Hexagram

“ My own opinion, as I have intimated, is, that the second account of the origin of the trigrams and hexagrams is the true one. However the idea of the whole and divided lines arose in the mind of the first framer, we must start from them; and then, manipulating them in the manner described, we arrive, very easily, at all the lineal figures, and might proceed to multiply them to billions. ”

“ The endowments of an ordinary man were sufficient for such a work. It was possible even to shorten the operation by proceeding at once from the trigrams to the hexagrams, according to what we find in Section i, paragraph 2:—
'A strong and a weak line were manipulated together (till there were the 8 trigrams) , and those 8 trigrams were added each to itself and to all the others (till the 64 hexagrams were formed) .'
Who first multiplied the figures to 64?It is a moot question who first multiplied the figures from the trigrams universally ascribed to Fû-hsî to the 64 hexagrams of the Yî. ”

“ If no chamfer at all is to be given, a hexagon head may still be distinguished from a square one, providing that the view giving three sides of the head, as in Figure 158, is shown, because the two sides D and E being half the width of the middle one C, imparts the information that it is a hexagon head. If, however, the view showing but two of the sides and a corner in front is given, and no chamfer is used, it could not be known whether the head was to be hexagon or square, unless an end view be given, as in Figure 161. ”

“ A is a hexagon, and if one of its sides be measured, it will [138] be found that it measures the same as along line 1 from O B to the diagonal line 45 degrees, which distance is shown by a thickened line.
At 1-1/2 is shown a seven-sided figure, whose diameter is 3 inches, and radius 1-1/2 inches, and if from the point at 1-1/2 (along the thickened horizontal line) , to the diagonal marked 49 degrees, be measured, it will be found exactly equal to the length of a side on the polygon. ”

“ A form, if of general character, that is, if composed of faces each of which is inclined to all three axes, will comprise more faces the higher the symmetry. Thus, in the cubic system, the form shown in Fig. 21, the hexakis octahedron, comprises as many as forty-eight faces, all covered by the form symbol {321}; while in the rhombic system the highest number of faces in a form is eight, in the monoclinic only four, and in the triclinic system two. It will also have become clear that the law of rational indices limits the number of forms possible of any one type. ”

“ Such a minor change I have attempted by occasionally shifting, in the first foot of the hexameter, the accent from the first syllable to the second. In the current English hexameter, it is on the first. Mr Spedding, who proposes radically to subvert the constitution of this hexameter, seems not to understand that anyone can propose to modify it partially; he can comprehend revolution in this metre, but not reform. ”

“ A regular hexagon, or six-sided figure, is easy enough, for everybody knows that all you have to do is to describe a circle and then, taking the radius as the length of one of the sides, mark off the six points round the circumference. But a pentagon is quite another matter. So, as my puzzle has to do with the cutting up of a regular pentagon, it will perhaps be well if I first show my less experienced readers how this figure is to be correctly drawn. Describe a circle and draw the two lines H B and D G, in the diagram, through the centre at right angles. ”

“ Are all squares similar? are all squares equal? are all triangles equal? are all triangles similar? What is the difference between a square and oblong? What is the difference between a square and a triangle? What is the difference between a square and a rhombus? What kind of corners has a rhombus? In what is a square like a rhombus? How do you describe a triangle? What is the name of the triangles you have learnt about? They will answer right-angled, equal-legged triangles. Then give them each a hexagon, and ask them what kind of corners it has? Whether any one is more blunt than another? ”

“ Section 66 All the angles of a square, summed together, equal 360 degrees. If one draws a line diagonally from one corner of the square to the other corner the square has been divided into a triangle. If so, it is shown to contain 180 degrees twice. A pentagon contains 180 degrees three times, for it can be divided into three triangles. A hexagon contains 180 degrees four times. The general rule is that if one subtracts two sides from the total number of sides of any shape the result is the number of times in contains 180 degrees. ”

“ The same inquiry is suggested by the problem of ascertaining in what cases the hypergeometric series F (α, β, γ, x) is the expansion of an algebraic (irrational) function of x. In order to explain the meaning of the question, suppose that the plane of x is divided along the real axis from -∞ to 0 and from 1 to +∞, and, supposing logarithms not to enter about x = 0, choose two quite definite integrals y1, y2 of the equation, say
y1 = F (λ, μ, 1 − λ1, x) , y2 = xλ1 F (λ + λ1, μ + λ1, 1 + λ1, x) ,
with the condition that the phase of x is zero when x is real and between 0 and 1. ”

“ Mathematicians know that there is not a fourth way possible in which a plane may be cut into little spaces that shall be equal, similar, and regular, without useless spaces. Of the three figures, the hexagon is the most proper for convenience and strength. Bees, as if they knew this, make their cells regular hexagons.
Again, it has been demonstrated that, by making the bottoms of the cells to consist of three planes meeting in a point, there is a saving of material and labour in no way inconsiderable. ”

