Diameter

“ For since the apparent diameter is reciprocally as the distance of the eye, when the distance of the eye is nothing or when the eye is close to the object at C, the apparent diameter will be the reciprocal of nothing, or infinite. ”

“ Now, if a parallelogram be drawn with a constant diameter, it is seen that by increasing the inclination of one of the sides to the diameter (and even decreasing its length) the inclination of the other side to the diameter may also be in creased. The most favourable case for such a change is when the side whose inclination is increased is nearly as long as the diameter. ”

“ 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. ”

“ 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. ”

“ For an iron ball, whose diameter is given, supposing a 9-pounder, which is nearly 4 inches, say, the cube root of 2.08 of 9 pounds is, to 4 inches, as the cube root of the given weight is to the diameter sought. Or, if 4 be divided by 2.08, the cube root of 9, the quotient 1.923 will be the diameter of a 1-pound shot; which being continually multiplied by the cube root of the given weight, gives the diameter required. ”

“ When great refinement of gauge diameter is necessary, it is obvious that all the gauges in a set should be adjusted to diameter while under an equal temperature, because a plug measuring an inch in diameter when at a temperature of, say, 60° will be of more than an inch diameter when under a temperature of, say, 90°. ”

“ In designing for forms in the round it is necessary to detail the ornament on a segment or a stretching-out of the area. This is easily done in the case of a cylinder of which the height is evident, and the extreme width and circumference easily obtainable. If the object is not in existence for direct measurement the width can be determined from the diameter as expressed in the drawing. As this diameter is about one-third of the circumference a parallelogram three times the width of the diameter will provide, in the flat, the complete area on which detail has to be drawn. ”

“ If 1,994 thousandths be the radius or half diameter, 3,988 will be the whole diameter of the column. Six figures are separated, because there are three in the multiplier, and 3 in the multiplicand. The three right hand decimals are unimportant. ”

“ The fourth, That next to the least is lesser, than that farther off: The fifth, That two onely on each side of the diameter are equall betweene themselves. All which are manifest, out of that same argument of equalitie, that is the center the beginning of decreasing, and the [191] end of increasing. For looke how much farther off you goe from the center, or how much nearer you come unto it, so much lesser or greater doe you make the inscript.
Let there be in a circle; many inscripts, of which one, to wit, ae, let it be the diameter: I say, that it is of them all the greatest or longest. ”

“ To geometricians, like you and Theaetetus, I can have no difficulty in explaining that man is a diameter, having a power of two feet; and the power of four-legged creatures, being the double of two feet, is the diameter of our diameter. ”

“ In modern man, where the transverse and antero-posterior diameters are the same as in the skull of La Chapelle-aux-Saints, the vertical diameter would be much greater, which would increase the capacity to 1,800 cubic centimeters and even to 1,900 cubic centimeters. ”

“ Thanks to the investigations of modern philosophers, we are enabled to magnify objects to 2,000 times their diameter with perfect distinctness; that is to say, the surface of the object appears to occupy 4,000,000 times its natural extent. Under such a power a hair would appear about six inches thick, a fine needle would look like a street post, and a grain of sand like a mass of rock. ”

“ Therefore, it is proven that the circumference is greater than three times the diameter. The actual measurement is said to be approximately 22/7. The amount that the diameter is smaller than the circumference is then approximately 7/22. ”

“ By these experiments it is proved, that the increase and decrease of the diameters occasioned by different apertures is not proportional to the diameters of the spurious disks.
(9.) But that the change of the apertures acts more on the small, and less on the large ones.
14th Experiment.
No. 1 of a set of globules, has the largest spurious diameter. No. 3 is larger than No. 2; whereas No. 2 has the largest real diameter. It is inclined to a greenish colour. No. 3 is now reddish, and is larger than No. 1 , which is at present less than No. 2. No. 1 grows bigger, and is now the largest. ”

“ Fig. 17 Within a given circle to describe any Regular Polygon—say a Pentagon.
Draw the diameter A F and divide it into the same number of parts as the required polygon is to have sides—in this case it will be five parts. To divide the diameter into the number of equal parts, draw a line A X any angle to A F. Set off any convenient measurement five times on this line. Join point 5 to F, and draw the lines 4, 4 ́, 3, 3 ́, &c., parallel to 5 F to meet the diameter. With A and F as centre and A F as radius describe arcs intersecting at L. From{181}
Fig. ”

“ 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. ”

