Book VIII treats of the sphere. Just as the circle may be defined either

as a plane surface or as the bounding line which is the locus of a point in a plane at a given distance from a fixed point, so a sphere may be defined either as a solid or as the bounding surface which is the locus of a point in space at a given distance from a fixed point. In higher mathematics the circle is defined as the bounding line and the sphere as the bounding surface; that is, each is defined as a locus. This view of the circle as a line is becoming quite general in elementary geometry, it being the desire that students may not have to change definitions in passing from elementary to higher mathematics. The sphere is less frequently looked upon in geometry as a surface, and in popular usage it is always taken as a solid.

Analogous to the postulate that a circle may be described with any given point as a center and any given line as a radius, is the postulate for constructing a sphere with any given center and any given radius. This postulate is not so essential, however, as the one about the circle, because we are not so concerned with constructions here as we are in plane geometry.

A good opportunity is offered for illustrating several of the definitions connected with the study of the sphere, such as great circle, axis, small circle, and pole, by referring to geography. Indeed, the first three propositions usually given in Book VIII have a direct bearing upon the study of the earth.

THEOREM. _A plane perpendicular to a radius at its extremity is tangent to the sphere._

The student should always have his attention called to the analogue in plane geometry, where there is one. If here we pass a plane through the radius in question, the figure formed on the plane will be that of a line tangent to a circle. If we revolve this about the line of the radius in question, as an axis, the circle will generate the sphere again, and the tangent line will generate the tangent plane.

THEOREM. _A sphere may be inscribed in any given tetrahedron._

Here again we may form a corresponding proposition of plane geometry by passing a plane through any three points of contact of the sphere and the tetrahedron. We shall then form the figure of a circle inscribed in a triangle. And just as in the case of the triangle we may have escribed circles by producing the sides, so in the case of the tetrahedron we may have escribed spheres by producing the planes indefinitely and proceeding in the same way as for the inscribed sphere. The figure is difficult to draw, but it is not difficult to understand, particularly if we construct the tetrahedron out of pasteboard.

THEOREM. _A sphere may be circumscribed about any given tetrahedron._

By producing one of the faces indefinitely it will cut the sphere in a circle, and the resulting figure, on the plane, will be that of the analogous proposition of plane geometry, the circle circumscribed about a triangle. It is easily proved from the proposition that the four perpendiculars erected at the centers of the faces of a tetrahedron meet in a point (are concurrent), the analogue of the proposition about the perpendicular bisectors of the sides of a triangle.

THEOREM. _The intersection of two spherical surfaces is a circle whose plane is perpendicular to the line joining the centers of the surfaces and whose center is in that line._

The figure suggests the case of two circles in plane geometry. In the case of two circles that do not intersect or touch, one not being within the other, there are four common tangents. If the circles touch, two close up into one. If one circle is wholly within the other, this last tangent disappears. The same thing exists in relation to two spheres, and the analogous cases are formed by revolving the circles and tangents about the line through their centers.

In plane geometry it is easily proved that if two circles intersect, the tangents from any point on their common chord produced are equal. For if the common chord is _AB_ and the point _P_ is taken on _AB_ produced, then the square on any tangent from _P_ is equal to _PB_ × _PA_. The line _PBA_ is sometimes called the _radical axis_.

Similarly in this proposition concerning spheres, if from any point in the plane of the circle formed by the intersection of the two spherical surfaces lines are drawn tangent to either sphere, these tangents are equal. For it is easily proved that all tangents to the same sphere from an external point are equal, and it can be proved as in plane geometry that two tangents to the two spheres are equal.

Among the interesting analogies between plane and solid geometry is the one relating to the four common tangents to two circles. If the figure be revolved about the line of centers, the circles generate spheres and the tangents generate conical surfaces. To study this case for various sizes and positions of the two spheres is one of the most interesting generalizations of solid geometry.

An application of the proposition is seen in the case of an eclipse, where the sphere _O'_ represents the moon, _O_ the earth, and _S_ the sun. It is also seen in the case of the full moon, when _S_ is on the other side of the earth. In this case the part _MIN_ is fully illuminated by the moon, but the zone _ABNM_ is only partly illuminated, as the figure shows.[92]

THEOREM. _The sum of the sides of a spherical polygon is less than 360°._

In all such cases the relation to the polyhedral angle should be made clear. This is done in the proofs usually given in the textbooks. It is easily seen that this is true only with the limitation set forth in most textbooks, that the spherical polygons considered are convex. Thus we might have a spherical triangle that is concave, with its base 359°, and its other two sides each 90°, the sum of the sides being 539°.

