Perimeter

“ Three feet being the average Perimeter it follows that, in a Polygon of three hundred sides, each side will be no more than the hundredth part of a foot in length, or little more than the tenth part of an inch; and in a Polygon of six or seven hundred sides the sides are little larger than the diameter of a Spaceland pin-head. It is always assumed, by courtesy, that the Chief Circle for the time being has ten thousand sides. ”

“ Shell-mantle one and a half times as long as broad, its frontal perimeter (fig. 4a) heptagonal, with seven concave sides, its sagittal perimeter octagonal, its equatorial perimeter hexagonal (fig. 4b) , the corners of the polygons are marked by the sixteen prominent styles. ”

“ Taking the next wheels in Fig. 12, it has been shown that the 3.33 lbs. delivered from a to the perimeter of b, becomes 2.49 lbs. at the perimeter of c, and it has also been shown that c makes two revolutions to one of a, and its diameter being 40 inches, the distance this 2.49 lbs. will move through in one revolution of a will therefore be equal to twice its circumference, which is (40 dia. × 3.1416 = 125.666 cir., and 125.666 × 2 = 251.332) 251.332 inches. Now 2.49 lbs. moving through 251.332 gives when brought to inch lbs. 627.67 inch lbs., thus 251.332 × 2.49 = 627.67. ”

“ Another inset shows the equivalence of the decagon and a rectangle which has one side equal to the perimeter of the decagon and the other equal to half of the altitude of each triangle composing the decagon. Small triangles divided horizontally in half can be fitted into this figure, with one of the upper triangles divided in half lengthwise.
Thus we demonstrate that the surface of a regular polygon may be found by multiplying the perimeter by half the hypotenuse.
SOME THEOREMS BASED ON EQUIVALENT FIGURES
A. All triangles having the same base and altitude are equal. ”

“ Let the letters a b c denote the three angular points of a rectilineal triangle. If the point did move continuously over the lines ab, bc, ca, that is, over the perimeter of the figure, it would be necessary for it to move at the point b in the direction ab, and also at the same point b in the direction bc. These motions being diverse, they cannot be simultaneous. ”

“ The course of the style, which corresponds to a kilometre, ought to be experimentally determined for each vehicle, for the perimeter of wheels is not always the same. But it is clear that, if from each kilometre-stone to another we obtain five millimetres, for example, for the course of the style, this length will always be found to be traversed each kilometre by the same vehicle. Our apparatus is then a measurer of distances, and dispenses with the necessity of attending to ”

“ The Surface of every Round Solid is equal to a Rectangle, whose Base is the Circumference of the Figure, by the Rotation whereof the Solid is begot, and the Height equal to the Circumference, which the Center of Gravity of the Perimeter of the Figure describeth. ”

“ The fourth Booke, which is of a Figure.
1. A figure is a lineate bounded on all parts.
So the triangle aei. is a figure; Because it is a plaine bounded on all parts with three sides. So a circle is a figure: Because it is a plaine every way bounded with one periphery.
2. The center is the middle point in a figure.
In some part of a figure the Center, Perimeter, Radius, Diameter and Altitude are to be considered. The Center therefore is a point in the midst of the figure; so in the triangle, quadrate, and circle, the center is, aei. ”

“ It is mathematically equal to the product of the length of the circumference, multiplied by half the radius. To square a circle of a given diameter in mètres, is the same as giving the number of squares, of a mètre in each side, of which the surface is the equivalent. If, the diameter being given, the exact circumference were known by a sort of inspiration, the superficial extent of the circular space would be deducible from the two numbers, by the mere multiplication of the numerical length of the circumference by the fourth of the diameter, or half the radius. ”

“ One degree, therefore, is the 360th part of a circumference, and thus we have a measure independent of all dimensions. For example, on a round table of 36 yards in circumference, one degree will be marked by one tenth of a yard; on a pond of 360 yards in circumference, one degree will be equal to one yard.
The degree, therefore, may be more or less, but it is always the 360th part of the circumference of a circle. Let it be quite understood that, whether an angle is to be on a sheet of paper, or in the skies, the divisions do not change. ”

