wholly above the axis of x. To find its form in the neighbourhood of the origin, give x a small value, x≈ ±01 or ±0 - 01, then y is very much smaller, = OO1 and 0 - 0001 in the two cases respectively; this shows that the curve touches the axis of x at the origin. Moreover, x may be as large as we please, but when it is large, y is much larger; for instance, z = 10, y = 100. The curve is a parabola (fig. 2).
(3.) y=x 3 . Here x being positive y is positive, but x being negative y is also negative: the curve passes through the origin, and also through the points (ï, 1) and ( — 1, — 1). Moreover, when x is small, = 0 - l for example, then not only is y, ≡≡ 0*001, very much smaller than x, but it is also very much smaller than y was for the last-mentioned curve y≈x 2 , that is, in the neighbourhood of the origin the present curve approaches more closely the axis of x. The axis of x is a tangent at the origin, but it is a tangent of a peculiar kind (a stationary or inflexional tangent), cutting the curve at the origin, which is an inflexion. The curve is the cubical parabola (fig. 3).
(4.) y i ≈x- l.ae-3.ae-4. Here y=0 forae=l, =3, =4. Whenever x- l.z-3.z- 4 is positive, y has two equal and opposite values; but when x- 1.x - 3.x - 4, is negative, then y is imaginary. In particular, for x less than 1, or between 3 and 4, y is imaginary, but for x between 1 and 3, or greater than 4, y has two values. It is clear that for x somewhere between 1 and 3, y will attain a maximum, the values of x and y may be found approximately by trial. The curve will consist of an oval and infinite branch, and it is easy to see that, as shown in fig. 4, the curve where it cuts the axis of x cuts H at right angles. It may be further remarked that, as x increases from 4, the value of y will increase more and more rapidly; for instance, x=5, y 2 ≈8, a:=10, y 2 = 378, &c., and it is easy to see that this implies that the curve has on the infinite branch two inflexions as shown.
(5.) y 2 ≈x-c.x-b.x-a, where α≥∙δ>c (that is, a nearer to + ∞, c to - oo ). The curve has the same general form as in the last figure, the oval extending between the limits x=c, x=b, the infinite branch commencing at the point x≈a.
(6.) y i =(x-c) 2 (x-a). Suppose that in the last-mentioned curve, y 2 ≈x-c.x-b.x-a, b gradually diminishes, and becomes ultimately = c. The infinite branch (see fig. 5) changes its form, but not in a very marked manner, and it retains the two inflexions. The oval lies always between the values x≈c, x=b, and therefore its length continually diminishes; it is easy to see that its breadth will also continually diminish; ultimately it shrinks up into a mere point. The curve has thus a conjugate or isolated point, or acnode. For a direct verification observe that x≈c, y≈0, so that (c, 0) is a point of the curve, but if x is either less than c, or between c and a, y 2 is negative, and y is imaginary.
(7.) y — (x — c) (x - α) 2 . If in the same curve b gradually increasesand becomes ultimately ≈a, the oval and the infinite branch change each of them its form, the oval extending always between the values x=c, x≈b, and thus continually approaching the infinite branch, which begins at x≈a. The consideration of a few numerical examples, with careful drawing, would show that the oval and the infinite branch as they approach sharpen out each towards the other (the two inflexions on the infinite branch coming always nearer to the point (α, 0) so that finally, when b becomes ≈a, the curve has the form shown in fig. 6, there being now a double point or node (crunode) at A, and the inflexions on the infinite branch having disappeared.
In the last four examples the curve is one of the cubical curves called the divergent parabolas: 4 is a mere numerical example of 5, and 6, 7, 8 are in Newton’s language the parabola cum ovali, punctata, and nodata respectively. When a, b, c are all equal, or the form is y 2 = (x- c) 3 , we have a cuspidal form, Newton’s parabola cuspidata, otherwise the semicubical parabola.
(8. ) As an example of a curve given by an implicit equation, suppose the equation is x 3 +y 3 - 3aψ = 0; this is a nodal cubic curve, the node at the origin, and the axes touching the two branches respectively (fig. 7). An easy mode of tracing it is to express x, y each of them in terms of a variable θ, 3θ 30“
X = l + 0 3 ’ = 1 + ⅛ 3 ’ * t * s instructive to trace the curve directly from its equation.
5. It may be remarked that the purely algebraical process, which is in fact that employed in finding a differential coefficient , if applied directly to the equation of the curve, determines the point consecutive to any given point of the curve, that is, the direction of the curve at such given point, or, what is the same tliing, the direction of the tangent at that point. In fact, if α, β are the coordinates of any point on a curve fox, y) = 0, then writing in the equation of the curve x = a + h, y = β + k, and in the resulting equation ∕(α + 7i, β + ⅛) = 0 (developed in powers of h and ⅛), omitting the term ∕(α, /3), which vanishes, and the terms containing the second and higher powers of h, k, we have a linear equation Åh + B½∙ = 0, which determines the ratio of the increments h, k. Of course, in the analytical development of the theory, we translate this into the notation of the differential calculus; but the question presents itself, and is thus seen to be solvable, as soon as it is attempted to trace a curve from its equation.
Geometry is Descriptive, or Metrical.
6. A geometrical proposition is either descriptive or metrical : in the former case it is altogether independent of the idea of magnitude (length, inclination, &c.); in the latter case it has reference to this idea. It is to be noticed that, although the method of coordinates seems to be by its inception essentially metrical, and we can hardly, except by metrical considerations, connect an equation with the curve which it represents (for instance, even assuming it to be known that an equation Az + By + C = 0 represents a line, yet if it be asked what line, the only form of answer is, that it is the line cutting the axes at distances from the origin - C÷A, -C÷B respectively), yet in dealing by this method with descriptive propositions, we are, in fact, eminently free from all metrical considerations.
7. It is worth while to illustrate this by the instance of the well-known theorem of the radical centre of three circles. The theorem is that, given any three circles A, B, C (fig. 8), the common chords αα', ββ> Tf' of the three pairs of circles meet in a point.
The geometrical proof is metrical throughout :—
Take 0 the point of intersection of αα', ββ', and joining this with γ', suppose that 7'O does not pass through 7, but that it meets the circles A, B in two distinct points 7 1 , 7 2 respectively. We have then the knoλvn metrical property of intersecting chords of a circle; viz., in circle C where αα', ββ' are chords meeting at a point 0,
0α.0α' = 0j8.0j8', where, as well as in what immediate follows 0α, &c., denote, of course, lengths or distances.
Similarly in circle A,
0 j 8.0j3'-0γι∙0y, and in circle B, 0α. 0α' = 0γ 2 .0γ'.
Consequently 0γ 1 .0γ' = 0γ 2 .0γ', that is, 0γ 1 = 0γ 2 , or the points γ 1 and γ 2 coincide; that is, they each coincide with 7.
We contrast this with the analytical method :—
Here it only requires to be known that an equation kx + By + C = 0 represents a line, and an equation x 2 + y 2 + kx + By + C = 0 represents a circle. A, B, C have, in the two cases respectively, metrical significations; but these we are not concerned with. Using S to denote the function x 2 + y 2 + Ax + By+C, the equation of a circle is S = 0, where S stands for its value; more briefly, we say the equation is S, = x 1 + y i + kx + By + 0, = 0. I <et the equation of any other circle be S', ≈x 2 +y 2 + A⅛ + B'y + C ’ = 0; the equation S-S'=O is a linear equation (S-S ’ is in fact = (A-A') z + (B-B') y + G-C'), and it thus represents a fine; this equation is satisfied by the coordinates of each of the points of intersection of the two circles (for at each of these points S = 0 and S ’ =0, therefore also S-S'≈≈O); hence the equation S-S'=∙O is that of the line joining the two points of intersection of the two circles, or say it is the equation of the common chord of the two circles. Considering then a third circle S", — ae 2 + y 2 + k!'x + B''y + C" = 0, the equations of the common chords are S-S'=O, S-S"=O, S'-S"-*O (each of these a linear equation); at the intersection of the first and second of these lines S≈=S' and S = S", therefore also S'=S", or the equation of the third line is satisfied by the coordinates of the point in question; that is, the three chords intersect in a point 0, the coordinates of whiβh are determined by the equations S = S' = S".