THEOREM. _The sum of the angles of a spherical triangle is greater than 180° and less than 540°._

It is for the purpose of proving this important fact that polar triangles are introduced. This proposition shows the relation of the spherical to the plane triangle. If our planes were in reality slightly curved, being small portions of enormous spherical surfaces, then the sum of the angles of a triangle would not be exactly 180°, but would exceed 180° by some amount depending on the curvature of the surface. Just as a being may be imagined as having only two dimensions, and living always on a plane surface (in a space of two dimensions), and having no conception of a space of three dimensions, so we may think of ourselves as living in a space of three dimensions but surrounded by a space of four dimensions. The flat being could not point to a third dimension because he could not get out of his plane, and we cannot point to the fourth dimension because we cannot get out of our space. Now what the flat being thinks is his plane may be the surface of an enormous sphere in our three dimensions; in other words, the space he lives in may curve through some higher space without his being conscious of it. So our space may also curve through some higher space without our being conscious of it. If our planes have really some curvature, then the sum of the angles of our triangles has a slight excess over 180°. All this is mere speculation, but it may interest some student to know that the idea of fourth and higher dimensions enters largely into mathematical investigation to-day.

THEOREM. _Two symmetric spherical triangles are equivalent._

While it is not a subject that has any place in a school, save perhaps for incidental conversation with some group of enthusiastic students, it may interest the teacher to consider this proposition in connection with the fourth dimension just mentioned. Consider these triangles, where [L]_A_ = [L]_A'_, _AB_ = _A'B'_, _AC_ = _A'C'_. We prove them congruent by superposition, turning one over and placing it upon the other. But suppose we were beings in Flatland, beings with only two dimensions and without the power to point in any direction except in the plane we lived in. We should then be unable to turn [triangle]_A'B'C'_ over so that it could coincide with [triangle]_ABC_, and we should have to prove these triangles equivalent in some other way, probably by dividing them into isosceles triangles that could be superposed.

Now it is the same thing with symmetric spherical triangles; we cannot superpose them. But might it not be possible to do so if we could turn them through the fourth dimension exactly as we turn the Flatlander's triangle through our third dimension? It is interesting to think about this possibility even though we carry it no further, and in these side lights on mathematics lies much of the fascination of the subject.

THEOREM. _The shortest line that can be drawn on the surface of a sphere between two points is the minor arc of a great circle joining the two points._

It is always interesting to a class to apply this practically. By taking a terrestrial globe and drawing a great circle between the southern point of Ireland and New York City, we represent the shortest route for ships crossing to England. Now if we notice where this great-circle arc cuts the various meridians and mark this on an ordinary Mercator's projection map, such as is found in any schoolroom, we shall find that the path of the ship does not make a straight line. Passengers at sea often do not understand why the ship's course on the map is not a straight line; but the chief reason is that the ship is taking a great-circle arc, and this is not, in general, a straight line on a Mercator projection. The small circles of latitude are straight lines, and so are the meridians and the equator, but other great circles are represented by curved lines.

THEOREM. _The area of the surface of a sphere is equal to the product of its diameter by the circumference of a great circle._

This leads to the remarkable formula, _a_ = 4[pi]_r_^2. That the area of the sphere, a curved surface, should exactly equal the sum of the areas of four great circles, plane surfaces, is the remarkable feature. This was one of the greatest discoveries of Archimedes (_ca._ 287-212 B.C.), who gives it as the thirty-fifth proposition of his treatise on the "Sphere and the Cylinder," and who mentions it specially in a letter to his friend Dositheus, a mathematician of some prominence. Archimedes also states that the surface of a sphere is two thirds that of the circumscribed cylinder, or the same as the curved surface of this cylinder. This is evident, since the cylindric surface of the cylinder is 2[pi]_r_ × 2_r_, or 4[pi]_r_^2, and the two bases have an area [pi]_r_^2 + [pi]_r_^2, making the total area 6[pi]_r_^2.

