which is used in Book VI. The 7th, 8th and 9th books are purely 8. It is next proved that two triangles which have the ikzee sides arithmetical, whilst the roth contains a most ingenious treatment of the one equal respectively to those of the other are identically equal, of geometrical irrational quantities. These four books will be other, those being equal which are opposile equal sides. This is Prop. 8. excluded from our survey. The remaining three books relate to Prop. 7 containing only a first step towards its proof. figures in space, or, as it is generally called, to “solid geometry.” These theorems allow now of the solution of a number of probe The 7th, 8th, 9th, roth, 13th and part of the 11th and 12th lems, viz. : To bisect a given angle (Prop. 9). books are now generally omitted from the school editions of the To bisecl a given finite straight line (Prop. 10). Elements. In the first four and in the 6th book it is to be under To draw a straight line perpendicularly to a given straight line stood that all figures are drawn in a plane. through a given point in il (Prop. 11), and also through a given point not in it (Prop. 12). Book I. OF EUCLID'S “ ELEMENTS." The solutions all depend upon properties of isosceles triangles. $ 6. According to the third postulate it is possible to draw in $9. The next three theorems relate to angles only, and might have any plane a circle which has its centre at any given point, and its been proved before Prop. 4, or even at the very beginning. The radius equal to the distance of this point from any other point first (Prop. 13) says, The angles which one straight line makes with given in the plane. This makes it possible (Prop. 1) to construct another straight line on one side of il either are two right angles or on a given line AB an equilateral triangle, by drawing first a circle are together equal to two righl angles. This theorem would have with Å as centre and AB as radius, and then a circle with B as been unnecessary, if Euclid had admitted the notion of an angle centre and BA as radius. The point where these circles intersect, such that its two limits are in the same straight line, and had besides that they intersect Euclid quietly assumes--is the vertex of the defined the sum of two angles. , required triangle. Euclid does not suppose, however, that a circle Its converse (Prop. 14) is of great use, inasmuch as it enables us may be drawn which has its radius equal to the distance betwecn in many cases to prove that two straight lines drawn from the same any two points unless one of the points be the centre. This implies point are one the continuation of the other. So also is also that we are not supposed to be able to make any straight line Prop. 15. If two straight lines cut one another, the vertical or opposite equal to any other straight line, or to carry a distance about in space. angles shall be equal. Euclid therefore nakt solves the problem: It is required along a 10. Euclid returns now to properties of triangles. Of great given straight line from a point in it to set off a distance equal to importance for the next steps (though alterwards superseded by a the length of another straight line given anywhere in the plane. more complete theorem) is This is done in two steps. It is shown in Prop. 2 how a straight line Prop. 16. If one side of a triangle be produced, the exterior angle may be drawn from a given point equal in length to another given shall be greater than eilher of the interior opposile angles., straight line not drawn from that point. And then the problem Prop. 17. Any two angles of a triangle are together less than two itseli is solved in Prop. 3, by drawing first through the given point right angles, is an immediate consequence of it: By the aid of these some straight line of the required length, and then about the same two, the following fundamental properties of triangles are easily point as centre a circle having this length as radius. This circle proved: will cut off from the given straight line a length equal to the required Prop. 18. The greater side of every triangle has the greater angle one. Nowadays, instead of going through this long process, we opposite to it; take a pair of compasses and set off the given length by its aid. Its converse, Prop. 19. The greater angle of every triangle is sube This assumes that we may move a length about without changing it. tended by the greater side, or has the greater side opposite to it; But Euclid has not assumed it, and this proceeding would be fully Prop. 20. Any two sides of e triangle are together greater than the justified by his desire not to take for granted more than was necessary, third side; if he were not obliged at his very next step actually to make this And also Prop. 21. If from the ends of the side of a triangle there assumption, though without stating it. be drason two straight lines lo e poin! within the triangle, these shall 8.7. We now come (in Prop: 4) to the first theorem.. It is the be less than the other two sides of the triangle, bul shall contain a grealer fundamental theorem of Euclid's whole system, there being only a angle. very few propositions (like Props. 13, 14, 15, 1.), except those in the $11. Having solved two problems (Props. 