A Syllabus of Plane Algebraical Geometry

Systematically Arranged, with Formal Definitions, Postulates, and Axioms.

Part I. Containing Points, Right Lines, Rectilinear Figures, Pencils, and Circles.

Introduction

In teaching the subject of Algebraical Geometry, I have found the advantage of giving formal definitions of such words as “axis,” “ordinate,” and “radius-vector,” as giving the learner a clearer notion of the elements of the subject than he could gather for himself from the ordinary treatises.

In drawing up these definitions I found that several of the geometrical terms borrowed from Euclid could not be used in Algebraical Geometry without some modification of their definitions as given by him, whilst other terms were wanted for which he had given no definition at all; hence I was led to prefix to my Definitions of Algebraical Geometry a modernised version of Euclid’s Definitions. The Propositions also of the subject appeared to me to be wanting in uniformity; for example, the relation between an equation of the first degree and a right line is discussed by one writer in the form “to find the equation to a straight line,” (thus assuming the possibility of such an equation being found;) by another, in the form “we proceed to enquire the geometrical signification of a single equation between the co-ordinates.” To remedy such defects as these I endeavoured to reduce the subject to a uniform series of formal Propositions. Such was the origin of the present Syllabus.

I hope it will not be thought presumptuous in me if I here state in order those defects which appear to me to exist in the modern treatises, and for which I have attempted to furnish a remedy; next, the method I have pursued in doing so; and lastly, the defects which still remain unremedied, and to which those, who are desirous of forwarding towards perfection this beautiful science, would do well to turn their attention.

The defects which I have attempted to remedy relate to three main points in the subject; I. Definitions, II. Enunciations, and III. Treatment of Subject.

I. Definitions. Many of the technical terms employed by the various writers on the subject are left undefined, or only with such definitions as may be gathered from a page or more of explanatory matter; others are inadequately defined, (as, for instance, the angle employed in Polar Co-ordinates, the definition of which, as given in Mr. Salmon’s Conic Sections, p. 9, will not apply to the case of a negative radius-vector); and even when a correct and formal definition is furnished, it is only to be found in the passage where the term first occurs, a method likely to give much unnecessary trouble in referring to it afterwards.

II. Enunciations. These, in the present treatises, are only given along with their deinonstrations, which again are interspersed with much explanatory and illustrative matter, all liable to confuse the student who desires to review the subject, briefly and connectedly, as a whole. Moreover there is a great want of uniformity in the language employed, the data being put first in some, and the quæsita in others; while some propositions have no “general enunciation” at all, but begin at once with the “particular enunciation.”

III. Treatment of Subject. The propositions are in too many cases made to depend on each other, so that if we wish to make out the proof of any particular theorem by itself, it is necessary to refer back to the proposition from which it is deduced, and from that perhaps to another, and so at last to build up the required proof. This method would not be so liable to objection if there were any received text-book on the subject, answering to Euclid in Geometry, and furnishing a recognised series of consecutive propositions which could be appealed to as authority; but, as the case now stands, when one writer may prove A from B, and another B from A, the whole system is illogical and therefore unsatisfactory.

Again, an unphilosophical method is employed in arriving at various of the necessary formulæ; it is a method neither analytical nor synthetical, but begins in synthesis, and concludes with a kind of analysis, introduced to prove the lawfulness of the assumption made, but giving no information as to how it was originally arrived at. As an instance, we may take the conversion of the equation $A x + B y + C = 0$ , into the form of $x . cos a + y . sin a - ρ = 0$ , where the rule is laid down “divide by $\sqrt{A^{2} + B^{2}}$ ,” and we are afterwards told that we may assume $\frac{A}{\sqrt{A^{2} + B^{2}}} = cos a$ , and $\frac{B}{\sqrt{A^{2} + B^{2}}} = sin a$ “since the sum of the squares of these two quantities $= 1$ .” Again, the focus of the ellipse is introduced by arbitrarily taking two points on the major axis, at a distance from the centre equal to $\pm \sqrt{a^{2} - b^{2}}$ , and discussing their properties; but no reason is given why these, rather than any other points, are so chosen for discussion.

A third deficiency is to be found in the notation adopted, where the symbols and language are not adapted to the expansion, which is necessary to meet the larger requirements of Algebraical Geometry of three dimensions. For instance, in the equations, $\frac{x - x^{'}}{l} = \frac{y - y^{'}}{m}$ , and $y = t x + b$ , when referred to rectangular Cartesian axes, it is usual to define “l” and “m” as “the cosine and sine of the angle which the line makes with the X-axis,”¹ and “t” as “the tangent of the same angle.” On coming to oblique axes, it is found necessary to modify both definitions; “l” and “m” become “the ratios which the sines of the angles, made by the line with the two axes respectively, bear to the sine of the angle between the axes,” and “t,” “the ratio between these two sines themselves.” On examination it appears that the definitions for rectangular axes are included as a particular case of these, and it may now be thought that we have got definitions sufficiently broad for the whole subject, but on coming to Solid Geometry we have to return to the cosines for rectangular axes, (as otherwise the phrase “the sine of the angle between the axes” would be unmeaning,) while for oblique axes we are forced upon the entirely new theory of projections, unless we choose to adopt the following definition for “l,” “the ratio which the sine of the angle made by the line with the intersection of two planes, (namely, the plane in which the line itself and the X-axis lie, and that in which the Y-axis and Z-axis lie,) bears to the sine of the angle made by the X-axis with the same intersection”! This instance will shew how necessary it is to consider the future requirements of Solid Geometry in dealing with Plane Geometry.

The method I have pursued in endeavouring to remedy these defects may be most conveniently discussed under the same three headings.

I. As to Definitions. In those borrowed from Euclid I have altered no word of his, except where it was necessary, either to bring the language into uniformity with the other definitions, or to meet the new requirements of Algebraical Geometry. In the rest, I have always taken his as the model, being careful at the same time to provide for the subsequent expansion necessary in Solid Geometry. And these definitions I have placed all together, at the commencement of each Book.

II. As to Enunciations. These I have given by themselves, without their proofs, to afford the student a convenient means of reviewing the subject, and testing his knowledge of it, by taking each enunciation in succession, and considering whether or no his unassisted memory can supply the demonstration. Where any difficulty is found in doing this, he is recommended to add a marginal reference to the place where the demonstration may be found, whether in the text-book which he employs on the subject, or in the Appendix to this Syllabus. I have endeavoured to reduce all these enunciations to one uniform shape, always placing the “data” before the “quæsita” or “demonstranda.” Lastly, I have endeavoured to make all the enunciations “general;” and I trust it will be found that they have thus gained in uniformity what they have, in some instances, lost in brevity and clearness.

III. As to Treatment of Subject. I have endeavoured, for reasons which I have already mentioned, to render each proposition, as far as possible, independent of others; and being thus released from the necessity of stringing them together in a consecutive series, I was able to adopt what seemed the more natural principle of arrangement, of beginning with those which involved the fewest and simplest conditions, and so going on to the more complicated. Whether I was right in attempting the further simplification of separating Problems from Theorems, I am now doubtful; it introduced several difficulties which would not have otherwise arisen; still, the great advantages of clearness and convenience of reference made the experiment a plausible one.

Next, I have introduced every new formula in a method as purely analytical as I found possible. For instance, in arriving at the equation to a right line, I first investigate the method of representing a point (the original purpose for which the system of co-ordinates is introduced); then, on finding that two equations are necessary for this, it naturally follows that we should consider the geometrical signification of one such equation taken by itself. Similarly, the rule already alluded to for reducing the equation $A x + B y + C = 0$ , to the form $x . cos a + y . sin a - ρ = 0$ , is found at once by simply applying the test that two equations shall represent the same line. And similarly, the co-ordinates of the “focus” of an ellipse may be easily found by solving the Problem “to find a point whose distance, from any given point on the curve, shall be a simple function of the abscissa of that given point.”

Another innovation will be found in the method of treating what is called by Mr. Salmon “abridged notation.” It appeared to me that the symbols “α” and “β,” employed in that system, are in no other sense abridged forms of “ $x . cos a + y . sin a - p$ &c., than that in which “x” is an abridged form of “ $ρ . cos θ$ ,” (the relation between a Cartesian and a Polar System,) or of “ $x^{'} . cos θ - y^{'} . sin θ$ ,” (the relation between two rectangular Cartesian Systems inclined at an angle θ). In other words, it appeared to me simpler, and I have certainly found it to be so in practice, to constitute a new System of reference, which I have called the “Distantial” System, and only to introduce the fact that $α = x . cos a + y . sin a - p$ , as a formula of transformation from a Distantial to a Cartesian System.

The distinction introduced between an infinitely small quantity and absolute zero needs perhaps a word of apology, as it is not noticed in other treatises, and indeed belongs more properly to the subject of Differential Calculus. Still it is, at worst, superfluous, and those who think it merely fanciful will find that it can easily be omitted, as nothing in the book is made to depend upon it.

Apology is also due for the number of new words I have introduced; I have not done so in any case without an apparent necessity, to avoid the constant repetition of a cumbrous periphrasis. I will here enumerate the new words introduced, with reference to the pages where their definitions may be found. I have endeavoured to make them, as far as possible, suggest their meaning by their derivations:

divisional-angles, and divisional-ratio 18
direction-angles 21
abscissa-ratio, ordinate-ratio, and co-ordinate-ratio 22
sagittal-line, sagittal-angles, and sagit 22, 23
interceptive, sagittal, and directional equations 29
fixed-radius 30
vectorial-line 30
distantial system, lines of reference, vertex of reference, principal portion, and referent 36
interceptor 44
co-radical circles 96

It remains that I should enumerate those defects in the subject, for which I have not been able here to furnish a remedy.

In the first place, there is something unphilosophical in the very groundwork of the science. It professes to prove, from independent sources, many of the propositions of Pure Geometry, while it is so entirely dependent upon that science for its merest elements, that even the equation to a straight line cannot be investigated without assuming Euclid, Book VI, Prop. 4. As long as Algebraical Geometry requires the previous proof of so large a range of geometrical propositions, it cannot fairly be employed to prove any theorems which fall within that range.

Secondly, there are terms employed in the science in more senses than one; for the secondary meaning a new word should be introduced. For example, while we have “superficies” to denote a surface of indefinite extent, and “figure" to denote a limited portion of one—while we have “space” to denote indefinite extent of three dimensions, and “solid” to denote a limited portion of space—the word “line” is made to do double duty. Another example may be found in the word “direction,” discussed in the Appendix, p. 124.

Thirdly, there are several instances of two or more words being used in precisely the same sense; these require to be desynonymised, or, if that be impossible, all the synonymous terms, but one, should be excluded from use, and reserved for the possible future requisitions of the subject. As instances of this may be mentioned,

denote, represent, indicate, express;
contain, involve, require, postulate;
constant, fixed, invariable, determinate;
given, known, determined.

Fourthly, some simple rules are required for ascertaining, in any given problem, whether it may best be solved by the Cartesian, Polar, or Distantial System.

Many defects in the present Syllabus have been discovered and rectified while the sheets were passing through the press; many more, in an attempt so new, and (so to speak) experimental, have no doubt escaped my notice; there is also a great hiatus in the subject of Trigonometry, the whole of which needs to be systematised in the same way. Still I have thought it better to publish it in its present incomplete state, in order that by bringing it under the notice of more mathematicians than would otherwise have seen it, and so getting more suggestions for its improvement than could otherwise have been obtained, means may be afforded, whether to myself or to some other writer, of hereafter reducing the whole subject to a complete and uniform system, which shall occupy, with regard to Algebraical Geometry, the same position which is occupied by that of Euclid with regard to Pure Geometry.

General Definitions, &c.

Plane Algebraical Geometry is the science of applying the symbols and operations of Algebra and Arithmetic to the subject-matter of Plane Geometry. This subject-matter consists of Points, Lines, and Figures; their relations to each other, viz. distances, and angles; and the properties of all these, viz. magnitude, direction, and position.

This science requires as its data, theoretically, only the sciences of Algebra and Arithmetic, with the definitions, postulates, and axioms of Plane Geometry: in its present state it requires, practically, Geometrical proofs of theorems which range through the first six Books of Euclid.

The present syllabus commences with two Algebraical theorems of special use in the subject, which are placed here for convenience of reference: next are given the definitions, postulates, and axioms of Plane Geometry; these are essentially the same as are found prefixed to the first six Books of Euclid, but with some alterations and additional theorems, required by the nature of the subject. After this the subject of Plane Algebraical Geometry commences. Proofs are supplied with some of the propositions, when they are not to be found in the ordinary treatises on the subject, or at least not in the form here given.

Algebra

Preliminary Theorems

I.

If $\frac{a}{b} = \frac{c}{d} = \frac{e}{f} = &c.$ , then each of these fractions is equal to $\frac{a + c + e + &c.}{b + d + f + &c.},$ and to $\frac{\sqrt{a^{2} + c^{2} + e^{2} + &c.}}{\sqrt{b^{2} + d^{2} + f^{2} + &c.}}$ .

II.

In the equation $a x^{2} + b x + c = 0$ ,
if $c = 0$ , one of the roots $= 0$ ;
if $b = 0$ , the roots are equal with opposite signs;
if $a = 0$ , one of the roots $= \frac{1}{0}$ ;
if $a = b = c = 0$ , the roots are indeterminate;
if $b^{2} - 4 a c$ be positive, the roots are real and unequal;
if $b^{2} - 4 a c$ be negative, the roots are imaginary;
if $b^{2} - 4 a c$ $= 0$ , the roots coincide.