THEOREM. _The area of a spherical triangle is equal to the area of a lune whose angle is half the triangle's spherical excess._

This theorem, so important in finding areas on the earth's surface, should be followed by a considerable amount of computation of triangular areas, else it will be rather meaningless. Students tend to memorize a proof of this character, and in order to have the proposition mean what it should to them, they should at once apply it. The same is true of the following proposition on the area of a spherical polygon. It is probable that neither of these propositions is very old; at any rate, they do not seem to have been known to the writers on elementary mathematics among the Greeks.

THEOREM. _The volume of a sphere is equal to the product of the area of its surface by one third of its radius._

This gives the formula _v_ = (4/3)[pi]_r_^3. This is one of the greatest discoveries of Archimedes. He also found as a result that the volume of a sphere is two thirds the volume of the circumscribed cylinder. This is easily seen, since the volume of the cylinder is [pi]_r_^2 × 2_r_, or 2[pi]_r_^3, and (4/3)[pi]_r_^3 is 2/3 of 2[pi]_r_^3. It was because of these discoveries on the sphere and cylinder that Archimedes wished these figures engraved upon his tomb, as has already been stated. The Roman general Marcellus conquered Syracuse in 212 B.C., and at the sack of the city Archimedes was killed by an ignorant soldier. Marcellus carried out the wishes of Archimedes with respect to the figures on his tomb.

The volume of a sphere can also be very elegantly found by means of a proposition known as Cavalieri's Theorem. This asserts that if two solids lie between parallel planes, and are such that the two sections made by any plane parallel to the given planes are equal in area, the solids are themselves equal in volume. Thus, if these solids have the same altitude, _a_, and if _S_ and _S'_ are equal sections made by a plane parallel to _MN_, then the solids have the same volume. The proof is simple, since prisms of the same altitude, say _a_/_n_, and on the bases _S_ and _S'_ are equivalent, and the sums of _n_ such prisms are the given solids; and as _n_ increases, the sums of the prisms approach the solids as their limits; hence the volumes are equal.

This proposition, which will now be applied to finding the volume of the sphere, was discovered by Bonaventura Cavalieri (1591 or 1598-1647). He was a Jesuit professor in the University of Bologna, and his best known work is his "Geometria Indivisilibus," which he wrote in 1626, at least in part, and published in 1635 (second edition, 1647). By means of the proposition it is also possible to prove several other theorems, as that the volumes of triangular pyramids of equivalent bases and equal altitudes are equal.

To find the volume of a sphere, take the quadrant _OPQ_, in the square _OPRQ_. Then if this figure is revolved about _OP_, _OPQ_ will generate a hemisphere, _OPR_ will generate a cone of volume (1/3)[pi]_r_^3, and _OPRQ_ will generate a cylinder of volume [pi]_r_^3. Hence the figure generated by _ORQ_ will have a volume [pi]_r_^3 - (1/3)[pi]_r_^3, or (2/3)[pi]_r_^3, which we will call _x_.

Now _OA_ = _AB_, and _OC_ = _AD_; also (_OC_)^2 - (_OA_)^2 = (_AC_)^2, so that (_AD_)^2 - (_AB_)^2 = (_AC_)^2, and [pi](_AD_)^2 - [pi](_AB_)^2 = [pi](_AC_)^2.

But [pi](_AD_)^2 - [pi](_AB_)^2 is the area of the ring generated by _BD_, a section of _x_, and [pi](_AC_)^2 is the corresponding section of the hemisphere. Hence, by Cavalieri's Theorem,

(2/3)[pi]_r_^3 = the volume of the hemisphere. [therefore] (4/3)[pi]_r_^3 = the volume of the sphere.

In connection with the sphere some easy work in quadratics may be introduced even if the class has had only a year in algebra.

For example, suppose a cube is inscribed in a hemisphere of radius _r_ and we wish to find its edge, and thereby its surface and its volume.

If _x_ = the edge of the cube, the diagonal of the base must be _x_[sqrt]2, and the projection of _r_ (drawn from the center of the base to one of the vertices) on the base is half of this diagonal, or (_x_[sqrt]2)/2.