22, 23), he returns to two şth book and the first half of ihe uith, which do not depend upon triangles which have two sides of the one equal respectively to two it. It is stated very accurately, though somewhat clumsily, as sides of the other. It is known (Prop. 4) that if the included angles follows: are equal then the third sides are equal; and conversely (Prop. 8), If two triangles have two sides of the one equal to two sides of the if the third sides are equal, then the angles included by the first other, cach to each, and have also the angles contained by those sides sides are equal. From this it follows that if the included angles are equal to one another, they shall also have their bases or third sides not equal, the third sides are not equal; and conversely, that if the equal; and the two triangles shall be equal; and their other angles third sides are not equal, the included angles are not equal. Euclid shall be equal, each to each, namely, those to which the equal sides are now completes this knowledge by proving, that "it the included opposite. angles are not equal, then the third side in that triangle is the greater That is to say, the triangles are identically" equal, and one which contains the greater angle"; and conversely, that if the third may be considered as a copy of the other. The proof is very simple. sides are unequal, that triangle contains the greater angle which contains The first triangle is taken up and placed on the second, so that the the greater side. These are Prop. 24 and Prop. 25. parts of the triangles which are known to be equal fall upon each $ 12. The next theorem (Prop. 26) says that if two triangles have other. It is then easily seen that also the remaining parts of one one side and two angles of the one equal respectively lo one side and coincide with those of the other, and that they are therefore equal. two angles of the other, vis. in both triangles either the angles adjacent This process of applying one figure to another Euclid scarcely uses lo the equal side, or one angle adjacent and one angle opposite it, then again, though many proofs would be simplified by doing so. The the two triangles are identically equal. process introduces motion into geometry, and includes, as already, This theorem belongs to a group with Prop. 4 and Prop. 8. Its stated, the axiom that figures may be moved without change of first case might have been given immediately after Prop. 4, but the shape or size. second case requires Prop. 16 for its proof. If the last proposition be applied to an isosceles triangle, which § 13. We come now to the investigation of parallel straight lines, has two sides equal, we obtain the theorem (Prop. 5), tf two sides i.e. of straight lines which lie in the same plane, and cannot be made of a triangle are equal, then the angles opposite these sides are equal. to meet however far they be produced either way. The investigation Euclid's proof is somewhat complicated, and a stumbling-block which starts from Prop. 16, will become clearer if a few names be to many schoolboys. The proof becomes much simpler if we consider explained which are not all used by Euclid. If two straight lines the isosceles triangle ABC (AB = AC) twice over, once as a triangle be cut by a third, the latter is now generally called a "transversal " BAC, and once as a triangle CAB; and now remember that AB, AC of the figure. It forms at the two points where it cuts the given lines in the first are equal respectively to AC, AB in the second, and the four angles with each. Those of the angles which lie between the angles included by these sides are equal. Hence the triangles are given lines are called interior angles, and of these, again, any two equal, and the angles in the one are equal to those in the other, viz. which lie on opposite sides of the transversal but one at each of the those which are opposite equal sides, i.e. angle ABC in the first two points are called " alternate angles." equals angle ACB in the second, as they are opposite the equal We may now state Prop. 16 thus:-1 two straight lines which sides AC and AB in the two triangles. meet are cut by a transversal, their alternate angles are unequal. For There follows the converse theorem (Prop. 6). If two angles in the lines will form a triangle, and one of the alternate angles will a triangle are equal, then the sides opposite them are equal,--.e. the be an exterior angle to the triangle, the other interior and opposite triangle is isosceles. The proof given consists in what is called a reductio ad absurdum, a kind of proof often used by Euclid, and From this follows at once the theorem contained in Prop., 27. principally in proving the converse of a previous theorem. It If two straight lines which are cut by a transversal make allernale assumes that the theorem to be proved is wrong, and then shows angles equal, the lines connot meet, however far they be produced, that this assumption leads to an absurdity, i.e. to a conclusion hence they are parallel. This proves the existence of parallel which is in contradiction to a proposition proved before-that lines. therefore the assumption made cannot be true, and hence that Prop: 28 states the same fact in different forms. If a straigh! the theorem is true. It is often stated that Euclid invented this line, folling on two other straight lines, make the exterior angle equal kind of proof, but the method is most likely much older. to the interior