Scalene triangle

“ Earth alone of these elements is not subject to change, the assumed cause being the peculiarity of its constituent triangles. For he thinks that in all the other elements the figures employed are homogeneous, the scalene triangle out of which they are all put together being one and the same, whereas for earth a triangle of peculiar shape is employed; the element of fire is a pyramid, of air an octahedron, of water an icosahedron, of earth a cube. ”

“ Without any knowledge of the kinds of triangles the student discovers by examining the various shapes of many triangles that there is a group in which none of the sides are equal. For the lack of a better name he terms these non-equilateral (scalene) . Further observation discloses another group in which two of the sides are equal. These he names bi-equilateral (isosceles) . Finally a third group is designated as tri-equilateral (equilateral) . This process is classification. Division would consist in separating the genus triangle into the three kinds—scalene, isosceles, equilateral. ”

“ The Pattern (Fig. 277) .—A distance equal to the hypotenuse of triangle AD is set off upon any vertical line. These points are lettered A and 1. With point 1 as a center and a radius equal to the hypotenuse of triangle 1–B, an arc is drawn bearing away from point A. This is intersected by an arc drawn from point A, whose radius is equal to line AB of Fig. ”

“ When you hit upon the right one and leave it about, you'll start seeing things sooner or later. But your chances are small. It is not an isosceles triangle, but a scalene. The original pentagram was a very rough effort, far from symmetrical, and only by a fluke did it contain this dangerous triangle.
"How did I discover all this? It began with my investigation of the haunting of a cottage in Norfolk. I connected the phenomena with a small glass prism which had been lying about the place (the former occupant was a spectroscopist—until he went mad and was put away) . ”

“ Rectilinear figures cannot be divided into triangles and quadrilaterals because there are rectilinear figures which have more than four sides. On the other hand, triangles can be divided into equilateral, isosceles and scalene, since no other kind of triangle can exist. ”

“ Here there are three similar planes of symmetry intersecting in the triad axis, and perpendicular to them is a fourth plane of symmetry; at the intersection of the three vertical planes with the horizontal plane are three similar dyad axes; there is no centre of symmetry.
The general form is bounded by twelve scalene triangles and is a ditrigonal bipyramid. Like the general form of the last class, this has two sets of indices {hkl, pqr}, (hkl) for faces above the equatorial plane of symmetry and (pqr) for faces below: with hexagonal axes there would be only one set of indices. ”

“ If the measure of the sine line of the angle under consideration is equal to the measure of the sine line of one of the other angles they both must be acute, for it is impossible for them both to be obtuse, as we have shown; and it is likewise impossible for one to be acute and the other obtuse, for if this were so these angles would be like an acute and obtuse angle that share one sine line, the two of which already total 180 degrees without the inclusion of the third angle of the triangle, and all three angles of a triangle must total no more than 180 degrees (as was written in section 63) . ”

“ 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. ”

“ If P, Q, R, be any three forces acting along BC, CA, AB, respectively, the line of action of the resultant is determined by the consideration that the sum of the moments about any point on it must vanish. Hence in “trilinear” co-ordinates, with ABC as fundamental triangle, its equation is Pα + Qβ + Rγ = 0. If P : Q : R = a : b : c, where a, b, c are the lengths of the sides, this becomes the “line at infinity,” and the forces reduce to a couple.
The sum of the moments of the two forces of a couple is the same about any point in the plane. ”

“ It is useless to consider here separately spherical trigonometry, which cannot give rise to any special philosophical consideration; since, essential as it is by the importance and the multiplicity of its uses, it can be treated at the present day only as a simple application of rectilinear trigonometry, which furnishes directly its fundamental equations, by substituting for the spherical triangle the corresponding trihedral angle. ”

“ If a, b, c, be the sides, and A, B, C, the angles of a spherical triangle, right-angled at A, then . This is frequently called "Geber's Theorem." Radical and bold as were his innovations in spherical trigonometry, in plane trigonometry he followed slavishly the old beaten path of the Greeks. ”

“ Calculation of Triangulation.
The surface of Great Britain and Ireland is uniformly covered by triangulation, of which the sides are of various lengths from 10 to 111 miles. The largest triangle has one angle at Snowdon in Wales, another on Slieve Donard in Ireland, and a third at Scaw Fell in Cumberland; each side is over a hundred miles and the spherical excess is 64′′. The more ordinary method of triangulation is, however, that of chains of triangles, in the direction of the meridian and perpendicular thereto. ”

