Logarithmic spiral

“ We reach a limiting case of the logarithmic spiral when the chambers are arranged in a straight line; and the eye will tend to associate with this limiting case the much more numerous forms in which the spiral angle is small, and the shell only exhibits a gentle curve with no succession of enveloping whorls. This constitutes the Nodosarian type (Fig. 87, p. 262) ; and here again, we must postulate some force which has tended to keep the chambers in a rectilinear series: such for instance as gravity, acting on a system of “hanging drops.” ”

“ When we say here, for short, that the shell is a logarithmic spiral, we merely mean that it is essentially related to one: that it can be inscribed in such a spiral, or that cor­re­spon­ding points (such, for instance, as the centres of gravity of successive chambers, or the extremities of successive septa) wall always be found to lie upon such a spiral. ”

“ In the main, the logarithmic spiral is char­ac­ter­is­tic, not of the living tissues, but of the dead. And for the same reason, it will always or nearly always be accompanied, and adorned, by a pattern formed of “lines of growth,” the lasting record of earlier and successive stages of form and magnitude.
It is evident that the spiral curve of the shell is, in a sense, a vector diagram of its own growth; for it shews at each instant of time, the direction, radial and tangential, of growth, and the unchanging ratio of velocities in these directions. ”

“ By this characteristic we recognize the ‘logarithmic spiral.’ Geometricians give this name to the curve which intersects obliquely, at angles of unvarying value, all the straight lines or ‘radii vectores’ radiating from a centre called the ‘Pole.’ The Epeira’s construction, therefore, is a series of chords joining the intersections of a logarithmic spiral with a series of radii. ”

“ It follows from this property of the function that we cannot have for log x a series which shall be convergent for all values of x, as is the case with sin x and cos x, for such a series could only represent a uniform function, and in fact the equation
log (1 + x) = x − 1/2x2 + 1/3x3 − 1/4x4 + ... is true only when the analytical modulus of x is less than unity. The exponential function, which may still be defined as the inverse of the logarithmic function, is, on the other hand, a uniform function of x, and its fundamental properties may be stated in the same form as for real values of x. ”

“ When the radii of these spheres are plotted down on a diagram as ordinates, and a series of integers as abscissas, the general form of the logarithmic curve becomes apparent; and close scrutiny has shown that either the logarithmic curve, or some curve lying very close to it, expresses the real law of nature. ”

“ The tables contain seven-figure logarithms to 108,000, and to 1200 to 20 places, some antilogarithms to 20 places, hyperbolic logarithms from 1 to 10 at intervals of ·01 and to 1200 at intervals of unity to 7 places, logistic logarithms, log sines and tangents to every second of the first two degrees, and natural and log sines, tangents, secants, and versed sines for every minute of the quadrant to 7 places. The natural functions occupy the left-hand pages and the logarithmic the right-hand. ”

“ The only other logarithmic canon to every second that has been published forms the second volume of Shortrede’s Logarithmic Tables (1849) . In 1784 the French government decided that new tables of sines, tangents, &c., and their logarithms, should be calculated in relation to the centesimal division of the quadrant. ”

“ The analogies that exist between relativity theory and Lobachevskian geometry are in any case interesting. The formulas of recent mechanics for are reduced to the formulas of Newtonian mechanics. Similarly also the Lobachevskian geometry, if we take the so called radius of curvature as infinite, goes over into the euclidean geometry. ”

“ A. Gernerth, Fünfstellige gemeine Logarithmen (Vienna, 1866) , gives logarithms to 10,800 and a ten-second canon. There are sixty lines on the page, so that the double page contains log sines, cosines, tangents, and cotangents extending over a minute. C. Bremiker, Logarithmisch-trigonometrische Tafeln mit fünf Decimalstellen (10th edition by A. Kallius, Berlin, 1906) , which has been already referred to, gives logarithms to 10,009 and a logarithmic canon to every hundredth of a degree (sexagesimal) , in a handy volume ”

“ Accordingly, we are led to believe that the spirals studied thus far have diameters 20 times or 100 times, or in some cases several thousand times, the diameter of our solar system. It is difficult to avoid the conclusion that in general they are immensely more massive than is our solar system. The spiral which has been assumed as the forerunner of our system must have been of diminutive size as compared with the larger and brighter spirals which we see to-day. ”

“ In finding the base we may pursue another method, namely:—
Add the logarithm of the versed sine of the given angle to the logarithms of the given sides, and the sum will be the logarithm of the difference between the versed sine of the difference of the sides and the versed sine of the base required. This difference being consequently known, add to it the versed sine of the difference of the sides, and the sum will be the versed sine of the base required. ”

“ They are generally to be found in collections of logarithmic tables, but rarely exceed a page in extent, and are very easy to construct. Schrön and Bruhns both give the first hundred multiples of the modulus and its reciprocal to 10 places, and Bremiker (in his edition of Vega and in his six-figure tables) and Dupuis to 7 places. C. F. Degen, Tabularum Enneas (Copenhagen, 1824) , gives the first hundred multiples of the modulus to 30 places.
Antilogarithms.—In the ordinary tables of logarithms the natural numbers are integers, while the logarithms are incommensurable. ”

“ Archimedes then passes to the spiral which, he says, presents another sort of problem, having nothing in common with the foregoing. After a definition of the spiral he enunciates the main propositions about it which are to be proved in the treatise. The spiral (now known as the Spiral of Archimedes) is defined as the locus of a point starting from a given point (called the “origin”) on a given straight line and moving along the straight line at uniform speed, while the line itself revolves at uniform speed about the origin as a fixed point. ”