Plane Geometry

Definitions

I.

A Point is that which has position only, without length, breadth, or thickness.

II.

A Line is that which has position and length, without breadth or thickness.

III.

A finite Line is that which is contained between two points.

IV.

A right Line is that in which any two points being taken, the contained finite line lies evenly between them.

V.

The distance between two Points is the length of the shortest line which can be drawn from one to the other.

VI.

The distance between a Point and a Line is the length of the shortest line which can be drawn from the Point to the Line,

VII.

A Superficies, (or Surface,) is that which has position, length, and breadth, without thickness.

VIII.

A Figure is a Surface contained by one or more lines.

IX.

The magnitude of a Figure is called Area.

X.

A Plane Surface, (or Plane,) and a Plane Figure, are those in which any two points being taken, the right line joining them lies wholly in that Surface or Figure.

XI.

A Pencil consists of two or more Lines, all lying in the same plane, and all terminated in the same point.

XII.

The Rays of a Pencil are the lines of which it is composed.

XIII.

A Transversal of a Pencil is a Line intersecting all its Rays.

XIV.

When two Lines are terminated in the same point, then, if one of them he supposed to have originally coincided with the other, and to have revolved, away from it, about the point of termination, the declination of the two Lines from each other, (being the same in magnitude whichever Line is supposed to have revolved,) is called the angle between them.

XV.

The magnitude of an angle is called Aperture.

XVI.

When a Line, meeting another Line, makes the adjacent angles equal to each other, each of these angles is called a right angle, and each of the Lines is said to be perpendicular to the other.

XVII.

An obtuse angle is that which is greater than a right angle.

XVIII.

An acute angle is that which is less than a right angle.

XIX.

Parallel Lines are such as are in the same plane, and which, being produced ever so far both ways, do not meet.

XX.

A rectilinear Figure is a plane Figure contained by right lines.

XXI.

A Vertex of a rectilinear Figure is the point where two adjacent sides meet each other.

XXII.

A trilateral Figure, (or Triangle,) is a rectilinear Figure which has three sides.

XXIII.

A quadrilateral Figure is a rectilinear Figure which has four sides.

XXIV.

A multilateral Figure, (or Polygon,) is a rectilinear Figure which has more than four sides.

XXV.

An equilateral Triangle is that which has three equal sides.

XXVI.

An isosceles Triangle is that which has two sides equal.

XXVII.

A scalene Triangle is that which has three unequal sides.

XXVIII.

A right-angled Triangle is that which has a right angle.

XXIX.

An obtuse-angled Triangle is that which has an obtuse angle.

XXX.

An acute-angled Triangle is that which has three acute angles.

XXXI.

A Parallelogram is a quadrilateral figure which has its opposite sides parallel.

XXXII.

A Diameter, (or Diagonal,) of a Parallelogram is the line joining two opposite vertices.

XXXIII.

All quadrilateral Figures, which are not parallelograms, are called Trapeziums.

XXXIV.

A Rectangle is a quadrilateral Figure which has all its angles right angles.

XXXV.

A Square is a quadrilateral Figure which has all its angles right angles and all its sides equal.

XXXVI.

A Circle is a plane Figure contained by one line, which is called the Circumference, and is such that the distances of all points on the circumference from a certain point within the figure are equal to one another.

XXXVII.

And this point is called the Centre of the Circle.

XXXVIII.

An Arc of a Circle is any portion of the circumference.

XXXIX.

A Radius of a Circle is a right Line drawn from the centre, and terminated by the circumference.

XL.

A Chord of a Circle is a right Line drawn within it, and terminated both ways by the circumference.

XLI.

A Diameter of a Circle is a right Line drawn through the centre, and terminated both ways by the circumference.

XLII.

A Sector of a Circle is the Figure contained by two radii and the portion of the circumference between them.

XLIII.

A Segment of a Circle is the Figure contained by a chord and the portion of the circumference which it cuts off.

XLIV.

A Semicircle is the Figure contained by the diameter of a Circle and either of the two portions into which it divides the circumference.

XLV.

An angle in a Segment of a Circle is the angle contained between two right lines drawn from any point in the circumference of the segment, to the extremities of the chord which cuts off that circumference.

XLVI.

A right Line and a Circle are said to touch each other, when the Line meets the Circle, and being produced does not cut it.

XLVII.

Circles are said to touch one another, which meet, but do not cut one another.

XLVIII.

A rectilinear Figure is said to be inscribed in another rectilinear Figure, when all its vertices are upon the sides of that other Figure.

XLIX.

A rectilinear Figure is said to be circumscribed about another rectilinear Figure, when all its sides pass through the vertices of that other Figure.

L.

A rectilinear Figure is said to be inscribed in a Circle, when all its vertices are upon the Circumference of the Circle.

LI.

A rectilinear Figure is said to be circumscribed about a Circle, when all its sides touch the Circle.

LII.

A Circle is said to be inscribed in a rectilinear Figure, when it touches all the sides of the Figure.

LIII.

A Circle is said to be circumscribed about a rectilinear Figure, when its circumference passes through all the vertices of the Figure.

LIV.

A Locus is a number of Points, all fulfilling some one and the same condition.

LV.

The ratio between two things of the same kind is the relation which one bears to the other in respect of magnitude, the comparison being made by considering what multiple, part, or parts, one is of the other: or, (which is the same thing,) it is the number expressing such multiple, part, or parts.

LVI.

Four things of the same kind, or two of one kind and two of another, are said to be in proportion, when the first of them has to the second the same ratio which the third has to the fourth.

LVII.

Three things of the same kind are said to be in proportion, when the first of them has to the second the same ratio which the second has to the third.

N.B. In future, by the word “line” will be meant “right line,” and by the word “figure” will be meant “plane figure,” unless when it is otherwise stated.

Postulates

I.

Let it be granted that a line can be drawn from any one Point to any other Point.

II.

That a finite Line can be produced to any length.

III.

That from the greater of two finite Lines, distances, Figures, or angles, a part can be cut off equal to the less.

IV.

That a finite Line, a distance, a Figure, or an angle, can be divided into any number of equal or unequal parts.

V.

That a line can be drawn from any Point, perpendicular to any Line of unlimited length.

VI.

That a line can be drawn from any Point, making any angle with any Line of unlimited length.

VII.

That a line can be drawn from any Point, parallel to any Line.

VIII.

That a Circle can be drawn about any Centre, at any distance from that Centre.

Axioms

A. Of magnitude

I.

Things which are equal to the same thing are equal to one another.

II.

If equals be added to equals, the wholes are equal.

III.

If equals be taken from equals, the remainders are equal.

IV.

If equals be added to unequals, the wholes are unequal.

V.

If equals be taken from unequals, the remainders are unequal.

VI.

Things which are double of the same thing are equal to one another.

VII.

Things which are halves of the same thing are equal to one another.

VIII.

Equimultiples of equals are equal.

IX.

Things, whose equimultiples are equal, are themselves equal.

X.

The whole is greater than its part.

XI.

Two right Lines cannot enclose a space.

XII.

Of all lines which can be drawn from one of two Points to the other, that which is right is shorter than any other. Hence, the distance between two Points is the length of the right line joining them.

XIII.

All right angles are equal to one another.

XIV.

When two Lines are terminated in the same point, by the words “the angle between them” may be meant either of two angles. For, if a circle be supposed to be drawn about the point of termination, intersecting both Lines, then, whichever of the two Lines is supposed to have revolved, this revolution may be supposed to have been made through either of the two portions into which the circle is thus divided.

N.B. The word “angle” is used by Euclid to denote such declinations only as are less than the sum of two right angles: it is here used, in a more extended sense, to denote any declination whatever.

B. Of direction

I.

A Line has either of two opposite directions, since it may be supposed to have been drawn either way.

II.

The distance between two Points has either of two opposite directions, since it may be measured from either Point.

III.

But the distance of one Point from another, (which is the same thing as the distance between them measured from one only,) has one direction only.

IV.

For the same reasons, the distance between a Point and a Line has either of two opposite directions.

V.

But the distance of a Point from a Line, and the distance of a Line from a Point, have each of them one direction only.

VI.

Each of the two angles between two Lines terminated in the same point has either of two opposite directions, since it may be measured from either Line.

VII.

But each of the two angles which one such Line makes with the other, (which is the same thing as the angle between them measured from one only,) has one direction only.

C. Of position

I.

The position of a Line is fixed, if the positions of two points on it are fixed, or if the position of one point on it, and the angle which it makes with a fixed Line, are fixed.

II.

The position of a rectilinear Figure is fixed, if the positions of its vertices, or of its sides, are fixed.

III.

The position of the circumference of a Circle is fixed, if the position of its Centre, and the length of its radius, are fixed.

IV.

The position of a Circle is fixed, if the position of its circumference is fixed.

Preliminary Theorems

I.

From the same Point there cannot be drawn two lines, both perpendicular to the same Line.

II.

Of all the lines which can be drawn from a Point, and terminated by a Line of unlimited length, that which is drawn perpendicular to the Line is shorter than any other.

Corollary. The distance between a Point and a Line is the length of the right line drawn from the Point, perpendicular to the Line, and terminated by it.

Plane Algebraical Geometry

Book I. Representation of Magnitude Only

Definitions

I.

The unit-Line is a Line of known length.

II.

The unit-distance is a distance equal to the length of the unit-Line.

III.

The unit-Figure is a square each of whose sides is equal to the unit-Line.

IV.

The unit-angle is an angle of known aperture.

N.B. In future, when two Lines are terminated in the same point, by the words “the angle between them” will be meant “the angle measured from one to the other the shortest way round,” unless when it is otherwise stated.

Postulates

I.

Let it be granted that a Line, a distance, a Figure, or an angle, may be represented by the symbol representing the ratio which it bears to the unit-Line the unit-distance, the unit-Figure, or the unit-angle.

II.

That the signs +, −, =, :, ::, &c., which, when prefixed to symbols of number, denote the addition, subtraction, &c. of the numbers which the symbols represent, may also be prefixed to symbols of magnitude, to denote the addition, subtraction, &c. of the magnitudes which those symbols represent.

Axioms

I.

If, when + and − are used to denote the addition and subtraction of magnitudes, a symbol of magnitude be affected with the sign −, it has no geometrical meaning when taken by itself, since a magnitude cannot be subtracted from nothing.

II.

And if several symbols of magnitude be connected by the be to signs + and −, and those affected with the sign − be together greater than those affected with the sign +, the result has no geometrical meaning, since the greater of two magnitudes cannot be subtracted from the less.

Propositions

I.

Given the lengths of two adjacent sides of a Rectangle: to represent its area.

Corollary. Hence, Given the length of one side of a Square: to represent its area.

II.

Given the lengths of the two sides which contain the right angle of a right-angled Triangle: to represent the area of the Triangle.

III.

Given the length of one side of a Triangle, and the distance between that side, (produced if necessary,) and the opposite vertex: to represent the area of the Triangle.

Book II. Representation of Direction Only.

Definitions

I.

One of the two directions of a Line is called its positive direction: and all parallel Lines have the same positive direction.

II.

That direction of a distance, which is measured in the positive direction of the line along which it is measured, is called positive.

III.

That side of a line, on which those Points, whose distances from it are positive, lie, is called its positive side.

IV.

One of the two directions of revolution from a Line, (i. e. one of the two directions in which a line, coinciding with it, may revolve away from it,) is called its positive direction of revolution.

V.

That direction of an angle, which is measured in the positive direction of revolution of the Line from which is is measured, is called positive.

VI.

The sign +, prefixed to a symbol representing the length of a Line, or a distance, or the aperture of an angle, denotes that such length, distance, or aperture, is measured in a positive direction.

Postulate

Let it be granted that the words “positive” and “negative,“ and the signs + and −, denoting contrary operations, may be used to denote opposite directions.

Axioms

I.

That direction of a Line, which is opposite to its positive direction, is its negative direction: and all parallel Lines have the same negative direction.

II.

That direction of a distance, which is measured in the negative direction of the line along which it is measured, is negative.

III.

That side of a Line, on which those Points, whose distances from it are negative, lie, is its negative side.

IV.

That direction of revolution from a Line, which is opposite to its positive direction of revolution, is its negative direction of revolution.

V.

That direction of an angle, which is measured in the negative direction of revolution of the Line from which it is measured, is negative.

VI.

The sign −, prefixed to the symbol representing the length of a Line, or a distance, or the aperture of an angle, denotes that such length, distance, or aperture, is measured in a negative direction.

Book III. Representation of Magnitude and Direction, i. e. Trigonometry

Trigonometry may be defined as that branch of Algebraical Geometry which is concerned with magnitude and direction, without regard to position. Some Definitions, &c. are here given, which are not included in the ordinary treatises on the subject, though properly belonging to it: for the rest, the reader is referred to the treatises themselves.

Definitions

I.

In a Pencil of 3 Rays, the angles which any one of them makes with the other two are called its divisional-angles, and the ratio between the sines of those angles is called its divisional-ratio, with regard to those other two Rays.

II.

In a Pencil of 4 Rays, the ratio between the divisional-ratios of any two of them with regard to the other two, (the divisional-angles being taken in the same order,) is called the anharmonic ratio of the Pencil with regard to those other two Rays.

III.

A harmonic Pencil is a Pencil of 4 Rays whose anharmonic ratio is −1.

