Oval

“ It is noteworthy that the oval hole is preferred by eastern peoples. The Greek auloi preserved in the British Museum possess oval holes, as do the pipes of Egypt, the arghool pipes, the Lady Maket pipes; and in truth the oval is the form naturally derived by cutting upon a circular surface, and it is also well adapted to the fingers; nothing but a formality for elaborating could have induced the modern habit of making round holes. ”

“ Suppose that within the oval any point is given, about which as a pole a right line is perpetually revolving, with an uniform motion, while in that right line a moveable point going out from the pole, moves always forward with a velocity proportional to the square of that right line within the oval. ”

“ The eye rests with equal delight on what is not leaf and on what is leaf,—on the broad, free, open sinuses, and on the long, sharp, bristle-pointed lobes. A simple oval out line would include it all, if you connected the points of the leaf ”

“ So that in a rough way it is a solid sound earth, surrounded by an oval body of water: the long axis of the oval representing the high tides, which, as they follow the moon, slide completely around the earth once in every twenty-four hours. ”

“ The flints in general form round or oval masses, but of irregular shape with many projections, and the masses lie in regular bands parallel to the stratification. The tremendous earth movement which has bent the strata into a great curve has compressed the vertical portion into about half its original thickness, and has made the chalk of our downs extremely hard. ”

“ It seems scarcely open to doubt that the oval forms are the older, though what their age may be is not so clear, nor have any descriptions of their contents been published which would enable us to form distinct opinion on the subject. ”

“ But the puzzle is to show how I was able to draw this spiral with so much exactitude without using anything but a pair of compasses and the sheet of paper on which the diagram was made. How would you proceed in such circumstances?
184.—HOW TO DRAW AN OVAL.
Can you draw a perfect oval on a sheet of paper with one sweep of the compasses? It is one of the easiest things in the world when you know how. ”

“ In any oval the diameter of this circle is equal to the width of the oval. The rectangle has for its dimensions the width of the oval, and the difference between the width and the length of the oval.
Example.—What is the area in square inches of an oval profile 4" wide and 14" long?
Area of rectangle = 4×10 = 40 sq. in.
Combined areas = 40+12.57 = 52.57 sq. in. Ans.
Problem 47A.—What are the areas of the following sizes of oval profiles?
(a) 31⁄2"×15" (b) 41⁄2"×141⁄2 (c) 3"×11" (d) 33⁄4"×151⁄2 (e) 6"×133⁄4" Problem 47B.−An 8" round pipe is to be "ovaled down" to a width of 35⁄8". ”

“ The forward triangle cut off by the road is a section of a field from the dark earth of which myriad bright-green blades of fall-sown rye are sprouting. A straggling line of piled rocks, too low to be called a wall, separates this field from the road. ”

“ Whether a Branch (suppose) of three Circles, cut at Spring, the sap ascending, or another of the same bigness, will at Michaelmas following, if cut again, be found to have increased one or more Circles than it had in the Spring: and whether at Spring or Fall, or at other season, it be found to have a Circle or half a Circle of Pricks next or betwixt the Barks, or a Circle of Wood next the inner Bark onely, or both? But here the Comparison is to be made with distinction, For it must be inquired, Whether some Trees shoot new Tops every year until a certain Age, and after not? ”

“ The distance thither from our house is rather more than a mile. The intermediate region is level low ground, bounded on each side, at some distance, by a ridge of mountain. These two ridges gradually converge, until they pass the Spring about one hundred yards, where a third ridge brings a sweep immediately across the line of their direction, and closes that end of the valley. The space about the Spring is a perfect level, amply extensive, and admirably adapted for improvements on a large and handsome scale. ”

“ One line, about two hundred feet in height, is regularly anticlinal, with the strata dipping away on both sides, at an angle of 15 degrees, from a central “valley of elevation,” about three hundred yards in width. A second narrow steep ridge, only sixty feet high, is uniclinal, the strata throughout dipping westward; those on both flanks being inclined at an angle of from ten to fifteen degrees; whilst those on the ridge dip in the same direction at an angle of between thirty and forty degrees. ”

