2. The above demonstration may be regarded as the analysis of the following problem, viz. To determine the triangle, the rectangle of whose sides is equivalent to the square of any line drawn from the vertex to the base, together with the rectangle of the parts into which it divides the base; and the above result shows that the triangle sought must be isosceles. By reversing the steps of the reasoning, and proceeding synthetically, it will result, that if the triangle be isosceles, the above property must always obtain. For suppose the triangle ABC to be isosceles, and let it be circumscribed by the circle ABGEC, and let any line AFG, be drawn from the vertex of the triangle to the circumference, while ADE bisects the angle A: join GE. Then since the arcs AB, AC are equal, it follows that the sum of the arcs AB, BG is equal to the sum of the arcs AC, BG, but these last are intercepted by the sides of the angle AFC: this angle is therefore equal to an angle at the circumference, which is subtended by an arc equal to this sum, that is to say, the angle AFC is equal to the angle AEG, consequently the angles GFD, DEG are together equal to two right angles; therefore (Prop. XVIII. B. III.) a circumference may be circumscribed about the quadrilateral FE, and consequently (Prop. XXVI. Cor. 2.), AG∙AFAE·AD, or which is the same thing, AF+AF FG AD2+AD DE; whence (Prop. XXIV.) AF+BF·FC=AD2+BD.DC= AB2+AC2 (Prop. XXIX.). Hence in an isosceles triangle the square of a side is equivalent to the square of any line drawn from the vertex to the base, together with the rectangle of the parts into which it divides the base; and here again we are very readily conducted to the property of the right angled triangle, demonstrated at proposition X. book 1.

For if ABD be a triangle right angled at D, and if DC be taken equal to BD, and AC be drawn, then AD bisects the angle A, and the base BC, of the isosceles triangle ABC; and, consequently, AB-AD2+BDo.

3. The student may exercise his ingenuity in demonstrating the above property of the isosceles triangle independently of any propositions besides those which the two first books. furnish.

PROPOSITION XXXI. THEOREM.

The diagonal and side of a square are incommensurable.

Let ABCD be a square, the diagonal AC is incommensurable with its side AB.

From the point C as a centre, with the radius CB, describe the semicircle. FBE.

:

D

F

E

Then, the angle B being right, AB is a tangent to the circumference; consequently (Prop. XXVI. Cor. 1.) AE AB AB: AF; and, there-fore (Prop. XIX. B. V.), AE, AB are incommensurable, and, consequently, AC, AB are also incommensurable; for if these had a common measure, the same also would measure their sum AE (Prop. XVII. B. V.), which has been proved to be incommensurable with AB; hence the diagonal of a square is incommensurable with its side.

Scholium.

A

B

It appears from the above demonstration that it would be in vain to attempt to express accurately by numbers the side and diagonal of a square; a fact which might, indeed, have been inferred from the third corollary to Proposition X. Book I. For, representing the side of a square by unity, double the square of the side will be 2; and, consequently, the diagonal will be expressed by the square root of 2. Now 2 is a surd expression, that is to say, its numerical value can never be accurately found, although it may be approximated to sufficiently near for every practical purpose. This circumstance affords a striking instance of the insufficiency of numbers to answer rigorously all the purposes of geometry. of geometry. We cannot, for instance, take upon ourselves to say that any two lines that may be promiscuously proposed, shall be susceptible of accurate numerical representation, without first inquiring whether these lines are commensurable or not; since, for aught we know to the contrary, one of the proposed lines may be equal to the side, and the other to the diagonal of the same square, or else they may be similarly related to each other. The reasonings of geometry, however, are quite independent of any proviso of this kind. That triangles and rhomboids of equal altitudes are to each other as their bases, is a truth which proposition I. of this book establishes as indisputably, when these bases are incommensurable, as when they are commensurable; if, indeed, it did not, that proposition, however completely it might satisfy the demands of practice, would, in a scientific. point of view, be very defective; for, as the above proposition shows, it is possible for the bases of these triangles and rhom

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.

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:: AB AD; therefore (Prop.

:

D

F

B

E

XIII. B. V.) AE-AB: AB A :: AB-AD: AD. But, by 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 ke 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.

I