Axiom

In a Pencil of 4 Rays, if the two Rays, whose divisional-ratios are considered, be produced through the point of intersection, and if the Lines so formed lie in the same pair of the four angles made by producing the other two lines through the point of intersection, the anharmonic-ratio of the Pencil is positive; but if they lie, one in one pair and one in the other pair, it is negative.

Theorem

If a Pencil of 4 Rays, terminated in a point O, be intersected by a Transversal in the points A, B, C, D, the ratio $\frac{A B . C D}{A C . B D}$ is constant, and is equal to the anharmonic ratio of the Pencil with regard to the two Rays $O A$ , $O D$ .

Book IV. Representation of Position

Definition

The Plane of reference is a Plane of known position, extending indefinitely in all directions, in which are supposed to lie all the Points, Lines, and Figures, to be treated of. Their positions may be represented by various systems, of which three are here discussed, the Cartesian, the Polar, and the Distantial.

N.B. In future, when two Lines intersect in a point, by the words “the angle between them” will be meant “the angle between those portions of them which are drawn, from the point of intersection, in the positive directions of the two Lines respectively,” unless when it is otherwise stated.

Chapter I. Cartesian System

Definitions

I.

The Axes are two lines of known position, intersecting in a point, which is called the Origin.

II.

One of these lines is called the X-axis, the other, the Y-axis. The direction from left to right of the X-axis, (taking its position as horizontal,) and the direction from below upwards of the Y-axis, are called positive.

III.

Of any other Line, either direction at pleasure may be called positive.

IV.

If the axes intersect at right angles, they are called rectangular; if not, they are called oblique, and the angle contained between them is sometimes represented by ω.

V.

If from any Point two lines be drawn, parallel to the two axes respectively, and each intersecting that axis to which it is not parallel, the distances of the points of intersection from the origin are called the co-ordinates of the Point.

VI.

Of these distances, that which is measured along the X-axis is called the abscissa of the Point; the other, its ordinate. They are represented, if unknown, by x, y; if known, by $x^{'}$ , $y^{'}$ , or by $x_{1}$ , $y_{1}$ , or other such symbols.

VII.

The angles which any Line makes with the axes are called the direction-angles of the Line.

VIII.

Of these angles, that which it makes with the X-axis is called the X-direction-angle of the Line; the other, its Y-direction-angle. Their apertures, if known, may be presented by λ, μ.

IX.

That direction of revolution from the X-axis, in which the right-hand portion of a line coinciding with it would revolve upwards, is called its positive direction of revolution: and that direction of revolution from the Y-axis, in which the upper portion of a line coinciding with it would revolve to the right-hand, is called its positive direction of revolution.

X.

If a Line pass through the origin, the ratio which the abscissa of any point on it bears to the distance of the same point from the origin is called the abscissa-ratio of the Line. If known, it is sometimes represented by l.

XI.

And the ratio which the ordinate of any point on it bears to the distance of the same point from the origin is called the ordinate-ratio of the Line. If known, it is sometimes represented by m.

XII.

And the ratio which the ordinate of any point on it bears to the abscissa of the same point is called the co-ordinate-ratio of the Line. If known, it is sometimes represented by t.

XIII.

All these ratios, taken together, are called the directional ratios of the Line.

XIV.

If a Line do not pass through the origin, its directional ratios are the same as those of a Line, parallel to it, and passing through the origin.

XV.

The intercepts of a Line are the distances of the points, where it intersects the axes, from the origin. They are represented, if unknown, by $x_{0}$ , $y_{0}$ ; if known, by a, b, or other such symbols.

XVI.

The sagittal-line of a Line is the line drawn through the origin, perpendicular to it.

XVII.

The sagittal-angles of a Line are the direction-angles of its sagittal-line. Their apertures, if known, are represented by α, β, or other such symbols.

XVIII.

The sagit of a Line is its distance from the origin. If known, it is represented by p, q, or other such symbols.

N.B. In future, by the words “the sagittal angle of a Line” will be meant “the X-direction-angle of its sagittal line.”

Postulates

I.

Let it be granted that the position of any Point may be represented by representing the lengths and directions of its co-ordinates. That, if these be represented, one Point, and one only, is represented. And that, when these are given or found, the Point may be said to be given or found.

II.

That the position of any Line may be represented by re presenting the positions of two points on it; or the position of one point on it, together with its co-ordinate ratio; or the lengths and directions of its intercepts; or the aperture and direction of either of its sagittal angles, together with the length and direction of its sagit. And that, if these be represented, one Line, and one only, is represented.

III.

That the position of any rectilinear Figure may be represented by representing the positions of its vertices, or of its sides. And that, if these be represented, one Figure, and one only, is represented.

IV.

That the position of the circumference of any Circle may be represented by representing the position of the centre and the length of the radius. And that, if these be represented, one circumference, and one only, is represented.

V.

That the position of any Circle may be represented by representing the position of its circumference. And that, if this be represented, one Circle, and one only, is represented.

Axioms

I.

The direction from right to left of the X-axis, and the direction from above downwards of the Y-axis, are negative.

II.

Of any other Line, passing through the origin, that direction which is opposite to its positive direction is negative.

III.

That direction of revolution from the X-axis, in which the right-hand portion of a line coinciding with it would revolve downwards, is negative: and that direction of revolution from the Y-axis, in which the upper portion of a line coinciding with it would revolve to the left-hand, is negative.

IV.

The abscissa-ratio and ordinate-ratio of a Line have the same sign, in whichever portion of the Line the point, from which they are estimated, be taken: their signs depend only on the position of the Line, and on which of its two directions is considered positive.

V.

The co-ordinate-ratio of a Line has the same sign, in whichever portion of the Line the point, from which it is estimated, be taken, and whichever of the two directions of the Line be considered positive: its sign depends only on the position of the Line.

VI.

If a Line, passing through the origin, lie in the angle contained between the right-hand portion of the X-axis and the upper portion of the Y-axis, its co-ordinate-ratio is positive; if otherwise, negative.

VII.

The ratio which the ordinate-ratio of a Line bears to its abscissa-ratio is equal to its co-ordinate-ratio. (i. e. $\frac{m}{l} = t$ .)

VIII.

The Y-direction-angle of a Line is equal to the angle between the axes minus the X-direction-angle of the same Line. (i. e. $μ = ω - λ$ .) And the Y-sagittal-angle of a Line is equal to the angle between the axes minus the X-sagittal-angle of the same Line. (i. e. $β = ω - α$ .)

IX.

When the axes are rectangular, the abscissa-ratio of a Line is equal to the cosine of its X-direction-angle. (i. e. $l = cos λ$ .)

X.

And the ordinate-ratio of a Line to the cosine of its Y-direction-angle, or, (which is the same thing) to the sine of its X-direction-angle. (i. e. $m = cos μ = sin λ$ .)

XI.

And the co-ordinate-ratio of a Line to the tangent of its X-direction-angle. (i. e. $t = tan λ$ .)

Propositions

A. Transformation of Co-ordinates

I.

Given the co-ordinates of a Point, when referred to one pair of axes, and the position of a new pair of axes with regard to the old: to find its co-ordinates, when referred to the new pair of axes, or, (which is the same thing,) to find two Equations expressing the values of its old co-ordinates in terms of the new, (from which Equations the values of its new co-ordinates may be found, if required.)

(a) New axes parallel to old, but not with same origin.
(b) with same origin, but not parallel.
(1.) both pairs of axes rectangular.
(2.) old rectangular, new oblique.
(3.) old oblique, new rectangular.
(4.) both oblique.
(c) not parallel to old, nor with same origin.

II.

Given an Equation, containing x, or y, or both, which is known to be true of the co-ordinates of a certain Point, when referred to one pair of axes, and the position of a new pair of axes with regard to the old: to find an Equation which shall be true of the co-ordinates of the same Point, when referred to the new pair of axes.

B. Equations

I.

Two simple equations, containing x, or y, or both, represent a Point.

II.

Conversely, any Point may be represented by two Equations.

III.

Two Equations of a higher degree, containing x, or y, or both, represent a finite number of Points.

IV.

One Equation, containing x, or y, or both, represents an infinite number of Points, all fulfilling one and the same condition, i. e. it represents a Locus.

V.

Conversely, any Locus may be represented by one Equation, provided that the condition to be fulfilled by every Point on it is capable of being expressed in terms of the co-ordinates of that Point.

N.B. Problems of this nature are discussed in Book VI.

VI.

The Locus of any simple Equation is, (with one exception, mentioned in Corollary 2,) a Line. And by one such Equation, one Line, and one only, is represented.

(a) Given Equation containing x only, or y only.
(b) x and y, but no constant term.
(c) x, y, and a constant term.

Corollaries. 1. In the Equation $y = t x + b$ , the geometrical meaning of t is “the co-ordinate-ratio of the Line,” and of b, “the Y-intercept of the Line.”

2. In the Equation $A x + B y + C = 0$

if $A = B = C = 0$ , the locus is the Plane of reference,
if A and B are infinitely small but C finite, the Locus is a Line, situated altogether at an infinite distance from the origin,
if $A = B = 0$ , but C is finite, the Equation is self-contradictory.

N.B. The Locus of a Quadratic Equation is discussed in Book VIII.

VII.

Conversely, any Line may be represented by one simple Equation.

(a) Given Line parallel to X-axis, or to Y-axis.
(b) parallel to neither axis, but passing through origin.
(c) and not passing through origin.

VIII.

Given a simple Equation, containing x, or y, or both: to draw the Line represented by it.

(a) Given Equation containing x only, or y only.
(b) x and y, but no constant term.
(c) x, y, and a constant term.

Corollary. Hence, if one such Equation be given or found, the Line may be said to be given or found.

IX.

Given certain magnitudes fixing the position of a Line: to find an Equation to it in terms of those magnitudes. (See post. 2.)

(a) Given Y-intercept, and co-ordinate-ratio.
(b) both intercepts.
(c) sagit-angles, and sagit.
(d) co-ordinates of a point on it, and abscissa-ratio and ordinate-ratio.

X.

Given an Equation to a Line, and the co-ordinates of a Point: to find what relation must exist among the given constants for the Point to lie on the Line.

XI.

Given two Equations to Lines: to find what relation must exist among the given constants for the two Equations to represent the same Line.

XII.

Given an Equation to a Line, in any one of the five forms, to find an Equation to the same Line, in the sagittal form.

The five forms of the Equation to a Line are

1. General. $A x + B y + C = 0$
2. Explicit. $y = t x + b$ .
3. Interceptive. $\frac{x}{x_{0}} + \frac{y}{y_{0}} = 1$ .
4. Sagittal. $x . cos α + y . sin α - p = 0$
or, $x . cos α + y . cos β - p = 0$ . (oblique axes.)
5. Directional. $\frac{x - x^{'}}{l} = \frac{y - y^{'}}{m} = δ$ .

N.B. In Equation 5, δ may be considered as having the nature of a co-ordinate of any Point on the Line; its Geometrical meaning being “the distance of any Point on the Line from the fixed point whose co-ordinates are $x^{'}$ , $y^{'}$ .” Hence any one of the 3 Equations of which N^o. 5 consists is sufficient by itself to represent the Line.

Chapter II. Polar System

Definitions

I.

The Fixed-radius is a line of known position, passing through a point of known position, which is called the Pole.

II.

The direction from left to right of this line, (taking its position as horizontal,) is called positive.

III.

Of any other Line, either direction at pleasure may be called positive.

IV.

The angle which any Line makes with the fixed-radius is called the direction-angle of the Line.

V.

That direction of revolution from the fixed-radius, in which the right-hand portion of a line coinciding with it, would revolve upwards is called its positive direction of revolution.

VI.

The vectorial-line of a Point is a line passing through it and the pole.

VII.

The vectorial-angle of a Point is the direction-angle of its vectorial-line. Its aperture, if unknown, is represented by θ; if known, by $θ^{'}$ , α, or other such symbols.

VIII.

The radius-vector of a Point is its distance from the pole. It is represented, if unknown, by ρ; if known, by $ρ^{'}$ , a, or other such symbols.

IX.

The vectorial-angle and radius-vector of a Point, taken together, are called its co-ordinates.

X.

The intercept of a Line is the distance of the point, where it intersects the fixed-radius, from the pole. It is represented, if unknown, by $ρ_{0}$ ; if known, by a, b, or other such symbols.

XI.

The sagittal-line of a Line is the line drawn through the pole, perpendicular to it.

XII.

The sagittal-angle of a Line is the direction-angle of its sagittal-line. Its aperture, if known, is represented by α, β, or other such symbols.

XIII.

The sagit of a Line is its distance from the pole. If known, it is represented by p, q, or other such symbols.

Postulates

I.

Let it be granted that the position of any Point may be represented by representing the aperture and direction of its vectorial-angle, together with the length and direction of its radius-vector. And that, if these be represented, one Point, and one only, is represented.

II.

That the position of any Line may be represented by re presenting the positions of two points on it; or the position of one point on it, together with the aperture and direction of its direction-angle; or the aperture and direction of its sagittal-angle, together with the length and direction of its sagit. And that, if these be represented, one Line, and one only, is represented.

For remainder of Postulates see Chap. I. Post. 3, 4, 5.

Axioms

I.

The direction from right to left of the fixed-radius is negative.

II.

That direction of revolution from the fixed-radius, in which the right-hand portion of a line coinciding with it, would revolve downwards, is negative.

