BOOK II.
§ 20. The propositions in the second book are very different in character from those in the first; they all relate to areas of rectangles and squares. Their true significance is best seen by stating them in an algebraic form. This is often done by expressing the lengths of lines by aid of numbers, which tell how many times a chosen unit is contained in the lines. If there is a unit to be found which is contained an exact number of times in each side of a rectangle, it is easily seen, and generally shown in the teaching of arithmetic, that the rectangle contains a number of unit squares equal to the product of the numbers which measure the sides, a unit square being. the square on the unit line. If, however, no such unit can be found, this process requires that connexion between lines and numbers which is only established by aid of ratios of lines, and which is therefore at this stage altogether inadmissible. But there exists another way of connecting these propositions with algebra, based on modern notions which seem destined greatly to change and to simplify mathematics. We shall introduce here as much of it as is required for our present purpose.
At the beginning of the second book we find a definition according to which " a rectangle is said to be ' contained ' by the two sides which contain one of its right angles "; in the text this phraseology is extended by speaking of rectangles contained by any two straight lines, meaning the rectangle which has two adjacent sides equal to the two straight lines.
We shall denote a finite straight line by a single small letter, a, b, c, . x, and the area of the rectangle contained by.two lines a and b by ab, and this we shall call the product of the two lines a and b. It will be understood that this definition has nothing to do with the definition of a product of numbers.
We define as follows:
The sum of two straight lines a and b means a straight line c which may be divided in two parts equal respectively to a and b. This sum is denoted by a+b.
The difference of two lines a and b (in symbols, a–b) means a line c which when added to b gives a; that is,
a–b=c if b+c=a.
The product of two lines a and b (in symbols, ab) means the area
of the rectangle contained by the lines a and b. For aa, which means the square on the line a, we write a2.
§ 21. The first ten of the fourteen propositions of the second book may then be written in the form of formulae as follows:
i. a(b+c+d+ . . .)=ab+ac+ad+ ..
2. ab+ac=a2 if b+c=a.
3. a(a+b) =a2+ab.
4. (a+b)2=a2+2ab+b2.
5. (a+b) (a–b)+b2=a2.
6. (a+b) (a–b)+b2=a2.
7. a2+(a–b)2=2a(a– b) +P.
8. 4(a+b)a+b2= (2a+b)2.
9. (a+b)2+(a – b)2=2a2+2b2. To. (a+b)2+(a – b)2=2a2+2b2.
It will be seen that 5 and 6, and also 9 and To, are identical. In Euclid's statement they do not look the same, the figures being arranged differently.
If the letters a, b, c, . denoted numbers, it follows from algebra that each of these formulae is true. But this does not prove them in our case, where the letters denote lines, and their products areas without any reference to numbers. To prove them we have to discover the laws which rule the operations introduced, viz. addition and multiplication of segments. This we shall do now; and we shall find that these laws are the same with those which hold in algebraical addition and multiplication.
§ 22. In a sum of numbers we may change the order in which the numbers are added, and we may also add the numbers together in groups and then add these groups. But this also holds for the sum of segments and for the sum of rectangles, axa little consideration shows. That the sum of rectangles has always a meaning follows from the Props. 4345 in the first book. These laws about addition are reducible to the two
a+b=b+a (I),
a+(b+c) =a+b+c . . (2);
or, when expressed for rectangles,
ab+ed=ed+ab . (3),
ab+(cd+ef) =ab+cd+ef . . (4).
The brackets mean that the terms in the bracket have been added together before they are added to another term. The more general cases for more terms may be deduced from the above.
For the product of two numbers we have the law that it remains unaltered if the factors be interchanged. This also holds for our geometrical product. For if ab denotes the area of the rectangle which has a as base and b as altitude, then ba will denote the area of the rectangle which has b as base and a as altitude. But in a rectangle we may take either of the two lines which contain it as base, and then the other will be the altitude. This gives
ab=ba . . (5).
