Rhombus

“ Third kind, the Rhombus.—Its sides are equal: let each of them be five, and let its diagonals be, the one eight and the other six yards. You may then compute the area, either from one of the diagonals, or from both. As you know them both, you multiply the one by the moiety of the other, the product is the area: that is to say, you multiply eight by three, or six by four; this yields twenty-four yards, which is the area. If you know only one of the diagonals, then you are aware, that there are two triangles, two sides of each of which have every one five yards, while the third is the diagonal. ”

“ On AC the diagonal of a square ABCD, a rhombus AEFC is described of the same area as the square, and having its acute angle at A: if AF be joined, shew that the angle BAC is, divided into three equal angles. ”

“ Similarly a mathematician should know that a rhombus is so called from its resemblance to a form of spinning-top (ῥομβος from ῥεμβω, to spin) and that, just as a parallelogram is a figure formed by two pairs of parallel straight lines, so a parallelepiped is a solid figure bounded by three pairs of parallel planes (παραλληλος, parallel, and επιπεδος, plane) ; incidentally, in the latter case, he will be saved from writing ‘parallelopiped’, a monstrosity which has disfigured not a few textbooks of geometry. Another good example is the word hypotenuse ”

“ Of all parallelograms which can be formed with diameters of given lengths the rhombus is the greatest.
452. Shew from I. 18 and I. 32 that if the hypotenuse BC of a right-angled triangle ABC be bisected at D, then AD, BD, CD are all equal.
453. If two equal straight lines intersect each other any where at right angles, the quadrilateral formed by joining their extremities is equal to half the square on either straight line.
454. Inscribe a parallelogram in a given triangle, in such a manner that its diagonals shall intersect at a given point within the triangle. ”

“ An obtuse-angled triangle is that which has an obtuse angle:
29. An acute-angled triangle is that which has three acute angles.
Of four-sided figures,
30. A square is that which has all its sides equal, and all its angles right angles:
31. An oblong is that which has all its angles right angles, but not all its sides equal:
32. A rhombus is that which has all its sides equal, but its angles are not right angles:
33. A rhomboid is that which has its opposite sides equal to one another, but all its sides are not equal, nor its angles right angles:
34. ”

“ For when the square is multiplied into itself, it becomes a biquadratic; but when the square is multiplied into a cube, it becomes the product of a quadratic and cube; but when a cube is multiplied into a cube, it becomes the product of cube multiplied by cube. Wherefore all the numbers are seven; so that the generation of things produced may be from the hebdomad—which is number, monad, power, cube, biquadratic, product of quadratic multiplied by cube, product of cube multiplied by cube. ”

“ Let the square be represented by a quadrangle, the sides of which are unknown to us, though they are equal among themselves, as also the angles. This is the quadrate A D, which comprises the three roots and the four of numbers mentioned in this instance. In every quadrate one of its sides, multiplied by a unit, is its root. We now cut off the quadrangle H D from the quadrate A D, and take one of its sides H C for three, which is the number of the roots. ”

“ In other words, he has managed to dispose the sixty-four first numbers on the squares of a chessboard in such a way that the sum of the numbers in every line, every column, and in each of the two diagonals, shall be the same; and more, that if one substitutes for all the numbers their squares, the square still remains magic." I at once set to work to solve this problem, and, although it proved a very hard nut, one was rewarded by the discovery of some curious and beautiful laws that govern it. The reader may like to try his hand at the puzzle. ”

“ For some purposes, 123/55 may be sufficient, as its square only differs from 5 by 4/3025; for others, the fraction first given might be necessary, or one whose square is even nearer to 5. We proceed to shew how to find the square root of a number, when it has one, and from thence how to find fractions whose squares shall be as near as we please to the number, when it has not. We premise, what is sufficiently evident, that of two numbers, the greater has the greater square; and that if one number lie between two others, its square lies between the squares of those others. ”

“ 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. ”

“ Can you construct a square of sixteen different numbers so that it shall be magic (that is, adding up alike in the four rows, four columns, and two diagonals) , whether you turn the diagram upside down or not? You must not use a 3, 4, or 5, as these figures will not reverse; but a 6 may become a 9 when reversed, a 9 a 6, a 7 a 2, and a 2 a 7. The 1, 8, and 0 will read the same both ways. Remember that the constant must not be changed by the reversal. ”

“ Our puzzle, therefore, resolves itself into forming a magic square with nine different prime numbers.
In Diagram A we have a magic square in prime numbers, and it is the one giving the smallest constant sum that is possible. As to the little trap I mentioned, it is clear that Diagram A is barred out by the words "every basket contained plums," for one plum is not plums. And as we were referred to the baskets, "as shown in the illustration," it is perfectly evident, without actually attempting to count the plums, that there are at any rate more than 7 plums in every basket. ”

“ Fig. 3 affords an interesting practical demonstration of the truth of this theorem. If the reader will copy this figure, cut out the squares on the two shorter sides of the triangle and divide them along the lines AD, BE, EF, he will find that the five pieces so obtained can be made exactly to fit the square on the longest side as shown by the dotted lines. The size and shape of the triangle ABC, so long as it has a right angle at C, is immaterial. ”

