Triangle

“ Thus should we mention the word triangle, and form the idea of a particular equilateral one to correspond to it, and should 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, though it be true with relation to that idea, which we had formed. If the mind suggests not always these ideas upon occasion, it proceeds from some imperfection in its faculties ”

“ 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. ”

“ 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. ”

“ But he that hath the use of words, when he observes that such equality was consequent, not to the length of the sides, nor to any other particular thing in his triangle but only to this, that the sides were straight, and the angles three and that that was all for which he named it a triangle, will boldly conclude universally that such equality of angles is in all triangles whatsoever, and register his invention in these general terms: every triangle hath its three angles equal to two right angles. ”

“ 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. ”

“ The character of the acute-angled triangle is this: if you multiply every one of its two short sides by itself, and add the products, their sum is more than the long side alone multiplied by itself. The definition of the obtuse-angled triangle is this: if you multiply its two short sides each by itself, and then add the products, their sum is less than the product of the long side multiplied by itself.
The rectangular triangle has two cathetes and an hypotenuse. It may be considered as the moiety of a quadrangle. You find its area by multiplying one of its cathetes by the moiety of the other. ”

“ The representation of a triangle is always limited to a certain size and figure. When we imagine a triangle, it is always with such or such extension and with greater or smaller angles. The imagination representing an obtuse angled triangle sees something very different from an acute or right angled triangle. But the idea of the triangle in itself is not subject to any particular size or figure ”

“ The young reader who knows nothing of the elements of geometry will get some idea of the fascinating character of that science. The triangle ABC in Fig. 27 is what we call a right-angled triangle, because the side BC is at right angles to the side AB. Now if we build up a square on each side of the triPg 32angle, the squares on AB and BC will together be exactly equal to the square on the long side AC, which we call the hypotenuse. This is proved in the case I have given by subdividing the three squares into cells of equal dimensions. ”

“ We have only to select a number of such triangles and a system of trigonometry ensues, in which base, perpendicular, and hypotenuse of every triangle is a whole measure without fractions, and in which the nomenclature for every angle is clear and simple. ”

“ It may now be pointed out that the triangle may be used to divide circles much more quickly than they could be divided by stepping around them with compasses. Suppose, for example, that we require to divide a circle into eight equal parts, and we may do so as in Figure 172, line a being marked from the square, and lines b, c and d from the triangle of forty-five degrees; the lines to be inked in to form an octagon need not be pencilled, as their location is clearly defined, being lines joining the ends of the lines crossing the circle, as for example, lines e, f. ”

“ If the exterior angle of a triangle, made by producing one of its sides, be bisected by a straight line which also cuts the base produced, the segments between the dividing straight line and the extremities of the base shall have the same ratio which the other sides of the triangle have to one another; and if the segments of the base produced have the same ratio which the other sides of the triangle have to one another, the straight line drawn from the vertex to the point of section shall bisect the exterior angle of the triangle. ”

“ Two triangles are all that are absolutely necessary for a beginner. The first is that shown in Figure 3, [20] which is called a triangle of 45 degrees, because its edge A is at that angle to edges B and C. That in Figure 4 is called a triangle of 60 degrees, its edge A being at 60 degrees to B, and at 30 degrees to C. The edges P and C are at a right angle or an angle of 90 degrees in both figures; hence they are in this respect alike. By means of these triangles alone, a great many straight line drawings may be made with ease without the use of a drawing square ”

“ Here are such propositions as that the three bisectors of the sides of a triangle are concurrent, needed in determining the position of the centre of gravity of a triangle; propositions in the circle needed in Practical Geometry and Mechanics; properties of the centres of similitudes, and the theories of inversion and reciprocations so useful in certain electrical questions. The proofs are always neat, and in many cases exceedingly elegant. ”

“ If a triangle of given form move so that its three sides pass through three fixed points, the locus of any point in its plane is a circle.
14. The angle A and the area of a triangle ABC are given in magnitude: if the point A be fixed in position, and the point B move along a fixed line or circle, the locus of the point C is a circle.
15. One of the vertices of a triangle of given form remains fixed; the locus of another is a right line or circle; find the locus of the third.
16. Find the area of a triangle— (1) in terms of its medians; (2) in terms of its perpendiculars. ”

“ With us the representation of the triangle is almost always simultaneous, excepting the case of exceedingly large triangles, much larger than we usually see, in which case, especially when we are unaccustomed to consider such, it seems necessary to go on extending the lines successively. ”

“ 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 ”

“ Draw an isosceles triangle whose equal sides shall each be less than the third side.
Write its two names within it.
Draw an oblique straight line twice as long as any short measure or unit.
At one end draw a straight line perpendicular to it, and three times as long as the same measure. ”

“ 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. ”

“ What more easy than for any one to look a little into his own thoughts, and there try whether he has, or can attain to have, an idea that shall correspond with the description that is here given of the general idea of a triangle—which is neither oblique nor rectangle, equilateral, equicrural nor scalenon, but all and none of these at once? ”

“ In this case, if we take the triangle together with the line at infinity, we get as the projection a triangle ABC, and some other line j which cuts the sides a, b, c of the triangle in the points Al, Bl, Cl. ”

“ The Triangle
Beneath Andromeda. Three lines compose
The Triangle. On two sides measured equal,
The third side less. It is not difficult
To be discerned. More luminous than many.
Frothingham's Aratos.
The Triangle is an asterism of considerable antiquity, and was evidently more noticed by the ancients than by us. ”

“ If it is triangle, the quality of having two right angles belongs to the particular kinds of triangles as a result of its belonging essentially to triangle, and the demonstration in regard to triangle is a universal demonstration. ”

“ The triangulum characteristicum is similar to the triangle formed by the tangent, the ordinate of the point of contact, and the sub-tangent, as well as to that between the ordinate, normal, and sub-normal. ”

“ Then, with my compass at E, I sight another line at an angle of sixty degrees from my last. I am making what is called an equilateral triangle; that is, a triangle with equal sides and equal angles. Each angle must measure sixty degrees. With two angles and one side, we can always get the other two sides; and the other angle will be where those two sides meet. They will meet at C. Now, since the sides are of equal length, the distance from D to C is the same as from D to E,—that is, four rods and thirty links, just the distance we wish to go; C, then, is the place for your corner stake. ”

“ THE MOVEMENT OF THE TRIANGLE The life of the spirit may be fairly represented in diagram as a large acute-angled triangle divided horizontally into unequal parts with the narrowest segment uppermost. The lower the segment the greater it is in breadth, depth, and area. ”

“ Pure triangular composition is a form rarely seen, as, in most cases where the lines of the triangle are detected as the first conception, other lines or points have been added to destroy or modify them. ”

“ If a minor triangle is produced which has all its angles acute and its three sides equal, then necessarily the second and third sides of the minor triangle will be equal, and likewise the sides of the major triangle frequently referred to will be equal. ”

“ Now, the most elementary geometry teaches us that every triangle having these angles will have the lengths of its sides in the same invariable proportions one to another whatever may be the size of the triangle itself, and it has been found convenient to calculate these proportions once for all, not merely for angle 20°, but for every angle up to 90°. ”

“ 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. ”