THEOREM VIII.

448. Of all equivalent polygons having the same number of sides, the regular polygon has the least perimeter.

HYPOTHESIS. Let P be a regular polygon, and M any equivalent irregular polygon having the same number of sides as P. CONCLUSION. The perimeter of P is less than that of M.

PROOF. Let N be a regular polygon having the same perimeter and the same number of sides as M; then, by 447,

But, of two regular polygons having the same number of sides, that which has the less surface has the less perimeter; therefore the perimeter of P is less than that of N or of M.

THEOREM IX.

449. If a regular polygon be constructed with a given perimeter, its surface will be the greater, the greater the number of its sides.

PROOF. Let P be the regular polygon of three sides, and Q the regular polygon of four sides, constructed with the same given perim

eter.

In any side AB of P take any arbitrary point D; the polygon P may be regarded as an irregular polygon of four sides, in which the sides AD, DB, make a straight angle with each other; then, by 447, the irregular polygon P of four sides is less than the regular isoperimetric polygon Q of four sides.

In the same manner it follows that is less than the regular isoperimetric polygon of five sides, and so on.

THEOREM X.

450. Of equivalent regular polygons, the perimeter will be the less, the greater the number of sides.

P

R

HYPOTHESIS. Let P and Q be equivalent regular polygons, and let Q have the greater number of sides.

CONCLUSION. The perimeter of P will be greater than that of Q. PROOF. Let R be a regular polygon having the same perimeter as Q and the same number of sides as P; then, by 449,

Q> R, or P> R;

therefore the perimeter of P is greater than that of R or of Q.

BOOK V.

RATIO AND PROPORTION.

Multiples.

451. NOTATION. In Book V., capital letters denote magnitudes.

Magnitudes which are or may be of different kinds are denoted by letters taken from different alphabets.

The small Italic letters m, n, p, q, denote whole numbers.

452. A greater magnitude is said to be a Multiple of a lesser magnitude when the greater is the sum of a number of parts. each equal to the less; that is, when the greater contains the less an exact number of times.

453. A lesser magnitude is a Submultiple, or Aliquot Part, of a greater magnitude when the less is contained an exact number of times in the greater.

454. When each of two magnitudes is a multiple of, or exactly contains, a third magnitude, they are said to be Commensurable.

455. If there is no magnitude which each of two given magnitudes will contain an exact number of times, they are called Incommensurable.

456. REMARK. It is important the student should know, that of two magnitudes of the same kind taken at hazard, or one being given, and the other deduced by a geometrical construction, it is very much more likely that the two should be incommensurable than that they should be commensurable.

To treat continuous magnitudes as commensurable would be to omit the normal, and give only the exceptional case. This makes the arithmetical treatment of ratio and proportion radically incomplete and inadequate for geometry.

PROBLEM I

457. To find the greatest common submultiple or greatest common divisor of two given magnitudes, if any exists.

Let AB and CD be the two magnitudes.

From AB, the greater, cut off as many parts as possible, each equal to CD, the less. If there be a remainder FB, set it off in like manner as often as possible upon CD. Should there be a second remainder HD, set it off in like manner upon the first remainder, and so on.