Arrangements suggested by these Reflections.

FIRST DIVISION. Relations of the angles about a point. Dependent on the number of divergent lines.

Subdivisions.

SECOND DIVISION. Relations of any number of points in space. Subdivisions. Dependent on the number of points.

These general principles of arrangement admit of modification whenever principles of a more partial kind tend to further-the sole object of classification-the ready acquirement and use of knowledge.

Such a subordinate principle arises from the facility with which graphic models can be delineated on plane surfaces; and hence the angles formed by divergent lines that lie in one plane will form the subject of a separate section.

CHAPTER I.

DETAILED ANALYSIS OF THE RELATIONS OF direction; AND OF THE RELATIONS PECULIAR TO THREE-TO FOUR-AND TO A GREATER NUMBER OF POINTS.

SECTION I.-Relations of three Divergent Lines that lie in one Plane. -The relations of direction are the same with the relations of angles that are formed about a common point-relations of three directions that lie in one plane-converse relations of the type of closed figures-transformations of the converse relations-tables of the most useful relations of angles about a point and in one plane. Page 289.

SECTION II.-Relations of Three Points, or Plane Trigonometry.—Notation best adapted to the relations of three points-relations expressed in terms of the opposite sides and angles-relations of the three sides-relations of the angles in terms of the sides—any three parts except the three angles will determine the remainder-logarithmic expressions-two sides and included angle-three sides-ambiguity of some of the formula-properties common to all plane triangles-case which admits two solutions-properties of particular triangles-relations that the parts of triangles have with lines drawn in and about the latter. Page 303.

APPENDIX TO SECTION II.-Examples. Page 316.

SECTION III.-Relations of Three Divergent Lines.-Equations of condition fulfilled by three lines that lie in one plane-relations of the angles formed by three divergent lines—inclinations of the planes included in the relations of three divergent lines-when one of these inclinations is a right angle-without the restriction of the last article-notation proper to three divergent lines-opposite solid angles-use of opposite solid angles in

analysis-table of the formulæ used when one solid angle is 90°-apply to the case wherein a plane angle is right-Napier's rules-three parts, in the most general case, determine the remainder-enumeration of data-two plane angles and an opposite solid angle-or the converse-two plane angles and the included solid angle-two solid angles and the interjacent plane angleanother solution of the two preceding cases-three sides-three angles— properties common to every case of three divergent lines-particular relations of three divergent lines-ambiguous cases-appendix. Page 324.

APPENDIX TO SECTION III.-Examples.

Page 355.

SECTION IV.-Relations of two Points restricted to a Given Distance and a Given Plane.-Equation of the circle-a straight line cannot intersect a circle in more than two points-line which is a tangent to a circle—the product of conjugate secants is independent of their direction—the arc of a circle intercepted by two straight lines that diverge from the centre measures their inclination-the sine, cosine, &c. of an angle may be expressed in relations of the arc that measures it-circumference compared with the radiusthe inclination of two secants is measured by the difference of the arcs intercepted between them-when the secants intersect in the circumference, their inclination is measured by half the arc intercepted-to find the radius of a circle that shall pass through three given points-to find the radius of a circle that shall be inscribed in a given triangle-differentials of the trigonometrical functions-imaginary formulæ connecting the arc with the trigonometrical functions of it-formulæ of Demoivre-formulæ of Euler-expansions of cos. x and sin. x. Page 367.

SECTION V.-Relations of Two Points restricted to a Given Distance. -Equation of the sphere-the section of a sphere by a plane is a circle-the tangent plane to any point of a spherical surface is at right angles to the radius which passes through that point-if through any point secants are drawn to the sphere, their properties will be the same as those of the secants of the circle-solid angles at the centre of the sphere are measured by the portion of the spherical surface intercepted by them-the solid angle formed by two secant planes is measured by the difference of the spherical surfaces which they intercept-the shortest distance between two points on a sphere is the arc of a great circle intercepted between them-the extremities of the perpendicular drawn from the centre of a sphere to a circle of the latter are every where equally distant from the circle-the angles and sides of a spherical polygon have the same relations as the parts of the solid angle which the polygon subtends at the centre-relations common to all spherical trianglesrelations of particular spherical triangles-formulæ for determining the parts of a spherical triangle when one of those parts is 90-polar spherical triangle -formulæ for determining the parts of any spherical triangle-measure of the surface of a spherical triangle-spherical trigonometry includes plane trigonometry as a particular case-equality by symmetry. Page 385.

SECTION VI.-Systems of Primordial Elements that have reference to the Sphere; additional Theorems for transforming co-ordinates.—The position of a point on the surface of a sphere is referred to the centre, to a great circle, and to a point arbitrarily chosen in the latter-of primary and secondary circles-distance of two points in terms of their spherical coordinates-relative directions of points expressed in terms of their spherical co-ordinates-transformation of spherical co-ordinates-transformation of polar systems-equations of transformation used by Euler. Page 400.

