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In like manner two areas are equal to one another, when they can be made to coincide in every part, that is, when one can be made exactly to cover the other, and no more. For example, all the pages of this book are exactly equal to one another. But areas may be equal also, when they are not exactly alike, (as the pages of the same book are,) but can be made so by a different arrangement of the parts of one or both. For it is evident, that this page might be cut up into many parts, without at all altering the total area; and those parts might be arranged so as to form a great variety of plane figures having precisely the same area, but with a different boundary. Thus, if we have a square and a triangle, and we can cut up the square, so as with the parts exactly to cover the triangle, the area of the square is equal to that of the triangle. Or, again, two triangles, which to the eye appear unequal, may yet be equal, and shall be so, if by a different arrangement of parts they can be made to coincide.
The Equality of Angles has been already defined in (8).
22. ADDITION, SUBTRACTION, &c. of LINES, AREAS, and ANGLES.
It follows from (21) that lines, areas, and angles, may be added together, subtracted from each other, multiplied, or divided, like other magnitudes. Thus c
if AB, CD be two straight lines equal to one another, 'produce CD indefinitely towards D, then by applying AB to it so that A is upon D, and AB upon DE, we find DE equal to AB, and .. CE is plainly twice AB. Again, if EF- AB, then CF = three times AB, and so on. And thus we multiply the line AB. Obviously also CD is one-third of the line CF; that is, a line may be divided. Again, that lines may be added or subtracted is plain enough, for AB + CD=CD+DE = CE; and AB taken from CE = CD.
The same principle, viz. that 'magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another', leads to the conclusion that, in like manner areas and angles may be added, subtracted, multiplied, or divided.
Thus, for instance, if AB, BD be in the same straight line, so that ABC, BCD, ACD are three distinct triangles, it is plain that the two areas ABC, BCD, exactly cover the same space as the area ACD, and .. the two areas ABC, BCD may be added together, and their sum will be the area, or triangle, ACD. Similarly, if the area BCD be subtracted from the area ACD, the difference is the area ABC.
Again, if area ABC= area BCD, then area ACD is double of the area ABC; and area ABC=half of area ACD. Angles likewise are magnitudes which may be added, subtracted, &c. Thus, ACB + ‹ BCD= ‹ ÅCD. And 4 BCD taken from ACD leaves 4 ACB. Also if
▲ ACB = 4 BCD, then ‹ ACD is double of ▲ ACB.
QUESTIONS ON THE PRECEDING DEFINITIONS, &c.
(1) What does Geometry treat of? To what properties of bodies is it restricted?
(2) Define a 'line'; can it be exhibited in practice? If not, why not?
(3) How many different kinds of lines are there? Give an example of each.
(4) Define a point'. Can it be exhibited to the eye? If not, why not? Give an example of a 'point'.
(5) Define 'superficies', 'surface', or 'area'. Give an example.
(6) How many kinds of 'surfaces' are there? Give an example of each. By what general rule are they distinguished from each other?
(7) What is meant by the line AB'? Is it the same as the line BA?
(8) What is an 'Angle'? Is it a magnitude admitting of increase or decrease? Exhibit two angles; and say which is the greater, and why.
(9) What is meant by the angle ABC'? Is it the same as 'the angle CBA'? Is it the same as 'the angle ACB'?
(10) Define a right angle, and exhibit it. Can one right angle be greater than another right angle? What is the way of determining whether one angle is greater than another?
(11) What are the names by which certain angles are distinguished? Exhibit an angle of each sort.
(12) Explain clearly the difference between the angle ABC and the triangle ABC.
(13) By what names are triangles distinguished according to their form? Exhibit a triangle of each sort.
(14) Does the magnitude of an angle depend upon the magnitude of the lines by which it is formed?
(15) How many lines are necessary to form an angle? How many to form a triangle?
(16) How many angles are there in a triangle? Does the magnitude of a triangle depend upon the magnitude of the lines which form its three angles?