Propositions

A. Transformation of Co-Ordinates

I.

Given the co-ordinates of a Point, when referred to one fixed-radius and pole, and the position of a new fixed-radius and pole with regard to the old: to find its co-ordinates, when referred to the new fixed-radius and pole, or, (which is the same thing,) to find two Equations expressing the values of its old co-ordinates in terms of the new, (from which Equations the values of its new co-ordinates may be found, if required.)

(a) New fixed-radius parallel to old, but not with same pole.
(b) with same pole, but not parallel.
(c) neither parallel, nor with same pole.

II.

Given the co-ordinates of a Point, when referred to a pair of Cartesian axes, and the position of a Polar fixed-radius and pole with regard to those Cartesian axes: to find its co-ordinates, when referred to the fixed-radius and pole, or, (which is the same thing,) to find two Equations expressing the values of its Cartesian co-ordinates in terms of its Polar co-ordinates, (from which Equations the values of its Polar co-ordinates may be found, if required.)

(a) Pole coinciding with origin, and fixed-radius with X-axis.
(b) but not fixed-radius with X-axis.
(c) Neither coinciding.

III.

Given the coordinates of a Point, when referred to a fixed-radius and pole, and the position of a pair of Cartesian axes with regard to that fixed-radius and pole: to find its co-ordinates, when referred to the Cartesian axes, or, (which is the same thing,) to find two Equations representing the values of its Polar co-ordinates in terms of its Cartesian co-ordinates, (from which Equations the values of its Cartesian co-ordinates may be found, if required.)

(a) Origin coinciding with pole, and X-axis with fixed radius.
(b) but not X-axis with fixed-radius.
(c) Neither coinciding.

IV.

Given an Equation, containing x, or y, or both, which is known to be true of the co-ordinates of a certain Point, referred to a Cartesian System, and the position of a Polar System with regard to that Cartesian System: to find an Equation which shall be true of the co-ordinates of the same Point, referred to the Polar System.

V.

Given an Equation, containing ρ, or θ, or both, which is known to be true of the co-ordinates of a certain Point, referred to a Polar System, and the position of a Cartesian System with regard to that Polar System: to find an Equation which shall be true of the co-ordinates of the same Point, referred to the Cartesian System.

B. Equations

I.

Given the Polar co-ordinates of a Point: to exhibit its position geometrically.

Corollary. Hence, if these be given or found, the Point may be said to be given or found.

II.

Any Point may be represented by two Equations.

III.

One Equation, containing ρ, or θ, or both, represents an infinite number of Points, all fulfilling one and the same condition, i. e. it represents a Locus.

IV.

Conversely, any Locus may be represented by one Equation, provided that the condition to be fulfilled by every Point on it is capable of being expressed in terms of the co-ordinates of that Point.

N.B. Problems of this nature are discussed in Book VI.

V.

Given an Equation to a Line: to draw the Line.

Corollary. Hence, if one such Equation be given or found, the Line may be said to be given or found.

V.

The Locus of any Equation, containing $ρ . cos θ$ , or $ρ . sin θ$ , or both, (but not involving the co-ordinates in any other form,) is (with one exception mentioned in the Corollary) a Line. And by one such Equation one Line, and one only, is represented.

Corollary. In the Equation $(A . cos θ + B . sin θ) . ρ + C = 0$ ,

if $A = B = C = 0$ , the Locus is the Plane of reference,
if A and B are infinitely small, but C finite, the Locus is a Line, situated altogether at an infinite distance from the pole,
if $A = B = 0$ , but C is finite, the Equation is self-contradictory.

VI.

Conversely, any Line may be represented by one simple Equation.

VII.

Given an Equation to a Line: to draw the Line.

Corollary. Hence, if one such Equation be given or found, the Line may be said to be given or found.

VIII.

Given the sagit-angle and sagit of a Line: to find an Equation to it in terms of them. (See Post. 2.)

IX.

Given an Equation to a Line, and the co-ordinates of a Point: to find what relation must exist among the given constants for the Point to lie on the Line.

X.

Given two Equations to Lines: to find what relation must exist among the given constants for the two Equations to represent the same Line.

XI.

Given an Equation to a Line, in the General form: to find an Equation to the same Line, in the Sagittal form.

The most useful forms of the Equation to a Line, in the Polar System, are

1. General. $(A . cos θ + B . sin θ) . ρ + C = 0$ .
2. Sagittal. $ρ . cos (θ - α) = p$ .

Chapter III. Distantial System

This system is again divided into three kinds, the Bilinear, the Trilinear, and the Multilinear.

Section I. Bilinear System

Definitions

I.

The Lines of reference are two lines of known position, intersecting in a point, which is called the Vertex of reference.

II.

One of these is called the α-line, the other, the β-line. One side of the α-line, and one side of the β-line, are called positive; and that portion of each line of reference, which lies on the negative side of the other, is called its principal portion.

III.

Of any other Line, passing through the vertex of reference, either portion at pleasure may be called principal.

IV.

The referents of a Point are its distances from the lines of reference.

V.

Of these distances that which is measured from the α-line is called the α-referent of the Point; the other, its β-referent. They are represented, if unknown, by α, β; if known, by $α^{'}$ , $β^{'}$ , or other such symbols. If these symbols are used in the same expression with α or β representing angles, they may be conveniently distinguished thus, $\bar{α}$ , $\bar{β}$ .

VI.

If a Line pass through the vertex of reference, the angles which its principal portion makes with the principal portions of the lines of reference are called the divisional-angles of the Line.

VII.

Of these angles, that which its principal portion makes with the principal portion of the α-line is called the α-divisional-angle of the Line; the other, its β-divisional-angle.

VIII.

That direction of revolution from either line of reference, in which a line, coinciding with it, would revolve, about the vertex of reference, away from the negative side of its principal portion, is called the positive direction of revolution of that line of reference.

IX.

If a Line pass through the vertex of reference, the ratio which the β-referent of any point on it bears to the α-referent of the same point, (being equal to the ratio which the sine of the β-divisional-angle of the Line bears to the sine of its α-divisional-angle,) is called the divisional-ratio of the Line. If known, it is represented by k, or l, or other such symbol.

X.

If a Line do not pass through the vertex of reference, its divisional-angles, and its divisional-ratio, are the same as those of a Line, parallel to it, and passing through the vertex of reference.

XI.

The intercepts of a Line are the α-referent of the point where it intersects the β-line, together with the β-referent of the point where it intersects the α-line. They are called respectively the α-intercept, and the β-intercept, and are represented, if unknown, by $α_{0}$ , $β_{0}$ ; if known, by a, b, or other such symbols.

Postulates

I.

Let it be granted that the position of any Point may be represented by representing the lengths and directions of its referents. And that, if these be represented, one Point, and one only, is represented.

II.

That the position of any Line may be represented by representing the positions of two points on it; or the position of one point on it, together with its divisional-ratio. And that, if these be represented, one Line, and one only, is represented.

For remainder of Postulates see Chap. I. Post. 3, 4, 5.

Axioms

I.

That direction of revolution from either line of reference, in which a line, coinciding with it, would revolve, about the vertex of reference, away from the positive side of its principal portion is negative.

II.

The divisional-ratio of a Line has the same sign, in whichever portion of the Line the point, from which it is estimated be taken, and whichever of the two portions of the Line be called principal: its sign depends only on the position of the Line.

III.

If a Line, passing through the vertex of reference, lie in the angle contained between the principal portions of the lines of reference, its divisional-ratio is positive; if other wise, negative.

IV.

When the lines of reference are perpendicular to each other, they may be considered as Cartesian rectangular axes, by taking the lines themselves as axes, the vertex of reference as origin, and the directions in which the principal portions of the lines lie with regard to the vertex as the negative directions of the two axes respectively. In this case the referents of a Point will coincide with its Cartesian co-ordinates, and the divisional-ratio of a Line with its Cartesian co-ordinate-ratio.

Propositions

A. Transformation of co-ordinates

I.

Given the referents of a Point, referred to a Bilinear System: to find its co-ordinates referred to the same lines considered as a Cartesian System, or, (which is the same thing,) to find two Equations expressing the values of its referents in terms of its co-ordinates, (from which Equations the values of its co-ordinates may be found, if required.)

II.

Given an Equation, containing α, or β, or both, which is known to be true of the referents of a Point, referred to a Bilinear System: to find an Equation which shall be true of the co-ordinates of the same Point, referred to the same lines considered as a Cartesian System.

B. Equations

I.

Given the referents of a Point: to exhibit its position geometrically.

Corollary. Hence, if these be given or found, the Point may be said to be given or found.

II.

Two simple Equations, containing α, or β, or both, represent a Point.

III.

Conversely, any Point may be represented by two Equations.

IV.

Two Equations of a higher degree, containing α, or β, or both, represent a finite number of Points.

V.

One Equation, containing α, or β, or both, represents an infinite number of Points, all fulfilling one and the same condition, i. e. it represents a Locus.

VI.

Conversely, any Locus may be represented by one Equation, provided that the condition to be fulfilled by every Point on it is capable of being expressed in terms of the referents of that Point.

N.B. Problems of this nature are discussed in Book VI.

VII.

The Locus of any simple Equation, containing α, or β, or both, is, (with one exception, mentioned in Corollary 2,) a Line. And by one such Equation one Line, and one only, is represented.

Corollaries. 1. In the Equation $β = k . a + b$ , the geometrical meaning of k is “the divisional-ratio of the Line,” and of b, “the β-referent of the point where the Line intersects the α-line.”

2. In the Equation $A α + B β + C = 0$

if $A = B = C = 0$ , the locus is the Plane of reference,
if A and B are infinitely small but C finite, the Locus is a Line situated altogether at an infinite distance from the vertex of reference,
if $A = B = 0$ , but C is finite, the Equation is self-contradictory.

N.B. The Locus of a Quadratic Equation is discussed in Book VIII.

VIII.

Conversely, any Line may be represented by one simple Equation.

IX.

Given a simple Equation, containing α, or β, or both: to draw the Line represented by it.

(a) Given Equation containing α only, or β only.
(b) α and β, but no constant term.
(c) α, β, and a constant term.

Corollary. Hence, if one such Equation be given or found, the Line may be said to be given or found.

X.

Given certain magnitudes fixing the position of a Line: to find an Equation to it in terms of those magnitudes.

(a) Given β-intercept, and divisional-ratio.
(b) both intercepts.

XI.

Given an Equation to a Line, and the referents of a Point: to find what relation must exist among the given constants for the Point to lie on the Line.

XII.

Given two Equations to Lines: to find what relation must exist among the given constants for the two Equations to represent the same Line.

The most useful forms of the Equation to a Line, in the Bilinear System, are

1. General. $A α + B β + C = 0$ .
2. Explicit. $β = k α + b$ .
3. Interceptive. $\frac{α}{α_{0}} + \frac{β}{β_{0}} = 1$ .

Section II. Trilinear System

Definitions

I.

The Lines of reference are 3 lines of known position, intersecting in 3 points, which are called the Vertices of reference, and so forming a triangle, which is called the Triangle of reference.

II.

These lines are called respectively the α-line, the β-line, and the γ-line. Their outer sides are called positive; and those portions of them, which constitute the sides of the triangle of reference, are called their principal portions.

III.

Of any other Line, passing through a vertex of reference, either portion at pleasure may be called principal.

IV.

The lengths of the sides of the triangle of reference are represented by a, b, and c respectively; and the apertures of the opposite angles by A, B, and C respectively; and the area of the triangle by M.

V.

The referents of a Point are its distances from the lines of reference.

VI.

Of these distances, that which is measured from the α-line is called the α-referent of the Point; the others, its β-referent and γ-referent respectively. They are represented, if unknown, by α, β, γ; if known, by $α^{'}$ , $β^{'}$ , $γ^{'}$ , or other such symbols.

VII.

If a Line pass through a vertex of reference, its divisional-angles, and its divisional-ratio, with regard to the lines of reference which intersect in that vertex, are the same as if those lines constituted a Bilinear System.

VIII.

That direction of revolution, from any line of reference, about either of the two vertices of reference which lie on it, in which a line, coinciding with it, would revolve away from the inner side of its principal portion, is called its positive direction of revolution with regard to that vertex of reference.

IX.

If a Line do not pass through a vertex of reference, and if each of the points where it intersects a line of reference be joined to the opposite vertex of reference, the joining lines are called the interceptors of the Line.

X.

Of these, that which is drawn from the point where the Line intersects the α-line is called its α-interceptor; the others, its β-interceptor and γ-interceptor respectively.

Postulates

I.

Let it be granted that the position of any Point may be represented by representing the lengths and directions of any two of its referents. And that, if these be represented, one Point, and one only, is represented.

II.

That the position of any Line, passing through a vertex of reference, may be represented by representing its divisional-ratio with regard to the lines of reference which intersect in that vertex; and that the position of any Line, not passing through any vertex of reference, may be represented by representing the positions of two points on it. And that, if these be represented, one Line, and one only, is represented.

For remainder of Postulates see Chap. I. Post. 3, 4, 5.

Axioms

I.

The inner sides of the lines of reference are negative.

II.

That direction of revolution, from any line of reference, about either of the two vertices of reference which lie on it, in which a line, coinciding with it, would revolve away from the outer side of its principal portion, is its negative direction of revolution with regard to that vertex of reference.

III.

The positive direction of revolution of a line of reference, with regard to one of the two vertices of reference which lie on it, is opposite to the positive direction of revolution of the same line, with regard to the other vertex of reference.

