boids to be incommensurable. Hence the great impropriety of confounding in books on geometry the expressions product and rectangle, since the terms of a product must be commensurable, while the sides of a rectangle may be incommensurable. Whatever is shown to be true of the rectangle of two lines must necessarily be true of the product of the numbers representing its sides, in the particular case when those sides are commensurable, or capable of such numerical representation. But it is evident that we cannot, conversely, from this particular case infer the general proposition in which it is included, without violating one of the most obvious rules of logic.

PROPOSITION XXXII. PROBLEM.

To divide a given straight line into two parts, such that the greater part may be a mean proportional between the whole line and the other part.

Let AB be the proposed line. Draw the perpendicular BC equal to half AB, and from C as a centre, with the radius CB, describe the semicircle DBE, make AF equal to AD, and AB will be divided in F; so that AB: AF:: AF: FB.

For, since AB is perpendicular to CB, it is a tangent, and, consequently (Prop. XXVI. Cor. 1.) AE: AB::

D

AB: AD; therefore (Prop. XIII. B. V.) AE-AB: AB A :: AB-AD: AD. But, by

E

construction, AB DE and AD=AF; so that in this last proportion the first and third terms are respectively the same as AF, FB; therefore putting these in their place, the proportion is AF: AB:: FB: AD, or AF; therefore, by inversion, AB AF:: AF: FB.

Cor. Since AB-DE, the proportion AE: AB:: AB: AD furnishes us with the method of performing the following similar problem, viz. To increase a given line, so that it may be a mean proportional between the whole and the part added, nothing more being necessary, after having performed the above construction, than to add DA to AB.

Scholium.

Lines, divided as the above problem directs, are said to be divided in extreme and mean proportion; and it is obvious,

from proposition XIX. book V., that a line so divided is incommensurable with its parts. Hence it appears that incommensurable lines may be found at pleasure, or that if any line be proposed, one incommensurable thereto may always be discovered; a fact which seems to illustrate in some measure the propriety of the remarks subjoined to the preceding proposition.

BOOK VII.

DEFINITIONS.

POLYGONS carry particular names according to the number of their sides, those of three and of four sides-triangles and quadrilaterals-have been already considered.

1. Polygons of five sides are called pentagons, those of six sides hexagons, those of seven sides heptagons, those of eight octagons, and so on.

2. Polygons, which are at once equilateral and equiangular, are called regular polygons.

Two regular polygons of the same number of sides are similar.

For the sides being equal in number, the angles are also equal in number; and the sum of the angles of the one polygon is equal to the sum of the angles of the other (Prop. XVII. B. I.); and, since the polygons are each equiangular, it follows that any angle in the one polygon is the same submultiple of their sum; and equi-submultiples of equal magnitudes being equal, the angles of the two polygons are all equal to each other; and it is obvious that the sides containing any angle in the one polygon are to each other as the sides containing any angle in the other, for each polygon is equilateral; therefore (Def. I. B. VI.) the polygons are similar.

A circle may be inscribed in, or circumscribed about, any regular polygon, and the circles so described have a common centre.

If the polygon ABCDEF be regular, then from a common centre a circle may be inscribed in, and circumscribed about, it.

This has already been shown to be true of the equilateral triangle and the square (Prop. XXIII. Schol., and Prop. XXV. Cor. B. IV.); we shall therefore consider the annexed polygon to have more than four sides.

Then, the angles of this polygon being each greater than a right angle (Prop. XVII. Cor. 4. B. I.), if the sides BA, EF be produced, they will meet as at G, forming the isosceles triangle GAF, and if BE be drawn, the triangle GBE will be also isosceles: hence, in the quadrilateral ABEF, the angles A, E are together equal to the angles F, B, and, consequently (Prop. XVIII. Cor. 1.

B

G

AF

C

D

E

B. III.), a circle may be circumscribed about it; therefore the points B, A, F, E all lie in the same circumference. By reasoning in a similar way it may be shown that the points A, F, E, D all lie in the same circumference, which circumference must be identical with the former (Prop. VIII. B. III.). Proceeding in this way it will appear that the same circumference must also pass through the next point C, and so on completely round the polygon.

Again, since the sides of the polygon are so many equal chords of the circumscribing circle, they must all be equally distant from the centre (Prop. VI. B. III.); and, consequently, the circumference described from the same centre, with this common distance as radius, must touch every side of the polygon.

Cor. I. It has been shown that the triangles GAF, GBE, having the common angle G, are both isosceles; it, therefore, follows that in a regular polygon the diagonal, cutting off three sides, is parallel to the middle one, and, further, that the diagonal, cutting off five, or indeed any odd number of sides, must be parallel to the middle side; for if the sides intercepted by the diagonals, cutting off three and five sides, are parallel, the diagonals themselves must be parallel by Prop. XXIX. Book I., and if the intercepted sides are not parallel, they meet when produced, and form, with the diagonals, two similar isosceles triangles; so that in this case the diagonals are parallel, and in the same way is it to be shown that the diagonal, cutting off seven sides, is parallel to the middle one, and so on; therefore, generally, the diagonal, which cuts off an uneven number of sides from a regular polygon, is parallel to the middle one of those sides.

Cor. 2. The angles formed at the centre of the circle, by lines drawn from it to the extremities of the sides of the polygon, are all equal, being subtended by equal chords.

Scholium.

It may be remarked that in regular polygons the centre of the inscribed and circumscribed circles is called also the centre of the polygon, and that the perpendicular from the centre to a side, that is, the radius of the inscribed circle, is called the apothem of the polygon.

PROPOSITION III. THEOREM. (Converse of Prop. II.)

If from a common centre circles can be inscribed within, and circumscribed about a polygon, that polygon is regular.

Suppose that from the point O, as a centre, circles can be described in, and about, the polygon in the margin; this polygon is regular.

For, supposing these circles to be described, the inner one will touch all the sides of the polygon; these sides are, therefore, equally distant from its centre (Prop. IX. B. III.), and, consequently, being chords of the outer circle they are equal, and, therefore, include equal angles (Prop. XIV. Cor. 4. B. III.). Hence the

B

F

E

polygon is at once equilateral and equiangular, that is (Def. 2.), it is regular.

PROPOSITION IV. THEOREM.

The surface of every polygon in which a circle may be inscribed, is equivalent to the rectangle of half the radius of that circle, and the perimeter of the polygon.

Let O be the centre of the circle inscribed in the polygon ABCD, &c. Draw from O lines to the extremities of the sides, thus dividing the polygon into as many triangles as it has sides.