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112

BOOK II.

DEFINITIONS.

1. A rectangle (or rectangular parallelogram) is said to be contained by any two of its conterminous sides.

Thus the rectangle ABCD is said to be contained by AB and BC; or by BC and CD; or by CD and DA ; or by DA and AB.

D

The reason of this is, that if the lengths of any two conterminous sides of a rectangle are given, the rectangle can be constructed; or, what comes to the same thing, that if two conterminous sides of one rectangle are respectively equal to two conterminous sides of another rectangle, the two rectangles are equal in all respects. The truth of the latter statement may be proved by applying the one rectangle to the other.

2. It is oftener the case than not, that the rectangle contained by two straight lines is spoken of when the two straight lines do not actually contain any rectangle. When this is so, the rectangle contained by the two straight lines will signify the rectangle contained by either of them, and a straight line equal to the other, or the rectangle contained by two other straight lines respectively equal to them.

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Thus ABEF (fig. 1) may be considered the rectangle contained by AB and CD, if BE CD; CDEF (fig. 2) may be considered the rectangle contained by AB and CD, if DE

=

=

AB; and EFGH (fig. 3) may be considered the rectangle contained by AB and CD, if EF = AB and FG = CD.

3. As the rectangle and the square are the figures which the Second Book of Euclid treats of, phrases such as 'the rectangle contained by AB and AC,' and 'the square described on AB,' will be of constant occurrence. It is usual, therefore, to employ abbreviations for these phrases. The abbreviation which will be made use of in the present text-book* for the rectangle contained by AB and BC' is AB.BC, and for 'the square described on AB, AB2.

4. When a point is taken in a straight line, it is often called a point of section, and the distances of this point from the ends of the line are called segments of the line.

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Thus the point of section D divides AB into two segments AD and BD.

In this case AB is said to be divided internally at D, and AD and BD are called internal segments.

The given straight line is equal to the sum of its internal segments; for AB = AD + BD.

5. When a point is taken in a straight line produced, it is also called a point of section, and its distances from the ends of the line are called segments of the line.

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Thus D is called a point of section of AB, and the segments into which it is said to divide AB are AD and BD.

* In certain written examinations in England, the only abbreviation allowed for the rectangle contained by AB and BC' is rect. AB, BC, and for the square described on AB,' sq. on AB; the pupil, therefore, if preparing for these examinations, should practise himself in the use of such abbreviations.

In this case, AB is said to be divided externally at D, and AD, BD are called external segments.

The given straight line is equal to the difference of its external segments; for AB: AD - BD, or BD – AD.

=

6. When a straight line is divided into two segments, such that the rectangle contained by the whole line and one of the segments is equal to the square on the other segment, the straight line is said to be divided in medial section.*

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Thus, if AB be divided at H into two segments AH and BH, such that AB · BH = AH2, AB is said to be divided in medial section at H.

It will be seen that AB is internally divided at H; and in general, when a straight line is said to be divided in medial section, it is understood to be internally divided. But the definition need not be restricted to internal division.

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Thus, if AB be divided at H' into two segments AH' and BH', such that AB BH' = AH'2, AB in this case also may be said to be divided in medial section.

7. The projection † of a point on a straight line is the foot of the perpendicular drawn from the point to the straight line.

A

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Thus D is the projection of A on the straight line BC.

8. The projection of one straight line on another straight

* The phrase, 'medial section,' seems to be due to Leslie. See his Elements of Geometry (1809), p. 66.

+ Sometimes the adjective 'orthogonal' is prefixed to the word projection, to distinguish this kind from others.

line is that portion of the second intercepted between perpendiculars drawn to it from the ends of the first.

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Thus the projections of AB and CD on EF are, in fig. 1, GH and KL; in fig. 2, AH and KD.

While the straight line to be projected must be limited in length, the straight line on which it is to be projected must be considered as unlimited.

9. If from a parallelogram there be taken away either of the parallelograms about one of its diagonals, the remaining figure is called a gnomon.

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Thus if ADEB is a ||m, BD one of its diagonals, and HF, CK ms about the diagonal BD, the figure which remains when HF or CK is taken away from ADEB is called a gnomon. In the first case, when HF is taken away, the gnomon ABEFGH (inclosed within thick lines) is usually, for shortness' sake, called AKF or HCE; in the second case, when CK is taken away, the gnomon ADEKGC would similarly be called AFK or CHE.

The word 'gnomon' in Greek means, among other things, a carpenter's square,* which, when the || ADEB is a square or a

* Another less known figure was, from its shape, called by the ancient geometers, the shoemaker's knife.' See Pappus, IV. section 14.

·

rectangle, the figure AKF resembles. The only gnomons mentioned by Euclid in the second book are parts of squares.

The more general definition given by Heron of Alexandria, that a gnomon is any figure which, when added to another figure, produces a figure similar to the original one, will be partly understood after the fourth proposition has been read.

PROPOSITION 1. THEOREM.

If there be two straight lines, one of which is divided internally into any number of segments, the rectangle contained by the two straight lines is equal to the rectangles contained by the undivided line and the several segments of the divided line.

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Let AB and CD be the two straight lines,

and let CD be divided internally into any number of segments CE, EF, FD:

it is required to prove AB.CD = AB. CE + AB· EF + AB. FD.

From C draw CG 1 CD and

through G draw GH || CD,

=

AB;

and through E, F, D draw EK, FL, DH || CG.

Then CH = CK EL + FH;

=

I. 11, 3

I. 31

I. Ax. 8

Const., I. 34

GC.CE+KE. EF + LF. FD.

that is, GC. CD

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.. AB. CD = AB. CE + AB. EF + AB. FD.

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