IV.

If a Line pass through a vertex of reference, its divisional-ratio has the same sign, in whichever portion of the Line the point from which it is estimated, be taken, and whichever of, the two portions of the Line be called principal: its sign depends only on the position of the Line.

V.

If a Line, passing through a vertex of reference, lie within the triangle of reference, its divisional-ratio, with regard to the lines of reference which intersect in that vertex, is positive; if otherwise, negative.

Propositions

A. Transformation of co-ordinates

I.

The expression $a α + b β + c γ$ is constant in value, whatever be the position of the Point, and represents double the area of the triangle of reference. (i. e. $a α + b β + c γ = - 2 M$ .)

Corollary. Hence, the expression $α . sin A + β . sin B + γ . sin C$ , (being equal to the former expression multiplied throughout by one or other of the equivalent fractions $\frac{sin A}{a}$ , $\frac{sin B}{b}$ , $\frac{sin C}{c}$ ,) is also constant in value, and represents double the area of the triangle of reference multiplied by $\frac{sin A}{a}$ . (i. e. $α . sin A + β . sin B + γ . sin C = - 2 M . \frac{sin A}{a}$ .)

II.

Given any two of the referents of a Point: to find its third referent.

III.

Given an Equation, containing α, β, and γ, which is known to be true of the referents of a Point: to find an Equation, containing two only of these three symbols, which shall be true of the referents of the same Point.

IV.

Given a non-homogeneous Equation, containing α, β, γ, (any or all of them,) with a constant term, which is known to be true of the referents of a Point: to find a homogeneous Equation, (i. e. one not containing a constant term,) which shall be true of the referents of the same Point.

V.

Given the referents of a Point, referred to a Trilinear System: to find its co-ordinates referred to two of the three lines of reference, considered as a Cartesian System, or, (which is the same thing,) to find three Equations expressing the values of its referents in terms of its co-ordinates, (from which Equations the values of its co-ordinates may be found, if required.)

VI.

Given an Equation, containing α, β, γ, (any or all of them,) which is known to be true of the referents of a Point, referred to a Trilinear System: to find an Equation which shall be true of the co-ordinates of the same Point, referred to two of the three lines of reference, considered as a Cartesian System.

B. Equations

I.

Given any two of the referents of a Point: to exhibit its position geometrically.

Corollary. Hence, if these be given or found, the Point may be said to be given or found.

II.

Two simple equations, containing α, β, γ, (any or all of them,) represent a Point.

III.

Conversely, any Point may be represented by two Equations.

IV.

Two Equations of a higher degree, containing α, β, γ, (any or all of them,) represent a finite number of Points.

V.

One Equation, containing α, β, γ, (any or all of them,) represents an infinite number of points, all fulfilling one and the same condition, i. e. it represents a Locus.

VI.

Conversely, any Locus may be represented by one Equation, provided that the condition to be fulfilled by every Point on it is capable of being expressed in terms of the referents of that Point.

N.B. Problems of this nature are discussed in Book VI.

VII.

The Locus of any simple Equation, containing α, β, γ, (any or all of them,) is, (with two exceptions, mentioned in the Corollaries,) a Line. And by one such Equation, one Line, and one only, is represented.

Corollaries. 1. In the Equation $A α + B β + C γ + D = 0$

if $A = B = C = D = 0$ , the Locus is the Plane of reference,
if A, B, and C, are infinitely small but D finite, the Locus is a Line, situated altogether at an infinite distance from the triangle of reference,
if $A = B = C = 0$ , but D is finite, the Equation is self-contradictory.

2. The Locus of each of the Equations $a α + b β + c γ = 2 M,$ $α . sin A + β . sin B + γ . sin C = 2 M . \frac{sin A}{a},$ is the Plane of reference.

N.B. The Locus of a Quadratic Equation is discussed in Book VIII.

VIII.

Conversely, any Line may be represented by one simple Equation.

IX.

Given a homogeneous simple Equation, containing α, β, and γ: to find the Equations to the interceptors of the Line represented by it.

X.

Given a simple Equation, containing α, β, γ, (any or all of them:) to draw the Line represented by it.

(a) Given Equation containing one referent only.
(b) homogeneous, and containing two referents.
(c) homogeneous, and containing three referents.
(d) non-homogeneous, and containing two, or, three, referents.

Corollary. Hence, if one such Equation be given or found, the Line may be said to be given or found.

XI.

Given an Equation to a Line, and any two of the referents of a Point: to find what relation must exist among the given constants for the Point to lie on the Line.

XII.

Given two Equations to Lines: to find what relation must exist among the given constants for the two Equations to represent the same Line.

The most useful forms of the Equation to a Line, in the Trilinear System, are

1. General. $A α + B β + C γ + D = 0$ .
2. Homogeneous. $A α + B β + C γ = 0$ .

Section III. Multilinear System

Definition

The Lines of reference are 4 or more lines of known position, intersecting in as many points, which are called the Vertices of reference.

This System, being exactly analogous to the Trilinear System and of little practical use, need not be further discussed here.

Book V. Discussion of Points, Right Lines, Rectilinear Figures, and Pencils

This Book contains two Divisions, Problems and Theorems.

The object of each Problem is to represent algebraically some required magnitude, Point, or Line, in terms of the algebraical symbols representing certain given magnitudes, Points, or Lines. The expression, so found, is called a formula. Some of these formulæ are intended to be worked out merely as models for the solution of other Problems of the same kind. Others are intended to be used in solving Problems, proving Theorems, and investigating Loci. This latter class should be committed to memory, for which purpose the formulæ themselves will be given hereafter in a separate list.

The object of each Theorem is to prove some fact concerning certain given magnitudes, Points, or Lines. In proving a theorem we may employ whatever system is most convenient for the purpose; hence the Theorems are not classed according to the three systems. But since, whatever system be employed, formulæ in that system are required, therefore the Problems are again divided into three Chapters, on the Cartesian, Polar, and Distantial Systems.

Each Chapter is divided, according as the Problems involve the consideration of

A. Points and Finite Lines.
B. Lines which are the Loci of Simple Equations.
C. Figures.
D. Pencils.

Each part is again divided, according as the “quæsita” of the Problems involve

(α) Magnitudes.
(β) Points.
(γ) Lines. ${\begin{cases} 1. Indeterminate. \\ 2. Determinate. \end{cases}$

Definitions

I.

When a Point, situated in the same line with two other Points, lies between them, it is said to divide the line joining them internally; when otherwise, externally.

II.

When a Point, situated in the same plane with a rectilinear Figure, lies within the Figure, then, if it be joined to the vertices of the Figure, the joining lines are said to divide the Figure internally; when without it, externally.

III.

When a Line, situated in the same plane, and terminated at the same point, with two other Lines, lies within the angle contained between them, it is said to divide that angle internally; when without it, externally.

Division I. Problems

Chapter I. Cartesian System

N.B. In this Chapter, some of the Problems result in the same formula, whether the axes employed be rectangular or oblique; these are marked thus *. In the others, when the formula for oblique axes is worth being worked out separately, the Proposition is marked thus §.

A. Discussion of Points and finite Lines

(α) Magnitudes required

§

Given 2 Points: to find the distance between them.

(a) One of the given Points at origin.
(b) Neither .

(β) Points required

I. *

Given 2 Points: to find the Point dividing internally the line joining them, into 2 parts, having a given ratio.

Corollary. Given 2 Points: to find the Point bisecting the distance between them.

II. *

Given 2 Points: to find the Point dividing externally the line joining them, into 2 parts, having a given ratio.

B. Discussion of Lines which are the Loci of simple Equations

(α) Magnitudes required

I. §

Given the co-ordinate-ratio of a Line: to find its abscissa-ratio and ordinate-ratio.

II. *

Given a Line: to find its intercepts.

III. §

Given a Line and a Point: to find the distance between them.

Corollary. Hence, given the referents of a Point, referred to a Distantial System, and the Cartesian Equations of the lines of reference, referred to a Cartesian System: to find its co-ordinates, referred to that Cartesian System, or, (which is the same thing, to find Equations expressing the values of its referents in terms of its co-ordinates, (from which Equations the values of its co-ordinates may be found, if required.)

Hence also, given an Equation, containing symbols of referents, which is known to be true of the referents of a Point, referred to a Distantial System, and the Cartesian Equations of the lines of reference, referred to a Cartesian System: to find an Equation which shall be true of the co-ordinates of the same Point, referred to that Cartesian System.

Hence also, given an Equation to a Line, referred to a Distantial System, and the Cartesian Equations of the lines of reference, referred to a Cartesian System: to find an Equation to the same Line, referred to that Cartesian System.

IV. *

Given a Line, a Point, and the abscissa-ratio and ordinate-ratio, (or else the co-ordinate-ratio,) of a Line passing through the Point: to find the distance between the given Point and the point where the two Lines intersect.

Corollary. To find what relation must exist among the given constants for this distance to be

(1) $= 0$ , (i. e. Point on first given Line,)
(2) $= \frac{1}{0}$ , (i. e. Lines parallel)
(3) indeterminate. (i. e. Lines coincident.)

V. *

Given a Line and 2 Points: to find the ratio in which the given Line divides, (whether internally or externally,) the line joining the given Points.

VI.

Given 2 Lines: to find the angle between them.

Corollary. To find what relation must exist among the given constants for this angle to be

(1) $= 0$ ,
(2) a right angle,
(3) a given angle.

(β) Points required

I. *

Given 2 Lines: to find their Point of intersection.

Corollary. To find what relation must exist among the given constants for this point to be

(1) having one or both of its co-ordinates $= \frac{1}{0}$ , (i. e. Lines parallel.)
(2) indeterminate. (i. e. Lines coincident.)

II. *

Given 3 Lines: to find what relation must exist among the given constants for them to intersect in the same point.

(γ) Lines required

(1) Indeterminate (i. e. fulfilling one condition only.)

I. *

Given a Point: to find a Line passing through it.

II. *

Given a Line: to find a Line parallel to it.

III.

Given a Line: to find a Line perpendicular to it.

IV.

Given a Line: to find a Line making a given angle with it.

V. *

Given 2 Lines: to find a Line passing through their point of intersection.

(2) Determinate (i. e. fulfilling two conditions.)

I. *

Given 2 Points: to find the Line passing through them.

II. *

Given 3 Points: to find what relation must exist among the given constants for them to lie on the same line.

III. *

Given 2 similar Simple Equations, involving respectively the co-ordinates of 2 unknown Points, and having the same constants and signs: to find the Line passing through those 2 unknown Points.

C. Discussion of Figures

(α) Magnitudes required

I. §

Given the vertices of a Triangle: to find its area.

(a) One of the given vertices at origin.
(b) None .

Corollary. Given 3 Points: to find what relation must exist among the given constants for them to lie on the same line. (2nd method. see above.)

II. §

Given the sides of a Triangle: to find its area.

(a) One of the given lines coinciding with Y-axis.
(b) None .

Corollary. Given 3 Lines: to find what relation must exist among the given constants for them to intersect in the same point. (2nd method. See p. 54.)

III. §

Given the vertices of a quadrilateral, or multilateral, Figure: to find its area.

IV. §

Given the sides of a quadrilateral, or multilateral, Figure: to find its area.

(β) Points required

I. *

Given the vertices of a Triangle: to find the Point, which being joined to the 3 vertices, the Triangle shall be divided internally into 3 parts having given ratios.

Corollary. Given the vertices of a Triangle: to find the Point, which being joined to the 3 vertices, the Triangle shall be trisected.

II. *

Given the vertices of a Triangle: to find the Point, which being joined to the 3 vertices, the Triangle shall be divided externally into 3 parts having given ratios.

D. Discussion of Pencils

(α) Magnitudes required

I.

Given a Line passing through the origin, (and so forming a Pencil with the axes): to find its divisional-ratio with regard to the axes.

II.

Given 2 Lines passing through the origin, (and so forming a Pencil with the axes): to find the anharmonic ratio of the Pencil so formed, with regard to the axes.

Corollary. To find what relation must exist among the given constants, for this Pencil to be harmonic.

III.

Given 3 Lines, having their Equations of the form

$A x + B y + C = 0$ , (1)
$A^{'} x + B^{'} y + C^{'} = 0$ , (2)
$A x + B y + C + k . (A^{'} x + B^{'} y + C^{'}) = 0$ , (3)

(and so forming a Pencil): to find the divisional-ratio of the third Line, with regard to the two first.

IV.

Given 4 Lines, having their Equations of the form

$A x + B y + C = 0$ , (1)
$A^{'} x + B^{'} y + C^{'} = 0$ , (2)
$A x + B y + C + k . (A^{'} x + B^{'} y + C^{'}) = 0$ , (3)
$A x + B y + C + l . (A^{'} x + B^{'} y + C^{'}) = 0$ , (4)

(and so forming a Pencil): to find the anharmonic ratio of the Pencil so formed, with regard to the two first Lines.

Corollary. To find what relation must exist among the given constants for this Pencil to be harmonic.

Chapter II. Polar System

Any formulæ required in this system may easily be deduced from the corresponding formulæ in the Cartesian System, by the transformation of co-ordinates. Some, however, can be more easily found directly, without such transformation: those here given are generally of this kind.

The subjects of Figures and Pencils are not here discussed, as they can be much more conveniently treated by other Systems.

A. Discussion of Points and finite Lines

(α) Magnitudes required

Given 2 Points: to find the distance between them.

