the included angle in the other, the rhomboid will be double the triangle (Prop. XXVII. Cor. 3. B. I.).

Cor. 2. Rectangles contained by equal lines are equal.

PROPOSITION III. THEOREM.

Rhomboids which have the same base and equal altitudes are equivalent.

Let the rhomboids AC, AE, standing upon the same base AB, have equal altitudes; or, which amounts to the same thing, let the opposite sides DC, FE, lie in the same line DE parallel to the base (Prop. XIII. Cor. 1. B. I.); these rhomboids are equal.

For DC is equal to FE, each being P equal to AB (Prop. XXVII. B. I.); consequently DF is equal to CE: and since DA, AF, are respectively equal to CB, BE, the triangle ADF is equal to the triangle BCE. Take the former triangle from the quadrilateral ABED, and there will remain the rhomboid AE; take the latter triangle from the same space, and there will remain the rhomboid AC; these rhomboids are, therefore, equivalent.

B

Cor. 1. Rhomboids whose bases and altitudes are respectively equal are equivalent, for the equal bases being placed the one upon the other must coincide.

Cor. 2. Triangles whose bases and altitudes are respectively equal are equivalent, as they are the halves of equivalent rhomboids (Prop. XXVII. Cor. 3. B. I.).

Cor. 3. Every rhomboid is equivalent to a rectangle of equal base and altitude.

Cor. 4. A line bisecting the opposite sides of a rhomboid divides the rhomboid into two equal parts; and a line from the middle of any side of a triangle to the vertex of the posite angle divides the triangle into two equal parts (Cor. 1 and 2.).

*op-.

Cor. 5. Therefore a triangle is equivalent to a rhomboid of equal base and of half its altitude, or to one of equal altitude and of half its base.

Scholium.

1. It is very evident that the converse of the above proposi

tion is not true, that is to say, it cannot be inferred that two equivalent rhomboids shall have their bases and altitudes equal; for it has been shown (Prop. 1.) that the rhomboids AG, GC, are equivalent (see the diagram), where the base GF must be longer than the base GE, provided BA is longer than AD, for then the angle ADB being greater than ABD (Prop. XIX. B. I.), the angle EGB, which is equal to ADB, is greater than EBD; consequently EB is longer than EG, but EB is equal to GF, therefore GF is longer than GE.

2. It is however, true, that equivalent rhomboids upon the same base have equal altitudes, for if the altitude of one be supposed less than that of the other, and the side opposite its base be prolonged, a portion of the other rhomboid must be cut off thereby, and the remaining portion still be equal to the former rhomboid, by the proposition, which is absurd; the altitudes therefore are equal. Having shown this, we may further prove that equivalent rhomboids of equal altitudes have also equal bases, for they are equivalent to rectangles of the same bases and altitudes: now any side of a rectangle may be considered as the base; taking then those sides as bases which are equal to the altitude of the rhomboids, the other sides or altitudes are, as shown above, equal, and these altitudes are the bases of the rhomboids: the bases are therefore equal.

Cor. 6. Hence, equivalent triangles whose bases are equal, have equal altitudes; and equivalent triangles whose altitudes are equal, have equal bases (Cor. 5.).

PROPOSITION IV. THEOREM.

If there be two straight lines of which one is divided into parts, the sum of the rectangles contained by the undivided line, and the several parts of the other, will be equal to the rectangle contained by the two whole lines.

Let the lines be AB, AC, of which the former is divided into the parts AD, DE, EB, then the rectangles contained by AC, and each of these parts, are together equal to the whole rectangle AH, contained by AB, AC.

Let DF, EG, be parallel to AC, then A D the angles FDE, GEB, being each equal to the angle A, the rhomboids AF, DG, EH, are rectangles, and DF, EG, being each equal to AC (Prop. XXVII. Cor. o

B

G

H

4. B. 1.), these rectangles are contained by AC, and the several parts of AB, and as they make up the whole rectangle AH, they are together equal to it.

Cor. The square of a line is equivalent to twice the rectangle of the whole line and the half thereof.

PROPOSITION V. THEOREM.

If a straight line be divided into two parts, the square described upon the whole line shall be equivalent to the squares on the two parts, together with double the rectangle contained by those parts.