“ When the elevation is determined by sights the angle of elevation is the angle between the line of sight and the axis of the piece, when these lines are in the same vertical plane,—or the angle between the line of sight and a plane containing the axis of the piece and a horizontal line intersecting it at right angles, when they are not. The graduations of tangent scales and fixed breech-sights give this angle in degrees. The graduation of the pendulum hausse gives the angle correctly only when the line of sight is horizontal. ”

“ As the horizontal contours of any portion of the earth's surface must depend on the vertical, or, in other words, the various features which are marked on the map—the coast line, plains, rivers, vallies, lakes, hills, mountains, &c., being consequent on the different elevations of the various portions of the surface above the level of the sea ”

“ For instance, no mountain in the Alps produces a more vigorous impression of peakedness than the Matterhorn. In Professor 180 Forbes's work on the Alps, it is spoken of as an "obelisk" of rock, and represented with little exaggeration in his seventh plate under the outline Fig. 31. Naturally, in glancing, whether at the plate or the mountain, we assume the mass to be a peak, and suppose the line a b to be the steep slope of its side. ”

“ This range is remarkable for the uniformity of outline which characterizes the different peaks, particularly of their summits, the ridges extending in the same general direction, sometimes hardly diverging from a straight line for a distance of fifty or sixty miles. Where the mountain chains are parallel, the ridges are also in parallel lines, preserving their general direction, and, to a wonderful extent, a uniformity of distance between them. When one curves round in a new direction, all curve with it. ”

“ If we draw a circle about this pole on a common terrestrial globe, and intersect it by spirals to represent the direction of the wind, we shall see that the wind enters all parts of this circle from the southwest, and, consequently, that a whirl ought to be created thereby, in which the ascending column of air revolves from right to left, or against the hands of a watch. ”

“ For in a right sphere the needle is parallel to the plane of the horizon; but when the cœlestial pole is vertically overhead, or when the pole of the earth is itself the place of the region, then the needle is perpendicular to the horizon. ”

“ To lay off the directions of these resultants, from any convenient point, , say, in the vertical through , as a center, describe an arc, , with a convenient radius, . With the same radius and as a center, describe the arc, , cutting the trial planes (produced if necessary) at , , ..., . Let be the point where the line of natural slope from cuts the arc, . On laying off the chords, , , ..., equal to the chords, , , ..., respectively, it will follow that , , ..., will make angles, , below the normals to the planes, , , ..., respectively. ”

“ The general observer, if he takes his stand on the edge of hills by which this basin is bounded, will see the enclosing character of the ridge, as well as the less conspicuous circle of somewhat elevated land occupying the central portion of the field, and which is separated by a valley or plain from the surrounding ridge. ”

“ In the third place, they recommend those veins which extend from XII north to XII south, through the slope of a mountain which faces east; whose hangingwalls are in the west, whose footwalls are in the east; whose heads rise toward the east; and whose rock seams raise their heads toward the north. Therefore they devote all their energies to those veins, and give very little or nothing to those whose heads, or the heads of whose rock seams rise toward the south or west. ”

“ RIDGES.—Among the Maria which exhibit the most remarkable arrangement of ridges is the Mare Humorum, in the south-eastern quadrant. Here, if it be observed under a rising sun, a number of these objects will be seen extending from the region north of the ring-mountain Vitello in long undulating lines, roughly concentric with the western border of the "sea," and gradually diminishing in altitude as they spread out, with many ramifications, to a distance of 200 miles or more towards the north. ”

“ Wave motion consists only in a vertical movement of the particles of water by which a crest and trough is formed alternately, the crest being as much above the normal horizontal line as the trough is below it ”