“ The agreement between geometrical and arithmetical thought is shown also in that we do not habitually follow the chain of reasoning back to the axioms in arithmetical, any more than in geometrical discussions. On the contrary we apply, especially in first attacking a problem, a rapid, unconscious, not absolutely sure combination, trusting to a certain arithmetical feeling for the behavior of the arithmetical symbols, which we could dispense with as little in arithmetic as with the geometrical imagination in geometry. ”

“ The three famous problems of antiquity, the duplication of the cube, the quadrature of the circle, and the trisection of the angle, have all been proved by modern mathematics to be insoluble by the rule and compass, which are the instruments assumed in the postulates of Euclid. Yet the problem of the trisection is frequently attacked by men of some mathematical education. I think it was about 1870 that I received from Professor Henry a communication coming from some institution of learning in Louisiana or Texas. ”

“ Besides these four elementary rules they cause children to be thoroughly at home in the principle of fractions, to multiply and divide fractions—as real things; all in good time it will become easy enough to let written figures represent them—to go through the rule of three, square root, and cube root. As a child has instilled into him the principles of arithmetic, so he acquires insensibly the groundwork of geometry, the sister science. ”

“ If there is a unit to be found which is contained an exact number of times in each side of a rectangle, it is easily seen, and generally shown in the teaching of arithmetic, that the rectangle contains a number of unit squares equal to the product of the numbers which measure the sides, a unit square being the square on the unit line. ”

“ Mathematicians know him as the reputed discoverer of the famous theorem about the square on the hypotenuse of a right-angled triangle (= Euclid I. 47) ; but he was also the first to make geometry a part of a liberal education and to explore its first principles (definitions, etc.) ”

“ But surely, were a solid geometrician, thoroughly versed in the abstract nature and fundamentals of numbers, to set his mind upon introducing a still more useful calculus into the world, instead of ten, he would select such a perfect square, or cube number, as by continual bisection, or halving, would at length terminate in unity, and be better adapted to the subdivisions of measures, weights, coins, etc. ”

“ 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. ”

“ 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. ”

“ In mathematics, effort is constantly being made to set up one-to-one correspondences between simple notions and more complicated ones, or between the well-explored fields of research and fields less known. Thus, by means of the mechanism employed in analytic geometry, algebraic theorems are made to yield geometric ones, and vice versa. In geometry we get at the properties of the conic sections by means of the properties of the straight line, and cubic surfaces are studied by means of the plane.
[pg 3] ”

“ By cutting the same parallelogram in two, along a diagonal A C, it may be easily shown that the two triangles can be placed exactly one upon the other, and that, consequently, they have equal areas. These lessons constitute a series of classical theorems in geometry which the child can try with his fingers and learn without even giving them the form of theorems. I might show the same as to the area of the trapeze and with many other theorems, but my purpose is only to present as many examples as will make my idea understood, without going into details. ”

“ Thus in the history of geometry the recognised principles of the science were temporarily disorganised by the admission of incommensurable magnitudes which was forced upon the early Greek mathematicians by the discovery that the side and diagonal of a square have no common measure, and the discrepancy was only removed when it became possible to revise the principles of the theory of numbers itself. ”

“ Every number is either a triangular number or the sum of two or three triangular numbers; either a square or the sum of two, three, or four squares; either a pentagonal number or the sum of two, three, four, or five pentagonal numbers; similarly for polygonal numbers in general. The proof of this and other theorems is promised by Fermat in a future work which never appeared. This theorem is also given, with others, in a letter of 1637 (?) ”

“ The celebrated interminable fraction 3.14159..., which the mathematician calls , is the ratio of the circumference to the diameter. But it is thousands of things besides. It is constantly turning up in mathematics: and if arithmetic and algebra had been studied without geometry, must have come in somehow, though at what stage or under what name must have depended upon the casualties of algebraical invention. ”

“ All types of series are capable of formal definition, and their properties can be deduced from the principles of symbolic logic by means of the Algebra of Relatives. The notion of a limit, which is fundamental in the greater part of higher mathematics, used to be defined by means of quantity, as a term to which the terms of some series approximate as nearly as we please. But nowadays the limit is defined quite differently, and the series which it limits may not approximate to it at all. This improvement also is due to Cantor, and it is one which has revolutionised mathematics. ”

“ They teach further the solution of problems leading to equations of the first and second degree, to determinate and indeterminate equations, not by single and double position only, but by real algebra, proved by means of geometric constructions, and including the use of letters as symbols for known numbers, the unknown quantity being called res and its square census. ”

“ Asymptotes of the hyperbola.—The asymptotes coincide with the diagonals of the parallelogram formed on any two conjugate diameters.—The portions of a secant comprised between the hyperbola and its asymptotes are equal.—Application to the tangent and to its construction.
The rectangle of the parts of a secant, comprised between a point of the curve and the asymptotes, is equal to the square of half of the diameter to which the secant is parallel. ”