It further appears that if the two circles S = 0, S = 0' do not intersect in any real points, they must be regarded as intersecting in two imaginary points, such that the line joining them is the real line represented by the equation S - S' = 0; or that two circles, whether their intersections be real or imaginary, have always a real common chord (or radical axis), and that for any three circles the common chords intersect in a point (of course real) which is the radical centre. And by this very theorem, given two circles with imaginary intersections, we can, by drawing circles which meet each of them in real points, construct the radical axis of the first-mentioned two circles.
8. The principle employed in showing that the equation of the common chord of two circles is S - S' = 0 is one of very extensive application, and some more illustrations of it may be given.
Suppose S=-0, S'=0 are lines (that is, let S, S' now denote linear functions kx + By+C, A⅛ + B'y + C'), then S-fcS'=O (⅛ an arbitrary constant) is the equation of any line passing through the point of intersection of the two given lines. Such a line may be made to pass through any given point, say the point (a ’ 0 , y t p ; i.c., if S o , S' o arc what S, S' respectively become on writing for (x, y) the values (x 0 , y 0 ), then the value of k is Zr=S 0 ÷S' 0 . The equation in factis SS' o -S o S'=O; and starting from this equation we at once verify it a posteriori ; the equation is a linear equation satisfied by the values of (ae, y) which make S —0, S'=0; and satisfied also by the values (a ’ 0 , 1∕ 0 ); and it is thus the equation of the line in question.
If, as before, S = 0, S' = 0 represent circles, then (Jc being arbitrary) S-Ä S' = 0 is the equation of any circle passing through the two points of intersection of the two circles; and to make this pass through a given point (α∙ 0 , y 0 ) we have again fc=S 0 ÷S' 0 . In the particular case Ä=l, the circle becomes the common chord (more accurately, it becomes the common chord together with the line infinity, but this is a question which is not here gone into).
If S denote the general quadric function, S≈ax 2 + 2hxy + by 2 + 2fy + 2gx + c, = (a, b, c,f, g, h)(x, y, 1) 2 , then the equation S = 0 represents a conic; assuming this, then, if S'= 0 represents another conic, the equation S - ⅛S= 0 represents any conic through the four points of intersection of the two conics.
Returning to the equation Az + By + C = 0 of a line, if this pass through two given points (x 1 , y 1 ), (ar 2 , y 2 ), then we must have Aaj 1 + By 1 + C=O, Az 2 + By 2 + C = 0, equations which determine the ratios A: B: C, and it thus appears that the equation of the line through the two given points is x ⅛fι - yj) - y( x ι ~ a∙ 2 )+ x l y∙2 - ¾Vι = o; or what is the same thing—
x, y, 1 = 0; aei, Vi, 1 a ⅛> y i D
9. The object still being to illustrate the mode of working with coordinates, we consider the theorem of the polar of a point in regard to a circle. Given a circle and a point 0 (fig. 9), we draw through 0 any two lines meeting the circle in the points A, A' and B, B' respectively, and then taking Q as the intersection of the lines AB' and A'B, the theorem is that the locus of the point Q is a right line depending only upon 0 and the circle, but independent of the particular lines OAA' and OBB'.
Taking 0 as the origin, and for the axes any two lines through O at right angles to each other, the equation of the circle will be
≈ 2 + y 2 + 2Az + 2By + C = 0; and if the equation of the line OAA' is taken to be y≈mx, then the points A, A' are found as the intersections of the straight line with the circle; or to determine x we have ae 2 (l + m 2 ) + 2x{k + Bm) + C = 0.
If (a⅛, y 1 ) are the coordinates of A, and (a⅛, y 2 ) of A', then the roots of this equation are x 1 , x 2 , whence easily
1_ + J_= _ 2 A + Bm x 1 x 2 C
And similarly, if the equation of the line OBB' is taken to be y = m'x, and the coordinates of B, B' to be (a⅛, y 3 ) and (x i , y i ) respectively, then
2_ 2. = - 2 À + IW a∙ 3 + aq C'
We have then χ(y 1 - y ì) ~ V( x ι ~ x a)+ x i l L ~ x Aι= 0 »
- 2/3) - !∕(≈ c S “⅜) + X A3 - x 3Vi = 0 » as the equations of the lines AB' and A'B respectively; for the first of these equations, being satisfied if we write therein (x l , y 1 ) or ( a ∙ y 4 ) for (x, y), is the equation of the line AB', and similarly the second equation is that of the line A'B. Reducing by means of the relations y 1 - mx 1 = 0, y 2 - mx 2 = 0, y 3 - m'x 3 = 0, y i - m'x i = 0, the two equations become x(mx 1 - m ’ x i ) - y(x 1 - x i ) + (∞' - m)x 1 x i =0, x(mx 2 - m'x 3 ) - y(x 2 - a⅛) + (m' - m)x 2 x 3 =C, and if we divide the first of these equations by wι 1 τn 4 , and the second by m 2 m 3 , and then add, we obtain
÷(⅛÷⅛M⅛÷⅜)M⅛÷⅛√-HB
+ 2»i'- 2∕∕ι = 0, or, what is the same thing, (⅜ 1 +½)to->"⅛)-(⅛ 1 +⅛ t )to-~≈>+ 2 '"'- 2 ''-β.
which by what precedes is the equation of a line through the point Q. Substituting herein for A + —, — + — their foregoing values, a ⅛ a∙ 2 a¾ x 4 the equation becomes
- (A + B∞)(y-wιct∙) + (A + Bm')(y - mx) + m ’ -m≈Q ; that is, (m - mj(Ax + By + C)=0; or finally it is Ax + By + C = 0, showing that the point Q lies in a line the position of which is independent of the particular lines OAA z , OBB' used in the construction. It is proper to notice that there is no correspondence to each other of the points A, A' and B, B'; the grouping might as well have been A, A' and B', B; and it thence appears that the line Arr + By + C = 0 just obtained is in fact the line joining the point Q with the point R which is the intersection of AB and A'B'.
10. The equation Az + By + C = 0 of a line contains in appearance 3, but really only 2 constants (for one of the constants can be divided out), and a line depends accordingly upon 2 parameters, or can be made to satisfy 2 conditions. Similarly, the equation (a, b, c, f i g, lιfyc 1 y, l) 2 = 0 of a conic contains really 5 constants, and the equation (-)(z, y, l) 3 = 0 of a cubic contains really 9 constants. It thus appears that a cubic can be made to pass through 9 given points, and that the cubic so passing through 9 given points is completely determined. There is, however, a remarkable exception. Considering two given cubic curves S = 0, S' = 0, these intersect in 9 points, and through these 9 points we have the whole series of cubics S - &S' = 0, where k is an arbitrary constant: k may be determined so that the cubic shall pass through a given tenth point (/¢ = S o ÷ S' o , if the coordinates are (x w y 0 ), and S o , S' o denote the corresponding values of 8, S'). The resulting curve SS' 0 - S'S 0 = 0 may be regarded as the cubic determined by the conditions of passing through 8 of the 9 points and through the given point (r 0 , y 0 ); and from the equation it thence appears that the curve passes through the remaining one of the 9 points. In other words, we thus have the theorem, any cubic curve which passes through 8 of the 9 intersections of two given cubic curves passes through the 9th intersection.
The applications of this theorem are very numerous; for instance, we derive from it Pascal’s theorem of the inscribed hexagon. Consider a hexagon inscribed in a conic. The three alternate sides constitute a cubic, and the other three alternate sides another cubic. The cubics intersect in 9 points, being the 6 vertices of the hexagon, and the 3 Pascalian points, or intersections of the pairs of opposite sides of the hexagon. Drawing a line through two of the Pascalian points, the conic and this line constitute a cubic passing through 8 of the 9 points of intersection, and it therefore passes through the . remaining point of intersection—that is, the third Pascalian point; and since obviously this does not lie on the conic, it must lie on the line—that is, we have the theorem that the three Pascalian points (or points of intersection of the pairs of opposite sides) lie on a line.
Metrical Theory.
