John Casey

“ In Modern Geometry a curve is considered as made up of an infinite number of points, which are placed in order along the curve, and then the secant through two consecutive points is a tangent. Euclid’s definition for a tangent is quite inadequate for any curve but the circle, and those derived from it by projection (the conic sections) ; and even for these the modern definition is better.
iv. Circles are said to touch one another when they meet, but do not intersect. There are two species of contact:—
1. When each circle is external to the other.
2. When one is inside the other. ”

“ A geometrical magnitude which has three dimensions, that is, length, breadth, and thickness, is a solid; that which has two dimensions, such as length and breadth, is a surface; and that which has but one dimension is a line. But a point is neither a solid, nor a surface, nor a line ”

“ The volume of the frustum of a cone, made by a plane parallel to the base, is equal to the sum of the three cones whose bases are the two ends of the frustum, and the circle whose diameter is a mean proportional between the end diameters, and whose common altitude is equal to one-third of the altitude of the frustum. ”

“ If from any point in the circumference of the circle, circumscribed about a regular polygon of n sides, lines be drawn to its angular points, the sum of their squares is equal to 2n times the square of the radius. ”

“ They are useful as indicating, by one word, the whole enunciation of a theorem.
Every Proposition in the Fifth Book is a Theorem.
PROP. I.—Theorem.
If any number of magnitudes of the same kind (a, b, c, &c.) , be equimultiples of as many others (a′, b′, c′, &c.) , then the sum of the first magnitudes (a + b + c, &c.) shall be the same multiple of the sum of the second which any magnitude of the first system is of the corresponding magnitude of the second system. ”

“ In any right-angled triangle the square on the sum of the hypotenuse and perpendicular, from the right angle on the hypotenuse, exceeds the square on the sum of the sides by the square on the perpendicular. ”

“ If a triangle of given form move so that its three sides pass through three fixed points, the locus of any point in its plane is a circle.
14. The angle A and the area of a triangle ABC are given in magnitude: if the point A be fixed in position, and the point B move along a fixed line or circle, the locus of the point C is a circle.
15. One of the vertices of a triangle of given form remains fixed; the locus of another is a right line or circle; find the locus of the third.
16. Find the area of a triangle— (1) in terms of its medians; (2) in terms of its perpendiculars. ”

“ Exercises.
1. Prove that the angles at the base are equal without producing the sides. Also by producing the sides through the vertex.
2. Prove that the line joining the point A to the intersection of the lines CF and BG is an axis of symmetry of the figure.
3. If two isosceles triangles be on the same base, and be either at the same or at opposite sides of it, the line joining their vertices is an axis of symmetry of the figure formed by them.
4. Show how to prove this Proposition by assuming as an axiom that every angle has a bisector. ”

“ A Problem is a proposition in which something is proposed to be done, such as a line to be drawn, or a figure to be constructed, under some given conditions.
The Solution of a problem is the method of construction which accomplishes the required end.
The Demonstration is the proof, in the case of a theorem, that the conclusion follows from the hypothesis; and in the case of a problem, that the construction accomplishes the object proposed.
The Enunciation of a problem consists of two parts, namely, the data, or things supposed to be given, and the quaesita, or things required to be done. ”

“ Dem.—If the angle AGH be not equal to GHD, one must be greater than the other. Let AGH be the greater; to each add BGH, and we have the sum of the angles AGH, BGH greater than the sum of the angles BGH, GHD; but the sum of AGH, BGH is two right angles; therefore the sum of BGH, GHD is less than two right angles, and therefore (Axiom xii.) the lines AB, CD, if produced, will meet at some finite distance: but since they are parallel (hyp.) they cannot meet at any finite distance. Hence the angle AGH is not unequal to GHD—that is, it is equal to it. ”

“ If one angle of a triangle be equal to the sum of the other two, the triangle can be divided into the sum of two isosceles triangles, and the base is equal to twice the line from its middle point to the opposite angle. ”

“ A ratio of greater inequality is increased by diminishing its terms by the same quantity, and diminished by increasing its terms by the same quantity.
2. A ratio of lesser inequality is diminished by diminishing its terms by the same quantity, and increased by increasing its terms by the same quantity.
3. If four magnitudes be proportionals, the sum of the first and second is to their difference as the sum of the third and fourth is to their difference (componendo et dividendo) . ”

“ If the plane faces of a polyhedron be equal and similar rectilineal figures, it is called a regular polyhedron.
iii. A pyramid is a polyhedron of which all the faces but one meet in a point. This point is called the vertex; and the opposite face, the base.
iv. A prism is a polyhedron having a pair of parallel faces which are equal and similar rectilineal figures, and are called its ends. The others, called its side faces, are parallelograms.
v. A prism whose ends are perpendicular to its sides is called a right prism; any other is called an oblique prism. ”

“ A cone is the solid figure described by the revolution of a right-angled triangle about one of the legs, which remains fixed, and which is called the axis. The other leg describes the base, which is a circle.
xi. A sphere is the solid described by the revolution of a semicircle about a diameter, which remains fixed. The centre of the sphere is the centre of the generating semicircle. Any line passing through the centre of a sphere and terminated both ways by the surface is called a diameter. ”

“ The difference between the square on one of the equal sides of an isosceles triangle, and the square on any line drawn from the vertex to a point in the base, is equal to the rectangle contained by the segments of the base. ”

“ A Lemma is an auxiliary proposition required in the demonstration of a principal proposition.
A Secant or Transversal is a line which cuts a system of lines, a circle, or any other geometrical figure.
Congruent figures are those that can be made to coincide by superposition. They agree in shape and size, but differ in position. ”

“ A right line drawn from the vertex and turning about it in the plane of the angle, from the position of coincidence with one leg to that of coincidence with the other, is said to turn through the angle, and the angle is the greater as the quantity of turning is the greater. Again, since the line may turn from one position to the other in either of two ways, two angles are formed by two lines drawn from a point. ”

“ We have certainly seen nowhere so good an introduction to Modern Geometry, or so copious a collection of those elementary propositions not given by Euclid, but which are absolutely indispensable for every student who intends to proceed to the study of the Higher Mathematics. ”

“ What is the difference between a chord and a secant?
4. When does a secant become a tangent?
5. What is the difference between a segment of a circle and a sector?
6. What is meant by an angle in a segment?
7. If an arc of a circle be one-sixth of the whole circumference, what is the magnitude of the angle in it? ”

“ Professor Casey’s aim has been to collect within reasonable compass all those propositions of Modern Geometry to which reference is often made, but which are as yet embodied nowhere....We can unreservedly give the highest praise to the matter of the book. In most cases the proofs are extraordinarily neat....The notes to the Sixth Book are the most satisfactory. Feuerbach’s Theorem (the nine-points circle touches inscribed and escribed circles) is favoured with two or three proofs, all of which are elegant. ”