Triglyph

“ For this reason the frieze contains more essential differences than those distinguishing the architrave, and is bound to emphasize them more sharply, especially in the case where architecture, although executed in stone materials, follows more stringently the fundamental type of the wood construction. This is supplied us by the distinction between triglyphs and metopes. In other words triglyphs are the tops of the beams which are divided into three spaces, the metopes are the rectangular spaces between the separate triglyphs. ”

“ The uppermost member of the entablature, the CORNICE, consists principally of a projecting portion, the CORONA, on whose inclined under surface or soffit are rectangular projections, the so-called MUTULES (best seen in the frontispiece) , one over each triglyph and each metope. Three rows of six guttae each are attached to the under surface of a mutule. Above the cornice, at the east and west ends of the building, come the triangular PEDIMENTS or gables, formed by the sloping roof and adapted for groups of sculpture. ”

“ And the wife stares at the old man, and then at the blond organist. And the blond organist stares at the mother of his beautiful children, and then at the bearded interloper. And they all stare at each other and feel very embarrassed.
The story is familiar to you? Well, perhaps so. It is the story of the eternal triangle, the most useful [82] pattern of geometrical forms in the construction of a romantic pattern.
Heigho! the trouble with human triangles is that they are never equilateral. Two sides together are invariably greater than the third side. ”

“ II It is a common custom to refer to the usual complication between one man and two ladies, or one lady and two men, or a lady and a man and a nobleman, or—well, any of those problems—as the triangle. But they are never unqualified triangles. They are always isosceles—never equilateral. So, upon the coming of Nevada Warren, she and Gilbert and Barbara Ross lined up into such a figurative triangle; and of that triangle Barbara formed the hypotenuse. ”

“ I was unable to think of any letter that could ever have been intended for the picture of a heart, but the triangle I knew to be the letter #A.# This was originally written without the cross-bar from prop to prop, and the two feet at the bottom of the props were not separated as now, but joined; so that the letter formed a true triangle. It was meant by the primitive man to be a picture of his primitive house, this house being, of course, hut-shaped, and consisting of a conical roof without walls. ”

“ If we are not of those who begin by invoking a divine Trinity, we must none the less invoke a human Trinity; and see that triangle repeated everywhere in the pattern of the world. For the highest event in history to which all history looks forward and leads up, is only something that is at once the reversal and the renewal of that triangle. Or rather it is the one triangle superimposed so as to intersect the other, making a sacred pentacle of which, in a mightier sense than that of the magicians, the fiends are afraid. ”

“ The triangle, or delta, is the symbol of Deity for this reason. In geometry a single line cannot represent a perfect figure; neither can two lines; three lines, however, constitute the triangle or first perfect and demonstrable figure. Hence this figure symbolizes the Eternal God, infinitely perfect in his nature. But the triangle properly refers to God only in his quality as an Eternal Being, its three sides representing the Past, the Present, and the Future. Some Christian symbologists have made the three sides represent the Father, Son, and Holy Ghost ”

“ The secret of the inverted triangle is that it is upheld by invisible hands, and it is the full programme of the Red Triangle that appeals so irresistibly to the men. If we were merely out to run a canteen, [12] others could perhaps have done the canteen work as well, or nearly as well, as the Y.M.C.A. Others could run lectures for the troops, and others cater for their spiritual needs, but it has been left to the Y.M.C.A. to formulate the appeal to the whole man—Body, Mind, and Spirit—and the appeal to every man, irrespective of creed or party. ”

“ But we shall win. Everything will be cleared up. There will be no more mysteries, not even the mystery of the golden triangle. . . . Oh, I don’t say that I shall bring you a beautiful triangle of eighteen-carat gold! We mustn’t allow ourselves to be fascinated by words. It may be a geometrical arrangement of the bags of gold, a triangular pile . . . or else a hole in the ground dug in that shape. No matter, we shall have it! And the bags of gold shall be ours! And Patrice and Coralie shall appear before monsieur le maire and receive my blessing and live happily ever after! ”

“ Can a triangular harmony form the substance of the elements? Is fire, as Plato affirms, an acute triangle, and light and heat properties of this triangle? Air and water, are they rectangular and equilateral triangles? Is the form of the terrestrial element a square, because, being the least perfect of all the four elements, it recedes as much as possible from a triangle without losing its essence? Do the male and female embrace only to complete the triangle of generation? These platonic ideas have two very different [334] aspects. ”

“ Thus there is no conception of triangle in general. When the mind thinks of this figure, it always conceives one or more definite and particular triangles, which it accepts as representative of all possible or actual triangles. It thinks of one or of several triangles with a mental reservation, that the varieties of which that figure is susceptible are not exhausted by the specimens of which it is thinking. ”

“ In fig. 226 there is a mint mark or counter mark of the same design as the triskelion, except that it has but two arms or rays (diskelion) .
Perrot and Chipiez, [227] speaking of Lycia, say:
The device of many of her coins is the “triskelis” or so-called “triquetra” (literally, three-cornered, triangular) , a name derived from three serpents’ heads, which usually figure in the field, much after the fashion of those supporting the famous tripod at Delphi, [228] consecrated by the Greeks to Apollo after the battle of Platæa. ”

“ What he said was that there is not an infinite number of worlds, nor yet one, nor yet five, but one hundred and eighty-three, arranged in a triangle with sixty worlds in each side. Of the three left over, each is 140placed at one angle. Each world keeps a light touch on its neighbours while they revolve as in a dance. The area inside the triangle is the common hearth of all, and is called the “Plain of Truth”, and within it the formulae, and ideas, and patterns, |C| of things which have been and things which are, lie undisturbed. ”