11. The foundation of the metrical theory consists in the simple theorem that if a finite line PQ (fig. 10) be projected upon any other line OO' by lines perpendicular to OO', then the length of the projection P'Q' is equal to the length of PQ into the cosine of its inclination to P'Q'; or, what is the same thing, that the perpendicular distance P'Q' of any two parallel lines is equal to the inclined distance PQ into the cosine of the inclination. It at once follows that the algebraical sum of the projections of the sides of a closed polygon upon any line is = 0; or, reversing the signs of certain sides, and considering the polygon as consisting of two broken lines, each extending from the same initial to the same terminal point, the sum of the projections of the lines of the first set upon any line is equal to the sum of the projections of the lines of the second set. Observe that if any line be perpendicular to the line on which the projection is made, then its projection is = 0.
Thus, if we have a right-angled triangle PQR (fig. 11), where QR, RP, QP are = ξ, η, p respectively, and whereof the baseangle is = α, then projecting successively on the three sides, we have
ξ = p cosα, ∙η = p sinα, ρ = ξ cosa + rç sin a; and we thence obtain p 2 = ξ 2 + n 2 J cos 2 α + sin 2 α=l.
And again, by projecting on a line Qaq, inclined at the angle « to QR, we have p cos (α - a') = ξ cos a' + η sin a'; and by substituting for ξ, η their foregoing values, cos (α - a') = cos a cos a' + sin a sin a.
It is to be remarked that, assuming only the theory of similar triangles, we have herein a proof of Euclid, Book I., Prop 47; in fact, the same as is given Book VI., Prop. 31; and also a proof of the trigonometrical formula for cos (α - a'). The formulae for cos (a + a') and sin (a ± a') could be obtained in the same manner.
Draw PT at right angles to Qz 1 , and suppose QT, TP=ξ 1 , ∙η 1 respectively, so that we have now the quadrilateral QRPTQ, or, what is the same thing, the two broken lines QRP and QTP, each extending from Q to P. Projecting on the four sides successively, we have [table]
where the third equation is that previously written p cos (α - a') = ξ cos a + ∙η sin a. Equations of Right Line and Circle.—Transformation of Coordinates.
12. The required formulae are really contained in the foregoing results. For, in fig. 11, supposing that the axis of x is parallel to QR, and taking a, b for the coordinates of Q, and (x, y) for those of P, then we have ξ, η = x - a, y -b respectively; and therefore x— a = p cos a, j∕-δ=psi∩β, p 2 =(x-α) 2 + (y-δ) 2 .
Writing the first two of these in the form x—a y-b. . = ’~ ( = p), cos α sm a we may regard Q as a fixed point, but P as a point moving in the direction Q to P, so that a remains constant, and then, omitting the equation (=p), we have a relation between the coordinates x, y of the point P thus moving in a right line,—that is, we have the equation of the line through the given point (a, b) at a given inclination α to the axis of x. And, moreover, if, using this equation ( = p), we write z=α + ρ cose, y=b + p sin a, then we have expressions for the coordinates x, y of a point of this line, in terms of the variable parameter p.
Again, take the point T to be fixed, but consider the point P as moving in the line TP at right angles to QT. If instead of ξ∣ we take p for the distance QT, then the equation ξ 1 = ξ cos a! + η sin α' will be
(z-a) cosa' + (y- b) sina'≈p; that is, this will be the equation of a line such that, its perpendicular distance from the point (a, b) is≈p, and that the inclination of this distance to the axis of x is = α'.
From either form it appears that the equation of a line is, in fact, a linear equation of the form Az + By + C = 0. It is important to notice that, starting from this equation, we can determine conversely the α but not the (a, b) of the form of equation which contains these quantities; and in like manner the α' but not the (a, b) or p of the other form of equation. The reason is obvious. In each case (a, b) denote the coordinates of a point, fixed indeed, but which is in the first form any point of the line, and in the second form any point whatever. Thus, in the second form the point from which the perpendicular is let fall may be the origin. Here (a, b) = (0, 0), and the equation is x cosα' + y sina'-j5 = 0. Comparing this with Ax + By + C = 0, we have the values of cos α', sin a, and p.
13. The equation p 2 =(z-<z) 2 + (y-5) 2 is an expression for the squared distance of the two points («, b) and (x, y). Taking as before the point Q, coordinates (a, b), as a fixed point, and writing c in the place of p, the equation (x - á) 2 + (y - 5) 2 = c 2 expresses that the point (x, y) is always at a given distance c from the given point (a, b) ; viz., this is the equation of a circle, having (a, b) for the coordinates of its centre, and c for its radius.
The equation is of the form x 2 + y 2 + 2 Ax + 2By + C = 0, and here, the number of constants being the same, we can identify the two equations; we find a = - A, b = - B, c 2 = A 2 + B 2 - C, or the last equation is that of a circle having - A, — B for the. coordinates of its centre, and s∕A 2 4-B 2 - C for its radius.
14. Drawing (fig. 11) Qy 1 at right angles at Qz 1 , and taking Qz 1 , Qy 1 as a new set of rectangular axes, if instead of ξ l , η l we write x l , y l , we have x χ , y 1 as the new coordinates of the point P; and writing also α in place of a (a now denoting the inclination of the axes Qz 1 and Oz), we have the formulae for transformation between two sets of rectangular axes. These are x - a = x 1 cos a - y x sin a , y - b = z 1 sin a + y l cos α, and x 1 ≈ (x-a) cos a + (y - b) sin a, y l — -(x-a) sin a + (y- b) cos a, each set being obviously at once deducible from the other one. In these formulae (a, b) are the zy-coordinates of the new origin Q 1 , and α is the inclination of Qz 1 to Ox. It is to be noticed that Qz 1 , Qy 1 are so placed that, by moving 0 to Q, and then turning the axes Oz 1 , Oy 1 round Q (through an angle α measured in the sense Oz to Oy), the original axes Ox, Oy will come to coincide with Q- c υ Qy i respectively. This could not have been done if Qy 1 had been drawn (at right angles always to Qz 1 ) in the reverse direction, we should then have had in the formulae -y 1 instead of y 1 . The new formulae which would be thus obtained are of an essentially distinct form: the analytical test is that in the formulae as written down we can, by giving to α a proper value (in fact α = 0), make the (z-α) and (y-b) equal to x x and y 1 respectively; in the other system we could only make them equal to x x , - y v or - x x , y 1 respectively. But for the very reason that the second system can be so easily derived from the first, it is proper to attend exclusively to the first system,—that is, always to take the new axes so that the two sets admit of being brought into coincidence.
In the foregoing system of two pairs of equations, the first pair give the original coordinates x, y in terms of the new coordinates x x , y x ; the second pair the new coordinates x 1 , y x in terms of the original coordinates x, y. The formulae involve (a, b), the original coordinates of the new origin; it would be easy instead of these to introduce (a χ , b χ ), the new coordinates of the origin. Writing (a, b) = (0, 0), we have, of course, the formulae for transformation between two sets of rectangular axes having the same origin, and it is as well to write the formulae in this more simple form; the subsequent transformation to a new origin, but with axes parallel to the original axes, can then be effected without any difficulty.
15. All questions in regard to the line may be solved by means of one or other of the foregoing forms—
Az+By+C = 0, y = Ax+ B , x-a = y-b os α sin a (z - a) cos a' + (y - b) sin a , -p= 0; or it may be by a comparison of these different forms: thus, using the first form, it has been already shown that the equation of the line through two given points (x χ , y χ ), (⅜ ⅜) is χ (yι - y i ) - y( χ ι - aes)+ χ dL - χ Aι= θ> or, as this may be written, y ~ vι ^⅛^ x ~ x i>'
A particular case is the equation
-+4=ι. a b representing the line through the points (a, 0) and (0, b), or, what is the same thing, the line meeting the axes of x and y at the distances from the origin a and b respectively. It
A
may be noticed that, in the form Ax + By + C = 0 , -
denotes the tangent of the inclination to the axis of x, or we may say that B ÷ λ ∕A 2 + B 2 and - A ÷ JA 2 + B 2 denote respectively the cosine and the sine of the inclination to the axis of x. A better form is this: A ÷ JA 2 + B 2 and B ÷ JA 2 + B 2 denote respectively the cosine and the sine of inclination to the axis of x of the perpendicιdar upon the line. So of course, in regard to the form y = Ax + B, A is here the tangent of the inclination to the axis of x; 1 ÷ JA 2 + 1 and A ÷ sÃ- + 1 are the cosine and sine of this inclination, &c. It thus appears that the condition in order that the lines Ax + By + C = 0 and A'z + By - C' = 0 may meet at right angles is AA' + BB' = 0; so when the equations are y = Ax + B, y = A'z + B', the condition is A A' + 1=0, or say the value of A' is = - 1 ÷ A.