“ But then a plaine may be drawne by them: Therefore both [56] of them shall be in the same plaine. And this plaine is geometrically to be conceived: Because the same plaine is not alwaies made the ground whereupon one oblique line, or two cutting one another are drawne, when a periphery is in a sphearicall: Neither may all peripheries cutting one another be possibly in one plaine.
And
9. With a right line given to describe a peripherie.
This fabricke or construction is taken out of the 3. Petition which is thus. Having a center and a distance given to describe, make, or draw a circle. ”

“ The apparent vertical indicated by the plummet-line or determined at the level of the sea, is simply the direction of weight, which may be materially affected by local attractions due to an irregular distribution of the masses composing the soil. ”

“ We usually specify a slope by its inclination to a level plane which is parallel to the plane of the horizon, or at right angles to the direction of gravity. At any given spot the level is, physically considered, definite and unalterable. ”

“ The angle of slope on either side is a little more than fifty-one degrees, a slope which is often called perpendicular, and, moreover, as the strata are horizontal, there are several vertical walls of rock, which sweep around the entire north and west faces, and apparently make impassable barriers. ”

“ On each profile across the continental margin is a series of benches and changes in gradient which range from the shelf break to slight changes in gradient on the continental slope.
If a field geologist enters a new area of sedimentary rocks where road cuts do not exist he invariably goes to the stream valleys, and here he gets his first and best view of the geologic section. The stream's gradient is adjusted to the resistance of the rocks over which it cuts, and the form of the valley-side slopes reveals the nature of the underlying rocks even if they are grassed over. ”

“ For example, the windings of a crooked stream, or a road, in perspective, are much exaggerated in retiring into the distance; they must therefore be straightened out in the sketch more and more, as they are more removed. 2nd. In looking at variously placed hills from a somewhat elevated station, the eye will in some cases look directly, or perpendicularly, at the face of some slopes, while in others, the surface of the slope, if prolonged, will pass through the eye, and will not be seen in its true dimensions, though its inclination may be judged. ”

“ When the fold is abrupt, so that the rocks on either side are made to incline at a great angle, ridges are formed, and this topographic structure of a country may be found even in a land of rains, though the ridges will usually be low, rounded, and more or less irregular, while in a dry climate they will be steep and regular, and will usually culminate above in a sharp edge ”

“ They stretched from north to south like an immense wall, occupying all that portion of the horizon lying to the north-west, west, and south-west. We could now see the surface of the plain, extending almost unvaried to the base of the first ridge, which rises by an abrupt ascent above the commencement of the snow. ”

“ If we compare the sloping strata to the roof of a house, a line down the slope of the roof will mark the direction of the dip, the ridge of the roof that of the strike. The strata we are considering dip towards the North ”

“ For so, by an infinite number of curves from every point of the equator dividing the sphere into two equal parts, and from every point of the surface from the æquator towards the North, and from the æquator towards the Southern pole, the whole force tends asunder toward the poles. ”

“ As I sounded through the ice I could determine the shape of the bottom with greater accuracy than is possible in surveying harbors which do not freeze over, and I was surprised at its general regularity. In the deepest part there are several acres more level than almost any field which is exposed to the sun wind and plough. In one instance, on a line arbitrarily chosen, the depth did not vary more than one foot in thirty rods; and generally, near the middle, I could calculate the variation for each one hundred feet in any direction beforehand within three or four inches. ”

“ While there are only two distinct classes of shape—the ‘Standard’ and the ‘Legged’ ones—so much variety is introduced into these that many appear to belong to totally different orders. And when I stop and think over some I know, I wonder where exactly they should be put. Certainly the ‘Valley’ shape, set up on its curved crane, like a big ‘C’ spring with a lamp-bowl atop, is neither a ‘Standard’ nor a ‘Legged’ lantern. ”

“ The natural slope of different materials runs from a nearly vertical plane in firm rock to planes as flat as one foot rise in a horizontal distance of four feet, or technically defined, a slope of one on four. ”