Zigzag

“ First, we have a succession of ellipses with their longer axes directed towards the light, each of which is described nearer and nearer to its source; then the loops are drawn out into a strongly pronounced zigzag line, with here and there a small loop still formed. At the same time that the movement towards the light is increased in extent and accelerated, that in the opposite direction is lessened and retarded, and at last stopped. The zigzag movement to either side is likewise gradually lessened, so that finally the course becomes rectilinear. ”

“ Well, it is perfectly true that so long as our zigzag path is formed of "steps" parallel to the sides of the square that path must be of the same length as the two sides. It does not matter if you have to use the most powerful microscope obtainable; the rule is always true if the path is made up of steps in that way. ”

“ An experienced front man will probably take these on a zigzag. To a less experienced walker, and to all beginners of energy and leg muscle, it is generally a temptation to cut the zigzags on the direct line, and so pass ahead. ”

“ The geometric figures present an endless variety; but we notice, as is shown by the cuts and plates, that triangles with an elongate acuminate apex and the zigzag are very common in the black-brown decorations. ”

“ Heliotropism.—When a plant which is strongly heliotropic (and species differ much in this respect) is exposed to a bright lateral light, it bends quickly towards it, and the course pursued by the stem is quite or nearly straight. But if the light is much dimmed, or occasionally interrupted, or admitted in only a slightly oblique direction, the course pursued is more or less zigzag; and as we have seen and shall again see, such zigzag movement results from the elongation or drawing out of the ellipses, loops, etc., which the plant would have described, if it had been illuminated from above. ”

“ A professor of mathematics in Geneva saw the numbers in a zigzag line which made turns at 10 and at 60, up to 116, and no further, and added to his description that when young he likened some sounds to colors: a grave sound was black, a less grave one, red ”

“ The region of ice and snow having been passed, tunnelling was no 96longer necessary. Four zigzags, each a mile long, sufficed to reach the first terrace, where another depôt was constructed; and a few years’ more labour, and about a dozen similar zigzags, accomplished the ascent of the tremendous middle precipice, thus bringing them within fifteen thousand feet of the cloud-screen.
As the great work neared its completion, the anxiety and excitement, not only of those actually engaged in it, but of the entire population of the planet, rose to a scarcely conceivable intensity. ”

“ If we have any doubt as to the security of a slope, which we shall have when the angle of the slope and quality of the snow are varying and the resultant stability is indeterminable, we must ascend or descend it vertically, if possible, or on very steep zigzags. ”

“ The track, cut, broken, and worn on the side of a wall of rock nearly 2,000 feet in abrupt elevation, is a series of rough narrow zigzags, rarely, if ever, wide enough for laden animals to pass each other, composed of broken ledges often nearly breast high, and shelving surfaces of abraded rock, up which animals have to leap and scramble as best they may. ”

“ A picture show, like everything else, should be taken on the zig-zag. The man who plans and cogitates the nearest way between two streets—that man is too busy, poor fellow, to know the sweets of the zig-zag. To go upon the zig-zag is to see more, and with greater entertainment. Who sees more stars, more lamp-posts, front-doors, and keyholes than other men—yea, even unto tenfold? He who goes home on the zig-zag. The zig-zag is the token, the mystic sign, of contentful ease and good fellowship the world over ”

“ Sol.—Draw the diagonals AC, BD intersecting in O (see diagram to Proposition vi.) . O is the centre of the required circle.
Dem.—Since ABC is an isosceles triangle, and the angle B is right, each of the other angles is half a right angle; therefore BAO is half a right angle. In like manner ABO is half a right angle; hence the angle BAO equal ABO; therefore [I. vi.] AO is equal to OB. In like manner OB is equal to OC, and OC to OD. Hence the circle described, with O as centre and OA as radius, will pass through the points B, C, D, and be described about the square. ”

“ The circle is, of course, a perfect illustration of Measure and Shape-Balance on a center. The balance is also symmetrical. We have a Harmony of Directions in the repetition of the same change of direction at every point of the outline, and we have a Harmony of Distances in the fact that all points of the outline are equally distant from the balance-center, which is unmistakably felt.
Fig. 167
The Ellipse is another example of Measure and Shape-Balance on a center. In this attitude it is also an illustration of Symmetry.
Fig. 168
In this case we still have balance but no symmetry. ”

“ Suppose, for instance, the mirror were cylindrical, and convex, and the object a circle placed directly in front of it. It will easily be seen that that diameter of the circle, which is parallel to the axis of the cylinder, will not be altered in the reflexion, but that the diameter perpendicular to that axis with all chords parallel to it, will have for their images portions of conic sections of less breadth than themselves, so that the image will appear diminished in breadth, and distorted into a form like that of the bowl of a spoon. ”

“ Suppose the circle whose centre is O and one of the points of intersection A to remain fixed, while the second circle turns round that point in such a manner that the second point of intersection B becomes ultimately consecutive to A; then, since the line OO′ always bisects AB, we see that when B ultimately becomes consecutive to A, the line OO′ passes through A. In consequence of the motion, the common chord will become in the limit a tangent to each circle, as in the second diagram.—Comberousse, Géométrie Plane, page 57. ”

“ Their self-similarity. Every inch of a straight line coincides with every other inch, and off a circle with every other off the same circle. Where, then, did Euclid fail? In not introducing the third curve, which has the same property—the screw. The right line, the circle, the screw—the representations of translation, rotation, and the two combined—ought to have been the instruments of geometry. With a screw we should never have heard of the impossibility of trisecting an angle, squaring the circle, etc.—De Morgan, A.
Quoted in Graves’ Life of Sir W. R. Hamilton, Vol. ”