“ The word perpendicular, as usually employed in astronomy, means the same thing, but in geometry, or in drafting, or in its use in the arts it means that a perpendicular [Pg 90] line is at right angles to some other line. Suppose you put a square upon a roof so that one leg of the square extends up and down on the roof, and the other leg projects outwardly from the roof. In this case the projecting leg is perpendicular to the roof. Never use the word vertical in this connection.
Signs to Indicate Measurements.—The small circle (°) is always used to designate degree. Thus 10° means ten degrees. ”

“ If any point be taken without a circle, and straight lines be drawn from it to the circumference, one of which passes through the centre; of those which fall on the concave circumference, the greatest is that which passes through the centre, and of the rest, that which is nearer to the one passing through the centre is always greater than one more remote; but of those which fall on the convex circumference, the least is that between the point without the circle and the diameter; and of the rest, that which is nearer to the least is always less than one more remote ”

“ For a plane surface, the ratio of the circumference of a circle to its diameter, both lengths being measured with the same rod, is, according to Euclidean geometry of the plane, equal to a constant value , which is independent of the diameter of the circle. On their spherical surface our flat beings would find for this ratio the value i.e. a smaller value than , the difference being the more considerable, the greater is the radius of the circle in comparison with the radius R of the “world-sphere.” ”

“ Since the straight diameter is easily measured with great accuracy, when he had the area he could readily have found the circumference by working backward the rule announced by Archimedes, viz.: that the area of a circle is equal to that of a triangle whose base has the same length as the circumference and whose altitude is equal to the radius. ”

“ If the circumference of a circus measuring 180 meters be divided into 360 places, each measuring 0.50 meters in width, then when the circus is full a person placed at the center will see each spectator occupying an angle of 1 degree. The angle does not alter with the distance, and whether it be measured at 1 meter, 10 meters, 100 kilometers, or in the infinite spaces of Heaven, it is always the same angle. Whether a degree be represented by a meter or a kilometer, it always remains a degree. ”

“ But, the circumference being deducible from the diameter only by approximation, the surface alluded to cannot be computed with mathematical rigour; yet the result can be obtained with all desirable precision by the aid of the ratios usually given for such purpose; for instance, the area of the space included within a circle of thirty-eight millions of leagues radius, may be determined within such a degree of precision that the probable error shall not exceed the space of a mite. ”

“ Is an arc of a hundred and seventy-nine degrees greater or less than a semi-circumference?
Can there be more than three hundred and sixty degrees in a circumference?
If the circumference of circle 1 were divided into degrees, each degree would be so small an arc that it would look like a dot.
If a degree were divided into sixty equal parts, each part would be called a minute.
If a minute were divided into sixty equal parts, each part would be called a second.
How many degrees in the large circle of Fig. 2?
How many in the smaller one?
Has a large circle any more degrees than a small circle? ”

“ I allude to the word "perpendicular." Many people think that the word "perpendicular" means the position of a plumb-line. It means no such thing, The proper word for describing that position is " vertical." Perpendicular is a relative word, and it ought not to be used without reference to something else. Vertical is an absolute word. Thus, in speaking of the word perpendicular, with reference to that trough of water or prism in the experimental case before us, I mean a line perpendicular, not to the horizon, but to the surface of the trough. ”

“ Place the point of angle on the centre of the semi-circle, or the centre of the semicircle on the point of the angle, and then by seeing how many degrees the opening of the angle measures on the graduated edge of the semicircle, you will find the size of the angle. If the sides of the angle do not extend to the circumference, you may extend them till they do. If they extend beyond the circumference, measure the angle where the graduated circle cuts the sides. ”