(a) One of the given Points at pole.
(b) Neither .

B. Discussion of Lines which are the Loci of simple Equations

(α) Magnitudes required

I.

Given a Line: to find its intercept.

II.

Given a Line and a Point: to find the distance between them.

III.

Given 2 Lines: to find the angle between them.

(β) Points required

Given 2 Lines: to find their Point of intersection.

(γ) Lines required

(1) Indeterminate (i. e. fulfilling one condition only.)

I.

Given a Point: to find a Line passing through it.

II.

Given a Line: to find a Line parallel to it.

III.

Given a Line: to find a Line perpendicular to it.

IV.

Given a Line: to find a Line making a given angle with it.

V.

Given 2 Lines: to find a Line passing through their point of intersection.

(2) Determinate (i. e. fulfilling two conditions.)

I.

Given 2 Points: to find the Line passing through them.

II.

Given 3 Points: to find what relation must exist among the given constants for them to lie on the same line.

III.

Given 2 similar Equations, having the form of an Equation to a Line, involving respectively the co-ordinates of 2 unknown Points, and having the same constants and signs: to find the Line passing through those 2 unknown Points.

Chapter III. Distantial System

Section I. Bilinear System

B. Discussion of Lines which are the Loci of simple Equations

(α) Magnitudes required

Given a Line: to find its intercepts.

(β) Points required

I.

Given 2 Lines: to find their Point of intersection.

II.

Given 3 Lines: to find what relation must exist among the given constants for them to intersect in the same point.

(γ) Lines required

(1) Indeterminate (i. e. fulfilling one condition only.)

I.

Given a Point: to find a Line passing through it.

II.

Given a Line: to find a Line parallel to it.

III.

Given a Line: to find a Line perpendicular to it.

IV.

Given 2 Lines: to find a Line passing through their point of intersection.

(2) Determinate (i. e. fulfilling two conditions.)

I.

To find the Line dividing internally the angle contained between the lines of reference, into 2 parts, whose sines shall have a given ratio.

Corollary. To find the Line bisecting the angle contained between the lines of reference.

II.

To find the Line dividing externally the angle contained between the lines of reference, into 2 parts, whose sines shall have a given ratio.

Corollary. To find the Line bisecting the supplement of the angle contained between the lines of reference.

III.

Given 2 Points: to find the Line passing through them.

IV.

Given 3 Points: to find what relation must exist among the given constants for them to lie on the same line.

V.

Given 2 similar Simple Equations, involving respectively the referents of 2 unknown Points, and having the same constants and signs: to find the Line passing through those two unknown Points.

D. Discussion of Pencils

(α) Magnitudes required

I.

Given 2 Lines passing through the vertex of reference, (and so forming a Pencil with the lines of reference:) to find the anharmonic ratio of the Pencil so formed, with regard to the lines of reference.

Corollary. To find what relation must exist among the given constants for this pencil to be harmonic.

II.

Given 4 Lines passing through the vertex of reference, (and so forming a Pencil:) to find the anharmonic ratio of the Pencil so formed, with regard to any two of its rays.

Corollary. To find what relation must exist among the given constants for this Pencil to be harmonic.

III.

Given 3 Lines, having their Equations of the form

$A α + B β + C = 0$ , (1)
$A^{'} α + B^{'} β + C^{'} = 0$ , (2)
$A α + B β + C + k . (A^{'} α + B^{'} β + C^{'}) = 0$ , (3)

(and so forming a Pencil:) to find the divisional-ratio of the third Line, with regard to the two first.

IV.

Given 4 Lines, having their Equations of the form

$A α + B β + C = 0$ , (1)
$A^{'} α + B^{'} β + C^{'} = 0$ , (2)
$A α + B β + C + k . (A^{'} α + B^{'} β + C^{'}) = 0$ , (3)
$A α + B β + C + l . (A^{'} α + B^{'} β + C^{'}) = 0$ , (4)

(and so forming a Pencil:) to find the anharmonic ratio of the Pencil so formed, with regard to the two first Lines.

Corollary. To find what relation must exist among the given constants for this Pencil to be harmonic.

Section II. Trilinear System

B. Discussion of Lines which are the Loci of simple Equations

(β) Points required

I.

Given 2 Lines: to find their Point of intersection.

II.

Given 3 Lines: to find what relation must exist among the given constants for them to intersect in the same point.

(γ) Lines required

(1) Indeterminate (i. e. fulfilling one condition only.)

I.

Given a Point: to find a Line passing through it.

II.

Given a Line: to find a Line parallel to it.

III.

Given 2 Lines: to find a Line passing through their point of intersection.

(2) Determinate (i. e. fulfilling two conditions.)

I.

Given 2 Points: to find the Line passing through them.

II.

Given 3 Points: to find what relation must exist among the given constants for them to lie on the same line.

III.

Given any 2 of the interceptors of a Line: to find the Line.

IV.

Given 2 similar Simple Equations, involving respectively the referents of 2 unknown Points, and having the same constants and signs: to find the Line passing through those 2 unknown Points.

C. Discussion of Figures

I.

Given the sides of a Triangle: to find the Line, passing through any vertex, and bisecting the opposite side.

II.

Given the sides of a Triangle: to find the Line, passing through any vertex, and parallel to the opposite side.

III.

Given the sides of a Triangle: to find the Line, passing through any vertex, and perpendicular to the opposite side.

IV.

Given the sides of a Triangle: to find the Line, passing through any vertex, and bisecting the angle formed at that vertex.

V.

Given the sides of a Triangle: to find the Line, passing through any vertex, and bisecting the external angle formed at that vertex.

D. Discussion of Pencils

I.

Given 3 Lines, having their Equations of the form

$A α + B β + C γ + D = 0$ , (1)
$A^{'} α + B^{'} β + C^{'} γ + D^{'} = 0$ , (2)
$A α + B β + C γ + D + k . (A^{'} α + B^{'} β + C^{'} γ + D^{'}) = 0$ , (3)

(and so forming a Pencil,) and the Cartesian Equations of the lines of reference, referred to a Cartesian System: to find the divisional-ratio of the third Line, with regard to the two first.

II.

Given 4 Lines, having their Equations of the form

$A α + B β + C γ + D = 0$ , (1)
$A^{'} α + B^{'} β + C^{'} γ + D^{'} = 0$ , (2)
$A α + B β + C γ + D + k . (A^{'} α + B^{'} β + C^{'} γ + D^{'}) = 0$ , (3)
$A α + B β + C γ + D + l . (A^{'} α + B^{'} β + C^{'} γ + D^{'}) = 0$ , (4)

(and so forming a Pencil:) to find the anharmonic ratio of the Pencil so formed, with regard to the two first Lines.

Corollary. To find what relation must exist among the given constants for this Pencil to be harmonic.

Division II. Theorems

N.B. The following Theorems are given merely as specimens of those which may be proved with the help of the foregoing formulæ.

Algebraical proofs might possibly be made for all the Theorems in the first six Books of Euclid, and it would be excellent practice for the Student to find them for himself. Since, however, the same Theorems must necessarily have been proved Geometrically in order to establish the elementary propositions of this Science in its present state, all such Algebraical proofs involve the Logical fallacy of arguing in a circle: they are therefore, philosophically speaking, valueless, and cannot properly be included in any formal treatise on the subject.

B. Discussion of Lines which are the Loci of simple Equations

(1) Indeterminate Lines

(β) In relation to Points

If an Equation to a Line contain one indeterminate constant only, and one power only of that indeterminate constant: the indeterminate Line represented by it either has a determinate direction, or passes through a determinate Point.

Corollaries. 1. If an Equation to a Line contain the Cartesian co-ordinates, or the referents, of an indeterminate Point, in the first degree only; and if this Point be known to lie on a given Line: the indeterminate Line represented by this Equation either has a determinate direction, or passes through a determinate Point.

2. If an Equation to a Line be of the form

$A x + B y + C = 0$ , (Cartesian)
or, $A α + B β + C = 0$ , (Bilinear)
or, $A α + B β + C γ = 0$ , (Trilinear)

where A, B, and C, are indeterminate; and if a second Equation be given of the form $a A + b B + c C = 0,$ where a, b, and c, are known quantities: the indeterminate Line represented by the first Equation either has a determinate direction, or passes through a determinate Point.

(2) Determinate Lines

(α) In relation to Magnitudes

(Cartesian System)

I.

The ratio wbich the sine of the X-direction-angle of a Line bears to the sine of its Y-direction-angle is equal to the ratio which its ordinate-ratio bears to its abscissa-ratio, or, (which is the same thing,) is equal to its co-ordinate ratio. (i. e. $\frac{sin λ}{sin μ} = \frac{m}{l} = t$ .)

II.

If the abscissa-ratio and ordinate-ratio of a Line be represented respectively by l and m, and the angle between the axes by ω, then $l^{2} + m^{2} + 2 l m . cos ω = 1 .$

Corollary. Given the abscissa-ratio of a Line: to find its ordinate-ratio, and vice versâ.

(β) In relation to Points

If, in any system, 3 Equations to Lines be such that, when their left-hand sides are combined by some one of the methods of addition, subtraction, multiplication, or division, and their right-hand sides similarly combined, the resulting Equation is an identity: the 3 Lines represented by these Equations either are all parallel, or else, intersect in the same Point.

C. Discussion of Figures

(α) In relation to Magnitudes

I.

If a Line intersect the 3 sides of a Triangle, (produced if necessary,) so as to divide them, (whether internally or externally,) in the ratios $\frac{m_{1}}{n_{1}}$ , $\frac{m_{2}}{n_{2}}$ , $\frac{m_{3}}{n_{3}}$ , (the ratios being affected with + or −, according as the division, in each case, is internal or external:) the ratio $\frac{m_{1} m_{2} m_{3}}{n_{1} n_{2} n_{3}} = - 1$ .

II.

If a Point, within or without a Triangle, be joined to its 3 vertices, and if the joining lines, (produced if necessary,) divide the opposite sides, (whether internally or externally,) in the ratios $\frac{m_{1}}{n_{1}}$ , $\frac{m_{2}}{n_{2}}$ , $\frac{m_{3}}{n_{3}}$ (the ratios being affected with + or − according as the division, in each case, is internal or external:) the ratio $\frac{m_{1} m_{2} m_{3}}{n_{1} n_{2} n_{3}} = 1$ .

(β) In relation to Lines which are the Loci of Simple Equations

I.

The 3 Lines, passing through the vertices of any Triangle, and bisecting the opposite sides, intersect in the same point.

II.

The 3 Lines, passing through the vertices of any Triangle, and perpendicular to the opposite sides, intersect in the same point.

III.

The 3 Lines, passing through the vertices of any Triangle, and bisecting the angles formed at those vertices, intersect in the same point.

IV.

The 3 Lines, passing through the vertices of any Triangle, one bisecting the angle formed at that vertex, and the other two bisecting the external angles formed at those vertices, intersect in the same point.

Book VI. Investigation of Loci

The general problem of this Book is, given certain conditions to be fulfilled by a variable Point, to find an Equation to the Locus; and this Equation can always be found, provided that an infinite number of Points exist, fulfilling the given conditions, and that those conditions are capable of being expressed in an Equation involving the co-ordinates, or referents, of the variable Point.

In finding the required Equation, any System may be employed that appears to be most convenient for the purpose. As a general rule, when two fixed Lines are given, the Cartesian or Bilinear Systems may be used; when three, the Trilinear; and when a variable angle is involved in the conditions, the Polar System.

The following Loci are given merely as specimens, and are divided according as the conditions are chiefly concerned with

A. Distances.
B. Intersections of Lines which are the Loci of Simple Equations.
C. Vertices of Figures.
D. Combinations of given Loci.

N.B. One of the sides of a Triangle is sometimes distinguished as the base: in this case, the opposite vertex is called the vertex, and its distance from the base, the altitude, of the Triangle.

A. Conditions of Distances

I.

Given a Point and the distance of the variable Point from it.

N.B. The properties of this Locus are discussed in Book VII.

II.

Given 2 Points and the sum, or difference, of the distances of the variable Point from them.

N.B. The properties of these Loci are discussed in Books X. and XI.

III.

Given 2 Points and the ratio between the distances of the variable Point from them.

(a) Given ratio one of equality.
(b) inequality.

IV.

Given a Line and the distance of the variable Point from it.

V.

Given a Line, a Point, and the sum, or difference, of the distances of the variable Point from them.

VI.

Given a Line, a Point, and the ratio between the distances of the variable Point from them.

(a) Given ratio one of equality.
(b) inequality.

N.B. The properties of these Loci are discussed in Books IX. X. and XI.

VII.

Given 2 Lines intersecting, and the sum, or difference, of the distances of the variable Point from them.

VIII.

Given 2 Lines intersecting, and the ratio between the distances of the variable Point from them.

(a) Given ratio one of equality.
(b) inequality.

Corollaries. 1. Given 2 Lines intersecting: to find the Line bisecting the angle contained between them.

2. Given 2 Lines intersecting: to find the Line bisecting the supplement of the angle contained between them.

IX.

Given 3 Lines intersecting, and that a variable Line is always parallel to one of the 3 Lines, and that the portion of it, intercepted between the other 2 Lines, is divided in a given ratio by the variable Point.

B. Conditions of Intersections of Lines

I.

Given 2 Equations to variable Lines, such that only one indeterminate constant is involved in them, and that the variable Point is their point of intersection.