The perpendicular distance of the point (a, b) from the line Ax + By + C = 0 is (Aα + B⅞ + C) ÷ ^∕A 2 + B 2 . In all the formulae involving JA 2 + B 2 or s ∕A 2 + 1, the radical should be written with the sign ±, which is essentially indeterminate: the like indeterminateness of sign presents itself in the expression for the distance of two points p= ± J{ x — a) 2 + (y - b) 2 ; if, as before, the points are Q, P, and the indefinite line through these is z'QPz, then it is the same thing whether we measure off from Q along this line, considered as drawn from z' towards z, a positive distance k, or along the line considered as drawn reversely from z towards z', the equal negative distance - k, and the expression for the distance p is thus properly of the form ± k. It is interesting to compare expressions which do not involve a radical: thus, in seeking for the expression for the perpendicular distance of the point (a, V) from a given line, let the equation of the given line be taken in the form, x cos a + y sin a -p = 0 (p being the perpendicular distance from the origin, α its inclination to the axis of x) : the equation of the line may also be written (z-α)cos α + (y-δ) sina-y> 1 = 0, and we have thence p 1 —p — a cos a - b sin α, the required expression for the distance p x : it is here assumed that p χ is drawn from (a, δ) in the same sense as p is drawn from the origin, and the indeterminateness of sign is thus removed.
16. As an instance of the mode of using the formulae, take the problem of finding the locus of a point such that its distance from a given point is in a given ratio to its distance from a given line.
We take (a, b) as the coordinates of the given point, and it is convenient to take (x, y) as the coordinates of the variable point, the locus of which is required: it thus becomes necessary to use other letters, sny (X, Y), for current coordinates in the equation of the given line. Suppose this is a line such that its perpendicular distance from the origin is =p, and that the inclination of p to the axis of X is = a; the equation is X cos α + Y sin a - p = 0. In the result obtained in § 15, writing (x, y) in place of (a, ò), it appears that the perpendicular distance of this line from the point (x, y) is =p-x cos a - y sin α; hence the equation of the locus is fox-<fa + (y-bf t =e {p- x cos a-y sin a), or say (x— α) s + (y - δ) 2 - e 2 (x cos a + y sin a -p) s = 0, an equation of the second order.
The Conics (Parabola, Ellipse, Hyperbola).
17. The conics or, as they were called, conic sections were originally defined as the sections of a right circular cone; but Apollonius substituted a definition, which is in fact that of the last example: the curve is the locus of a point such that its distance from a given point (called the focus) is in a given ratio to its distance from a given line (called the directrix); taking the ratio as e: 1, then e is called the eccentricity.
Take FD for the perpendicular from the focus F upon the directrix, and the given ratio being that of e : 1 (e≥,-= or < 1, but positive), and let the distance FD be divided at 0 in the given ratio, say we have 0D=m, OF= ein, where m is positive;— then the origin may be taken at 0, the axis Ox being in the direction OF (that is from 0 to F), and the axis Oy at right angles to it. The distance of the point (x, y) from F is = V(x - em)' 2 + y 2 , its distance from the directrix is =x+m; the equation therefore is (x- em) 3 +y 2 =e 2 (x + m) 2 ; or, what is the same thing, it is
(1 - e 2 > 2 - 2me (1 + e)x + y 2 =0.
If e 2 = l, or, since ¢ is taken to be positive, if ¢ = 1, this is . . , . 3∕ 2 -4wur=0, which is the parabola.
If e 2 not = 1, then the equation may be written (l-e s )faj- + -
∖ l-e∕ 1-e
Supposing e positive and < 1, then, writing f), the e equation becomes
(1 - c 2 )(x - α) 2 + y 2 =α 2 (l - e 2 ), that is,
(¾-α) 3 y 2 - a 2 a 2 (l-e 2 ) ’
or, changing the origin and writing δ 2 = α 2 (l - β 2 ) z this is ~2 ∙>fi - + y -≈l, a i b i which is the ellipse.
And similarly if e be positive and ≥ 1, then writing m ≈ a ^ e e ’ the equation becomes
(1-e 2 )(x + α) 2 + y 2 =α 2 (l-e 2 ), that is,
(¾÷α) 2 y 2
α 2 + α 2 (l-e 2 ) ’
or changing the origin and writing δ 2 =α 2 (e 2 -1), this is
α a δ a ’ which is the hyperbola.
18. The general equation ax 2 + 2hxy + by 2 + 2fy + 2gx + c = 0, or as it is written (a, b, c, f, g, h)(x, y, 1) 2 = 0, may be such that the quadric function breaks up into factors, = (ax +βy + y)(α⅛ + β'y +y'); and in this case the equation represents a pair of lines, or (it may be) two coincident lines. When it does not so break up, the function can be put in the form λ { (x - a') 2 + (y - b') 2 - e 2 (x cos a + y sin a - p) 2 } , or, equating the two expressions, there will be six equations for the determination of λ, a , b', e, p, a; and by what precedes, if a', V, e, p, a are real, the curve is either a parabola, ellipse, or hyperbola. The original coefficients (a, b, c,s g, h) may be such as not to give any system of real values for a b ’ , e,p,a; but when this is so the equation (a, b, c, f, g, h)(x, y, 1) 2 = 0 does not represent a real curve; the imaginary curve which it represents is, however, regarded as a conic. Disregarding the special cases of the pair of lines and the twice repeated line, it thus appears that the only real curves represented by the general equation (a, b, c,s g, h)(x, y, 1) 2 = 0 are the parabola, the ellipse, and the hyperbola. The circle is considered as a particular case of the ellipse.
The same result is obtained by transforming the equation (a, b, c, s g, lfi)(x, y, 1) 2 = 0 to new axes. If in the first place the origin be unaltered, then the directions of the new (rectangular) axes O<r 1 , Oy 1 can be found so that h χ (the coefficient of the term xgyst shall be =0; when this is done, then either one of the coefficients of αq 2 , ysl is = 0, and the curve is then a parabola, or neither of these coefficients is = 0, and the curve is then an ellipse or hyperbola, according as the two coefficients are of the same sign or of opposite signs.
19. The curves can be at once traced from their equations :—
y 2 = imx, for the parabola (fig. 13),
/v»2 λ Α cf + lP = 1’ f° 1 el ⅛ se 0⅛ ’
= 1, for the hyperbola (fig. 15); and it will be noticed how the form of the last equation puts in evidence the two asymptotes — = ±-∣- of the hyperbola. Referred to the asymptotes (as a set of oblique axes) the equation of the hyperbola takes the form xy≈c> and in particular, if in this equation the axes are at right angles, then the equation represents the rectangular hyperbola referred to its asymptotes as axes.
Tangent, Normal, Circle and Radius of Curvature, fyc.
20. There is great convenience in using the language and notation of the infinitesimal analysis; thus we consider on a curve a point with coordinates (x, y), and a consecutive point the coordinates of which are (x + dx, y + dish or again a second consecutive point with coordinates (x + dx + ⅛sh i x, y + dy + ⅛d 2 y), &c.; and in the final results the ratios of the infinitesimals must be replaced by differential coefficients in the proper manner; thus, if x, y are considered as given functions of a parameter θ, then dx, dy have in fact the values — dθ , γ-. dθ , and (only the ratio being really material) they may in the result be replaced by ⅛L ~γ λ ∙ This includes the case where the equation dfr dê of the curve is given in the form y = φ(x) ; θ is here = x, and the increments dx, dy are in the result to be replaced by 1, . So also with the infinitesimals of the higher orders d 2 x, &c.
