Inverse-square law

“ It is known to physical science as the "law of inverse squares," by which the forces of reciprocal attraction or repulsion, as the case may be, are not merely equivalent to the sum of the forces emitted by the two bodies concerned, but are equivalent to these two forces multiplied together and divided by the square of the distance between them, so that the resultant power continually rises in a rapidly-increasing ratio as the two reciprocally exciting bodies approach one another. ”

“ The simple law of inverse square does not apply, because a perturbation is a differential effect: the satellite or ocean when nearer to the perturbing body than the rest of the earth, is attracted more, and when further off it is attracted less than is the main body of the earth ”

“ We have not extended the laws of gravitation to small bodies, nor have we found that our electrical laws will work on a cosmic scale. It does not seem so very surprising that over a limited domain, in which the most important phenomena are of a restricted type, the conduct of nature should follow comparatively simple rules.
A tempting question is whether there may not be some laws of nature that are really simple, without relation to our mode of formulation, such as the law of the inverse square. ”

“ On the contrary, the operation of replacing x by an assigned number in any rational function of x is not, in the present sense, although it leads to a unique result, a definite operation; there is in fact no unique inverse operation corresponding to it. It is to be noticed that the question whether an operation is a definite operation or no may depend on the range of the objects on which it operates. For example, the operations of squaring and extracting the square root are definite inverse operations if the objects are restricted to be real positive numbers, but not otherwise. ”

“ So far we have treated only of the process by which we pass from special facts to general laws, that inverse application of deduction which constitutes induction. But the direct employment of deduction is often combined with the inverse. No sooner have we established a general law, than the mind rapidly draws particular consequences from it. In geometry we may almost seem to infer that because one equilateral triangle is equiangular, therefore another is so. In reality it is not because one is that another is, but because all are. ”

“ This is a practical test upon which merchants are willing to stake, and do stake, billions of dollars every day.
It is now well established that, like the diagonal and side of a square, the diameter and circumference of any circle are incommensurable quantities. But, as De Morgan says, "most of the quadrators are not aware that it has been fully demonstrated that no two numbers whatsoever can represent the ratio of the diameter to the circumference, with perfect accuracy. ”

“ It was claimed later on by Hooke that he had discovered a method demonstrating the truth of the theory of inverse squares, and after the full announcement of Newton's discovery a heated controversy was precipitated in which Hooke put forward his claims with accustomed acrimony. Hooke, however, never produced his demonstration, and it may well be doubted whether he had found a method which did more than vaguely suggest the law which the observations of Kepler had partially revealed. Newton's great merit lay not so much in conceiving the law of inverse squares as in the demonstration of the law. ”

“ If we should suppose the attraction by inverse squares to be absolute, with the superposition of a repulsion of much higher inverse power, and of far greater intensity, it is evident that the two curves of force would cross each other, and that at the intersection the resultant would involve a "change of sign." ”

“ You need only consider squares of one colour, for whatever can be done in the case of the white squares can always be repeated on the black, and they are here quite independent of one another. This equality, of course, is in consequence of the fact that the number of squares on an ordinary chessboard, sixty-four, is an even number. If a square chequered board has an odd number of squares, then there will always be one more square of one colour than of the other. ”

“ Many of these operations consist of two parts, the direct and the inverse. To extract a root is the inverse, with reference to that of raising any number to a higher power. So the integral is the inverse of the differential calculus; and the same observation applies to finite differences. In all these cases the inverse method is by far the more difficult of the two. ”

“ It will be seen that the constant of the adding square is n times that of the subtracting square derived from it, where n is the number of cells in the side of square. And the manner of derivation here is simply to reverse the two diagonals. Both squares are "associated"—a term I have explained in the introductory article to this department.
The third square is a multiplying magic. The constant, 216, is obtained by multiplying together the three numbers in any line. It is "associated" by multiplication, instead of by addition. ”

“ We proceed to indeterminate problems of the second degree: limiting ourselves to the consideration of the formula y2 = a + bx + cx2, where x is to be found, so that y may be a rational quantity. The possibility of rendering the proposed formula a square depends altogether upon the coefficients a, b, c; and there are four cases of the problem, the solution of each of which is connected with some peculiarity in its nature.
Case 1. Let a be a square number; then, putting g2 for a, we have y2 = g2 + bx + cx2. Suppose √ (g2 + bx + cx2) = g + mx ”

“ As regards the analytical treatment, the difficulty is of the same nature as that arising in the theory of curves in space, for a congruence is not in general the complete intersection of two complexes.
A Ruled Surface, Regulus or Skew is a configuration of lines which satisfy three conditions, and therefore depend on only one parameter. Such lines all lie on a surface, for we cannot draw one through an arbitrary point; only one line passes through a point of the surface; the simplest example, that of a quadric surface, is really two skews on the same surface. ”

“ Thus, regarding the coordinates of a line as homogeneous coordinates in five dimensions, we may say that line geometry is equivalent to geometry on a quadric surface in five dimensions. A linear complex is represented by a hyperplane section; and if two such complexes are in involution, the corresponding hyperplanes are conjugate with respect to the fundamental quadric. By projecting this quadric stereographically into space of four dimensions we obtain Klein’s analogy. In the same way geometry in a linear complex is equivalent to geometry on a quadric in four dimensions ”

“ General Sacco, dealing with fractional substitutions (Chapter XXII) , shows the same idea in a checkerboard which he describes as “frequential.” This square is simply filled with letters, used in proportions roughly approximating their normal frequencies; for ready finding, all repetitions of a letter are placed close together, but filled in on diagonals, which, to some extent, will prevent their being represented by consecutive numbers. ”