II.

Given a Triangle, and that a variable Line is always parallel to its base, and that the points, where it intersects the sides, (produced if necessary,) are joined to the opposite extremities of the base, and that the variable Point is the Point of intersection of the joining lines.

III.

Given a Triangle, and that a variable Line is always parallel to its base, and that from the points, where it intersects the sides, (produced if necessary,) Lines are drawn perpendicular to those sides, and that the variable Point is their point of intersection.

C. Conditions of Vertices of Figures

I.

Given the base, and the altitude, of a Triangle, and that the variable Point is its vertex.

II.

Given the base, and the vertical angle, of a Triangle, and that the variable Point is its vertex.

III.

Given the bases, and the sum of the areas, of any number of Triangles, and that the variable Point is their common vertex.

IV.

Given the 3 angles, and one vertex, of a Triangle, and that another vertex lies on a given Line, and that the variable Point is its third vertex.

V.

Given that 2 of the vertices of a Triangle lie on 2 given Lines, and that its 3 sides pass through 3 given Points, (which lie on the same Line,) and that the variable Point is its third vertex.

D. Conditions of Combinations of Given Loci

Given the Equations to 2 or more Loci, and that the variable Point lies on any one of them.

Corollaries. 1. Given a Quadratic Equation: to find what relation must exist among the given constants for it to represent 2 Lines.

2. Given a Quadratic Equation representing 2 Lines: to find 2 Simple Equations, respectively representing the same 2 Lines.

Book VII. The Circle

This Book contains three Divisions.

I. Equations to the Circle.
II. Problems.
III. Theorems.

Each of the two first Divisions contains three Chapters, on the Cartesian, Polar, and Distantial Systems. The third Division, and the separate Chapters of the second Division, are sub-divided as in Book V.

The word “Circle” is always employed by Euclid to denote a figure, and the word “circumference” to denote its bounding line: and his definitions of the Points and Lines connected with it are all based upon this notion. These definitions are given in this syllabus among those on “Plane Geometry.”

But in Algebraical Geometry the word “Circle” is employed to denote the bounding line itself, and the word “area” to denote the enclosed figure. This convention is adopted in order to preserve the analogy between this and some other curves, to be hereafter discussed, which do not contain any figure properly so called, and must therefore necessarily be treated as curved lines.

N.B. The words “semicircle” and “segment of a Circle” still retain the meanings given to them by Euclid.

Definitions

I.

A Circle is a Locus all the points of which lie in the same plane and at the same distance from a certain fixed Point.

II.

And this Point is called the Centre of the Circle.

III.

An Arc of a Circle is any portion of it.

IV.

A Radius of a Circle is a line drawn from the centre, and terminated by the Circle.

V.

A Chord of a Circle is the line joining any two points on it.

VI.

A Diameter of a Circle is the Locus of the centres of any system of parallel Chords.

N.B. We shall hereafter prove that this coincides with Euclid’s definition.

VII.

A Circle and a Line are said to touch each other, when they meet, but do not cut each other, i. e. when the Circle lies wholly on the same side of the Line.

VIII.

Two Circles are said to touch each other, when they meet, but do not cut each other, i. e. when each Circle lies wholly on the same side, (whether it be the outer side, or the inner side,) of the other.

IX.

When a Circle and a Line touch each other, the Line is called a Tangent of the Circle.

X.

If through the point, where a Line touches a Circle, a Line be drawn perpendicular to the Tangent, it is called a Normal of the Circle.

XI.

If from any Point without a Circle 2 Lines be drawn touching the Circle, the line joining the points of contact is called the Chord of contact of the Circle with regard to that Point.

XII.

If 2 Circles intersect in 2 points, the line joining the points of intersection is called the Chord of intersection of the 2 Circles.

XIII.

When a Line touches 2 Circles, it is called a common Tangent of the 2 Circles.

XIV.

When a common Tangent of 2 Circles is such that the 2 Circles lie on the same side of it, it is called direct; but when such that they lie, one on one side of it, and one on the other, it is called transverse.

XV.

The Point of intersection of the 2 direct common Tangents of 2 Circles, together with the Point of intersection of their 2 transverse common Tangents, are called the Centres of Similitude of the 2 Circles.

XVI.

A Circle is said to be inscribed in a rectilinear Figure, when it touches all the sides of that Figure.

XVII.

A Circle is said to be circumscribed about a rectilinear Figure, when it passes through all the vertices of that Figure.

XVIII.

A Circle is said to be escribed without a rectilinear Figure, when it touches one side of that Figure and the produced portions of the two adjacent sides, produced through the extremities of that side.

XIX.

Circles which have the same centre are called concentric.

N.B. In future, when a Chord of a Circle is spoken of as if it were an infinite Line, (e. g. when its Equation is said to be given or found,) that Line is intended, of which the Chord forms a part.

Axioms

I.

If there be given a Circle and a Point, and if through the Point a Line be drawn intersecting the Circle in 2 points: the distances of the 2 points of intersection from the given Point have the same sign, if the given Point lies without the Circle; but opposite signs, if it lies within the Circle.

II.

If there be given a Circle and 2 Points, and if the Line passing through them intersect the Circle in 2 points: the ratios, in which the Circle divides, (whether internally or externally,) the line joining the 2 given Points, have the same sign, if the given Points lie, either both within, or both without, the Circle; but opposite signs, if they lie, one within it, and one without it.

III.

If there be given a Circle and a Line intersecting it in 2 points, and if the Line be moved so as to make the 2 points of intersection approach each other: the nearer they approach, the less of the Circle will there be on one side of the Line, and the more of it on the other side of the Line, and finally, when they coalesce, the Circle will lie wholly on the same side of the Line, i. e. the Line will become a Tangent of the Circle.

Conversely, if there be given a Circle and a Line touching it: the Line may be moved so as to begin to intersect the Circle in 2 points, which, having at first coalesced in the point of contact, will then begin to recede from each other.

IV.

If there be given a Circle and a Point on it, and if through the Point a Line be drawn intersecting the Circle in another point, and if the Line be moved, (either by moving the Point through which it is drawn, or by changing its direction, or both,) so as to make the distance of the second point of intersection from the given Point become less and less: the less it becomes, the nearer will the 2 points of intersection approach each other, and finally, when it becomes $= 0$ , they will coalesce, i. e. the Line will become a Tangent of the Circle.

Conversely, if there be given a Circle and a Point on it, and if through the Point a Line be drawn touching the Circle: the Line may be moved, (either by moving the Point through which it is drawn, or by changing its direction, or both,) so as to begin to intersect the Circle in another point, the distance of which from the given Point, having at first been $= 0$ , will then begin to increase.

V.

If there be given a Circle and a Point without it, and if through the Point a Line be drawn intersecting the Circle in 2 points, and if the Line be moved, (either by moving the Point through which it is drawn, or by changing its direction, or both,) so as to make the distances of the 2 points of intersection from the given Point become more and more equal: the more equal they become the nearer will the 2 points of, intersection approach each other, and finally, when they become equal, (i. e. coincide, since they have the same sign,) these 2 points will coalesce, i. e. the Line will become a Tangent of the Circle.

Conversely, if there be given a Circle and a Point without it, and if through the Point a Line be drawn touching the Circle: the Line may be moved, (either by moving the Point through which it is drawn, or by changing its direction, or both,) so as to begin to intersect the Circle in 2 points, the distances of which from the given Point having at first coincided, will then begin to become more and more unequal.

VI.

If there be given a Circle and 2 Points without it, and if the Line passing through them intersect the Circle in 2 points, and if the Line be moved, (by moving either or both of the Points through which it is drawn, so as to make the ratios, in which the Circle divides, (whether internally or externally,) the line joining the 2 given Points, become more and more equal: the more equal they become, the nearer will the 2 points of intersection approach each other, and finally, when they become equal, (i. e. coincide, since they have the same sign,) these 2 points will coalesce, i. e. the Line will become a Tangent of the Circle.

Conversely, if there be given a Circle and 2 Points without it, and if the Line passing through them touch the Circle: the Line may be moved, (by moving either or both of the Points through which it is drawn,) so as to begin to intersect the Circle in 2 points, i. e. so that the Circle shall begin to divide the line joining the 2 given Points in 2 ratios, which, having at first coincided, will then begin to become more and more unequal.

VII.

If there be given a Circle, a Point, and the abscissa-ratio and ordinate-ratio, (or else the co-ordinate-ratio of a Line passing through the Point, and if the distances of the points, where the Line intersects the Circle, from the given Point, be found: then,

if these distances are
(1) real,
the Line meets the Circle;
if, being real, they are also
(α) equal with opposite signs,
the Line is a Chord, bisected at the given Point;
(β) equal with the same sign, (i. e. coincident,)
the Line is a Tangent;
but if they are
(2) imaginary,
the Line does not meet the Circle.

VIII.

If there be given a Circle and 2 Points, and if the ratios, in which the Circle divides, (whether internally or externally,) the line joining the 2 given Points, be found: then

if these ratios are
(1) real,
the Line meets the Circle;
if, being real, they are also
(α) equal with the same sign, (i. e. coincident,)
the Line is a Tangent;
but if they are
(2) imaginary,
the Line does not meet the Circle.

IX.

When 2 Circles touch each other, the distance between their centres is equal to the sum of their radii, if each Circle lies without the other; and to the difference of their radii, if one Circle lies within the other.

Propositions

Division I. Equations to the Circle

Chapter I. Cartesian System

I.

Given the co-ordinates of the centre, and the length of the radius, of a Circle: to find an Equation to it in terms of those magnitudes.

II.

The Locus of any Equation of the form $A x^{2} + A y^{2} + D x + E y + F = 0, (Rectangular,)$ $or, A x^{2} + 2 A . cos ω . x y + A y^{2} + D x + E y + F = 0, (Oblique,)$ is, (with one exception, mentioned in the Corollary,) a Circle. And by one such Equation one Circle, and one only, is represented.

Corollary. In the above Equation

if $F = 0$ , the Circle passes through the origin,
if $D = E = 0$ , the centre is at the origin,
if $D = E = F = 0$ , the radius $= 0$ , and the Circle coincides with the origin,
if $A = D = E = F = 0$ , the Locus is the Plane of reference,
if A, D, and E, are infinitely small, but F finite, the Locus is a Circle, situated altogether at an infinite distance from the origin,
if $A = D = E = 0$ , but F is finite, the equation is self-contradictory.

III.

Given an Equation to a Circle: to draw the Circle.

Corollary. Hence, if one such Equation be given or found, the Circle may be said to be given or found.

IV.

Given an Equation to a Circle, and the co-ordinates of a Point: to find what relation must exist among the given constants for the Point to lie on the Circle.

V.

Given two Equations to Circles: to find what relation must exist among the given constants for the two Equations to represent the same Circle.

The most useful forms of the Equation to a Circle, in the Cartesian System, are

1. General. $A x^{2} + A y^{2} + D x + E y + F = 0$ , (Rect.)
or, $A x^{2} + 2 A . cos ω . x y + A y^{2} + D x + E y + F = 0$ . (Obl.)

2. Reduced. $x^{2} + y^{2} + D x + E y + F = 0$ , (Rect.)
or, $x^{2} + 2 cos ω . x y + y^{2} + D x + E y + F = 0$ . (Obl.)

3. Positional. ${(x - a)}^{2} + {(y - b)}^{2} = r^{2},$ (Rect.)
or, ${(x - a)}^{2} + {(y - b)}^{2} + 2 (x - a) (y - b) . cos ω = r^{2}$ . (Obl.)
(α) Central. $x^{2} + y^{2} = r^{2}$ , (Rect.)
or, $x^{2} + 2 cos ω . x y + y^{2} = r^{2}$ . (Obl.)
(β) Vertical. $y^{2} = 2 r x - x^{2}$ . (Rect.)

N.B. Equation (α) is the particular form assumed by Equation 3 when the centre is at the origin; and Equation (β) is the form assumed by it when the centre is on the X-axis, and the Circle passes through the origin.

Chapter II. Polar System

I.

The Locus of any Equation of the form $A ρ^{2} + (D cos θ + E sin θ) . ρ + F = 0,$ is, (with one exception, mentioned in the Corollary,) a Circle. And by one such Equation one Circle, and one only, is represented.

Corollary. In the above equation

if $F = 0$ , the Circle passes through the pole,
if $D = E = 0$ , the centre is at the pole,
if $D = E = F = 0$ , the radius $= 0$ , and the Circle coincides with the pole,
if $A = D = E = F = 0$ , the Locus is the Plane of reference,
if A, D, and E, are infinitely small, but F finite, the Locus is a Circle, situated altogether at an infinite distance from the pole,
if $A = D = E = 0$ , but F is finite, the Equation is self-contradictory.

II.

Given the co-ordinates of the centre, and the length of the radius, of a Circle: to find an Equation to it in terms of those magnitudes.

III.

Given an Equation to a Circle: to draw the Circle.

Corollary. Hence, if one such Equation be given or found, the Circle may be said to be given or found.

IV.

Given an Equation to a Circle, and the co-ordinates of a Point: to find what relation must exist among the given constants for the Point to lie on the Circle.

V.

Given two Equations to Circles: to find what relation must exist among the given constants for the two Equations to represent the same Circle.