and opposite angle on the same side of the line, or make to it. - the interior angles on the same side together equal to two right angles, the same or on equal bases, in the same straight line, and on the same the two straighi lines shall be parallel to one another. side of it, are between the same parallels. Hence we know that," is two straight lines which are cut by a That the two cases here stated are given by Euclid in two separate transversal meet, their alternate angles are not equal "; and hence propositions proved separately is characteristic of his method. that, " is alternate angles are equal, then the lines are parallel." $ 18. To compare areas of other figures, Euclid shows first, in The question now arises, Are the propositions converse to these Prop. 42, how to draw a parallelogram which is equal in area lo a true or not? That is to say. If alternate angles are unequal, do given triangle, and has one of its angles equal to a given angle. If the the lines meet?" And it'the lines are parallel, are alternate given angle is right, then the problem is solved to draw a "rectangle angles necessarily equal ?" equal in area to a given triangle. The answer to either of these two questions implies the answer Next this parallelogram is transformed into another parallelogram, to the other. But it has been found impossible to prove that the which has one of its sudes equal to a given straight line, whilst its anglos negation or the affirmation of ei is true. remain unaltered. This may be done by aid of the theorem in The difficulty which thus arises is overcome by Euclid assuming Prop. 43. The complements of the parallelograms which are about that the first question has to be answered in the affirmative. This the diameter of any parallelogram are equal to one another. gives his last axiom (12), which we quote in his own words. Thus the problem (Prop. 44), is solved to construct a parallelogram Axiom 12.-If a straight line meet two straight lines, so as to make on a given line, which is equal in area to a given triangle, and which the two interior angles on the some side of it taken together less than has one angle equal to a given angle (generally a right angle). kvo right angles, these straight lines, being continually produced, shall As every polygon can be divided into a number of triangles, we at length meet on thal side on which are the angles which are less than can now construct a parallelogram having a given angle, say a two right angles. right angle, and being equal in arca to a given polygon. For each The answer to the second of the above questions follows from this, of the triangles into which the polygon has been divided, a paralleloand gives the theorem Prop. 29:- If a straight line fall on two parallel gram may be constructed, having one side equal to a given straight straight lines, it makes the alternate angles equal to one another, and line and one angle equal to a given angle. If these parallelograms the exterior angle equal to the interior and opposite angle on the same be placed side by side, they may be added together to form a single side, and also the two interior angles on the same side together equal parallelogram, having still one side of the given length. This is to troo right angles. done in Prop. 45. 14. With this a new part of elementary geometry begins. The Herewith a means is found to compare areas of different polygons. earlier propositions are independent of this axiom, and would be We need only construct two rectangles cqual in area to the given true even if a wrong, assumption had been made in it. Th all polygons, and having each one side of given length. By comparing relate to figures in a plane. But a plane is only one among an infinite the unequal sides we are enabled to judge whether the areas are number of conceivable surfaces. We may draw figures on any one equal, or which is the greater. Euclid does not state this consequence, of them and study their properties. We may, for instance, take a but the problem is taken up again at the end of the second book, sphere instead of the plane, and obtain a spherical" in the place of where it is shown how to construct a square equal in area to a given plane stereometry. "If on one of these surfaces lines and figures polygon. 46 is: To describe a square on a given straight line. could be drawn, answering to all the definitions of our plane figures, and if the axioms with the exception of the last all hold, then all $19. The first book concludes with one of the most important propositions up to the 28th will be true for these figures. This is theorems in the whole of geometry, and one which has been cele the case in spherical geometry if we substitute " shortest line" or brated since the earliest times. It is stated, but on doubtful authority, great circle" for "straight line," "small circle" for "circle," and that Pythagoras discovered it, and it has been called by his name. if, besides, we limit all figures to a part of the sphere which is less !! we call that side in a right-angled triangle which is opposite the than a hemisphere, so that two points on it cannot be opposite ends right angle the hypotenuse, we may state it as follows: of a diameter, and therefore determine always one and only one great Theorem of Pythagoras (Prop. 47). In every right-angled triangle circle. the square on the hypotenuse js equal to the sum of the squares of ihe For spherical triangles, therefore, all the important propositions other sides. 