In order further to multiply a sum by a number, we have in algebra the rule:—Multiply each term of the sum, and add the products thus obtained. That this holds for our geometrical products is shown by Euclid in his first proposition of the second book, where he proves that the area of a rectangle whose base is the sum of a number of segments is equal to the sum of rectangles which have these segments separately as bases. In symbols this gives, in the simplest case,
a(b+c) =ab+ac (6)
and (b+c)a=ba+ca '
To these laws, which have been investigated by Sir William Hamilton and by Hermann Grassmann, the former has given special names. He calls the laws expressed in
(I) and (3) the commutative law for addition;
(5) ,, multiplication;
(2) and (4) the associative laws for addition;
(6) the distributive law.
§ 23. Having proved that these six laws hold, we can at once prove every one of the above propositions in their algebraical form.
The first is proved geometrically, it being one of the fundamental laws. The next two propositions are only special cases of the first. Of the others we shall prove one, viz. the fourth :
(a+b)2= (a+b) (a+b) = (a+b)a+(a+b)b by (6).
(a+b)a=aa+ba by (6),
=aa+ab by (5);
and (a+b)b=ab+bb by (6).
Therefore (a+b)2 = as+ab+(ab+bb) )
=aa+(ab+ab)+bb
r by (4)r =aa+2ab+bb
)
This gives the theorem in question.
In the same manner every one of the first ten propositions is proved.
It will be seen that the operations performed are exactly the same as if the letters denoted numbers.
Props. 5 and 6 may also be written thus
(a+b) (a–b)=a=–b2.Prop. 7, which is an easy consequence of Prop. 4, may be transformed. If we denote by c the line a+b, so that c=a+b, a=c–b,
c2+(c – b)2=2c(c – b)+b2
=2c2 – 2bc+b2.
Subtracting c2 from both sides, and writing a for c, we get (a – b)2 =a2– 2ab+b2.
In Euclid's Elements this form' of the theorem does not appear, all propositions being so stated that the notion of subtraction does not enter into them.
§ 24. The remaining two theorems (Props. I2 and 13) connect the square on one side of a triangle with the sum of the squares on the other sides, in case that the angle between the latter is acute or obtuse. They are important theorems in trigonometry, where it is possible to include them in a single theorem.
§ 25. There are in the second book two problems, Props. 11 and 14.
If written in the above symbolic language, the former requires to find a line x such that a(a–x) =x2. Prop. 11 contains, therefore, the solution of a quadratic equation, which we may write x2+ax =a2. The solution is required later on in the construction of a regular decagon.
More important is the problem in the last proposition (Prop. 14). It requires the construction of a square equal in area to a given rectangle, hence a solution of the equation
x2 = ab.
In Book I., 4245, it has been shown howa rectangle may be constructed equal in area to a given figure bounded by straight lines. By aid of the new proposition we may therefore now determine a line such that the square on that line is equal in area to any given rectilinear_ figure, or we can square any such figure.
As of two squares that is the greater which has the greater side, it follows that now the comparison of two areas has been reduced to the comparison of two lines.
The problem•of reducing other areas to squares is frequently met with among Greek mathematicians. We need only mention the problem of squaring the circle (see CIRCLE).
In the present day the comparison of areas is performed in a simpler way by reducing all areas to rectangles having a common base. Their altitudes give then a measure of their areas.
The construction of a rectangle having the base u, and being equal in area to a given rectangle, depends upon Prop. 43, I. This therefore gives a solution of the equation
ab =ux,
where x denotes the unknown altitude.
BooK III.
§ 26. The third book of the Elements relates exclusively to properties of the circle. A circle and its circumference have been defined in Book I., Def. 15. We restate it here in slightly different words:
Definition.—The circumference of a circle is a plane curve such that all points in it have the same distance from a fixed point in the plane. This point is called the " centre " of the circle.
Of the new definitions, of which eleven are given at the beginning of the third book, a few only require special mention. The first,, which says that circles with equal radii are equal, is in part a theorem, but easily proved by applying the one circle to the other. Or it may be considered proved by aid of Prop. 24, equal circles not being used till after this theorem.
In the second definition is explained what is meant by a line which " touches " a circle. Such a line is now generally called a tangent to the circle. The introduction of this name allows us to state many of Euclid's propositions in a much shorter form.
For the same reason we shall call a straight line joining two points on the circumference of a circle a " chord.”
Definitions 4 and 5 may be replaced with a slight generalization by the following:
Definition.—By the distance of a point from a line is meant the length of the perpendicular drawn from the point to the line.