“ Induction is, in fact, the inverse operation of deduction, and cannot be conceived to exist without the corresponding operation, so that the question of relative importance cannot arise. Who thinks of asking whether addition or subtraction is the more important process in arithmetic? But at the same time much difference in difficulty may exist between a direct and inverse operation; the integral calculus, for instance, is infinitely more difficult than the differential calculus of which it is the inverse. ”

“ The sum of the digits is 59, the sum of which is 14, the sum of which is 5 (which I call the "digital root") , and therefore I know that the number cannot be a square, and for this reason. The digital root of successive square numbers from 1 upwards is always 1, 4, 7, or 9, and can never be anything else. In fact, the series, 1, 4, 9, 7, 7, 9, 4, 1, 9, is repeated into infinity. The analogous series for triangular numbers is 1, 3, 6, 1, 6, 3, 1, 9, 9. ”

“ A binary form which has a square factor has its discriminant equal to zero. This can be seen at once because the factor in question being once repeated in both differentials, the resultant of the latter must vanish.
Similarly, if a form in k variables be expressible as a quadratic function of k – 1, linear functions X1, X2, ... Xk – 1, the coefficients being any polynomials, it is clear that the k differentials have, in common, the system of roots derived from X1=X2=...=Xk – 1=0, and have in consequence a vanishing resultant. ”

“ If a1 is not a secant, but made to pass through any point Q in space, the ruled quadric surface determined by a1 will pass through Q. There will therefore be one line of the congruence passing through Q, and only one. For if two such lines pass through Q, then the lines S1Q and S2Q will be corresponding lines; hence Q will be a point on the cubic k, and an infinite number of secants will pass through it. Hence:—
Through every point in space not on the twisted cubic one and only one secant to the cubic can be drawn. ”

“ The property last mentioned, however, π would have if the squaring of the circle with ruler and compasses were possible.
#The verdict of mathematics.#
"It is impossible with ruler and compasses to construct a square equal in area to a given circle." These are the words of the final determination of a controversy which is as old as the history of the human mind. But the race of circle-squarers, unmindful of the verdict of mathematics, that most infallible of arbiters, will never die out so long as ignorance and the thirst for glory shall be united. ”

“ That the squares of the periodic times of the planets are proportional to the cubes of their distances from [Pg 145] the sun, is an uniformity derived from the laws of the causes (or forces) which produce the planetary motions; but that the square of any number is four times the square of half the number, is true independently of any cause. The only laws of resemblance, therefore, which we are called upon to consider independently of causation, belong to the province of mathematics. ”

“ The fact that, so long as we confine ourselves to numbers less than some given finite number, the proportion of squares tends towards zero as the given finite number increases, does not contradict the fact that the number of all finite squares is the same as the number of all finite numbers. ”

“ If we set to work to extract the square root of 2 by the usual arithmetical rule, we shall obtain an unending decimal which, taken to so-and-so many places, exactly fulfils the above conditions. We can equally well form a descending series of fractions whose squares are all greater than 2, but greater by continually smaller amounts as we come to later terms of the series, and differing, sooner or later, by less than any assigned amount. ”

“ Here the directions of the railways will resemble the moves of a rook in chess—that is, we may move in any direction parallel to the sides of the diagram, but not diagonally. Therefore the first town (or square) visited must be a black one; the second must be a white; the third must be a black; and so on. Every odd square visited will thus be black and every even one white. Now, we have 23 squares to visit (an odd number) , so the last square visited must be black. But Z happens to be white, so the puzzle would seem to be impossible of solution. ”

“ The reciprocal relation between differentiation and integration (§ 6) was first observed explicitly by Barrow in the book cited above. If the quadrature of a curve y = ƒ (x) is known, so that the area up to the ordinate x is given by F (x) , the curve Barrow’s Inversion-theorem. y = F (x) can be drawn, and Barrow showed that the subtangent of this curve is measured by the ratio of its ordinate to the ordinate of the original curve. The curve y = F (x) is often called the “quadratrix” of the original curve; and the result has been called “Barrow’s inversion-theorem.” ”

“ Since this is an incommensurable number, we seek for its limits by extraction of the square root, namely 1415 the greater limit and 1414 the less limit, or more accurately the greater, and the less; for as we reduce the difference of the limits we increase the accuracy. ”

“ Any device or game, which [page 10] will provide a square, is likely to be seized upon as the source of a transposition key. We find two widely-known examples of this in the “magic square” and the “knight’s tour.”
A magic square, as most of us understand this term, is made up of a series of numbers, such as 1 to 25, 1 to 36, which are so arranged in their cells (positions) that the added numbers of any row, column, or diagonal, will always give the same total. A square of given size will provide more than one magic square arrangement ”

“ Square each part, and multiply all that come before by twice that part. Hence a reverse rule for extracting the square root presents itself with more than usual simplicity. It is: To extract the square root of a number N, choose a number A, and see if N will bear the subtraction of the square of A; if so, take the remainder, choose a second number B, and see if the remainder will bear the subtraction of the square of B, and twice B multiplied by the preceding part A: if it will, there is a second remainder. ”

“ The general run of circle-squarers, hearing that the quadrature is not pronounced to be demonstratively impossible, imagine that the arithmetical quadrature is open to their ingenuity. Before attempting the arithmetical problem, they ought to acquire knowledge enough to read Lambert's [355] demonstration (last given in Brewster's [356] translation of Legendre's [357] Geometry) and, if they can, to refute it. ”