4. 8. 26; 5 and 6; and 18, 19 and 20 will hold good. And conversely This remark will be sufficient to show the impossibility of proving Prop. 48. If the square described on one of the sides of a triangle be Euclid's last axiom, which would mean proving that this axiom is equal to the squares described on the other sides, then the angle contained a consequence of the others, and hence that the theory of parallels by these two sides is a right angle, would hold on a spherical surface, where the other axioms do hold, On this theorom (Prop. 47) almost all geometrical measurement whilst parallels do not even exist. depends, which cannot be directly obtained. It follows that the axiom in question states an inherent difference between the plane and other surfaces, and that the plane is only BOOK II. fully characterized when this axiom is added to the other assump § 20. The propositions in the second book are very different in tions. character from those in the first; they all relate to areas of rectangles is. The introduction of the new axiom and of parallel lines leads and squares. Their true significance is best seen by stating them in to a new class of propositions. an algebraic form. This is often done by expressing the lengths of After proving (Prop. 30) that "two lines which are each parallel lines by aid of numbers, which tell how many times a chosen unit to a third are parallel to each other," we obtain the new properties is contained in the lines. If there is a unit to be found which is con of triangles contained in Prop. 32. of these the second part is the tained an exact number of times in each side of a rectangle, it is most important, viz. the theorem, The three interior angles of every easily seen, and generally shown in the teaching of arithmetic, that triangle are together equal to two righe angles. the rectangle contains a number of unit squares equal to the product As easy deductions not given by Euclid but added by Simson of the numbers which measure the sides, a unit square being the follow the propositions about the angles in polygons; they are given square on the unit line. If, however, no such unit can be found, in English editions as corollaries to Prop. 32. this process requires that connexion between lines and numbers These theorems do not hold for spherical figures. The sum of the which is only established by aid of ratios of lines, and which is thereinterior angles of a spherical triangle is always greater than two fore at this stage altogether inadmissible. But there exists another right angles, and increases with the area. way of connecting these propositions with algebra, based on modern 16. The theory of parallels as such may be said to be finished notions which seem destined greatly to change and to simplify with Props. 33 and 34, which state properties of the parallelogram, mathematics. We shall introduce here as much of it as is required j.e. of a quadrilateral formed by two pairs of parallels. They are for our present purpose. Prop. 33. The straight lines which join the extremities of two equal At the beginning of the second book we find a definition according and parallel straight lines towards the same parls are themselves equal to which "a rectangle is said to be contained by the two sides and parallel; and which contain one of its right angles "; in the text this phraseology Prop. 34. The opposile sides and angles of a paralldogram are is extended by speaking of rectangles contained by any two straight equal to one another, and the diameter (diagonal) bisects the parallelo- lines, meaning the rectangle which has two adjacent sides equal to gram, that is, divides it into two equal parts. the two straight lines. 17. The rest of the first book relates to areas of figures. We shall denote a finite straight line by a single small letter, The theory is made to depend upon the theorems a, b, c,... x, and the area of the rectangle contained by two lines Prop. 35. Parallelograms on the same base and between the same a and 6 by ab, and this we shall call the product of the two lines a. parallels are equal to one another; and and b. It will be understood that this definition has nothing to do Prop. 36. Parallelograms on equal bases and between the same with the definition of a product of numbers. parallels are equal to one another. We define as follows: As each parallelogram is bisected by a diagonal, the last theorems The sum of two straight lines a and 8 means a straight line e which hold also if the word parallelogram be replaced by " triangle," as is may be divided in two parts equal respectively to a and b. This sum done in Props. 37 and 38. is devoted by a+b. It is to be remarked that Euclid proves these propositions only que difference of two lines a and 6 (in symbols, a-b) means a line in the case when the parallelograms or triangles have their bases in c which when added to b gives a; that is, the same straight line. The theorems converse to the last form the contents of the next a-b-c if 6+c= a. three propositions, viz.: Props. 40 and 41.