INDICATIVE MOOD.

Future.

Cadrò or caderò, I shall or will
fall

cadrai or caderdi, thou wilt fall
cadrà or caderà, he will fall

cadrémo or caderémo, we will

fall

cadréte or caderéte, you will fall
cadranno or caderanno, they

will fall

Conditional Present.

From the point A draw A E perpendicular to BC (I. 12). Then, because the angle A EB is a right angle (Cons.), therefore the square of A B is equal to the squares of B E and EA together (I._47).

For the same reason the square of A c is equal to the squares of AE and EC together.

Therefore, adding equals to equals (I. Ax. 2), the squares of AB and AC together are equal to the squares of BE and EC together with twice the square of EA.

But because BC is divided into two equal parts in D and two unequal parts in E, therefore the squares of BE and EC are together equal to twice the square of BD together with twice the square of D E (II. 9).

Therefore the squares of AB and AC together are equal to Cadréi, or caderei, caderia, ca-twice the squares of B D, D E, and E A together. dria, I should or would fall cadrésti or caderésti, thou wouldst fall

cadrébbe or caderebbe, caderia,
cadría, he would fall

cadde, cadéo, cadette, or cade, cadrémmo or caderémmo, we

he fell

cadémmo, we fell

cadéste, you fell

caddero, cadéro, cader, cadettero,

cadérono, they fell

would fall

cadréste or caderéste, you would
fall

cadrébbero or caderebbero, cade-
riano, they would fall

But the squares of D E and EA are together equal to the square of AD (I. 47), and the doubles of these equals are equal. Therefore twice the squares of D E and EA are equal to twice the square of A D. Therefore the squares of AB and A c together are equal to twice the squares of B D and DA together. Q.E.D.

Prop. B. Theorem. The squares of the two diagonals of a parallelogram are together equal to the squares of its four sides.

Let ABCD be any parallelogram of which the diagonals are A C and BD cutting each other in E. Then the squares of a C and BD together are equal to the squares of AB, BC, CD, and DA together. By Exercise 2 to Proposition XXXIV. of the first book, the diagonals bisect each other. And because A B D is a triangle of which the side BD is bisected in E, therefore, by the last proposition, the squares of AB and AD together are equal to twice the squares of B E and A E together.

For the same reason the squares of BC and CD are together equal to twice the squares of BE and E C together. Therefore, adding equals to equals, the squares of AB, BC, CD, and DA together are equal to four times the square of B E together with twice the squares of AE and E C (I. Ax. I. 2).

But AE is equal to E C, because the diagonals bisect each other, and therefore the square of AE is equal to the square of EC. Therefore the squares of A B, BC, CD, and DA together are equal to four times the squares of B E and AE together.

But the square of any straight line is equal to four times the square of half the line (II. 4, Cor. 2). Therefore the square of BD is equal to four times the square of B E, and the square of a c is equal to four times the square of A E.

Therefore the square of AB, BC, CD, and D A are together equal to the squares of AC and BD together. Q.E.D.

Prop. c. Theorem. The squares of the four sides of a tra pezium are together equal to the squares of its two diagonals and four times the square of the straight line which joins the points of the bisection of the diagonals.

From each of these equals take the square of GM, then the rectangle A M, M B is equal to the square of H G (I. Ax. 3). But the number of days in which A can do the whole. Therefore he would the square of H G is equal to the square CDEF (Cons.). Therefore the rectangle A M, M B is equal to the square CDEF require 8 days to do the remaining (I. Ax, 1). Q.E.F.

408

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