21. The tangent at the point (x, y) is the line through this point and the consecutive point (x + dx,y + dy) ; hence, taking ξ, η as current coordinates, the equation is ¢-≈ = y~y dx dy 1 an equation which is satisfied on writing therein ξ, η = (x, y) or =(x + dx, y + dy). The equation may be written á ν-y d ^-^∙ being now the differential coefficient of y in regard to x ; and this form is applicable whether y is given directly as a function of x, or in whatever way y is in effect given as a function of x: if as before x, y are given each of them as a function of θ, then the value of is = — ÷ ^∙, which dx dð d∂ ’ is the result obtained from the original form on writing therein , , for dx, dy respectively.
So again, when the curve is given by an equation u = 0 between the coordinates (x, y), then — is obtained from ., ,. du du du λ t , . , the equation y + τ ^ j = θ∙ But here it is more elegant. dx dy dx o ’ using the original form, to eliminate dx, dy by the formula - i - dx + − dy; we thus obtain the equation of the tangent in the form
For example, in the case of the ellipse - .√ + = 1, the il· - o - equation is (ξ -x) + (η -y) = 0: or reducing by means of the equation of the curve the equation of the tangent is
α 2 δ 2 1 '
The normal is a line through the point at right angles to the tangent; the equation therefore is
(ξ - x)dx + (η - y)dy = 0, where dx, dy are to be replaced by their proportional values as before.
22. The circle of curvature is the circle through the point and two consecutive points of the curve. Taking the equation to be
(ξ-≡) 2 + ⅛-3) 2 =γ 2 , the values of α, β are given by dy(dx s + dy 2 ) -dx(dx 2 +dy i ) dxd 2 y - dyd 2 x > y ~ ? dxd i y - dyd 2 x ’ and we then have β , m , , x , (i to 2 +⅜ 2 ) 3
7,-(≈-α) +(y ß) >~(dxd ii y-dy (Pxst
In the case where y is given directly as a function of x, then, writing for shortness p = f γ- , q = , this is α j - dx ’ 1 dx 2
(1 +d 2 ) 3 y 2 = —- 3 --, or, as the equation is usually written, (1+p 2 )?
γ = ——— , the radius of curvature, considered to be positive or negative according as the curve is concave or convex to the axis of x.
It may be added that the centre of curvature is the intersection of the normal by the consecutive normal.
The locus of the centre of curvature is the evolute. If from the expressions of a, β regarded as functions of x we eliminate x, we have thus an equation between (α, β), which is the equation of the evolute.
Polar Coordinates.
23. The position of a point may be determined by means of its distance from a fixed point and the inclination of this distance to a fixed line through the fixed point. Say we have r the distance from the origin, and θ the inclination of r to the axis of x; r and θ are then the polar coordinates of the point, r the radius vector, and θ the inclination. These are immediately connected with the Cartesian coordinates x, y by the formulae x = r cos θ, y = r sin. θ∙, and the transition from either set of coordinates to the other can thus be made without difficulty. But the use of polar coordinates is very convenient, as well in reference to certain classes of questions relating to curves of any kind—for instance, in the dynamics of central forces—as in relation to curves having in regard to the origin the symmetry of the regular polygon (curves such as that represented by the equation r = cos mθ), and also in regard to the class of curves called spirals, where the radius vector r is given as an algebraical or exponential function of the inclination θ. TriUnear Coordinates.
24. Consider a fixed triangle ABC, and (regarding the sides as indefinite lines) suppose for a moment that p, q, r denote the distances of a point P from the sides BC, CA, AB respectively,—these distances being measured either perpendicularly to the several sides, or each of them in a given direction. To fix the ideas each distance may be considered as positive for a point inside the triangle, and the sign is thus fixed for any point whatever. There is then an identical relation between p, q, r : if a, b, c are the lengths of the sides, and the distances are measured perpendicularly thereto, the relation is ap + bq + cr = twice the z ,area of triangle. But taking x, y, z proportional to p, q, r, or if we please proportional to given multiples of p, q, r, then only the ratios of x, y, z are determined; their absolute values remain arbitrary. But the ratios of p, q, r, and consequently also the ratios of x, y, z determine, and that uniquely, the point; and it being understood that only the ratios are attended to, we say that (z, y, z) are the coordinates of the point. The equation of a line has thus the form ax + by + cz = 0, and generally that of a curve of the nth order is a homogeneous equation of this order between the coordinates, ( * §z, y, z) n = 0. The advantage over Cartesian coordinates is in the greater symmetry of the analytical forms, and in the more convenient treatment of the line infinity and of points at infinity. The method includes that of Cartesian coordinates, the homogeneous equation in x, y, z is in fact an equation in — , -, í√ z which two quantities may be regarded as denoting Cartesian coordinates; or, what is the same thing, we may in the equation tvrite z== 1. It may be added that if the trilinear coordinates (z, y, z) are regarded as the Cartesian coordinates of a point of space, then the equation is that of a cone having the origin for its vertex; and conversely that such equation of a cone may be regarded as the equation in trilinear coordinates of a plane curve.
General Po int-Coordinates. —Line-Coordina tes.
25. All the coordinates considered thus far are pointcoordinates. More generally, any two quantities (or the ratios of three quantities) serving to determine the position of a point in the plane may be regarded as the coordinates of the point; or, if instead of a single point they determine a system of two or more points, then as the coordinates of the system of points. But, as noticed under Curve, there are also line-coordinates serving to determine the position of a line; the ordinary case is when the line is determined by means of the ratios of three quantities ξ, η, ζ (correlative to the trilinear coordinates z, y, z). A linear equation aξ + bη + cζ= 0 represents then the system of lines such that the coordinates of each of them satisfy this relation, in fact, all the lines which pass through a given point; and it is thus regarded as the line-equation of this point; and generally a homogeneous equation ( * y fyξ,η,ζ) n = 0 represents the curve which is the envelope of all the lines the coordinates of which satisfy this equation, and it is thus regarded as the line-equation of this curve.
II. Solid Analytical Geometry (§§26-40).
26. We are here concerned with points in space,—the position of a point being determined by its three coordinates z, y, z. We consider three coordinate planes, at right angles to each other, dividing the whole of space into eight portions called octants, the coordinates of a point being the perpendicular distances of the point from the three planes respectively, each distance being considered as positive or negative according as it lies on the one or the other side of the plane. Thus the coordinates in the eight octants have respectively the signs
The positive parts of the axes are usually drawn as in fig. 16, which represents a point P, the coordinates of which have the positive values OM, MN, NP.
27. It may be remarked, as regards the delineation of such solid figures, that if we have in space three lines at right angles to each other, say Oα, Ob, Oc, of equal lengths, then it is possible to project these by parallel lines upon a plane in such wise that the projections Oa', Ob', Oc' shall be at given inclinations to each other, and that these lengths shall be to each other in given ratios: in particular the two lines Oct', Oc' may be at right angles to each other, and their lengths equal, the direction of Ob', and its proportion to the two equal lengths Oα', Oc' being arbitrary. It thus appears that we may as in the figure draw Oz, Oz at right angles to each other, and Oy in an arbitrary direction; and moreover represent the coordinates x, z on equal scales, and the remaining coordinate y on an arbitrary scale (which may be that of the other two coordinates z, z, but is in practice usually smaller). The advantage, of course, is that a figure in one of the coordinate planes xz is represented in its proper form without distortion; but it may be in some cases preferable to employ the isometrical projection, wherein the three axes are represented by lines inclined to each other at angles of 120°, and the scales for the coordinates are equal (fig. 17).
For the delineation of a surface of a tolerably simple form, it is frequently sufficient to draw (according to the foregoing projection) the sections by the coordinate planes; and in particularwhen the surface is symmetrical in regard to the coordinate planes, it is sufficient to draw the quartersections belonging to a single octant of the surface; thus fig. 18 is a' convenient representation of an octant of the wave surface. Or a surface may be delineated by means of a series of parallel sections, or (taking these to be the sections by a series of horizontal planes) say by a series of contour lines. Of course, other sections may be drawn or indicated, if necessary. For the delineation of a curve, a convenient method is to represent, as above, a series of the points P thereof, each point P being accompanied by the ordinates PN, which serves to refer the point to the plane of xy ; this is in effect a representation of each point P of the curve, by means of two points P, N such that the line PN has a fixed direction. Both as regards curves and surfaces, the employment of stereographic representations is very interesting.