Hence, by the Pythagorean Theorem,

_r_^2 = _x_^2 + ((_x_[sqrt]2)/2)^2 = (3/2)_x_^2

[therefore] _x_ = _r_[sqrt](2/3),

and the total surface is 6_x_^2 = 4_r_^2,

and the volume is _x_^3 = (2/3)_r_^3[sqrt](2/3).

FOOTNOTES:

[92] The illustration is from Dupin, loc. cit.

L'ENVOI

In the Valley of Youth, through which all wayfarers must pass on their journey from the Land of Mystery to the Land of the Infinite, there is a village where the pilgrim rests and indulges in various excursions for which the valley is celebrated. There also gather many guides in this spot, some of whom show the stranger all the various points of common interest, and others of whom take visitors to special points from which the views are of peculiar significance. As time has gone on new paths have opened, and new resting places have been made from which these views are best obtained. Some of the mountain peaks have been neglected in the past, but of late they too have been scaled, and paths have been hewn out that approach the summits, and many pilgrims ascend them and find that the result is abundantly worth the effort and the time.

The effect of these several improvements has been a natural and usually friendly rivalry in the body of guides that show the way. The mountains have not changed, and the views are what they have always been. But there are not wanting those who say, "My mountain may not be as lofty as yours, but it is easier to ascend"; or "There are quarries on my peak, and points of view from which a building may be seen in process of erection, or a mill in operation, or a canal, while your mountain shows only a stretch of hills and valleys, and thus you will see that mine is the more profitable to visit." Then there are guides who are themselves often weak of limb, and who are attached to numerous sand dunes, and these say to the weaker pilgrims, "Why tire yourselves climbing a rocky mountain when here are peaks whose summits you can reach with ease and from which the view is just as good as that from the most famous precipice?" The result is not wholly disadvantageous, for many who pass through the valley are able to approach the summits of the sand dunes only, and would make progress with greatest difficulty should they attempt to scale a real mountain, although even for them it would be better to climb a little way where it is really worth the effort instead of spending all their efforts on the dunes.

Then, too, there have of late come guides who have shown much ingenuity by digging tunnels into some of the greatest mountains. These they have paved with smooth concrete, and have arranged for rubber-tired cars that run without jar to the heart of some mountain. Arrived there the pilgrim has a glance, as the car swiftly turns in a blaze of electric light, at a roughly painted panorama of the view from the summit, and he is assured by the guide that he has accomplished all that he would have done, had he laboriously climbed the peak itself.

In the midst of all the advocacy of sand-dune climbing, and of rubber-tired cars to see a painted view, the great body of guides still climb their mountains with their little groups of followers, and the vigor of the ascent and the magnificence of the view still attract all who are strong and earnest, during their sojourn in the Valley of Youth. Among the mountains that have for ages attracted the pilgrims is Mons Latinus, usually called in the valley by the more pleasing name Latina. Mathematica, and Rhetorica, and Grammatica are also among the best known. A group known as Montes Naturales comprises Physica, Biologica, and Chemica, and one great peak with minor peaks about it is called by the people Philosophia. There are those who claim that these great masses of rock are too old to be climbed, as if that affected the view; while others claim that the ascent is too difficult and that all who do not favor the sand dunes are reactionary. But this affects only a few who belong to the real mountains, and the others labor diligently to improve the paths and to lessen unnecessary toil, but they seek not to tear off the summits nor do they attend to the amusing attempts of those who sit by the hillocks and throw pebbles at the rocky sides of the mountains upon which they work.

* * * * *

Geometry is a mountain. Vigor is needed for its ascent. The views all along the paths are magnificent. The effort of climbing is stimulating. A guide who points out the beauties, the grandeur, and the special places of interest commands the admiration of his group of pilgrims. One who fails to do this, who does not know the paths, who puts unnecessary burdens upon the pilgrim, or who blindfolds him in his progress, is unworthy of his position. The pretended guide who says that the painted panorama, seen from the rubber-tired car, is as good as the view from the summit is simply a fakir and is generally recognized as such. The mountain will stand; it will not be used as a mere commercial quarry for building stone; it will not be affected by pellets thrown from the little hillocks about; but its paths will be freed from unnecessary flints, they will be straightened where this can advantageously be done, and new paths on entirely novel plans will be made as time goes on, but these paths will be hewed out of rock, not made out of the dreams of a day. Every worthy guide will assist in all these efforts at betterment, and will urge the pilgrim at least to ascend a little way because of the fact that the same view cannot be obtained from other peaks; but he will not take seriously the efforts of the fakir, nor will he listen with more than passing interest to him who proclaims the sand heap to be a Matterhorn.