“ Other remarkable signs are the triad, in the shape of dots placed in a triangle, and the triskele, which are seen on many objects. There was evidently importance attached to the numbers “Three” and “Nine;” but it is impossible to tell what was the true meaning in the mythology of these people of the triad, which is very common on the jewels and other objects illustrated in this work, [228] and it is remarkable that some of the graves are made to represent the above signs. ”

“ But the whole secret really lies in just one triangle of that figure. The surface size is irrelevant, and the rest of the pentagram frame redundant. It's the angles of that one triangle which are important. Fashion a triangle with its three angles of sizes I could give you (though an error amounting to a second will suffice to make it impotent) and you will have a triangle of terror indeed. ”

“ It is a symbol of the moral law as set forth in the revealed will of God.
Every man must have his trestle board, because it is the duty of every man to work out the task which God, the chief Architect, has assigned to him.
TRIANGLE. A symbol of Deity.
This symbolism is found in many of the ancient religions.
Among the Egyptians it was a symbol of universal nature, or of the protection of the world by the male and female energies of creation.
TRIANGLE, RADIATED. A triangle placed within a circle of rays. In Christian art it is a symbol of God; then the rays are called a glory. ”

“ The conception of the triangle, with all the mathematical truth which it contains, is necessary, universal, and eternal. It has, therefore, its foundation in necessary being, or in the divine intelligence. But it is in God in an eminent mode, and formally only in the human intellect. Geometrical truth is founded in the essence of God, who is the archetype of the triangle and of every other geometrical figure. But that which the triangle imitates the human intellect cannot see ”

“ I direct it in like manner to the end of the base E, and then the light it receives is in the direction of the side of the triangle FE ; I turn it then to the same distant signal G. Therefore, by these observations, I have really and truly got, by the theodolite, the measure of the two angles of the triangle at E and F. Now, that is sufficient. Every person who has a knowledge of trigonometry, knows that if we have got the measures of the side EF, and of these two angles, we are able, either to construct the triangle on paper, or to determine, by calculation, the whole of its parts. ”

“ Now for a little puzzle in triangular numbers.
Sandy McAllister, of Aberdeen, practised strict domestic economy, and was anxious to train his good wife in his own habits of thrift. He told her last New Year's Eve that when she had saved so many sovereigns that she could lay them all out on the table so as to form a perfect square, or a perfect triangle, or two triangles, or three triangles, just as he might choose to ask he would add five pounds to her treasure. ”

“ Thus shou’d we mention the word triangle, and form the idea of a particular equilateral one to correspond to it, and shou’d we afterwards assert, that the three angles of a triangle are equal to each other, the other individuals of a scalenum and isosceles, which we overlooked at first, immediately crowd in upon us, and make us perceive the falshood of this proposition, tho’ it be true with relation to that idea, which we had form’d. If the mind suggests not always these ideas upon occasion, it proceeds from some imperfection in its faculties ”

“ Always the short side of the triangle was uppermost. As she explained all this to me, she pointed to the decorations on her golden armlets, upon the knob of her dagger-hilt and upon the band which encircled her right leg above the knee—always was the design partly made up of isosceles triangles, and when she explained the significance of this particular geometric figure, I at once grasped its appropriateness. ”

“ The three corners of the triangle are joined by three lines, each line made of a series of points which should symbolize each a linear series of cognate forms, and we have these three forms in the trapezohedron d, the three-faced octahedron e, and the four-faced cube f, each linking by a single unbroken series the three corner forms. These are placed each on its proper line on the diagram. They are each twenty-four-sided figures. In two, each side is an isosceles triangle, in one, a trapezium, a combination of two isosceles triangles. ”

“ Regarding what is written there in Moreh Nebukhim, "Can God make a triangle a square?" - Rabbeinu z"l said, "I believe that Hashem Yitbarakh can make a triangle into square. For God's ways are beyond our view, He is the Omnipotent, and there's nothing He can't do." ”

“ The figure drawn on the blackboard is the sensible object. The conception of a triangle is the intelligible object formed by abstraction, and universal. In this conception are contained the general notions of a point, a line, an angle; and in these notions are involved several self-evident principles or axioms. From these are demonstrated the various mathematical propositions of trigonometry. It is easy to see that, in the intellectual process of the pupil’s mind, the genesis and development of the act of cognition of mathematical truth is precisely what has been above described. ”

“ Triggs! You mustn't do anything of the sort. He's most kind and considerate, and if I am here I must do what he wants."
"But beetles and babies and potatoes, me dear," said Mr. Triggs. "That's more than a joke."
"Oh! you don't know what a joke a beetle can be," said Patricia, looking up and laughing in spite of herself at the expression of anxiety on Mr. Triggs's face.
Mr. Triggs mumbled something to himself.
"God bless my soul!" he exclaimed a moment after. "'Ere am I, forgetting what I come about. I've seen The Morning Post, me dear." ”

“ The person who wants a triangle of which two angles and the sum of the sides are given, grasps in his mind the determinateness of the form of this triangle and restricts his search for it to a certain group of triangles of the same form. Under very different circumstances, therefore, the simplicity, the intellectual perviousness, of the subject-matter of mathematics and natural science is felt, and promotes both the discipline and the self-confidence of the reason. ”

“ It will now be understood when I say that one of the keys to the puzzle was the fact that these are always the squares of triangular numbers—that is, the squares of 1, 3, 6, 10, 15, 21, 28, etc., any of which numbers we have seen will form a triangle.
Every whole number is either triangular, or the sum of two triangular numbers or the sum of three triangular numbers. That is, if we take any number we choose we can always form one, two, or three triangles with them. ”