“ Thus we are more apt to-day to omit the oblong and rhomboid as unnecessary, and to define "rhombus" in such a manner as to include a square. We use "parallelogram" to cover "rhomboid," "rhombus," "oblong," and "square." For "oblong" we use "rectangle," letting it include square. Euclid's definition of "square" [Pg 149] illustrates his freedom in stating more attributes than are necessary, in order to make sure that the concept is clear; for he might have said that it "is that which is equilateral and has one right angle." We may profit by his method, sacrificing logic to educational necessity. ”

“ 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. ”

“ 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. ”

“ My Pentagonal Sons were men of character and standing, and physicians of no mean reputation, but not great in mathematics, and, in that respect, unfit for my purpose. But it occurred to me that a young and docile Hexagon, with a mathematical turn, would be a most suitable pupil. Why therefore not make my first experiment with my little precocious Grandson, whose casual remarks on the meaning of 33 had met with the approval of the Sphere? ”

“ If the hexameter best helps the translator to the Homeric rapidity, what style may best help him to the Homeric plainness and directness? It is the merit of a metre appropriate to your subject, that it in some degree suggests and carries with itself a style appropriate to the subject; the elaborate and self-retarding style, which comes so naturally when your metre is the Miltonic blank verse, does not come naturally with the hexameter; is, indeed, alien to it. ”

“ The above hexameters are perfectly regular, both in being purely dactylic, and in having the regular cæsura, namely the end of a word at the beginning of the third dactyl, as—
We from the land that we love
We go as exiles away.
But these hexameters admit of irregularities in the same manner as the common English measures of which we have spoken. We may have dissyllable feet instead of trissyllable in any place in the line; thus in the fourth—
Tityrus, you in the shade of a chestnut idly reclining. ”

“ 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. ”

“ In the same manner it may be shewn that the other angles of the hexagon ABCDEF are each of them equal to the angle AFE or FED;
therefore the hexagon is equiangular.
And it has been shewn to be equilateral; and it is inscribed in the circle ABCDEF.
Wherefore an equilateral and equiangular hexagon has been inscribed in the given circle, q.e.f.
Corollary. From this it is manifest that the side of the hexagon is equal to the straight line from the centre, that is, to the semidiameter of the circle. ”

“ Needless to say, of course, observation exercises do not take the place of geometry, still less of trigonometry, but prepare the ground for these sciences. When the pupil reaches plain geometry, the wooden triangles are put aside, and observation is subordinated to geometrical construction. Meanwhile arithmetic is passing beyond exercises that deal merely with proportions, to powers, roots, and logarithms. In fact, without the concept of the square root, not even the Pythagorean Theorem can be fully grasped. ”

“ Comparing the Latin Hexameter and English heroic rhyme, the former has obviously the advantage in the following particulars. It is greatly preferable as to arrangement, by the latitude it admits in placing the long and short syllables. Secondly, the length of an Hexameter line hath a majestic air: ours, by its shortness, is indeed more brisk and lively, but much less fitted for the sublime. And, thirdly, the long high-sound{442}ing words that Hexameter admits, add greatly to its majesty. ”

“ Between the recognition of this rhythm as ideally the best, and the recommendation of it to the translator for instant practical use, there must come all that consideration of circumstances, all that pliancy in foregoing, under the pressure of certain difficulties, the absolute best, which I have said is so indispensable to the critic. The hexameter is, comparatively, still unfamiliar in England; many people have a great dislike to it. A certain degree of unfamiliarity, a certain degree of dislike, are obstacles with which it is not wise to contend. ”

“ 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. ”

“ The best way to introduce this theorem is to cut a parallelogram from paper, and then, with the class, separate it into two parts by a cut perpendicular to the base. The two parts may then be fitted together to make a rectangle. In particular, if we cut off a triangle from one end and fit it on the other, we have the basis for the proof of the textbooks. The use of squared paper for such a proposition is not wise, since it makes the measurement appear to be merely an approximation. The cutting of the paper is in every way more satisfactory. ”

“ This is the hexad axis or hexagonal axis, for which the angle of rotation is 60°, or one-sixth of 360°. There can be only one hexad axis of symmetry in any crystal (see figs. 77-80) .
Planes of Symmetry.—A regular octahedron can be divided into two equal and similar halves by a plane passing through the corners a1a3ā1ā3 and the centre O (fig. 3) . One-half is the mirror reflection of the other in this plane, which is called a plane of symmetry. Corresponding planes on either side of a plane of symmetry are inclined to it at equal angles. ”

“ 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. ”

“ It was Clough's hexameters, together with some "English Hexameter Translations" by Dr. Hawtrey and Mr. Lockhart (1847) , that chiefly encouraged Matthew Arnold to believe that the measure might be well adapted to a translation of Homer. "The hexameter, whether alone or with the pentameter, possesses a movement, an expression, which no metre hitherto in common use amongst us possesses, and which I am convinced English poetry, as our mental wants multiply, will not always be content to forego." ”