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HISTORICAL VIEW OF THE

ancient geometry, but which has become celebrated asaffording one of the highest triumphs to the modernanalysis, and has led to the most extensive applications,of which we shall give an account in the sequel.

The principal work of Pappus is his “ CollectionesMathematics:” in this, among a variety of other curiousdiscussions, we have a considerable space occupied inthe establishment of the doctrine of maxima and mini-ma, on geometrical grounds ; and a remarkable appli-cation of it to the form of the cells of bees. This, infact, involved the consideration of a class of problemswhich, in a more extended sense, have since been calledisoperimetrical.* The nature of these will be renderedintelligible by a very simple illustration.—Let a circlebe cut out in card, and its length of circumferencemeasured by, and marked upon, a piece of tape woundround it; let then several other regular figures, as atriangle, square, hexagon, &c. be also cut out; by alittle care in cutting, they may be gradually brought tosuch sizes, that the same tape will measure round theirperimeters, so that they shall be all of exactly the samelength as that of the circle; the perfect regularity of therespective figures being all the while carefully pre-served. This being effected, it will be very easy to see,by merely placing them successively upon one another,that although their lengths of perimeter are exactly thesame, yet their areas will be seen to be of palpablydifferent extent.

To investigate such points by mathemetical reasoningis a matter of some difficulty ; but the subject wastreated by Pappus with great success in a variety ofcases. It is found (and may be easily shown by theabove method of illustration) that, of all figures havingthe same perimeter, that will have the greatest areawhich has the greatest number of sides: hence thecircle (being regarded as a polygon of an infinite num-ber of sides) will include the greatest area of all iso-

irot t equal j circumference.