INDEX

Ahmes, 27, 254, 278, 306

Alexandroff, 164

Algebra, 37, 84

Al-Khowarazmi, 37

Allman, G. J., 29

Almagest, 35

Al-Nair[=i]z[=i], 171, 193, 214, 264

Al-Qif[t.][=i], 49

Analysis, 41, 161

Angle, 142, 155; trisection of, 31, 215

Anthonisz, Adriaen, 279

Antiphon, 31, 32, 276

Apollodotus (Apollodorus), 259

Apollonius, 34, 214, 231

Applied problems, 75, 103, 178, 186, 192, 195, 203, 204, 209, 215, 217, 242, 267, 295, 317

Appreciation of geometry, 19

Arab geometry, 37, 51

Archimedes, 34, 42, 48, 139, 141, 215, 276, 278, 314, 327, 328

Aristæus, 310

Aristotle, 33, 42, 134, 135, 137, 145, 154, 177, 209

Aryabhatta, 36, 279

Associations, syllabi of, 58, 60, 64

Assumptions, 116

Astrolabe, 172

Athelhard of Bath, 37, 51

Athenæus, 259

Axioms, 31, 41, 116

Babylon, 26, 272

Bartoli, 10, 44, 238

Belli, 10, 44, 172

Beltinus, 239, 241

Beltrami, 127

Bennett, J., 224

Bernoulli, 280

Bertrand, 62

Betz, 131

Bezout, 62

Bhaskara, 232, 268

Billings, R. W., 222

Billingsley, 52

Bion, 192, 239

Boethius, 43, 50

Bolyai, 128

Bonola, 128

Books of geometry, 165, 167, 201, 227, 252, 269, 289, 303, 321

Bordas-Demoulin, 24

Borel, 11, 67, 196

Bosanquet, 272

Bossut, 23

Bourdon, 62

Bourlet, 67, 165, 196

Brahmagupta, 36, 268, 279

Bretschneider, C. A., 30

Brouncker, 280

Bruce, W. N., 199

Bryson, 31, 32, 276

Cajori, 46

Calandri, 30

Campanus, 37, 51, 135

Cantor, M., 29, 46

Capella, 50, 135

Capra, 44

Carson, G. W. L., 18, 96, 114

Casey, J., 38

Cassiodorius, 50

Cataneo, 10, 44

Cavalieri, 136, 329

Chinese values of [pi], 279

Church schools, 43

Cicero, 34, 50, 259, 314

Circle, 145, 201, 270, 287; squaring the, 31, 32, 277

Circumference, 145

Cissoid, 34

Class in geometry, 108

Clavius, 121

Colleges, geometry in the, 46

Collet, 24

Commensurable magnitudes, 206, 207

Conchoid, 34

Condorcet, 23

Cone, 315

Congruent, 151

Conic sections, 33, 315

Continuity, 212

Converse proposition, 175, 190, 191

Crelle, 142

Cube, duplicating the, 32, 307

Cylinder, 313

D'Alembert, 24, 67

Dase, 279

Decagon, 273

Definitions, 41, 132

De Judaeis, 239, 241

De Morgan, A., 58

De Paolis, 67

Descartes, 38, 84, 320

Diameter, 146

Dihedral, 298

Diocles, 34

Diogenes Laertius, 259

Diorismus, 41

Direction, 150

Distance, 154

Doyle, Conan, 8

Drawing, 95, 221, 281

Duality, 173

Duhamel, 164

Dupin, 11, 217

Duplication problem, 32, 307

Dürer, 10

Educational problems, 1

Egypt, 26, 40

Eisenlohr, 27

Engel and Stäckel, 128

England, 14, 46, 58, 60

Epicureans, 188

Equal, 151, 153

Equilateral, 147

Equivalent, 151

Eratosthenes, 48

Euclid, 33, 42, 43, 44, 119, 125, 135, 156, 165, 167 ff., 201 ff., et passim; editions of, 47, 52; efforts at improving, 57; life of, 47; nature of his "Elements," 52, 55; opinions of, 8

