Geometer

“ Another frame which we impose on the world is space. Whence come the first principles of geometry? Are they imposed on us by logic? Lobachevski has proved not, by creating non-Euclidean geometry. Is space revealed to us by our senses? Still no, for the space our senses could show us differs absolutely from that of the geometer. Is experience the source of geometry? A deeper discussion will show us it is not. We therefore conclude that the first principles of geometry are only conventions; but these conventions are not arbitrary and if transported into another world ”

“ To these two-dimensioned geometers geodesies would possess many of the attributes of straight lines in ordinary space. If the surface to which the beings were confined were actually a plane, then the geometry would be the same as our geometry. They would find that only one straight line could be drawn between two points, that through a point only one parallel to a given line could be drawn, and that the ends of a line would never meet even though the line be prolonged to infinity.
We might also suppose that intelligent beings could exist on the surface of a sphere. ”

“ One inch, for example, in pure geometry, is always one cubic inch, but when used to measure a line, or extension in one direction, we take only one dimension of the unit, namely, the linear edge of the cube, and thus the operation not demanding either the breadth or the thickness of the unit, geometers have fallen into the error of supposing that a line is length without any breadth. These are the leading principles on which Mr. Smith attempts the audacious task of rearing a new fabric of geometrical science, without regard to the wisdom of antiquity or the universal traditions of the schools. ”

“ Geometers have long been accustomed to introduce in analytical investigations, in the place of the magnitudes themselves which they wished to study, their different powers, or their logarithms, or their sines, &c., in order to simplify the equations, and even to obtain them more easily. ”

“ Little advanced as geometers really are as yet in the science of integrations, they have nevertheless obtained, from their scanty abstract conceptions, the solution of a multitude of questions of the first importance in geometry, in mechanics, in thermology, &c. ”

“ So it is as rare for geometers to be men of subtle wit as it is for the latter to be geometers, because geometers like to treat these nicer matters geometrically, and so make themselves ridiculous; they like to commence with definition, and then go on to principles—a mode which does not at all suit this sort of reasoning. ”

“ Having thus recognized, as a general principle, the close and necessary connexion of the study of the properties of lines and surfaces with those researches which constitute the final object of geometry, it is evident that geometers, in the progress of their labours, must by no means constrain themselves to keep such a connexion always in view. ”

“ We should see that the first germ of this idea is found in the procedure constantly employed by the Greek geometers, under the name of the Method of Exhaustions, as a means of passing from the properties of straight lines to those of curves, and consisting essentially in substituting for the curve the auxiliary consideration of an inscribed or circumscribed polygon, by means of which they rose to the curve itself, taking in a suitable manner the limits of the primitive ratios. ”

“ Euclid, Archimedes, and Apollonius brought geometry to as high a state of perfection as it perhaps could be brought without first introducing some more general and more powerful method than the old method of exhaustion. A briefer symbolism, a Cartesian geometry, an infinitesimal calculus, were needed. The Greek mind was not adapted to the invention of general methods. Instead of a climb to still loftier heights we observe, therefore, on the part of later Greek geometers, a descent, during which they paused here and there to look around for details which had been passed by in the hasty ascent. ”

“ In determining the ratio of the areas between two curvilinear plane figures, say two circles, geometers first inscribed or circumscribed similar polygons, and then by increasing indefinitely the number of sides, nearly exhausted the spaces between the polygons and circumferences. From the theorem that similar polygons inscribed in circles are to each other as the squares on their diameters, geometers may have divined the theorem attributed to Hippocrates of Chios that the circles, which differ but little from the last drawn polygons, must be to each other as the squares on their diameters. ”

“ We give it in brief as an exercise for our juvenile readers to fill up. It reminds us of the old days when real geometers used to think it worth while seriously to demolish pretenders. Mr. Smith's fame is now assured: Sir W. R. Hamilton's brief and easy exposure will procure him notice in connection with this celebrated problem.
"It is to be shown that the perimeter of a regular polygon of 20 sides is greater than 31⁄8 diameters of the circle, and still more, of course, is the circumference of the circle greater than 31⁄8 diameters. ”

“ One of the most curious of these cases was that of a student, I am not sure but a graduate, of the University of Virginia, who claimed that geometers were in error in assuming that a line had no thickness. He published a school geometry based on his views, which received the endorsement of a well-known New York school official and, on the basis of this, was actually endorsed, or came very near being endorsed, as a text-book in the public schools of New York. ”

“ I am induced to think that, when geometers shall have exhausted the most important applications of our present transcendental analysis, instead of striving to impress upon it, as now conceived, a chimerical perfection, they will rather create new resources by changing the mode of derivation of the auxiliary quantities introduced in order to facilitate the establishment of equations, and the formation of which might follow an infinity of other laws besides the very simple relation which has been chosen, according to the conception suggested in the first chapter. ”

“ Thus if we draw a circle, placing a massive particle near the centre so as to produce a gravitational field, and measure with a rigid scale the circumference and the diameter, the ratio of the measured circumference to the measured diameter will not be the famous number but a little smaller. Or if we inscribe a regular hexagon in this circle its sides will not be exactly equal to the radius of the circle. Placing the particle near, instead of at, the centre, avoids measuring the diameter through the particle, and so makes the experiment a practical one. ”