28. In plane geometry, reckoning the line as a curve of the first order, we have only the point and the curve. In solid geometry, reckoning a line as a curve of the first order, and the plane as a surface of the first order, we have the point, the curve, and the surface; but the increase of complexity is far greater that would hence at first sight appear. In plane geometry a curve is considered in connexion with lines (its tangents); hut in solid geometry the curve is considered in connexion with lines and planes (its tangents and osculating planes), and the surface also in connexion with lines and planes (its tangent lines and tangent planes); there are surfaces arising out of the line—cones, skew surfaces, developables, doubly and triply infinite systems of lines, and whole classes of theories which have nothing analogous to them in plane geometry: it is thus a very small part indeed of the subject which can be even referred to in the present article.
In the case of a surface we have between the coordinates (x, y, z) a single, or say a onefold relation, which can be represented by a single relation fox, y, z) = 0 ; or we may consider the coordinates expressed each of them as a given function of two variable parameters p, q ; the form zπ= fox, y) is a particular case of each of these modes of representation; in other words, we have in the first mode f (x, y, z) = zf(x, y), and in the second mode x =p, y-q for the expression of two of the coordinates in terms of the parameters.
In the case of a curve we have between the coordinates (z, y, z) a twofold relation: two equations fox, y, z) = 0, φ(x, y, z) = 0 give such a relation; τ.e,, the curve is here considered as the intersection of two surfaces (but the curve is not always the complete intersection of two sur-faoes, and there are hence difficulties); or, again, the coordinates may be given each of them as a function of a single variable parameter. The form y — φχ, z = ψx, where two of the coordinates are given in terms of the third, is a particular case of each of these modes of representation.
29. The remarks under plane geometry as to descriptive and metrical propositions, and as to the non-metrical character of the method of coordinates when used for the proof of a descriptive proposition, apply also to solid geometry; and they might be illustrated in like manner by the instance of the theorem of the radical centre of four spheres. The proof is obtained from the consideration that S and S' being each of them a function of the form x 2 + y 2 + z 2 + ax + by + cz + d, the difference S-S' is a mere linear function of the coordinates, and consequently that S - S' = 0 is the equation of the plane containing the circle of intersection of the two spheres S = 0 and S' = 0.
Metrical Theory.
30. The foundation in solid geometry of the metrical theory is in fact the before-mentioned theorem that if a finite right line PQ be projected upon any other line 00' by lines perpendicular to OO', then the length of the projection P'Q' is equal to the length of PQ into the cosine of its inclination to P'Q'—or (in the form in which it is now convenient to state the theorem) the perpendicular distance P'Q' of two parallel planes is equal to the inclined distance PQ into the cosine of the inclination. Hence also the algebraical sum of the projections of the sides of a closed polygon upon any line is = 0; or, reversing the signs of certain sides and considering the polygon as made up of two broken lines each extending from the same initial to the same terminal point, the sum of the projections of the one set of lines upon any line is equal to the sum of the projections of the other set of lines upon the same line. When any of the lines are at right angles to the given line’ (or, what is the same thing, in a plane at right angles to the given line) the projections of these lines severally vanish.
31. Consider the skew quadrilateral QMNP, the sides QM, MN, NP being respectively parallel to the three rectangular axes Ox, Oy, Oz; let the lengths of these sides be ξ, η, ζ, and that of the side QP be = ρ ; and let the cosines of the inclinations (or say the cosine-inclinations) of p to the three axes be α, β, y, then projecting successively on the three sides and on QP we have ξ, η, i=p a > pβ, py, and p = aζ 4- β∙η + yζ , whence p 2 = i 2 + ^ 2 + ζ 2 , which is the relation between a distance p and its projections ξ, η, ζ upon three rectangular axes. And from the same equations we obtain a 2 + β 2 + γ 2 = 1, which is a relation connecting the cosineinclinations of a line to three rectangular axes.
Suppose we have through Q any other line QT, and let the cosine-inclinations of this to the axes be α', β', s, and δ be its cosine-inclination to QP; also let p be the length of the projection of QP upon QT; then projecting on QT we have p = a!ζ + β'∙η + γ'f, = pS .
And in the last equation substituting for ξ, η, s their values pa, pβ, py we find
δ = βα' + ββ' + γγ', which is an expression for the mutual cosine-inclination of two lines, the cosine-inclinations of which to the axes are a, β, y and α', β', γ' respectively. We have of course a 2 + β 2 +γ 2 =l, and α' 2 + β' 2 + y' 2 = 1; and hence also
1 - δ 2 = α 2 + β 2 + γ 2 )(a ,2 + β' 2 + γ^ 2 ) - (aa' + ββ' + γγ') 2 ,-
= (3γ' - 3'γ) 2 +(γa' - γ'β) 2 + (aβ' - a ’ β) 2 ; so that the sine of the inclination can only be expressed as a square root. These formulae are the foundation of spherical trigonometry.
The Line, Plane, and Sphere.
32. The foregoing formulae give at once the equations of these loci.
For first, taking Q to be a fixed point, coordinates (a, b, c y ) and the cosine-inclinations (α, β, y) to be constant, then P will be a point in the line through Q in the direction thus determined; or, taking (x, y, z) for its coordinates, these will be the current coordinates of a point in the line. The values of ξ, η, ¢ then are x - a, y -b, z-c, and we thus have x-a y-b z ~ c ( » ^cΓ~ β “7 k ~ p ' , which (omitting the last equation, =∕>) are the equations of the line through the point (a, b, c), the cosine-inclinations to the axes being α. β, y, and these quantities being connected by the relation α 2 + ⅛ 2 + γ 2 =l. This equation may be omitted, and then a, β, y, instead of being equal, will only be proportional to the cosineinclinations.
Using the last equation, and writing x, y, z = a + aρ, b + βp, c + yp, these are expressions for the current coordinates in terms of a parameter p, which is in fact the distance from the fixed point (a, b, c).
It is easy to see that, if the coordinates (x, y, z) are connected by any two linear equations, these equations can always be brought into the foregoing form, and. hence that the two linear equations represent a line.
Secondly, taking for greater simplicity the point Q to be coincident with the origin, and α', β', γ', p to be constant, then p is the perpendicular distance of a plane from the origin, and α',θj8', χ are the cosine-inclinations of this distance to the axes (α' 2 + j8^ + y 2 =l). P is any point in this plane, and taking its coordinates to be {x, y, z) then (ξ, y, ζ) are=(z, y, z), and the foregoing equation p = a 'ξ + β , η + yζ becomes a!x + β'y + fz=p , which is thθ equation of the plane in question.
If, more generally, Q is not coincident with the origin, then, taking its coordinates to be (a, b, c), and writing p 1 instead of p, the equation is a!(x - α) + βfiy -b) + γ'(z - ¢) =p 1 ; and we thence have p 1 =p - (aa + bβ' + cγ'), which is an expression for the perpendicular distance of the point (a, b, c) from the plane in question.
It is obvious that any linear equation Az + By+C⅛+D = 0 between the coordinates can always be brought into the foregoing fohn, and hence that such equation represents a plane.
Thirdly, supposing Q to be a fixed point, coordinates (a, b, c) and the distance QP, =p, to be constant, say this is=d, then, as before, the values of ξ, η, ζ are x-a, y-b, z—c, and the equation ξ t + rl t + f 2 =p 2 becomes
(z-α) 2 + (y-⅛) 2 + (z -c) 2 =d 2 , which is the equation of the sphere, coordinates of the centre = (a, b, c) and radius—d.
A quadric equation wherein the terms of the second order are z 2 + y 2 + z 2 , viz., an equation z 2 + y i + z 2 + Ax + By + Cz + D = 0, can always, it is clear, be brought into the foregoing form; and it thus appears that this is the equation of a sphere, coordinates of the centre - ⅛A, - ⅜ B, - ⅜C, and squared radius = ∣ (A 2 + B 2 + C 2 ) - D.