“ Axes.—Conjugate diameters perpendicular to each other are called axes, and the points where they cut the curve vertices of the conic.
In a circle every diameter is an axis, every point on it is a vertex; and any two lines at right angles to each other may be taken as a pair of axes of any circle which has its centre at their intersection. ”

“ Stars of different magnitudes impress on the plate images differing both in size and blackness. To determine the magnitude from the character of the image, the easiest quantity to measure is the diameter of the image, and when measurements of position are being made with a micrometer, it is a simple matter to record the diameter as well, in spite of the indefiniteness of the border. ”

“ This is because one fifth of the diameter fails to be represented four times in the circle's circumference. For example: if we multiply the perimeter of a square by one-fourth of any line one-fifth greater than one side, we can in like manner make the square's area to appear one-fifth greater than the fact, as is done by taking the diameter for the linear unit instead of the quadrant of the circle's circumference. ”

“ The diameter of the work is determined by the distance of the cutting edge of the tool from the line of centres, when such tool is in line with the work, or, in other words, is in position to operate upon the work. ”

“ Now, finding the real size of an object like the sun, whose apparent size we can so easily measure, is the same thing as finding his distance. Any one can tell how many times its own diameter the sun is removed from us. Take a circular disc an inch in diameter,—a halfpenny, for instance—and see how far away it must be placed to exactly hide the sun. The distance will be found to be rather more than 107 inches, so that the sun, like the halfpenny which hides his face, must be rather more than 107 times his own diameter from us. But 107 times 846,000 miles amounts to 90,522,000 miles. ”

“ In a circle the thing is most apparent: as in the Astrolabe the index may be put up and downe by all the points of the periphery. So in a speare and all rounds the thing is more easie to be conceived, where the diameters are equall: yet notwithstanding in other figures the thing is the same. Because the diameter is a right line inscribed by the center, whether from corner to corner, or side to side, the matter skilleth not. Therefore that there are in the same figure infinite diameters, it issueth out of the difinition of a diameter.
And
7. ”

“ After having drawn a semicircle and its diameter, draw a perpendicular radius. This radius forms a right angle with the diameter, and cutting the semicircle in two equal parts or quarters of circles, leaves 90 degrees for each of them. If 90 degrees of a circle make a right angle, 45 degrees will make half a right angle, &c.
Or by another method. A radius, if made a cord of the semicircle, will allow three cords, each of which will contain 60 degrees; halve these arcs, and you have arcs of 30 degrees; halve the arcs of 30 degrees, and you have 15 degrees ”

“ When the diameter of a molecule has been measured, as it has been approximately, and found to be about one fifty-millionth of an inch, then fifty million in a row would reach an inch, and the cube of fifty million is 125,000,000000,000000,000000, one hundred and twenty-five thousand million million millions. ”

“ The mean diameter of the earth is 7,916 English statute miles, but the actual diameter from pole to pole is a still meaner diameter, being 7,899 miles, while the equatorial diameter is 7,925-1/2 miles. ”

“ This, then, being thus disposed, and taking CM at 100,000 parts, I found by the calculation which will be given at the end, the semi-major diameter CP to be 105,032, and the semi-axis CS to be 93,410, the ratio of which numbers is very nearly 9 to 8; so that the spheroid was of the kind which resembles a compressed sphere, being generated by the revolution of an ellipse about its smaller diameter. I found also the value of CG the semi-diameter parallel to the tangent ML to be 98,779. ”

“ But, in order to get such a result as we can with our limited powers, if we begin with a density of 8·8 at the diameter of 6284·5 miles and fix the density of 3—which we have adopted above—at the diameter of 3200 miles, we shall get a mass somewhat less than one-half of the earth ”

“ Since figures as large as a million cannot be realized properly, it may be worth while to give an illustration of what is meant by a fiber one-millionth of an inch in diameter.
A piece of quartz an inch long and an inch in diameter would, if drawn out to this degree of fineness, be sufficient to go all the way round the world 658 times; or a grain of sand just visible—that is, one-hundredth of an inch long and one hundredth of an inch in diameter—would make one thousand miles of such thread. ”

“ Imagine a circle to be described with Berlin as centre, and the circumference to pass through Hamburg; then let the circumference of the circle be computed by multiplying its diameter with the value of π to 15 decimal places, and then conceive it to be actually measured. The deviation from the true length in so large a circle as this even could not be as great as the 18 millionth part of a millimetre.
An idea can hardly be obtained of the degree of exactness produced by 100 decimal places. But the following example may possibly give us some conception of it. ”