§ 27. From the definition of a circle it follows that every circle has a centre. Prop. 1 requires to find it when the circle is given, i.e. when its circumference is drawn.
To solve this problem a chord is drawn (that is, any two points in the circumference are joined), and through the point where this is bisected a perpendicular to it is erected. Euclid then proves, first, that no point off this perpendicular can be the centre, hence that the centre must lie in this line; and, secondly, that of the points on the perpendicular one only can be the centre, viz. the one which bisects the parts of the perpendicular bounded by the circle. In the second part Euclid silently assumes that the perpendicular there used does cut the circumference in two, and only in two points. The proof therefore is incomplete. The proof of the first part, however, is exact. By drawing two nonparallel chords, and the perpendiculars which bisect them, the centre will be found as the point where these perpendiculars intersect.
§ 28. In Prop. 2 it is proved that a chord of a circle lies altogether within the circle.
Prop.
But
we get
What we have called the first part of Euclid's solution of Prop. t may be stated as a theorem:
Every straight line which bisects a chord, and is at right angles to it, passes through the centre of the circle.
The converse to this gives Prop. 3, which may be stated thus
If a straight line through the centre of a circle bisect a chord, then it is perpendicular to the chord, and if it be perpendicular to the chord it bisects it.
An easy consequence of this is the following theorem, which is essentially the same as Prop. 4:
Two chords of a circle, of which neither passes through the centre, cannot bisect each other.
These last three theorems are fundamental for the theory of the circle. It is to be remarked that Euclid never proves that a straight line cannot have more than two points in common with a circumference.
§ 29. The next two propositions (5 and 6) might be replaced by a single and a simpler theorem, viz:
Two circles which have a common centre, and whose circumferences have one point in common, coincide.
Or, more in agreement with Euclid's form:
Two different circles, whose circumferences have a point in common, cannot have the same centre.
That Euclid treats of two cases is characteristic of Greek mathematics.
The next two propositions (7 and 8) again belong together. They may be combined thus:
If from a point in a plane of a circle, which is not the centre, straight lines be drawn to the different points of the circumference, then of all these lines one is the shortest, and one the longest, and these lie both in that straight line which joins the given point to the centre. Of all the remaining lines each is equal to one and only one other, and these equal lines lie on opposite sides of the shortest or longest, and make equal angles with them.
Euclid distinguishes the two cases where the given point lies within or without the circle, omitting the case where it lies in the circumference.
From the last proposition it follows that if from a point more than two equal straight lines can be drawn to the circumference, this point must be the centre. This is Prop. 9.
As a consequence of this we get
If the circumferences of the two circles have three points in common they coincide.
For in this case the two circles have a common centre, because from the centre of the one three equal lines can be drawn to points on the circumference of the other. But two circles which have a common centre, and whose circumferences have a point in common, coincide. (Compare above statement of Props. 5 and 6.)
This theorem may also be stated thus:
Through three points only one circumference may be drawn; or, Three points determine a circle.
Euclid does not give the theorem in this form. He proves, however, that the two circles cannot cut another in more than two points (Prop. to), and that two circles cannot touch one another in more points than one (Prop. 13).
§ 30. Propositions 11 and 12 assert that if two circles touch, then the point of contact lies on the line joining their centres. This gives two propositions, because the circles may touch either internally or externally.
§ 31. Propositions 14 and 15 relate to the length of chords. The first says that equal chords are equidistant from the centre, and that chords which are equidistant from the centre are equal;
Whilst Prop. 15 compares unequal chords, viz. Of all chords the diameter is the greatest, and of other chords that is the greater which is nearer to the centre; and conversely, the greater chord is nearer to the centre.
§ 32. In Prop. I6 the tangent to a circle is for the first time introduced. The proposition is meant to show that the straight line at the end point of the diameter and at right angles to it is a tangent. The proposition itself does not state this. It runs thus:
Prop. 16. The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle; and no straight line can be drawn from the extremity, between that straight line and the circumference, so as not to cut the circle.
Cbrollary.—The straight line at right angles to a diameter drawn through the end point of it touches the circle.