Cylinders, Cones, Ruled Surfaces.
33. A singly infinite system of lines or system of lines depending upon one variable parameter forms a surface; and the equation of the surface is obtained by eliminating the parameter between the two equations of the line.
If the lines all pass through a given point, then the surface is a cone; and, in particular, if the lines are all parallel to a given line, then the surface is a cylinder.
Beginning with this last case, suppose the lines are parallel to the line x=mz, y=nz, the equations of a line of the system are x=mz + a, y=nz + b, — where a, b are supposed to be functions of the variable parameter, or, what is the same thing, there is between them a relation f(a, b) = 0: we have a≈x-mz, b=y-nz, and the result of the elimination of the parameter therefore is fx-mz, y-nz) = Q, which is thus the general equation of the cylinder the generating lines whereof are parallel to the line x=mz, y=nz. The equation of the section by the plane z = 0 is fox, y) = 0, and conversely if the cylinder be determined by means of its curve of intersection with the plane 2=0, then, taking the equation of this curve to t>ej ∖ x, y) = 0, the equation of the cylinder is√(z-røz, y-nz)≈0. Thus, if the curve of intersection be the circle (x - α) 2 + {y-β) 2 = γ 2 , we have {x - mz - α) 2 + (y - nz - 3) 2 = γ 2 as the equation of an oblique cylinder on this base, and thus also (zα) 2 + (y- 3) 2 = γ 2 as the equation of the right cylinder.
If the lines all pass through a given point {a, b, c), then the equations of a line are x - a = a{z - c), y - b =β{z - c), where α, β are functions of the variable parameter, or, what is the same thing, there exists between them an equation ska, 3) = 0; the elimination of the parameter gives, therefore, f( X — -> — —ð^) = 0; and this equation, or, what is the same thing, any homogeneous equation f{x-a, y-b, z-c) = to, or, taking f to be a rational and integral function of the order n, say (*)(z -a, y-b, z-c) n =0, is the general equation of the cone having the point (α, b, c) for its vertex. Taking the vertex to be at the origin, the equation is (*)(z, y, z) n = 0: and, in particular, (*)(z, y, ≡) 2 =0 is the equation of a cone of the second order, or quadricone, having the origin for its vertex.
34. In the general case of a singly infinite system of lines, the locus is a ruled surface (or regulus). If the system be such that a line does not intersect the consecutive line, then the surface is a skew surface, or scroll; but if it be such that each line intersects the consecutive line, then it is a developable, or torse.
Suppose, for instance, that the equations of a line (depending on the variable parameter 0) are = 1 + ∣Q, α - c°°ii 1- δ) ’ r lP ’ ÍC" ý√2 g2
then, eliminating θ, we have - = i ~ > or 8a F i ½2 + ⅛≡ - 1 ’
the equation of a quadric surface, afterwards called the hyperboloid of one sheet; this surface is consequently a scroll. It is to be remarked that we have upon the surface a second singly infinite series of lines; the equations of a line of this second system (depending on the variable parameter φ) are x , z V y ∖ x z 1/ y ∖ a + c b) ’ a c~Φ ∖ + bsh
It is easily shown that any line of the one system intersects every line of the other system.
Considering any curve (of double curvature) whatever, the tangent lines of the curve form a singly infinite system of lines, each line intersecting the consecutive line of the system,—that is, they form a developable, or torse; the curve and torse are thus inseparably connected together, forming a single geometrical figure. A plane through three consecutive points of the curve (or osculating plane of the curve) contains two consecutive tangents, that is, two consecutive lines of the torse, and is thus a tangent plane of the torse along a generating line.
Transformation of Coordinates.
35. There is no difficulty in changing the origin, and it is for brevity assumed that the origin remains unaltered. We have, then, two sets of rectangular axes, Oz, Oy, Oz, and Oz 1 , Oy 1 , Oz 1 , the mutual cosine-inclinations being shown bv the diagram— [table]
that is, α, β, γ are the cosine-inclinations of Oz 1 to Oz, Oy, Oz; a, β', γ' those of Oy 1 , &c.
And this diagram gives also the linear expressions of the coordinates (z 1 , y 1 , z 1 ) or (z, y, z) of either set in terms of those of the other set; we thus have [table]
which are obtained by projection, as above explained. Each of these equations is, in fact, nothing else than the before-mentioned equation p = dξ + β'η + Jζ, adapted to the problem in hand.
But we have to consider the relations between the nine coefficients. By what precedes, or by the consideration that we must have identically z 2 + y 2 + z 2 = z 1 2 + y 1 2 + z 1 2 , it appears that these satisfy the relations— [table]
either set of six equations being implied in the other set. It follows that the square of the determinant [table]
is=l; and hence that the determinant itself is =±1. The distinction of the two cases is an important one: if the determinant is = +1, then the axes Oz 1 , Oy 1 , Oz 1 are such that they can by a rotation about 0 be brought to coincide with Oz, Oy, Oz respectively; if it is = -1, then they cannot. But in the latter case, by measuring x 1 ,y 1 , z 1 in the opposite directions we change the signs of all the coefficients and so make the determinant to be = +1; hence this case need alone be considered, and it is accordingly assumed that the determinant is = +1. This being so, it is found that we have a further set of nine equations, « = β ’ γ" - β"y, &c.; that is, the coefficients arranged as in the diagram have the values
[table]
36. It is important to express the nine coefficients in terms of three independent quantities. A solution which, although unsymmetrical, is very convenient in Astronomy and Dynamics is to use for the purpose the three angles θ, φ, τ of fig. 19; say θ = longitude of the node ; φ = inclination ’ and τ = longitude of aq from node.
The diagram of transformation then is [table]
But a more elegant solution (due to Rodrigues) is that contained, in the diagram [table]
÷(1 + Λ*- m 2 + f s )
The nine coefficients of transformation are the nine functions of the diagram, each divided by 1 + λ 2 + μ 2 + v 2 ; the expressions contain as they should do the three arbitrary quantities λ, μ, v; and the identity x 1 2 + y 1 2 + z 1 2 ≈x 2 + y 2 + z 2 can be at once verified. It maybe added that the transformation can be expressed in the quaternion form w⅛ +⅝ι + kz 1 = (l+Λ)(ix +jy +fe)(l+Λ)~ 1 where Λ denotes the vector iλ+jμ + kv. Quadric Surfaces (Paraboloids, Ellipsoid, Hyperboloids).
37. It appears by a discussion of the general equation of the second order (a, . . φr, y, z, l) 2 = 0 that the proper quadric surfaces 1 represented by such an equation are the following five surfaces (a and b positive) :—
fl.) 2 = 2α + ∣⅛ , e l⅛tic paraboloid.
(2.) z = — - , hyperbolic paraboloid.
(3 ∙)^+^--^= 1, ellipsoid.
(4.) ⅛+ ≈---⅛ = 1, hyperboloid of one sheet C* (r cr
(5.) ^^ + δ 2 ^ ~ c*” -1, h yP erboloid two sheets it is at once seen that these are distinct surfaces; and the equations also show very readily the general form and mode of generation of the several surfaces.
In the elliptic paraboloid (fig. 20), the sections by the planes of zx and zy are the parabolas β -⅛ β -⅛ 2α 2δ ’
having the common axes Oz; and the section by any plane z = y parallel to that of xy is the ellipse 7 -≤-⅛ r 2α 2⅛ ’
so that the surface is generated by a variable ellipse moving parallel to itself along the parabolas as directrices.
In the hyperbolic paraboloid (fig. 21) the sections by the planes of zx, zy are the parabolas z = — ,z=-⅛, hav- 2α 2b ing the opposite axes Oz, Oz , , and the section by a plane z≈y parallel to that of xy is the hyperbola γ -⅛-g, whichhas its transverse axis parallel to Ox or Oy according as y is positive or negative. The surface is thus generated by a variable hyperbola moving parallel to itself along the parabolas as direc trices. The form is best seen from fig. 22, which represents the sections by planes parallel to the plane of xy, or say the contour lines; the continuous lines are the sections above the plane of xy, and the dotted lines the sections below this plane. The form is, in fact, that of a saddle.
