ISOPERIMETRICAL figures, in geo• metry, are such as have equal perimeters or circumferences.
Isoperimetrical lines and figures have greatly engaged the attention of mathe• maticians at all times. The fifth book of Pappus's Collections is chiefly upon this subject ; where a great variety of curious and important properties are demon. strated, both of planes and solids, some of which were then old in his time, and many new ones of his own. Indeed, it seems, he has here brought together into this book all the properties relating ta isoperimetrical figures then known, and their different degrees of capacity. The analysis of the general problem concern ing figures, that, among all those of the same perimeter, produce maxima and minima, was given by Mr. James Ber noulli, from computations that involve the second and third fluxions. And several enquiries of this nature have been since prosecuted in like manner, but not al ways with equal success. Mr. Maclaurin, to vindicate the doctrines of fluxions from the imputation of uncertainty or obscurity, has illustrated this subject, which is considered as one of the most abstruse parts of this doctrine, by giving the resolution and composition of these problems by first fluxions only ; and in a manner that suggests a synthetic demon stration, serving to verify the solution. See Maclaurin's Fluxion. Mr. Crane also, in the Berlin Memoirs for 1752, has given a paper, in which he proposes to demonstrate, in general, what can be de monstrated only of regular figures in the elements of geometry, viz. that the circle is the greatest of all isoperimetrical figures, regular or irregular. We shall now mention a few of the properties of isoperimetrical figures.
1. Of isoperimetrical figures, that is the greatest that contains the greatest number of sides, or the most angles, and consequently a circle, is the greatest of all figures that have the same ambit as it has 2. Of two isoperimetrical triangles, having the same base, whereof two sides of one are equal, and of the other une qual, that is the greater whose two sides are equal.
3. Of isoperimetrical figures, whose sides are equal in number, that is the greatest which is equilateral and equian gular. From hence follows that common problem of making the hedging or walling that will wall in one acre, or even any determinate number of acres, a ; fence or wall in any greater number of acres whatever, b. In order to the solution of this problem, let the greater number, b, be supposed a square ; let x be one side of an oblong, whose area is a : then will a — be the other side ; and 2 a + 2 x will be the ambit of the oblong, which must be equal to four times the square root of b; that is, 2 + 2 x = b. Whence the value of x may be easily had, and you may make infinite numbers of squares and oblongs that have the same ambit, and yet shall have different given areas.
Let b Then 2 a+ 4 d a+2xx=2dx 2xx-2dx=--a a xx—dx =— — 2 a xx—dx+idd=— x= _2+ +dd+ d Thus if one side of the square be 10, and one side of an oblong be 19, and the other 1 ; then will the ambits of that square and oblong be equal, viz. each 40, and yet the area of the square will be 100, and of the oblong but 19.