The statement of the proposition and its whole treatment show the difficulties which the tangents presented to Euclid.
Prop. 17 solves the problem through a given point, either in the circumference or without it, to draw a tangent to a given circle.
Closely connected with Prop. 16 are Props. 18 and 19, which' state (Prop. 18), that the line joining the centre of a circle to the point 'of contact of a tangent is perpendicular to the tangent; and conversely (Prop. 19), that the straight line through the point of contact of, and perpendicular to, a tangent to a circle passes through the centre of the circle.
§ 33. The rest of the book relates to angles connected with a circle, viz. angles which have the vertex either at the centre or on the circumference, and which are called respectively angles at the centre and angles at the circumference. Between thesetwo kinds of angles exists the important relation expressed as follows :
Prop. no. The angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same arc.
This is of great importance for its consequences, of which the two following are the principal:
Prop. 21. The angles in the same segment of a circle are equal to one another;
Prop. 22. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.
Further consequences are:
Prop. 23. On the same straight line, and on the same side of it, there cannot be two similar segments of circles, not coinciding with one another;
Prop. 24. Similar segments of circles on equal straight lines are equal to one another.
The problem Prop. 25. A segment of a circle being given to describe the circle of which it is a segment, may be solved much more easily by aid of the construction described in relation to Prop. t, III., in § 27.
§ 34• There follow four theorems connecting the angles at the centre, the arcs into which they divide the circumference, and the chords subtending these arcs. They are expressed for angles, arcs and chords in equal circles, but they hold also for angles, arcs and chords in the same circle.
The theorems are:
Prop. 26. In equal circles equal angles stand on equal arcs, whether they be at the centres or circumferences;
Prop. 27. (converse to Prop. 26). In equal circles the angles which stand on equal arcs are equal to one another, whether they be at the centres or the circumferences;
Prop. 28. In equal circles equal straight lines (equal chords) cut off equal arcs, the greater equal to the greater, and the less equal to the less;
Prop. 29 (converse to Prop. 28). In equal circles equal arcs are subtended by equal straight lines.
§ 35. Other important consequences of Props. 2022 are:
Prop. 31. In a circle the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle;
Prop. 32. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles which this line makes with the line touching the circle shall be equal to the angles which are in the alternate segments of the circle.
36. Propositions 30, 33, 34, contain problems which are solved by aid of the propositions preceding them:
Prop. 30. To bisect a given arc, that is, to divide it into two equal parts ;
Prop. 33. On a given straight line to describe a segment of a circle containing an angle equal to a given rectilineal angle;
Prop. 34.. From a given circle to cut off a segment containing an an le equal to a given rectilineal angle.
37. If we draw chords through a point A within a circle, they will each be divided by A into two segments. Between these segments the law holds that the rectangle contained by them has the same area on whatever chord through A the segments are taken. The value of this rectangle changes, of course, with the position of A.
A similar theorem holds if the point A be taken without the circle. On every straight line through A, which cuts the circle in two points B and C, we have two segments AB and AC, and the rectangles contained by them are again equal to one another, and equal to the square on a tangent drawn from A to the circle.
The first of these theorems gives Prop. 35, and the second Prop. 36, with its corollary, whilst Prop. 37, the last of Book III., gives the converse to Prop. 36. The first two theorems may be combined in one:
If through a point A in the plane of a circle a straight line be drawn cutting the circle in B and C, then the rectangle AB.AC has a constant value so long as the point A be fixed; and if from A a tangent AD can be drawn to the circle, touching at D, then the above rectangle equals the square on AD.
Prop. 37 may be stated thus:
If from a point A without a circle a line be drawn cutting the circle in B and C, and another line to a point D on the circle, and AB.AC= AD2, then the line AD touches the circle at D.
It is not difficult to prove also the converse to the general proposition as above stated. This proposition and its converse may be expressed as follows:
If four points ABCD be taken on the circumference of a circle, and if the lines AB, CD, produced if necessary, meet at E, then
EA.EB =EC.ED;
and conversely, if this relation holds then the four points lie on a circle, that is, the circle drawn through three of them passes through the fourth.
That a circle may always be drawn through three points, provided that they do not lie in a straight line, is proved only later on in Book IV.
End of Article: BOOK II 

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