In the ellipsoid (fig. 23) the sections by the planes of zx, zy, and xy are each of them an ellipse, and the section by any parallel plane is also an ellipse. The surface may be considered as generated by an ellipse moving parallel to itself along two ellipses as directrices.
In the hyperboloid of one sheet (fig. 24), the sections by the planes of zx, zy are the hyperbolas ae 2 z i _ i ιj t z 2 1 7γ- 1, ^ γ --^ 1, having a common conjugate axis zOzt; the section by the plane of 1 The improper quadric surfaces represented by the general equation of the second order are (1) the pair of planes or plane-pair, including as a special case the twice repeated plane, and (2) the cone, including as a special case the cylinder. There is but one form of cone; but the βylinder may be parabolic, elliptic, or hyperbolic.
xy, and that by any parallel plane, is an ellipse; and the surface may be considered as generated by a variable ellipse moving parallel to itself along the two hyperbolas as directrices.
In the hyperboloid of two sheets (fig. 25), the sections by the planes of 2a5 and zy are the hyperbolas c 2 α 2 ’ c 8 δ 2 ’ having the common transverse axis sθ√; the section by any plane z — ± 7 parallel to that of xy, 7 being in absolute magnitude ≥∙ c, is the ellipse
α 2 δ 2 c 2 and the surface, consisting of two distinct portions or sheets, may be considered as generated by a variable ellipse moving parallel to itself along the hyperbolas as directrices.
The hyperbolic paraboloid is such (and it is easy from the figure to understand how this may be the case) that there exist upon it two singly infinite series of right lines. The same is the case with the hyperboloid of one sheet (ruled or skew hyperboloid, as with reference to this property it is termed). If we imagine two equal and parallel circular disks, their points connected by strings of equal length, so that these are the generating lines of a right circular cylinder, then by turning one of the disks about its centre through the same angle in one or the other direction, the strings will in each case generate one and. the same hyperboloid, and will in regard to it be the two systems of lines on the surface, or say the two systems of generating lines; and the general configuration is the same when instead of circles we have ellipses. It has been already shown analytically that the equation-^ + ' ! L·- =1 is satisfied by each of two pairs of linear relations between the coordinates.
Curves; Tangent, Osculating Plane, Curvature, fyc.
38. It will be convenient to consider the coordinates (ar, y, z) of the point on the curve as given in terms of a parameter θ, so that dx, dy, dz, d 2 x, &c., will be proportional to , j∣∙, ⅛ θ , &c. But only a part of the analytical formulae will be given, ξ, η, ζ are used as current coordinates.
The tangent is the line through the point (x, y, z) and the consecutive point (x + dx,y + dy, z + dzfo its equations therefore are dx dy = dz
The osculating plane is the plane through the point and two consecutive points, and contains therefore the tangent; its equation is [table]
or, what is the same thing, (ξ-x)(dy<Pz-dzdfy) + (y-y)(dzdpx-dxdpz) + (ζ-z)(dx<Py-dy<Px') = 0.
The normal plane is the plane through the point at right angles to the tangent. It meets the osculating plane in a line called the principal normal; and drawing through the point a line at right angles to the osculating plane, this is called the binormal. We have thus at the point a set of three rectangular axes—the tangent, the principal normal, and the binormaL
We have through the point and three consecutive points a sphere of spherical curvature,—the centre and radius thereof being the centre, and radius, of spherical curvature. The sphere is met by the osculating plane in the circle of absolute curvature,—the centre and radius thereof being the centre, and radius, of absolute curvature. The centre of absolute curvature is also the intersection of the principal normal by the normal plane at the consecutive point.
Surfaces; Tangent Lines and Plane, Curvature, ÿc.
39. It will be convenient to consider the surface as given by an equation fox, y, z) = 0 between the coordinates; taking (x, y, z) for the coordinates of a given point, and (x + dx, y + dy, z + dz) for those of a consecutive point, the increments dx, dy, dz satisfy the condition dx dy a dz ’ but the ratio of two of the increments, suppose dx: dy, τa⅛y be regarded as arbitrary. Only a part of the analytical formulae will be given, ξ, η, ζ are used as current coordinates.
We have through the point a singly infinite series of right lines, each meeting the surface in a consecutive point, or say having each of them two-point intersection with the surface. These lines lie all of them in a plane which is the tangent plane; its equation is f(i-≈) + ∣⅛-rt + g(f-⅛ = 0, as is at once verified by observing that this equation is satisfied (irrespectively of the value of dx: dy) on writing therein ξ, η, ζ = x + dx, y + dy, z + dz.
The line through the point at right angles to the tangent plane is called the normal; its equations are ξ-χ n-y = s-≈ df^ ds ds‘ dx dy dz
In the series of tangent lines there are in general two (real or imaginary) lines, each of which meets the surface in a second consecutive point, or say it has three-point intersection with the surface; these are called the chieftangents (Haupt-tangenten). The tangent-plane cuts the surface in a curve, having at the point of contact a node (double point), the tangents to the two branches being the chief-tangents.
In the case of a quadric surface the curve of intersection, qua curve of the second order, can only have a node by breaking up into a pair of lines; that is, every tangentplane meets the surface in a pair of lines, or we have on the surface two singly infinite systems of lines; these are τeal for the hyperbolic paraboloid and the hyperboloid of one sheet, imaginary in other cases.
At each point of a surface the chief-tangents determine two directions; and passing along one of them to a consecutive point, and thence (without abrupt change of direction) along the new chief-tangent to a consecutive point, and so on, we have on the surface a chief-tangent curve; and there are, it is clear, two singly infinite series of such curves. In the case of a quadric surface, the curves are the right lines on the surface.
40. If at the point we draw in the tangent-plane two lines bisecting the angles between the chief-tangents, these lines (which are at right angles to each other) are called the principal tangents. We have thus at each point of the surface a set of rectangular axes, the normal and the two principal tangents.
Proceeding from the point along a principal tangent to a consecutive point on the surface, and thence (without abrupt change of direction) along the new principal tangent to a consecutive point, and so on, we have on the surface a curve of curvature; there are, it is clear, two singly infinite series of such curves, cutting each other at right angles at each point of the surface.
Passing from the given point in an arbitrary direction to a consecutive point on the surface, the normal at the given point is not intersected by the normal at the consecutive point; but passing to the consecutive point along a curve of curvature (or, .what is the same thing, along a principal tangent) the normal at the given point is intersected by the normal at the consecutive point ; we have thus on the normal two centres of curvature, and the distances of these from the point on the surface are the two principal radii of curvature of the surface at that point; these are also the radii of curvature of the sections of the surface by planes through the normal and the two principal tangents respectively; or say they are the radii of curvature of the normal sections through the two principal tangents respectively. Take at the point the axis of z in the direction of the normal, and those of x and y in the directions of the principal tangents respectively, then, if if the radii of curvature be a, b (the signs being such that the coordinates of the two centres of curvature are z = a and z = b respectively), the surface has in the neighbourhood of the point the form of the paraboloid ,r ^ st ≈ ~ 2a + 2b ’ and the chief-tangents are determined by the equation
0 = -—μ^-. The two centres of curvature may be on
2α 2o the same side of the point or on opposite sides; in the former case a and b have the same sign, the paraboloid is elliptic, and the chief-tangents are imaginary; in the latter case a and b have opposite signs, the paraboloid is hyperbolic, and the chief-tangents are real.
The normal sections of the surface and the paraboloid by the same plane have the same radius of curvature; and it thence readily follows that the radius of curvature of a normal section of the surface by a plane inclined at an angle θ to that of zx is given by the equation
1 cos 3 fl ι sin 2 g pa b
The section in question is that by a plane through the normal and a line in the tangent plane inclined at an angle θ to the principal tangent along the axis of x. To complete the theory, consider the section by a plane having the same trace upon the tangent plane, but inclined to the normal at an angle φ ; then it is shown without difficulty (Meunier’s theorem) that the radius of curvature of this inclined section of the surface is = pcos<∕>. (a. CA.)