“ And astonishing as this statement may be, we have still to add that, in face of the actual facts, it may be regarded as even a moderate estimate of the abundance of spirals in the universe. We must remember that a spiral nebula is a flat object with long arms extending from it which lie nearly in the same plane. If we are actually to see that such an object is spiral, it is necessary for it to be turned squarely towards the earth. If the object be too much foreshortened, it is quite plain that we can hardly expect to detect its spiral character. ”

“ 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. ”

“ It is independent of the observer's mesh-system, but it depends on his gauge. It is obvious that the number expressing the radius of curvature of the world at a point must depend on the unit of length; so we cannot say that the curvatures at two points are absolutely equal, because they depend on the gauges assigned at the two points. Conversely the radius of curvature of the world provides a natural and absolute gauge at every point; and it will presumably introduce the greatest possible symmetry into our laws if the observer chooses this, or some definite fraction of it, as his gauge. ”

“ In relativity theory the parallelogram of velocities is not valid; in Lobachevskian geometry there is no parallelogram at all. In relativity theory which banned the absolute from physics, there exists an absolute speed c; in Lobachevskian geometry there exists an absolute length R etc.. ”

“ The Lorentz-Fitzgerald hypothesis of electron contraction led me to the assumption, whether this contraction could be interpreted as a consequence of geometrical anisotropy of space. It seemed to me that this contraction is analogues to the deformation of lengths in a very familiar interpretation of Lobachevskian geometry. [3] Now it seems that my assumption of the connection of non-euclidean geometry with relativity theory can be realized. ”

“ The earlier methods proposed were, like those of Briggs, purely arithmetical, and for a long time logarithms were regarded from the point of view indicated by their name, that is to say, as depending on the theory of compounded ratios. The introduction of infinite series into mathematics effected a great change in the modes of calculation and the treatment of the subject. Besides Napier and Briggs, special reference should be made to Kepler (Chilias, 1624) and Mercator (Logarithmotechnia, 1668) , whose methods were arithmetical, and to Newton, Gregory, Halley and Cotes, who employed series. ”

“ If, instead of referring to the pseudosphere, we merely define Transition to the projective method. distance and angle, in the Euclidean plane, as those functions of the coordinates which gave us distance and angle on the pseudosphere, we find that the geometry of our plane has become Lobatchewsky’s. All the points of the limiting circle are now at infinity, and points beyond it are imaginary. If we give our circle an imaginary radius the geometry on the plane becomes elliptic. Replacing the circle by a sphere, we obtain an analogous representation for three dimensions. ”

“ The two systems of logarithms for which extensive tables have been calculated are the Napierian, or hyperbolic, or natural system, of which the base is e, and the Briggian, or decimal, or common system, of which the base is 10; and we see that the logarithms in the latter system may be deduced from those in the former by multiplication by the constant multiplier 1/log10, which is called the modulus of the common system of logarithms. ”

“ The first tables of logarithmic sines and tangents were constructed by Edmund Gunter (1581-1626) , professor of astronomy at Gresham College, London; he was also the first to employ the expressions cosine, cotangent, and cosecant for the sine, tangent, and secant of the complement of an arc. A treatise by Albert Girard (1590-1634) , published at The Hague in 1626, contains the theorems which give areas of spherical triangles and polygons, and applications of the properties of the supplementary triangles to the reduction of the number of different cases in the solution of spherical triangles. ”

“ Since the scale of the instrument presents to view the proportion of logarithms to each other, while the slider represents the ratios of their respective numbers, it becomes a means of illustrating many points relative to the general theory of logarithms, whether to exhibit a series of logarithms formed according to any particular modulus, or by an inverted position of the slider to find the moduli of all different systems, and accordingly in that, for instance, in which the modulus is equal to the basis. ”

“ Barred spirals.—In the normal spiral the arms emerge from two opposite points on the periphery of the nuclear region. There is, however, a smaller group, containing about 20 per cent of all spirals, in which a bar of nebulosity extends diametrically across the nucleus. In these spirals, the arms spring abruptly from the ends of this bar. These nebulae also form a sequence, which parallels that of the normal spirals, the arms apparently unwind, the nuclei dwindle, the condensations form and work inward. ”

“ Hence I infer that the motions which we call magnetic attractions and repulsions may be quite analogous to such helical motions; also, that these motions exist in ether, and evidently may be either right-handed or left-handed. Wind up on a pencil a piece of wire twelve or fifteen inches long, making a loose spiral. Bring the two ends of the spiral together; and note first that one is twisted to the right, the other to the left. If they be twisted into each other, they will advance very easily ”

“ Before obtaining this result it is shown that spherical trigonometry is unchanged, and that the normals to a circle or a sphere still pass through its centre. When the radius of the circle or sphere becomes infinite all these normals become parallel, but the circle or sphere does not become a straight line or plane. It becomes what Lobatchewsky calls a limit-line or limit-surface. The geometry on such a surface is shown to be Euclidean, limit-lines replacing Euclidean straight lines. (It is, in fact, a surface of zero measure of curvature.) ”

“ If the earth were absolutely rigid—if every particle of the earth were absolutely at a fixed distance from every other particle—if under no stress of forces, and in no conceivable circumstance, the earth experienced even the minutest change of form—if the same could be said of the sun and of all the other planets—then Lagrange's prediction of the eternal duration of our system must be fulfilled.
But what are the facts of the case? Is the earth really rigid? We know from experiment that a rigid body in the mathematical sense of the word does not exist. Rocks are not rigid; steel is not rigid ”