SECTION VII.-Relations of any number of Divergent Lines.—Relations of divergent lines agree with those of spherical polygons--equations relative to the inclinations of divergent lines obtained from the sides of closed figures, by equating with zero the common denominator found in the values of the latter-in a plane, lines equally inclined to other lines form the same angle as the latter-applies also to planes which have a common intersection— perpendiculars to the sides of a closed figure, or to planes that include a solid angle, have for their inclinations the supplements of the internal angles of the latter-sum of the interior angles of a polygon--other relations of divergent lines-data required to determine the relations of divergent lines. Page 409.

SECTION VIII.-Relations of any number of Points.-Number of distances, of plane angles, of planes, and of solid angles involved in the relations of n points-conditions exist involving merely the directions of the points, and a similar remark applies to their distances-use in the analysis of closed figures of the equations deduced for divergent lines-number and nature of the data that assign the relations of n points-examples in the relations of four points. Page 422.

PART IV.

INDETERMINATE ANALYSIS.

PRELIMINARY REFLECTIONS.

The principle that led to the ideas of the circle and the sphere admits of a more extensive application. The hypothesis may be generalized. We may assume more than one point as assigned in space, and innumerable points as connected with them by given relations. Lastly, we may investigate the nature of a surface wherein all the latter points are found.

Proceeding in this way we shall be conducted to a peculiar science, teaching to arrange lines and surfaces, not by their apparent forms, but the connection which points they contain have with other points that are given.

Inquiries suggested by these Reflections.

Given any number of primordial elements, to find the points that have assigned relations with them.

CHAPTER I.

OF LINES AND SURFACES.

SECTION I. Of the Straight Line.-A straight line may be regarded as formed by an infinite number of points that have the same direction-equation of the straight line-equation of a straight line restricted to lie in a given plane-equations of lines that are parallel-when restricted to lie in a given plane-equations of lines that are perpendicular-when restricted to lie in a given plane-equation of a line that passes through a given point-equation of a line that passes through two given points-distance between a given point and line-when restricted to lie in a given plane. Page 437.

SECTION II. Of the Plane.-Equation of the plane-equations of parallel planes equation of a perpendicular to a plane-traces of a plane-equations of a straight line that lies in a given plane-equations of a line that is parallel to a given plane-intersection of two planes-projections of lines-the traces of a plane are at right angles to the projections of its perpendicular. Page 450.

PRELIMINARY REFLECTIONS TO SECTIONS III. AND IV.

In investigating the relations of a definite number of points, the number is, itself, a character, in terms of which the analysis can be arranged. But as this principle of classification manifestly fails when the number of points is infinite, we have yet to supply that deficiency.

Now the analysis of an infinite number of points proceeding by the equations they give rise to, we may adopt a principle of classification extensively used in algebra, and arrange the several steps of the process by the degrees of the resulting equations.

Inquiry suggested by these Reflections.

To discover the curve, or surface, formed by all those points the relations of which to known elements shall be expressed in an equation of the second degree.

SECTION III.—Of Plane Lines of the Second Order.-Points which lie in a plane are assigned by reference to two primordial elements-of the parabola-of the ellipse-of the hyperbola-connection of the equations discussed in this section-the curves discussed in this section referred to oblique coordinates-conjugate diameters-equation of the hyperbola referred to its asymptotes-polar equations of the curves discussed in this section-every equation of the second degree between two variables will be fulfilled by the co-ordinates of one or other of the curves discussed in this section. Page 461.

SECTION IV.-Of Surfaces of the Second Order.-Surfaces arranged by the degrees of their equations-equation of the cylinder-sections of the cylinder-equation of the cone-sections of the cone-surfaces of the second degree-ellipsoid-hyperboloid of one sheet-hyperboloid of two sheetsparaboloid. Page 489.

PRELIMINARY REFLECTIONS TO SECTIONS V., VI., VII. AND VIII.

The principle of arrangement that assigns the place of a curve by the degree of its equation, applies only to the curves the equations of which are algebraic. But the idea of a curve, or of a mathematical line of any kind, is derived, as explained in Part I., from the intersection of two surfaces; and thus every principle of arrangement that applies to the latter species of quantity, applies also to curve lines.

Inquiry suggested by these Reflections.

Of lines considered as the intersections of surfaces. Most commodious method of arranging curve surfaces.

SECTION V.-Of Lines considered as the Intersections of Surfaces.Remarks on the arrangement of lines and surfaces-lines regarded as the intersections of surfaces-of the helix-of the spiral described by the sunof curves arranged as the loci of points subjected to known motions. Page 511.

SECTION VI.—Method of arranging Lines and Surfaces by Parameters. -A parameter, in the arrangement of mathematical quantities, is a variable by means of which we pass from a subdivision to a division-parameters may be measured as co-ordinates-of simple and complex systems of lines-transformation of parameters-dependent parameters are co-ordinates of points wherein the lines or surfaces of the system intersect a line or surface that does not belong to it-two equations between three co-ordinates and a parameter, indicate a line lying in a given surface-two equations between three co-ordinates and two parameters, indicate a system of curves that do not lie in a surface. Page 527.