The same results might also be obtained from the differences of the preceding arcs.

Of the regular polygons, three only are susceptible of perfect adaptation, and capable therefore of covering, by their repeated addition, a plane surface. These are the equilateral triangle, the square, and the hexagon. The angles of an equilateral triangle are each two-thirds of a right angle, those of a square are right angles, and the angles of a hexagon are each equal to four-third parts of a right angle. Hence there may be constituted about a point, six equilateral triangles, four squares, and three hexagons. But no other regular polygon can admit of a like disposition. The pentagon, for instance, having each of its angles equal to six-fifths of a right angle, would not fill up the whole space about a point, on being repeated three times; yet it would do more than cover that space, if added four times. On the other hand, since each angle of a polygon which has more than six sides must exceed four-third parts of a right angle, three such polygons cannot stand round a point. Nor can the space about a point ever be bisected by the application of any regular polygons, of whatever number of sides; for their angles are always necessarily each less than two right angles.

ELEMENTS

OF

GEOMETRY.

BOOK V.

OF PROPORTION.

THE preceding Books treat of magnitude as concrete, or having mere extension; and the simpler properties of lines, of angles, and of surfaces, were deduced, by a continuous process of reasoning, grounded on the principle of superposition. But this mode of investigation, how satisfactory soever to the mind, is by its nature very limited and laborious. By introducing the idea of Number into geometry, a new scene is opened, and a far wider prospect rises into view. Magnitude, being considered as discrete, or composed of integrant parts, becomes assimilated to multitude; and under this aspect, it presents a vast system of rela

tions, which may be traced out with the utmost facility.

Numbers were at first employed, to denote the aggregation of separate, though kindred, objects; but the subdivision of extent, whether actually effected or only conceived to exist, bestowing on each portion a sort of individuality, they came afterwards to acquire a more comprehensive application. In comparing together two quantities of the same kind, the one may contain the other, or be contained by it; that is, the one may result from the repeated addition of the other, or it may in its turn produce this other by a successive composition. The one quantity is, therefore, equal, either to so many times the other, or to a certain aliquot part of it.

Such seems to be the simplest of the numerical relations. It is very confined, however, in its application, and is evidently, in this shape, insufficient altogether for the purpose of general comparison. But that object is attained, by adopting some intermediate term of reference. Though a quantity neither contain another exactly, nor be contained by it; there may yet exist a third and smaller quantity, which is at once capable of measuring them both. This measure corresponds to the arithmetical unit; and as number denotes the collection of units, so quantity may be viewed as the aggregate of its component measures.

But mathematical quantities are not all suscep

tible of such perfect mensuration. Two quantities may be conceived to be so constituted, as not to admit of any other quantity that will measure them completely, or be contained in both without leaving a remainder. Yet this apparent imperfection, which proceeds entirely from the infinite variety ascribed to possible magnitude, creates no real obstacle to the progress of accurate science. The measure or primary element, being assumed successively still smaller and smaller, its corresponding remainder must be perpetually diminished. This continued exhaustion will hence approach nearer than any assignable difference to its absolute term.

Quantities in general can, therefore, either exactly or to any required degree of precision, be represented abstractly by numbers; and thus the science of Geometry is at last brought under the dominion of Arithmetic.

It is obvious, that quantities of any kind must have the same composition, when each contains its measure the same number of times. But quan tities, viewed in pairs, may be considered as having a similar composition, if the corresponding terms of each pair contain its measure equally. Two pairs of quantities of a similar composition, being thus formed by the same distinct aggregations of their elementary parts, constitute a Próportion.