“ The bases of two or more triangles having a common vertex are given, both in magnitude and position, and the sum of the areas is given; prove that the locus of the vertex is a right line.
30. If the sum of the perpendiculars let fall from a given point on the sides of a given rectilineal figure be given, the locus of the point is a right line.
31. ABC is an isosceles triangle whose equal sides are AB, AC; B′C′ is any secant cutting the equal sides in B′, C′, so that AB′ + AC′ = AB + AC: prove that B′C′ is greater than BC.
32. ”

“ Of course, all the men must be used on every occasion.
137.—A STUDY IN THRIFT.
Certain numbers are called triangular, because if they are taken to represent counters or coins they may be laid out on the table so as to form triangles. The number 1 is always regarded as triangular, just as 1 is a square and a cube number. Place one counter on the table—that is, the first triangular number. Now place two more counters beneath it, and you have a triangle of three counters; therefore 3 is triangular. Next place a row of three more counters, and you have a triangle of six counters ”

“ Trilinear and Areal Coordinates.—Consider a fixed triangle ABC, and regard its sides as produced without limit. Denote, as in trigonometry, by a, b, c the positive numbers of units of a chosen scale contained in the lengths BC, CA, AB, by A, B, C the angles, and by Δ the area, of the triangle. We might, as in § 6, take CA, CB as axes of x and y, inclined at an angle C. Any point P (x, y) in the plane is at perpendicular distances y sin C and x sin C from CA and CB. Call these β and α respectively. ”

“ It is rarely that the terms of the fourth order are required. As a rule spheroidal triangles are calculated as spherical (after Legendre) , i.e. like plane triangles with a decrease of each angle of about ε /3; ε must, however, be calculated for each triangle separately with its mean measure of curvature k.
The geodetic line being the shortest that can be drawn on any surface between two given points, we may be conducted to its most important characteristics by the following considerations: let p, q be adjacent points on a curved surface ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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? ”

“ This process can be shortened by omitting the squaring of the hypotenuse, and just dividing the square of the base by the hypotenuse, with the remainder then subtracted by the hypotenuse, as is known in algebraic math. Half of this remainder is then the length of the vertical, as is shown in this formula:
( (a^2) /a) - (b^2) = ( (b^2) /a) -a
Section 120 All this applies to a right triangle. If the triangle is not a right triangle but an equilateral triangle there is only a need to know the measure of one side, and there is no need to know any of the angles. ”

“ To get the analogous proposition of plane geometry, pass a plane through the vertex so as to cut the base. We shall then have the sides and altitude of the triangle divided proportionally, and of course the section will merely be a line-segment, and therefore it is similar to the base line.
The cutting plane may pass through the vertex, or it may cut the pyramidal space above the vertex. In either case the proof is essentially the same.
Theorem. The volume of a triangular pyramid is equal to one third of the product of its base by its altitude, and this is also true of any pyramid. ”

“ 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. ”

“ Christoph Clavius in 1574 deduced the axiom from the assumption that a line whose points are all equidistant from a straight line is itself straight. John Wallis in 1663 showed that the postulate follows from the possibility of similar triangles on different scales. Girolamo Saccheri (1733) showed that it is sufficient to have a single triangle, the sum of whose angles is two right angles. Other equivalent forms may be obtained, but none shows any essential superiority to Euclid’s. Indeed plausibility, which is chiefly aimed at, becomes a positive demerit where it conceals a real assumption. ”

“ On a given finite right line (AB) to construct an equilateral triangle.
Sol.—With A as centre, and AB as radius, describe the circle BCD (Post. iii.) . With B as centre, and BA as radius, describe the circle ACE, cutting the former circle in C. Join CA, CB (Post. i.) . Then ABC is the equilateral triangle required.
Dem.—Because A is the centre of the circle BCD, AC is equal to AB (Def. xxxii.) . Again, because B is the centre of the circle ACE, BC is equal to BA. Hence we have proved.
But things which are equal to the same are equal to one another (Axiom i.) ; therefore AC is equal to BC ”