“ And thus this folded wave will continue to advance until the point N has reached the focus E. The curve AFE can be seen in smoke, or in flying dust, when a concave mirror is held opposite the sun. And it should be known that it is none other than that curve which is described [Pg 128] by the point E on the circumference of the circle EB, when that circle is made to roll within another whose semi-diameter is ED and whose centre is D. So that it is a kind of Cycloid, of which, however, the points can be found geometrically. ”

“ We may take any point along the major axis, as shown, for instance, in Fig. 149. Let B be this point, taken at random. Then describe the half circle (C) . We may, also, arbitrarily, take any point, as, for instance, D on the minor axis E, and by drawing the diagonal lines (F) we find marks on the circle (C) , which are the meeting lines for the large curve (H) , with the small curve (C) . In this case we have formed an ovate or an oval form. Experience will soon make perfect in following out these directions.
Focal Points.—The focal point of a circle is its center, and is called the focus. ”

“ Gradually it learns to get the same assurance from some distant point on the floor ahead: its eye has been educated to reason from a diagonal as much as from a level glance; and the ‘semicircular canal’ has been taught to interpret the message, and convert it into automatic balance. As it moves forward it shifts the point ahead, unconsciously obtaining that the third side of the triangle thus formed by its glance shall always demonstrate that the other two sides, the line of the floor and the vertical line through its own centre of gravity, are forming an approximate right angle. ”

“ Knowing that dinner preparations would occupy Zöga for some time after his arrival, we rode leisurely along, admiring the green level plain far below us. When wondering how we were to get down to it, we suddenly and unexpectedly came to a yawning chasm or rent running down through the edge of the plateau. It seemed about 100 feet deep, 100 feet wide, and was partially filled with enormous blocks of basalt which had toppled down from either side; where more, cracked and dissevered, still impended, as if they might fall with a crash at the slightest noise or touch. ”

“ 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. ”

“ Being given an obtuse-angled triangle, draw from the obtuse angle to the opposite side a line whose square shall be equal to the rectangle contained by the segments into which it divides the opposite side.
53. O is a point outside a circle whose centre is E; two perpendicular lines passing through O intercept chords AB, CD on the circle; then AB2 + CD2 + 4OE2 = 8R2.
54. The sum of the squares on the sides of a triangle is equal to twice the sum of the rectangles contained by each perpendicular and the portion of it comprised between the corresponding vertex and the orthocentre ”

“ If the medium is symmetrical with respect to a certain plane, each ellipsoid must have an axis at right angles to that plane. If the medium after a revolution of less than 180° about a certain axis is then equivalent to the medium in its first position, or symmetrical with it with respect to a plane at right angles to that axis, each ellipsoid must have an axis of revolution parallel to that axis. ”

“ Draw a common tangent to two circles. Hence, show how to draw a line cutting two circles, so that the intercepted chords shall be of given lengths.
PROP. XVIII.—Theorem
If a line (CD) touch a circle, the line (OC) from the centre to the point of contact is perpendicular to it.
Dem.—If not, suppose another line OG drawn from the centre to be perpendicular to CD. Let OG cut the circle in F. Then because the angle OGC is right (hyp.) the angle OCG [I. xvii.] must be acute. Therefore [I. xix.] OC is greater than OG; but OC is equal to OF [I. Def. xxxii.] ”

“ A radius drawn from the centre to the number 30 on the graduated semicircle, will form an angle of 30 degrees with the diameter of the semicircle. And so for any other number of degrees. It will be seen that any number of degrees less than 90 will make an acute angle, and more than 90 degrees will form an obtuse angle thus, in fig. 4, 30 degrees form an acute angle, and the remaining 150 degrees of the half circle form an obtuse angle. ”

“ D a horizontal stroke, longer and thicker than in kappa, runs to the left from the bottom of the vertical line.
Sigma retains its angular shape ( [Symbol] or Σ) only on inscriptions, as the Rosetta, and that long after the square shapes of omicron and theta were discarded. The uncial or semicircular form, however, arose early, and to this letter must be applied all that was said of epsilon as regards terminal points (a knob at the lower extremity occurs even in Cod. א, e.g. Acts ii. 31) , and its cramped shape in later ages. ”

“ The line joining the middle points of the diagonals of a quadrilateral circumscribed to a circle—
divides each pair of opposite sides into inversely proportional segments; is divided by each pair of opposite lines into segments which, measured from the centre, are proportional to the sides; is divided by both pairs of opposite sides into segments which, measured from either diagonal, have the same ratio to each other. ”

“ When, proceeding from this point, the rings without the cross and with the black spot in the centre were formed, this spot, on the heat being increased, divided itself into two, which removed themselves from the centre into the second and fourth quadrants, and after having passed through the figure in the circular light, closed into a circle with the arcs proceeding from the first and third quadrants, so as to produce the system of rings with a bright centre, which would have been obtained at the very beginning by turning the polarizing prism 90°. ”

“ No regard need be paid to the situation of the zeros of the plate.
The radial bar must be set home upon the base; and while one hand presses the plate firm against the rock, the other must be applied to the head of the screw g, and bring the quadrant round till the level, properly adjusted by turning it upon its axis, is horizontal. Its axis will now be parallel to the line of stretch, that is, to a line determined by the intersection of the plane of the stratification with the plane of the horizon. ”

“ For a five-sided figure it is best to step around the circumference of the circle with the compasses, but for a three-sided one, or trigon, the construction is as follows: It will be found that the compasses set to the radius of a circle will accurately divide it into six equal divisions, as is shown in Figure 178; hence every other one of these divisions will be the location for a corner of a trigon. ”