The most useful forms of the Equation to a Circle, in the Polar System, are

1. General. $A ρ^{2} + (D cos θ + E sin θ) . ρ + F = 0$ .
2. Reduced. $ρ^{2} + (D cos θ + E sin θ) . ρ + F = 0$ .
3. Positional. $ρ^{2} - 2 h ρ . cos (θ - α) + h^{2} = r^{2}$ .
(α) Central. $ρ = r$ .
(β) Vertical. $ρ = 2 r . cos θ$ .

N.B. Equation (α) is the particular form assumed by Equation 3 when the centre is at the pole; and Equation (β) is the form assumed by it when the centre is on the fixed-radius, and the Circle passes through the pole.

Chapter III. Distantial System

The Equations to a Circle in this System are, (generally speaking,) too complicated to be worth being discussed separately. Any Questions involving them may be solved by employing the Cartesian System and the Formulæ of transformation of co-ordinates. But, unless the circumstances of the Question lead to some simplification of the Equation, it is better to employ some other System of reference.

Division II. Problems on the Circle

Chapter I. Cartesian System

N.B. For the meaning of the symbols * and § see p. 51.

(α) Magnitudes required

I. *

Given a Circle: to find its intercepts.

II. §

Given a Circle, a Point, and the abscissa-ratio and ordinate-ratio, (or else the co-ordinate-ratio,) of a Line passing through the Point: to find the distances between the given Point and the points where the Line intersects the Circle.

Corollary. To find what relation must exist among the given constants for these distances to be

(1) real, (i. e. Line meeting Circle,)
and if real, to be also
(α) equal with opposite signs (i. e. Line a bisected Chord,)
(β) coincident (i. e. Line a Tangent,)
(2) imaginary, (i. e. Line not meeting Circle.)

III. §

Given a Circle and a Point: to find the distance between the given Point and the point where a Tangent of the Circle, passing through the given Point, meets the Circle.

Corollary. To find what relation must exist among the given constants for this distance to be

(1) $= 0$ , (i. e. given Point on Circle,)
(2) imaginary, (i. e. given Point within Circle.)

IV. §

Given a Circle and 2 Points: to find the ratios in which the Circle divides, (whether internally or externally,) the line joining the 2 given Points.

Corollary. To find what relation must exist among the given constants for these ratios to be

(1) real (i. e. Line, drawn through given Points, meeting Circle,)
and if real, to be also
(α) coincident, (i. e. Line a Tangent,)
(2) imaginary, (i. e. Line not meeting Circle.)

(β) Points required

I.

Given a Circle and a Line: to find their Points of intersection.

II.

Given 2 Circles: to find their Points of intersection.

(γ) Lines required

(1) Indeterminate

Given a Circle: to find a Tangent of it.

(2) Determinate

I. §

Given a Circle and a Point: to find the Line forming a Chord bisected at the given Point.

Corollary. To find what relation must exist among the given constants for this Line to be

(1) real (i. e. Point within, or on, Circle,)
and if real, to be also
(α) indeterminate, (i. e. Point at centre,)
(2) imaginary, (i. e. Point without Circle.)

II. §

Given a Circle, and a Point on it: to find the Tangent at the Point.

N.B. The position of the Point may be represented, either by its co-ordinates, or by the direction-angle of the Line passing through it and the centre of the Circle. This latter method has the advantage of employing one symbol instead of two.

III. §

Given a Circle, and a Point on it: to find the Normal at the Point.

IV. §

Given a Circle and a Point: to find the Tangents passing through the Point.

Corollary. To find what relation must exist among the given constants for these Tangents to be

(1) real (i. e. Point without, or on, Circle,)
and if real, to be also
(α) coincident (i. e. Point on Circle,)
(2) imaginary, (i. e. Point within Circle.)

V. §

Given a Circle and a Point: to find the Chord of contact of Tangents drawn from the Point.

N.B. Since the Equation to this Line is always real, whatever be the position of the Point, i. e. whether the Tangents passing through it be real or imaginary, and since the name “Chord of contact” is only applicable to it when the Tangents are real, a general name has been given to the Line, represented by the Equation so found, viz. “the Polar of the Point with regard to the Circle.” And the Point is called “the Pole of the Line with regard to the Circle.”

Corollary. Given a Circle and a Line: to find the Pole of the Line with regard to the Circle.

VI. *

Given 2 Circles: to find their Chord of intersection.

N.B. Since the Equation to this Line is always real, whether the Circles meet or not, and since the name “Chord of intersection” is only applicable to it when they do meet, a general name has been given to the Line represented by the Equation so found, viz. “the Radical Axis of the 2 Circles.”

VII.

Given 2 Circles, to find their common Tangents.

Corollary. To find what relation must exist among the given constants for these Tangents to be

(1) all 4 real, (i. e. each Circle lying wholly without the other,)
and if so, for
(α) the 2 transverse to be coincident, (i. e. the Circles touching externally,)
(2) the 2 direct real the 2 transverse imaginary, (i. e. one of the Circles lying, wholly or in part, within the other,)
and if so, for
(α) the 2 direct to be coincident, (i. e. one of the Circles touching the other internally,)
(3) all 4 imaginary, (i. e. one of the Circles lying wholly within, and not touching, the other.)

(δ) Circles required

(1) Indeterminate; one condition

I.

Given a Point: to find a Circle passing through it.

II.

Given a Line: to find a Circle having its centre on it.

III.

Given a Line: to find a Circle touching it.

(2) Indeterminate; two conditions

I.

Given 2 Points: to find a Circle passing through them.

II.

Given 2 Lines: to find a Circle touching them.

(3) Determinate

I.

Given the vertices of a Triangle: to find the circumscribed Circle.

II.

Given 3 Lines forming the sides of a Triangle: to find the inscribed Circle.

III.

Given 3 Lines forming the sides of a Triangle: to find the 3 escribed Circles.

IV.

Given 3 Circles: to find the 8 Circles, each touching all 3 of the given Circles.

(ε) Loci required

I.

Given a Circle, and the abscissa-ratio and ordinate-ratio, (or else the co-ordinate-ratio,) of a system of parallel Lines, and that the variable Point is the centre of any one of the Chords so formed, (so that its Locus is a Diameter of the Circle.)

Corollary. A Diameter of a Circle is a Line passing through the centre, and therefore answers to Euclid’s Definition.

II.

Given a Circle, and the length of a Chord, (of variable position,) drawn within it, and that the variable Point divides the Chord in a given ratio.

III.

Given a Circle, and the length of a Chord, (of variable position,) drawn within it, and that Tangents of the Circle are drawn at the extremities of the Chord, and that the variable Point is their Point of intersection.

IV.

Given a Circle, and the angle between 2 Tangents of variable position, and that the variable Point is their point of intersection.

V.

Given a Circle and a Point, and that a variable Line always passes through the given Point, and that at the points, where the Line intersects the given Circle, Tangents are drawn to the Circle, and that the variable Point is their point of intersection.

VI.

Given 2 Circles and the ratio between the distances from the variable Point to the points where Tangents, drawn from it to the 2 Circles, touch the Circles.

(a) Given ratio one of equality.
(b) inequality.

Chapter II. Polar System

Any formulæ required in this System may easily be deduced from the corresponding formulæ in the Cartesian System, by the transformation of co-ordinates. Some, however, can be more easily found directly, without such transformation: those here given are generally of this kind.

(α) Magnitudes required

Given a Circle: to find its intercepts.

(β) Points required

I.

Given a Circle and a Line: to find their Points of intersection.

II.

Given 2 Circles: to find their Points of intersection.

(γ) Lines required

Given 2 Circles: to find their Radical Axis.

(ε) Loci required

Given a Circle and a Point, and that a variable Line always passes through the given Point and meets the Circle, and that the variable Point lies on this Line, and is such that its distance from the given Point has a given relation to the distances, from the same Point, of the 2 points where the Line intersects the Circle.

(a) Distance the sum, or difference of the other two,
(b) an Arithmetic Mean between the other two,
(c) a Geometric
(d) Harmonic

Chapter III. Distantial System

It has been already stated that the Equations to a Circle in this System are, generally speaking, too complicated to be worth being discussed separately. And even when the circumstances of the case lead to some simplification of the result, it is generally better to arrive at it by means of the Cartesian System and the Formulæ of transformation of co-ordinates: some problems of this kind are here given.

Trilinear System

Circles required

I.

To find the Circle circumscribed about the triangle of reference.

Corollary. To find the 3 Tangents of this Circle at the vertices of reference.

II.

To find the Circle inscribed within the triangle of reference.

III.

To find the 3 Circles escribed without the triangle of reference.

IV.

Given that the triangle of reference is isosceles: to find the Circle which touches the 2 sides, and has the base as its Chord of contact.

Division III. Theorems on the Circle

(A) Determinate Circles

(α) In relation to Magnitudes

I.

If there be a Circle and a Point, and if through the Point any Line be drawn: the product of the distances between the given Point and the points where the Line intersects the Circle is constant, whatever be the position of the Line.

II.

If there be a Circle and a Point, and if through the Point any Line be drawn, and if the Polar of the Point be drawn: the line joining the first Point to the point where the 2 Lines intersect is divided by the Circle externally and internally in the same ratio. i. e. it is divided harmonically.

III.

If there be 2 Circles, and if their Radical Axis be drawn, and if from any Point on the Radical Axis Tangents be drawn to the Circles: the distances between this Point and the points of contact are equal.

(γ) In relation to Lines which are the Loci of Simple Equations

(1) Indeterminate Lines

I.

All Normals of a Circle pass through the centre.

Corollary. Hence, if a Line be drawn through the centre of a Circle, and if, through either of the points where it intersects the Circle, a second Line be drawn perpendicular to the first: this Line is a Tangent of the Circle.

II.

If an Equation to a Line be of the form $(x - A) . cos θ + (y - B) . sin θ = C,$ where A, B, and C, are given constants, but θ indeterminate, the indeterminate Line represented by it touches a determinate Circle.

(2) Determinate Lines

I.

If there be a Circle, and 2 Points such that the Polar of one of them passes through the other: the Polar of the other also passes through the first.

Corollary. If there be a Circle, and 2 Points such that the Polar of each of them passes through the other; and if these Polars intersect in a third Point: the Polar of the third Point passes through the first 2 Points.

II.

If there be 3 Circles, and if the Radical Axis of every 2 of them be drawn: these Radical Axes either are all parallel, or else intersect in the same point.

N.B. This point is called the Radical Centre of the 3 Circles.

III.

If 3 Equations to Circles be such that, when their left-hand sides are combined by some one of the operations of addition, subtraction, multiplication, or division, and their right-hand sides similarly combined, the resulting Equation is an identity: the 3 Circles represented by these Equations are either concentric or such that the same Line is Radical Axis to every 2 of them.

N.B. Such Circles may be called co-radical.

IV.

If there be 3 Circles; and if a Line be drawn through a centre of similitude of the first and second, and also through a centre of similitude of the second and third: this Line will also pass through a centre of similitude of the third and first.

N.B. Such a Line is called an Axis of similitude of the 3 Circles.

(B) Indeterminate Circles

(γ) In relation to Lines

I.

If an Equation to a Circle contain one indeterminate constant only, and one power only of that indeterminate constant: the Circles represented by it are either concentric or co-radical.

Corollary. Hence, if one such Equation be given or found, a system of concentric, or co-radical, Circles may be said to be given or found.

N.B. If the indeterminate constant enter into any term containing x or y, the Circles are co-radical; if otherwise, concentric.

II.

If there be a system of co-radical Circles: their centres lie on the same Line.

III.

If there be a system of co-radical Circles; and if from any Point on their Radical Axis Tangents be drawn to the Circles: the distances between this Point and the several points of contact are equal.

Corollaries. 1. Hence the Locus of these points of contact is a Circle.

2. Hence also, given a system of co-radical Circles, and a Point on their Radical Axis: to find the Circle on which lie the points of contact of all Tangents drawn, from the given Point to the Circles of the system.

IV.

If there be a system of concentric Circles, and a Point; and if the Polar of this Point, with regard to each Circle of the system, be drawn: these Polars are all parallel.

V.

If there be a system of co-radical Circles, and a Point; and if the Polar of this Point, with regard to each Circle of the system, be drawn: these Polars either are all parallel, or else pass through a determinate point.

Corollary. There are 2 Points, of either of which it is true that if its Polar, with regard to each Circle of the system, be drawn, these Polars coincide.

N.B. These Points are called by Poncelet the Limiting Points of the system of Circles.

(δ) In relation to Circles

I.

If there be a system of co-radical Circles; and if from any Point on their Radical Axis Tangents be drawn to the Circles: the Circle which is the locus of the points of contact passes through the Limiting Points of the system.

II.

If there be a system of concentric Circles, and another Circle; and if the Radical Axis belonging to this Circle and each several Circle of the system be drawn: these Radical Axes are all parallel.

III.

If there be a system of co-radical Circles, and another Circle; and if the Radical Axis belonging to this Circle and each several Circle of the system be drawn: these Radical Axes either are all parallel, or else pass through a determinate Point.

Formulæ

The following formulæ are those most frequently used in working Problems and Theorems, and are such as the Student ought to be able either to write down at once from memory, or at all events to work out in a few moments. To enable him to test, at pleasure, the accuracy of his memory, the results are given by themselves below.

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Appendix

Notes explanatory of the foregoing matter, as well as proofs of some of the propositions, are here given. The proofs of those propositions, on which nothing is said here, may generally be found in Mr. Salmon’s Treatise.

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(Or, which is the same thing, as “the cosines of the angles which the line makes with the two axes.”) ↩