bafes, demonftrates the equality of the vertical angles, but the other from the inequality of the bafes, fhews that the vertical angles are unequal. It is, however, common to these four (two of which are converfant with equality, I mean the fourth, and the eighth, but two about inequality, the prefent and the following; and two begin from angles, viz. the fourth, and the object of inveftigation in the prefent, but two from bafes, viz. the eighth, and the following propofition); it is common, I fay, to all thefe four, as well to the fourth and the eighth, as to the twenty-fourth and twenty-fifth, to have two fides equal to two, each to each. For thefe being unequal, all enquiry is fuperfluous, and subject to deception. And thus much for a univerfal fpeculation concerning the present theorem.

But let us now confider the conftruction of the elementary inftitutor, and add to it where deficient. For Euclid receiving two triangles, a bc, def, having the fides a b, a c, equal to the fides de,

df, each to each, and the angle at the point a, being greater than the angle at the point d, and willing to fhew that the bafe bc, is greater than the base ef, on the right line ed, and at a point in it d, conftitutes an angle e db, equal to the angle at the point a. For the

df,

angle at the point a, is greater than the angle at the point d, and he connects db, equal to a c. The right line, therefore, e b, produced to the point b, either falls above, or upon, or beneath the line e f. The inftitutor of the Elements, indeed, confiders it as lying above the line. But let it be upon the right line. Again, therefore, we may exhibit the fame. For the two a b, a c, are equal to the two d'e, dh, and

they contain equal angles. Hence, the base bc, is equal to the base e b. But e b is greater than eƒ; and on this account b c is greater than e f Again, let it be placed beneath ef. Connecting, therefore, e h, we must say, that fince ab, a c, are equal to de, dh, and they comprehend equal angles, bc is alfo equal to eh. Because, therefore, within the triangle de h, two right lines df, fe, are constructed on the fide de, they are less than the external fides. But dh, is equal to d ft for it is equal to a c Hence he is greater than ef: But be is equal to b c. And therefore, bc is greater than ef. The theorem, therefore, is exhibited according to every pofition.

Why then, as is the fourth theorem, he at the fame time demonftrated

that the areas of triangles are equal, does he not add in the present, that besides the inequality of the bases, the areas alfo are unequal? Against this doubt we must say, that there is not the fame proportion in equal, as in unequal angles and bafes. For when the angles and bafes are equal, the equality alfo of the triangles follows: but when they are unequal, it is not neceffary that the inequality of the areas fhould be confequent; fince the triangles may as well be equal, as unequal; and that may be greater, and likewife lefs, which contains the greater angle, and the greater bafe. On this account, therefore, the inftitutor of the Elements leaves the comparison of the triangles; to which we may add, that the contemplation of thefe, requires the doctrine of parallels.

But if it be requifite, that anticipating things which are afterwards exhibited, we at prefent make a comparison of areas, we must fay, that if the angles a, d, are equal to two right, the triangles may be fhewn to be equal: but when they are greater than two right, the leffer triangle will be that which contains the greater angie; and when they are less than two right, this will be the cafe with the greater triangle. For let the conftruction in the element be given, and produce

e d

e d, fd, to the points k, b; and let us fuppofe the angles ba c, edf, equal to two right. Because, therefore, the angle b a c, is equal to the angle e d g, the añgles e dig, e df, are equal to two right. But the angles ed g, k d g, are alfo equal to two right. g, are alfo equal to two right. Let the common. angle ed g be taken away, and the remainder e df, will be equal to the remainder k dg. But e dfis equal to b dk; for they are vertical angles. Hence, the angle k d g, is equal to the angle hd k. Anď because the angle g d h, is external to the triangle g df, it is equal to the two internal and oppofite angles at the points g and f. angles are equal to each other, because dg is equal to df. angleg d b, is double of the angle at the point g, and of the angle at the point f. The angle, therefore, at the point g, is equal to the angle g d k, and they are alternate; and confequently de is parallel tofg. The triangles, therefore, gde, fde, are upon the fame base de, and between the fame parallels de, gf; and are confequently equal. But the triangle g de, is equal to the triangle a bc; and fo

But these Hence, the

the

the triangle def, is not unequal to the triangle a b c. And here you may obferve, that we require three theorems belonging to the doctrine of parallels; one, indeed, affirming, that the external angle of every triangle is equal to the two internal and oppofite angles: but the other, that if a right line falling upon two right lines, makes the alternate angles equal, the right lines are parallel; and the third, that triangles conftituted upon the fame base, and between the fame parallels, are equal, which the inftitutor of the Elements alfo knowing, omits the comparison of triangles.

But let the angles ba c, e df, be greater than two right, and let the fame things be conftructed. Because, therefore, the angles bac, edf, i. e. the angles e dg, e df, are greater than two right; but the angles edg, gd k, are equal to two right, by taking away the common angle edg, the angle e df, is greater than the angle d k. Hence, the angle g db, is more than double of the angle g dk; and fo the angle g d k, is less than the angle at the point g. Let g dk be placed equal to dg l, and let el, and d, be connected:

g

h

g 1, therefore, is parallel to de; and hence, the triangles g de, Id es are equal. But the triangle Ide, is lefs than the trianglefde. The triangle, therefore, g de, is less than the triangle ƒ de. trianglefde. But the triangle g de, is equal to the triangle a bc; and hence, the triangle abc, is less than the triangle fd e, viz. is less than the triangle which contains the greater angle.