Circumference

“ 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. ”

“ 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 ”

“ 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? ”

“ And those who speak thus are acting precisely like a man who, having no idea of what a circle is, should declare that this requirement, that every point of the circumference should be an equal distance from the center, is exaggerated. ”

“ A man ignorant of geometry may perhaps lament, that the circumference of a circle does not happen to be exactly three times the diameter, or in some other assignable proportion to it, rather than such that we have not yet been able to explain what the ratio between them is ”

“ 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. ”

“ 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. ”

“ Indeed, the late Professor Newcomb stated that "ten decimal places are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimal places would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful microscope." ”

“ The first species grows to an immense size, and is very commonly twenty-seven feet in circumference six feet above the earth’s surface: they rise to the height of two hundred and thirty feet, and one hundred and twenty of that height without a limb. We have often found them thirty-six feet in circumference. One of our party measured one, and found it to be forty-two feet in circumference, at a point beyond the reach of an ordinary man. ”

“ 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. ”

“ In circles, in round plats, or in cylindrical vessels, he will see a beautiful, universal law pervading the whole—the increase of the circumference is proportional to the increase of the diameter, while the increase of the circle is as the square of the diameter. ”

“ Every mathematician knows that scores of methods, differing altogether from each other in process, all end in this mysterious 3.14159..., which insists on calling itself the circumference to a unit of diameter. A reader who is competent to follow processes of arithmetic may be easily satisfied that such methods do actually exist. I will give a sketch, carried out to a few figures, of three: the first two I never met with in my reading; the third is the old method of Vieta. ”

“ We see that a circle-squarer can advance, with the utmost confidence, the assertion that when the diameter is 1,000, the circumference is accurately 3,125: the mathematician declaring that it is a trifle more than 3,1411⁄2. If the squarer be right, the mathematician has erred by about a 200th part of the whole: or has not kept his accounts right by about 10s. in every 100l. ”

“ 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. ”

“ 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. ”

“ It is supposed that all the circumference of the earth is divided into 360 degrees; and each degree is of 17 leagues and a half, consequently the circumference of the earth is 6,300 leagues. Land leagues are of three miles and sea leagues of four miles. ”

“ Physicists became convinced long ago that the only fixed standard of linear measure is some portion of the earth's circumference. No intelligent Greek or Roman from the time of Plato had any doubts about the shape of the earth. ”

“ In like manner, of fifty young persons of the average of fourteen years and eleven months, the measurement in height was four feet five inches, and in circumference two feet three inches; while of forty-nine young persons employed on farms, of the average of fifteen years and six months, the measurement in height was four feet ten inches and eight-elevenths, and, in circumference, two feet three inches, being a difference of nearly six inches in height in favour of the agricultural labourers. ”

“ The greatest danger to be guarded against in considering all scales as continuous rather than discrete, is that careless thinkers may refine their calculations far beyond the accuracy which their original measurements would warrant. One should be very careful not to make such unjustifiable refinements in his statement of results as are often made by young pupils when they multiply the diameter of a circle, which has been measured only to the nearest inch, by 3.1416 in order to find the circumference. ”

“ Similarly, the distance a boy can run may be considered as depending on the time. Or, we may say that the area of a square is a function of the length of a side, and the volume of a sphere is a function of its diameter. ”

“ The same principles by which we measure an inch or an acre of ground will measure to millions in extent. A circle of an inch diameter has the same geometrical properties as a circle that would circumscribe the universe. ”

“ 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. ”