“ Through knowledge of the times one is able to determine other unknowns, as was seen in section 200. One who knows the measure of arc BC or angle BEG can determine, through use of the tables, how many units are needed to function as a secant for that angle, and after the measure of the maximum sine is determined, which is line BE, or half the diameter of the Earth, it can now be seen that the height of the mountain is the difference between the maximum sine and the secant line. ”

“ They take the place of straight lines in the intrinsic geometry of a surface. When we inquire whether the angles of a triangle add up to two right angles or not, we mean to speak of a triangle whose sides are geodesics. And when we speak of the distance between two points, we mean the distance along a geodesic.
The next step in our generalizing process is rather difficult: it is the transition to non-Euclidean geometry. We live in a world in which space has three dimensions, and our empirical knowledge of space is based upon measurement of small distances and of angles. ”

“ You say that you cannot explore the world without some kind of apparatus. If you explore with a scale, you find out the natural geometry; if you explore with a magnetic needle, you find out the magnetic field. What we may call the field of extension, or space-field, is just as much a physical quality as the magnetic field. You can think of them both existing together in the aether, if you like. The laws of both must be determined by experiment. Of course, certain approximate laws of the space-field (Euclidean geometry) have been familiar to us from childhood ”

“ To express this unmanageable mass of detail in a unified way, a world-geometry is found in which the tracks of greatest length are the actual tracks of the particles. It only remains to express the laws of this geometry in a concise form. The change from a mechanical to a geometrical theory of fields of force is not so fundamental a change as might be supposed. If we are now reducing mechanics to a branch of natural geometry, we have to remember that natural geometry is equally a branch of mechanics, since it is concerned with the behaviour of material measuring-appliances. ”

“ But these spheres are not to be rigid in the sense of Euclidean geometry; their radius is to increase (in the sense of Euclidean geometry) when they are moved away from S towards infinity, and this increase is to take place in exact accordance with the same law as applies to the increase of the radii of the disc-shadows L' on the plane.
After having gained a vivid mental image of the geometrical behaviour of our L' spheres, let us assume that in our space there are no rigid bodies at all in the sense of Euclidean geometry, but only bodies having the behaviour of our L' spheres. ”

“ The crystal is considered to be placed inside a sphere from the centre of which normals are drawn to all the faces of the crystal. The points at which these normals intersect the surface of the sphere are called the poles of the faces, and by these poles the positions of the faces are fixed. The poles of all faces in the same zone on the crystal will lie on a great circle of the sphere, which are therefore called zone-circles. The calculation of the angles between the normals of faces and between zone-circles is then performed by the ordinary methods of spherical trigonometry. ”

“ A doubly infinite number of geodesic surfaces can be drawn through every point. If a rigid body be divided into two parts by a geodesic plane, then no matter how the body be displaced the plane of section will still be geodesic. The plane of section may be made to pass through any point, and the body may then be given such an aspect as shall cause the section to coincide with any geodesic surface through the point, but this necessarily involves that the section shall fit each geodesic surface, in other words, that all the geodesic surfaces shall have a constant curvature. ”

“ When the standard quantity is greater than that to be measured, we often adopt the first mode, and subdivide the unit until we get a magnitude equal to that measured. The angles observed in surveying, in astronomy, or in goniometry are usually smaller than a whole revolution, and the measuring circle is divided by the use of the screw and microscope, until we obtain an angle undistinguishable from that observed. The dimensions of minute objects are determined by subdividing the inch or centimetre, the screw micrometer being the most accurate means of subdivision. ”

“ Our measurements of distance in space are found to be subject to certain laws—the laws of geometry. But it has now become impossible to regard the subject of space-geometry as complete in itself. Consider a triangle formed by three points (or events) in the four-dimensional world; if we happen to have drawn our instantaneous strata so that the three points lie in one stratum, then the triangle is a space-triangle and its properties fall within the scope of our classical geometry. ”

“ If we imagine constructions to be made with rigid rods in the surface (similar to that above with the marble slab) , we should find that different laws hold for these from those resulting on the basis of Euclidean plane geometry. The surface is not a Euclidean continuum with respect to the rods, and we cannot define Cartesian co-ordinates in the surface. Gauss indicated the principles according to which we can treat the geometrical relationships in the surface, and thus pointed out the way to the method of Riemann of treating multi-dimensional, non-Euclidean continua. ”

“ It is mathematically equal to the product of the length of the circumference, multiplied by half the radius. To square a circle of a given diameter in mètres, is the same as giving the number of squares, of a mètre in each side, of which the surface is the equivalent. If, the diameter being given, the exact circumference were known by a sort of inspiration, the superficial extent of the circular space would be deducible from the two numbers, by the mere multiplication of the numerical length of the circumference by the fourth of the diameter, or half the radius. ”

“ To realize what is meant when an angle is measured with a “probable error” of a few hundredths of a second of arc, it should be remembered that one second of arc on a circle 10 inches in diameter is less than 1/40000 of an inch. The accuracy which has been attained in the measurement of base lines is not easily realized by a layman. An engineer realizes the practical impossibility of measuring a line twice and obtaining356 precisely the same result to the finest unit of measurement. ”

“ To determine the Gaussian constants we proceed of course in the reverse direction, equating the observed values of the force components to the theoretical values involving g mn, &c. If we knew the values of the component forces at regularly distributed stations all over the earth’s surface, we could determine each Gaussian constant independently of the others. Our knowledge however of large regions, especially in the Arctic and Antarctic, is very scanty, and in practice recourse is had to methods in which the constants are not determined independently. ”