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33. Construct a rectangle whose sides are 1.3" and 2.3"; construct a parallelogram of equal area whose sides are 1.6" and 2.6". Measure the acute angle of the parallelogram. [Use Prop. 35.]

34. Draw a right-angled triangle in which the hypotenuse is 2" and one side is 1". Find the length of the other side by measurement and calculation.

35. Draw a square equal to the sum of two squares whose sides are respectively 1", 2". Measure its diagonal.

36. Verify the last result by calculation.

37. Draw a square equal to the difference of two squares whose sides are 22 and 33 mm. Measure its diagonal.

38. By means of Prop. 47 construct lines of the following lengths and measure them-(i) v13", (ü) V5", (iii) V41 mm., (iv) 129 mm., (v) v7", (vi) V23 cm., (vii) V33 cm., (viii) V43 cm.

39. Construct a triangle whose perimeter is 4 inches, and whose angles are 40°, 60°, 80°. Measure its longest side. [See Ex. 11, p. 102.]

40. Calculate algebraically from Book I. the perimeter of an equilateral triangle whose area is 2 sq. cm.

41. Calculate algebraically from Book I. the perimeter of a triangle whose sides are in the ratios 3: 4: 5, and whose area is 1 sq. mm.

42. Using question 8, show that the area of a triangle is double that of a rectangle contained by half the height and half the base of the triangle.

BOOK II.

ON RECTANGLES AND SQUARES.

DEFINITIONS.

1. A rectangle is a parallelogram which has one of its angles a right angle.

2. A rectangle is said to be contained by any two of its sides which contain a right angle.

3. In any parallelogram the figure formed by either of the parallelograms about a diagonal together with the two complements is called a gnomon.

. The shaded portion in the figure is a gnomon.

NOTES ON THE DEFINITIONS. 1. A rectangle is often defined as a right-angled parallelogram ; but if one angle of a parallelogram is a right angle, all its angles are right angles, and therefore it is sufficient to know that one angle is a right angle.

2. In speaking of rectangles the words contained by' are often omitted ; e. g. 'the rectangle AB, CD' or 'the rectangle under AB, CD' is taken to mean 'the rectangle contained by AB and CD.'

It is evident that both the shape and area of a rectangle are known if the lengths of two sides which contain a right angle are given ; and the rectangle AB, CD is the same as the rectangle CD, AB.

In Arithmetic and Algebra the area of a rectangle is represented by the product of the length into the breadth of the rectangle; and the propositions of Euclid, which deal with rectangles, can generally be represented either arithmetically or by means of algebraical formulæ.

Many writers also use various algebraical signs and abbreviations in propositions belonging to Pure Geometry. These are useful, because they are brief, and they may perhaps be used in Riders; but they should not be used in the ordinary propositions of Euclid. The following are examples of such signs and abbreviations :

AB.CD means the rectangle contained by AB and CD.
AB2

the square described on AB. +,

and their usual senses. There is, however, an important difference between problems and theorems in pure Geometry and those in Arithmetic and Algebra. In the latter, lengths, areas, etc. are always expressed in terms of some unit, as e. g. an inch, a square foot, etc.; but in Pure Geometry no such units are used.

The length of any line, as e. g. A.

-B, may be measured in two directions. We may start from A and measure AB; or we may start from B and measure BA. In works on Mathematics it is usual to express this difference by the signs + and Lines which are measured horizontally from left to right are considered positive; and lines measured in the reverse order are considered negative. This distinction is also observed in naming a line, so that AB BA; and the rectangle AQ, QB rectangle AQ, BQ. This convention is of less importance in Euclid ; but it is as well for the learner to be taught to observe it from the first, and so the propositions in this edition of Euclid are printed according to the rule, except in Prop. 6 of the second book. There the rule has been disregarded in order to make clear to the beginner the similarity of Prop. 6 to Prop. 5. Of course, this convention about signs does not affect squares, because all real square quantities are positive; and ABP = BA?.

PROP. 1.-Theorem.-If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two straight lines is equal to the sum of the rectangles contained by the undivided line and the several parts of the divided line.

Let X and AB be two straight lines, and let AB be divided into

any number of parts AC, CD, DB ; then the rectangle contained by X, AB shall be equal to the sum

of the rectangles contained by X, AC, by X, CD, and by X, DB.

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Construction. From A draw AE at right angles to AB, I. 11 and from AE cut off AF equal to X;

1. 3 through F draw FG parallel to AB,

I. 31 and through C, D, B draw CH, DK, BG parallel to AF. Proof.—Because the rectangle AG is equal to the sum of the

rectangles AH, CK, DG; and AG is equal to the rectangle contained by X, AB, for it is contained by AF, AB; and AF is equal to X; Constr.

and AH is equal to the rectangle contained by X, AC, for it is contained by AF, AC; and AF is equal to X; Constr.

and CK is equal to the rectangle contained by X, CD, for it is contained by CH, CD, and CH is equal to AF, which is equal to X;

I. 34 and similarly DG is equal to the rectangle contained by X, DB. Therefore the rectangle contained by X, AB is equal to the sum of the rectangles contained by X, AC, by X, CD, and by X, DB.

Q. E. D.

Prop. 1 may also be expressed briefly thus :(1) X.AB

X.AC + X.CD + X.DB; or (2) x (a + b + c) = xa + xb + xc, where x is supposed to denote

a straight line containing x units of length, and a, b, c lines

containing a, b, c units respectively. The sum of the perpendiculars let fall from any point within any equilateral figure upon its sides is the same, wherever the point be taken.

PROP. 2.-Theorem.-11 a straight line be divided into any two parts, the square on the whole line is equal to the sum of the rectangles contained by the whole line and each of the parts.

Let the straight line AB be divided into any two parts AC, CB ; then the square on AB shall be equal to the sum of the

rectangles contained by AB, AC, and by AB, CB.

[blocks in formation]

Construction.—On AB describe the square ADEB, 1. 46

and through C draw CF parallel to AD or BE. I. 31 Proof. Because the square AE is equal to the sum of the

rectangles AF, CE;
and AE is the square on AB ;

Constr. and AF is equal to the rectangle contained by AB, AC,

for it is contained by AD, AC; and AD is equal to AB; and CE is equal to the rectangle contained by AB, CB,

for it is contained by BE, CB; and BE is equal to AB. Therefore the square on AB is equal to the sum of the rectangles

contained by AB, AC and by AB, CB. Q. E. D.

Prop. 2 may also be expressed thus :(1) AB2

AB.AC + AB.CB; or (2) (a + b)2 (a + b)a + (a + b)b, where a and b denote the two

parts of a line which contains a +b units of length.

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