The square ABCD upon the line AB, is equivalent to the squares upon any two parts AE, EB, into which the line is divided, together with double the rectangle contained by them.

Let EG be parallel to BC, BH equal to BE, and HFI parallel to BA. Then the opposite sides of the figure EH being parallel, and the angle B being right, EH is a square. Again, because AC is a square, the lines IH, EG, parallel to its sides are equal (Prop. XXVII. Cor. 2 and A 4. B. I.); if then from each, the equals

D

G

I

H

B

FH, FE, be respectively taken, the remainders IF, FG, will be equal, and F being a right angle, FD is a square; hence the containing sides of the rectangle AF are equal to those of the rectangle FC, consequently the square AC includes the squares on DG, (or AE) EB, together with double the rectangle contained by AE, EB.

Cor. The square of a line is equivalent to four times the square of half the line.

PROPOSITION VI. THEOREM.

The square described on the difference of two lines is equivalent to the squares on the two lines diminished by twice their rectangle.

The square upon AB, the difference of the two lines AC, BC, is equivalent to the squares on these lines, diminished by twice their rectangle.

A

B

G

Let AD be the square on AB, BF the square on BC, and AI the square on AC, and let ED be produced to H. The adjacent angles GBC, CBD being right angles, BD is the continuation of GB, (Prop. IV. B. I.); CH for a similar reason is the continuation of FC, and the figure GH is a rectangle. Again, since DH is equal to FG, or GB, EH is equal to DG; also EK is equal to BC or GF, each being the excess of a side of the square AI above a side of the square AD; hence the rectangle EI is equal to the rectangle GH (Prop. II. Cor. 2.), consequently the square on AB is equivalent to the squares on AC, BC, diminished by twice the rectangle of AC, BC.

PROPOSITION VII.

E

THEOREM.

K

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H

The difference of the squares of any two lines is equivalent to the rectangle contained by the sum and difference of those lines,

The difference of the squares of the two lines AC, AB, is equivalent to the rectangle contained by AC, BC, their sum and difference.

B

Let AD be the square on AC, and AG the square on AB, and produce FG to H, then BH is the rectangle of AB, BC; also, since FE is equal to BČ, each being the excess of a side of the square AD, above a side of the square AG, the rectangle FD is contained by lines equal r to AC, BC; and the two rectangles BH, FD are therefore together equal to the rectangle contained by the sum AB, AC of the two lines, and their difference BC (Prop. IV. B. II.) and these rectangles make up the excess of AD above AG.

D

G

The sum of the squares on two lines is equivalent to half the square on their sum, together with half the square on their difference.

The squares on the two lines AB, AC, are together equiva

lent to half the square on their sum, BC their difference.

and half the square on

Let AG be the square on AB, and AD the square on AC, and through P, the middle of BC, let LH, parallel to EA, be drawn, meeting FG produced in H; make PI equal to PH, and draw IK parallel to PC.

A

E

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I

K

B

P

Since the angles at A are right, EAFis a straight line, so that EH is a rectangle, and it is contained by lines equal to the sum of AC, AB, and half that sum AP; it is therefore equal to half the square of the sum AC, AB (Prop. IV. Cor.). Again, IP, PC, being equal, by construction, to PH, PB, the rectangles PK, PG, are equal; hence the two squares AD, AG, are together equivalent to the two rectangles EH, LK; now, CK being equal to BG or BA, KD is equal to BC, for EC is a square, the rectangle LK is thus contained by lines equivalent to BC, and the half thereof PC, and is consequently equal to half the square on BC; it therefore follows that the squares AD, AG, are together equivalent to half the square on the sum of AC, AB, and half the square on BC, the difference of AC, AB.

G H

Cor. Hence, twice the sum of the squares of two lines is equivalent to the squares of their sum and difference.

A trapezium is equivalent to a rectangle contained by its altitude, and half the sum of its parallel sides.

The trapezium ABCD is equivalent to the rectangle contained by its altitude, and the sum of its parallel sides AB, DC.

The four sides of the quadrilateral AC are respectively equal to those of the quadrilateral FB, and at the same time. the angle A is equal to the angle F (Prop. XXVII. B. I.); therefore these quadrilaterals are equal (Prop. XXV. Cor. B. I.). Hence the trapezium AC is equivalent to half the rhomboid

D