“ The angle available for subdivision, to measure ranges between infinity and 250 yds., is only one-third of a degree. In a travel of 6 in. the prism renders accurate measurements possible within the required limits. To bring images of distant objectives into coincidence, the prism must be moved towards the eye-piece, and for near objectives in the opposite direction. The range scale is attached to the prism. A consequent advantage is that the accuracy of the instrument is not affected by back lash arising from wear, or irregularity in the actuating mechanism. ”

“ Even, however, if we confine ourselves to mile, yard, foot and inch, we have the following six connecting ratios: 3, 12, 36, 1,760, 5,280, 63,360.
In the metric system there is the meter, about ten per cent, longer than the yard, and its decimal derivatives, all evaluated at a glance by a shift of the decimal point. In English speaking countries, roads are measured in miles and furlongs, short distances in yards, houses or ships in feet, horses in hands and small objects in inches. ”

“ The construction and evolution of the system of electrical units now in general use began in Europe where the metric system originated. Two of the basic units, the centimeter and the gram, are metric units; the second of time is universally employed in time measurements.
The meter, 3.2808 feet = 39.37 inches, is theoretically the one ten-millionth part of the earth’s quadrant ”

“ Therefore, if we could easily measure ∠O and could measure the distance x, we could find the height of a [Pg 236] building y. In trigonometry we use a transit for measuring angles, but it is easy to measure them with sufficient accuracy for illustrative purposes by placing an ordinary paper protractor upon something level, so that the center comes at the edge, and then sighting along a ruler held against it, so as to find the angle of elevation of a building. ”

“ But the most easy and habitual manner of determining our want of perpendicularity, is by attending to the apparent motion of the objects within the sphere of distinct vision; for this apparent motion of objects, when we incline from our perpendicularity, or begin to fall, is as much greater than the real motion of the eye, as the diameter of the sphere of distinct vision is to our perpendicular height. ”

“ You may find the distance of this point, and the height, in the same manner, which I have shown in the acute-angled triangle; the whole computation is the same. Here is the figure:
We have above treated at length of the circles, of their qualities and their computation. The following (62) is an example: If a circle has seven for its diameter, then it has twenty-two for its circumference. Its area you find in the following manner: Multiply the moiety of the diameter, which is three and a half, by the moiety of the circumference, which is eleven. ”

“ The metre is a length of 40 inches nearly, and by applying to it exclusively the principle of decimal division, no measure corresponding to the ancient foot was provided. A unit of that denomination, though with slightly varied differences of length, was in universal use among all civilised nations, and the type of it is found in the dimensions of the human body. Perhaps for half of the occasions which arise in the life of every individual for a linear measure, the instrument to suit his purpose must be portable, and fit to be carried in his pocket. ”

“ The diameter or axis of the figure X being drawn, the infinitesimal elements taken are parallel sections of X in general, but not always, at right angles to the axis or diameter, so that the centres of gravity of all the sections lie at one point or other of the axis or diameter and their weights can therefore be taken as acting at the several points of the diameter or axis. ”

“ Run a line from A to C by standing at C in line with A and P. Then run two perpendiculars from A and C by any of the methods already given,—sighting on a protractor or along the edge of a book if no better means are at hand. Then sight from some point D, on CD, to P, putting a stake at B. Then run the perpendicular BE. Since DE : EB = BA : AP, and since we can measure DE, EB, and BA with the tape, we can compute the distance AP.
There are many variations of this scheme of measuring distances by means of similar triangles, and pupils may be encouraged to try some of them. ”

“ Draw a periphery by three points, which doe not fall in a right line.
As here, by aei, (First from a, to e, let a right line be drawne; And likewise from e to i. Then, by the 12 e v, let both these lines be cut into equall parts, by two infinite right lines: These halfing lines also shall meete: And in their meeting shall be the Center, by the 8 e. And therefore from that meeting unto any of the sayd points given is the ray of the periphery desired.)
10. If a diameter doe halfe an inscript, that is, not a diameter, it doth cut it perpendicularly: And contrariwise: 3 p iij. ”