(17) Does the word triangle mean 'three angles' in such a sense as to signify that the triangle is made up of the three angles, so as to be equal to them?
(18) Define parallel straight lines; and give an example.
If a straight line were drawn on the ceiling, and another on the floor, these two lines being produced ever so far both ways would never meet. Would they necessarily be parallel? Does the definition exclude such?
(19) How many kinds of parallelograms are there? What is the distinctive character of all, and of each? Exhibit each separately, and fully describe it.
(20) How many letters are used to denote a particular parallelogram, and where are they placed? Give an example.
(21) What is meant by the 'base' of a parallelogram? (22) Define a 'circle'; and explain clearly the difference between a circle and the circumference of a circle.
(23) How many letters are required to denote an arc of a circle? Why will not two serve, as in the case of a straight line? Where are the letters placed?
(24) What is the object of Euclid's three Postulates? (25) Upon what axiom does the Equality of geometrical magnitudes depend?
(26) Can one angle be equal to two other angles, or to three? Explain clearly.
(27) Is it possible for a triangle to be equal to a square? If so, say how.
(28) How many angles are there in a parallelogram? Is the parallelogram equal to the sum of its angles? (29) Is a semi-circle a line or an area? (30) Is an angle an area? If so, how do stand the statement at the end of (9) page 5?
(31) Can one triangle be added to another? If triangles be added together will the resulting sum necessarily be a triangle?
(32) Is a triangle equal to the sum of its three sides?
(33) What is the difference between an angle, and a corner? What is the geometrical name for the latter?
EXPLANATION OF TECHNICAL TERMS USED
(1) To 'describe' a certain geometrical figure, means to construct, or trace, it on a plane surface, as a board or sheet of paper.
(2) A 'given' line means a line 'given' sometimes in position, sometimes in magnitude, sometimes in both, according to circumstances; and the word 'given' means fixed or known.
(3) A proposition' is something proposed to be done; so that the heading of each separate article in the following section may be called a proposition. Sometimes 'propositions' are distinguished into two kinds; they are called problems, when something is proposed to be constructed or made; and they are called theorems, when some proposed statement is required to be proved.
(4) Corollary' signifies an after-conclusion beyond what is due, following obviously without any or much further proof from what has been already done or proved.
(5) à fortiori means by so much the more'. Thus, if A, B, C represent three geometrical magnitudes, and we know that A is greater than B, having proved that B is greater than C, we conclude, à fortiori, that A is greater than C.
(6) The 'converse' of a proposition is when the conclusion is turned into an assumption, and the previous assumption is made the conclusion. Thus to the proposition "The angles at the base of an isosceles triangle are equal to one another" the converse would be "Shew that, if the angles at the base of a triangle are equal to one another, the triangle is isosceles".
(7) reductio ad absurdum', (reducing to an absurdity), is a particular mode of demonstration often used by Euclid. It may be briefly explained by the following case:-Required to shew that two geometrical magnitudes, represented by A and B, are equal to one another. We argue thus. If A is not equal to B, then A and B must be unequal. Suppose them unequal, and proceeding upon this assumption we arrive, by means of acknowledged axioms and legitimate reasoning, at an absurd conclusion, such as, for instance, that a portion of a magnitude is greater than the whole. If then the supposition that A and B are unequal legitimately leads to such a conclusion, it is plain that that supposition cannot stand; and therefore the only alternative is that A = B.
(8) To produce' a given straight line is to continue or extend it, so that the part added may be in one and the same straight line with the given line. Thus a radius of a circle, continued through the centre to meet the circumference again, until it becomes a diameter, is said to be produced.
(9) Anaxiom' is a statement of an admitted truth, so plain and unquestionable as to need no demonstration, as long as words mean what they do; as that, for instance, "the whole of any magnitude is greater than a part of the same magnitude"—or, again, that 'two is greater than one'. Such truths do not specially belong to Geometry, but are practically interwoven with almost every operation of daily life.