ISOMERIA
, in Algebra, a term of Vieta, denoting the-freeing an equation from fractions; which is done by reducing all the fractions to one common denominator, and then multiplying each member of the equation by that common denominator, that is rejecting it out of them all.
ISOPERIMETRICAL Figures, are such as have equal perimeters, or circumferences.
It is demonstrated in geometry, that among Isoperimetrical figures, that is always the greatest which contains the most sides or angles. From whence it follows, that the circle is the most capacious of all figures which have the same perimeter with it.
That of two Isoperimetrical triangles, which have the same base, and one of them two sides equal, and the other unequal; that is the greater whose sides are equal.
That of Isoperimetrical figures, whose sides are equal in number, that is the greatest which is equilateral, and equiangular. And hence arises the solution of that popular problem, To make the hedging or walling, which will fence in a certain given quantity of land, also to fence in any other greater quantity of the same. For, let x be one side of a rectangle that will contain the quantity aa of acres; then will (aa)/x be its other side, and double their sum, viz, , will be the perimeter of the rectangle: let also bb be any greater number of acres, in the form of a square, then is b one side of it, and 4b its perimeter, which must be equal to that of the rectangle; and hence the equation , or , in which quadratic equation the two roots are , which are the lengths of the two dimensions of the rectangle, viz, whose area b2 is in any proportion less than the square a2, of the same perimeter. As, for example, if one side of a square be 10, and one side of a rectangle be 19, but the other only 1; such square and parallelogram will be Isoperimetrical, viz, each perimeter 40; yet the area of the square is 100, and of the parallelogram only 19.
Isoperimetrical lines and figures have greatly engaged the attention of mathematicians at all times. The 5th book of Pappus's Collections is chiefly upon this subject; where a great variety of curious and important properties are demonstrated, 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 to Isoperimetrical figures then known, and their different degrees of capacity.
The analysis of the general problem concerning figures that, among all those of the same perimeter, produce maxima and minima, was given by Mr. James Bernoulli, from computations that involve 2d and 3d fluxions. And several enquiries of this nature have been since prosecuted in like manner, but not always with equal success. Mr. Maclaurin, to vindicate the doctrine 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 demonstration, serving to verify the solution. See Maclaurin's Fluxions, p. 486; Analysis Magni problematis Isoperimetrici Act. Erud. Lips. 1701, p. 213; Mem. Acad. Scienc. 1705, 1706, 1718; and the works of John Bernoulli, tom. 1, p. 202, 208, 424, and tom. 2, p. 235; where is contained what he and his brother James published on this problem. Mr. John Bernoulli, in his first paper, considered only two small successive sides of the curve; whereas the true method of resolving this problem in general, requires the considering three such small sides, as may be perceived by examining the two solutions.
M. Euler has also published, on this subject, many profound researches, in the Petersburg commentaries; and there was printed at Lausanne, in 1744, a pretty large work upon it, intitled, Methodus inveniendi lineas curvas, maximi minimive proprietate gaudentes: sive Solutio problematis Isoperimetrici in latiss<*>o sensu accepti.
M. Cramer too, in the Berlin Memoirs for 1752, has given a paper, in which he proposes to demonstrate in general, what can be demonstrated only of regular figures in the elements of geometry, viz, that the circle is the greatest of all Isoperimetrical figures, regular or irregular.|
On this head, see also Simpson's Tracts, p. 98; and the Philos. Trans. vol. 49 and 50.
ISOSCELES Triangle, is a triangle that has two sides equal. In the 5th prop. of Euclid's 1st book, which prop. is usually called the Pons Asinorum, or Asses bridge, it is demonstrated, that the angles, a and b, at the base of the Isosceles triangle, are equal to each other; and that if the equal sides be produced, the two angles, c and d, below the base, will also be equal. It is also inferred, that every equilateral triangle is also equiangular.
Other properties of this figure are, that the perpendicular AP, from the vertex to the base, bisects the base, the vertical angle, and also the whole triangle. And that if the vertical angles of two Isosceles triangles be equal, the two triangles will be equiangular.