Eudemus, 33, 168, 171, 185, 216, 309

Eudoxus, 32, 41, 48, 227, 308, 314, 317

Euler, 38, 280, 318

Eutocius, 184

Exercises, nature of, 74, 103; how to attack, 160

Exhaustions, method of, 31

Extreme and mean ratio, 250

Figures in geometry, 104, 107, 113

Finaeus, 44, 239, 240, 243

Fourier, 142

Fourth dimension, 326

Frankland, 56, 117, 127, 135, 159

Fusion, of algebra and geometry, 84; of geometry and trigonometry, 91

Gargioli, 44

Gauss, 140, 274

Geminus, 126, 128, 149

Geometry, books of, 165, 167, 201, 227, 252, 269, 289, 303, 321; compared with other subjects, 14; introduction to, 93; modern, 38; of motion, 68, 196; reasons for teaching, 7, 15, 20; related to algebra, 84; textbooks in, 70

Gerbert, 43

Gherard of Cremona, 37, 51

Gnomon, 212

Golden section, 250

Gothic windows, 75, 221 ff., 274, 282

Gow, J., 29, 56

Greece, 28, 40

Gregoire de St. Vincent, 267

Gregory, 280

Grévy, 67

Gymnasia, geometry in the, 45

Hadamard, 164

Hamilton, W., 14

Harmonic division, 231

Harpedonaptae, 28

Harriot, 37

Harvard syllabus, 63

Heath, T. L., 49, 56, 119, 126, 127, 135, 149, 159, 170, 175, 228, 261

Hebrews, 26

Henrici, O., 11, 14, 25, 164, 196

Henrici and Treutlein, 68, 164, 196

Hermite, 281

Herodotus, 28

Heron, 35, 137, 139, 141, 209, 259, 267

Hexagon, regular, 272

High schools, geometry in the, 45

Hilbert, 119, 131

Hipparchus, 35

Hippasus, 273, 309

Hippias, 31, 215

Hippocrates, 31, 41, 281

History of geometry, 26

Hobson, 166

Hoffmann, 242

Holzmüller, 320

Hughes, Justice, 9

Hypatia, 36

Hypsicles, 34

Hypsometer, 245

Iamblichus, 273, 309

Illusions, optical, 100

Ingrami, 128

Instruments, 96, 178, 236

Introduction to geometry, 93

Ionic school, 28

Jackson, C. S., 12

Jones, W., 271

Junge, 259

Karagiannides, 128

Karpinski, 37

Kaye, G. B., 232

Kepler, 24, 149

Kingsley, C., 36

Klein, F., 68, 89

Kolb, 222

Lacroix, 24, 46, 62, 66

Langley, E. M., 291

Laplace, 101

Legendre, 10, 45, 62, 127, 128, 152

Leibnitz, 140, 150

Leon, 41

Leonardo da Vinci, 264

Leonardo of Pisa, 37, 43, 279

Lettering figures, 105

Limits, 207

Lindemann, 278, 281

Line defined, 137

Lobachevsky, 128

Loci, 163, 198

Locke, W. J., 13

Lodge, A., 14

Logic, 17, 104

Loomis, 164

Ludolph van Ceulen, 279

Lycées, geometry in, 45

M'Clelland, 38

McCormack, T. J., 11

Measured by, 208

Memorizing, 12, 79

Menæchmus, 33, 316

Menelaus, 35

Méray, 67, 68, 196, 289

Methods, 41, 42, 115

Metius, 279

Mikami, 264

Minchin, 14

Models, 93, 290

Modern geometry, 38

Mohammed ibn Musa, 37

Moore, E. H., 131

Mosaics, 274

Müller, H., 68

Münsterberg, 22

Napoleon, 24, 287

Newton, 24

Nicomedes, 34, 215

Octant, 242

Oenopides, 31, 212, 216

Optical illusions, 100

Oughtred, 37

Paciuolo, 86

Pamphilius, 185

Pappus, 36, 230, 263

Parallelepiped, 303

Parallels, 149, 181

Parquetry, 222, 274

Pascal, 24, 159

Peletier, 169

Perigon, 151

Perry, J., 13, 14

Petersen, 164

Philippus of Mende, 32, 185

Philo, 178

Philolaus, 309

[pi], 26, 27, 34, 36, 271, 278, 280

Plane, 140

Plato, 25, 31, 41, 48, 129, 136, 137, 309, 310

Playfair, 128

Pleasure of geometry, 16

Plimpton, G. A., 51, 52

Plutarch, 259

Poinsot, 311

Point, 135

Polygons, 156, 252, 269, 274

Polyhedrons, 301, 303, 310

Pomodoro, 179

_Pons asinorum_, 174, 265

Posidonius, 128, 149

Postulates, 31, 41, 116, 125, 292

Practical geometry, 3, 7, 9, 44

Printing, effect of, 44

Prism, 303

Problems, applied, 75, 103, 178, 186, 192, 195, 203, 204, 209, 215, 217, 242, 267, 295, 317

Proclus, 36, 47, 48, 52, 71, 127, 128, 136, 137, 139, 140, 149, 155, 186, 188, 197, 212, 214, 253, 258, 310, 311

Projections, 300

Proofs in full, 79

Proportion, 32, 227

Psychology, 12, 20

Ptolemy, C., 35, 278; king, 48, 49

Pyramid, 307

Pythagoras, 29, 40, 258, 272, 273, 310

Pythagorean Theorem, 28, 36, 258

Pythagorean numbers, 32, 36, 263, 266

Pythagoreans, 136, 137, 185, 227, 269, 309

Quadrant, 236

Quadratrix, 31, 215

Quadrilaterals, 148, 157

Quadrivium, 42

Questions at issue, 3

Rabelais, 24

Radius, 153

Ratio, 205, 227

Real problem defined, 75, 103

Reasons for studying geometry, 7, 100

Rebière, 25

Reciprocal propositions, 173

Recitation in geometry, 113

Recorde, 37

Rectilinear figures, 146

Reductio ad absurdum, 41, 177

Regular polygons, 269

"Rhind Papyrus," 27

Rhombus and rhomboid, 148

Riccardi, 47

Richter, 279

Roman surveyors, 247

Saccheri, 127

Sacrobosco, 43

San Giovanni, 192

Sauvage, 164

Sayre, 272

Scalene, 147

Schlegel, 68

Schopenhauer, 121, 265

Schotten, 46, 135, 149

Sector, 154, 156

Segment, 154

Semicircle, 146

Shanks, 279

Similar figures, 232

Simon, 38, 56, 135

Simson, 142

Sisam, 11

Smith, D. E., 25, 37, 51, 52, 119, 131, 135, 159

Solid geometry, 289

Speusippus, 32

Sphere, 321

Square on a line, 257

Squaring the circle, 31, 32, 277

Stäckel, 128

Stamper, 46, 83

Stark, W. E., 172, 238

Stereoscopic slides, 291

Stobæus, 8

Straight angle, 151

Straight line, 138

Suggested proofs, 81

Sulvasutras, 232

Superposition, 169

Surface, 140

Swain, G. F., 13

Syllabi, 58, 60, 63, 64, 66, 67, 80, 82

Sylvester II, 43

Synthetic method, 161

Tangent, 154

Tartaglia, 153

Tatius, Achilles, 272

Teaching geometry, reasons for, 7, 15, 20; development of, 40

Textbooks, 32, 33, 41, 70, 80, 82

Thales, 28, 168, 171, 185, 210

Theætetus, 48, 310

Theon of Alexandria, 36

Thibaut, 185

Thoreau, 246

Trapezium, 148

Treutlein, 68, 164

Triangle, 147

Trigonometry, 234

Trisection problem, 31, 215

Trivium, 42

Universities, geometry in the, 43

Uselessness of mathematics, 13

Veblen, 131, 159

Vega, 279

Veronese, 68

Vieta, 279

Vogt, 259

Wallis, 127, 280

Young, J. W. A., 25, 131, 159, 277

Zamberti, 52

Zenodorus, 34, 253

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Transcribers notes

On page 30: Pythagoras fled to Megapontum has been left as printed, though the author probably meant Metapontum.

On page 269: 100 B.C. has been left as it was printed, though it is probably a typo for 100 A.D.

End of Project Gutenberg's The Teaching of Geometry, by David Eugene Smith