--Equal triangles, on The product of two lines a and 6 (in symbols, 4b) means the area 1 of the rectangle contained by the lines a and b. For aa, which Prop. 7, which is an easy consequence of Prop. 4, may be transmeans the square on the line a, we write a'. formed. If we denote by ĉ the line a+b, so that $ 21. The first ten of the fourteen propositions of the second book c=a+b, d=c- b, may then be written in the form of formulae as follows: we get Prop. 1. 2(6+c+d+;: .)-ab+actadt ... C+(c- b) = 2cc-b)+ b 2. ab toc=ai if b+c=2. = 20 - 2bc+. 3. a(Q+b)=e' tab. Subtracting from both sides, and writing a for c, we get 4. (a+b) = a* + ab +84. (2-6)*=0:- 2ab +b. 5. (a+b) (a-b) +6= a. In Euclid's Elements this form of the theorem does not appear, 8. (a+b) la-6)+620. 7. 9+(a - b) = 2a(2-0) +67. all propositions being so stated that the notion of subtraction does not enter into them. $24. The remaining two theorems (Props. 12 and 13) connect 1o. (a+b)*+la-bjø=20° +26. 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 It will be seen that 5 and 6, and also 9 and 10, are identical. In obtuse. They are important theorems in trigonometry, where it is Euclid's statement they do not look the same, the figures being possible to include them in a single theorem. arranged differently. $ 25. There are in the second book two problems, Props. II and 14. If the letters a, b, c, ... denoted numbers, it follows from algebra If written in the above symbolic language, the former requires to that each of these formulae is true. But this does not prove them in find a line x such that ala-x)=x*. Prop. 11 contains, therefore, our case, where the letters denote lines, and their products areas the solution of a quadratic equation, which we may writer' tax=a* without any reference to numbers. To prove them we have to The solution is required later on in the construction of a regular discover the laws which rule the operations introduced, viz. addition decagon. and multiplication of segments. This we shall do now; and we shall More important is the problem in the last proposition (Prop. 14). find that these laws are the same with those which hold in algebraical It requires the construction of a square equal in area to a given addition and multiplication. rectangle, hence a solution of the equation 22. In a sum of numbers we may change the order in which ** = ab. 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 structed equal in area to a given figure bounded by straight lines. In Book I., 42-45, it has been shown howa rectangle may be conSum of segments and for the sum of rectangles, as a little considera- By aid of the new proposition we may therefore now determine a tion shows. That the sum of rectangles has always a meaning line such that the square on that line is equal in area to any given follows from the Props. 43-45 in the first book. These laws about rectilinear figure, or we can square any such figure. addition are reducible to the two As of two squares that is the greater which has the greater side, a+b=bta (1), it follows that now the comparison of two areas has been reduced a+b+c)=2+6+c : to the comparison of two lines. or, when expressed for rectangles, The problem of reducing other areas to squares is frequently met with among Greek mathematicians. We need only mention the abted=cd tab (3), problem of squaring the circle (see CIRCLE). ab+(cd ten) =abtod tej : In the present day, the comparison of areas is performed in a The brackets mean that the terms in the bracket have been added simpler way by reducing all areas to rectangles having a common together before they are added to another term. The more general base. Their altitudes give then a measure of their areas. cases for more terms may be deduced from the above. The construction of a rectangle having the base u, and being equal For the product of two numbers we have the law that it remains in area to a given rectangle, depends upon Prop. 43, 1. This therefore unaltered if the factors be interchanged. This also holds for our gives a solution of the equation 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 where x denotes the unknown altitude. 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 Book III. base, and then the other will be the altitude. This gives $26. The third book of the Elements relates exclusively to proab- be (5). perties of the circle. A circle and its circumference have been defined In order further to multiply a sum by a number, we have in algebrain Book I., Def. 15. We restate it here in slightly different words: the rule :--Multiply each term of the sum, and add the products Definition.--The circumference of a circle is a plane curve such thus obtained. That this holds for our geometrical products is that all points in it have the same distance from a fixed point in shown by Euclid in his first proposition of the second book, where the plane. This point is called the "centre" of the circle. he proves that the area of a rectangle whose base is the sum of a of the new definitions, of which eleven are given at the beginning number of segments is equal to the sum of rectangles which have of the third book, a few only require special mention. The first these segments separately as bases. In symbols this gives, in the which says that circles with equal radii are equal, is in part a theorem, simplest case, but easily proved by applying the one circle to the other. Or it c(b+c) =ab+ac ? and may be considered proved by aid of Prop. 24, equal circles not being (6). (+c)a=ba +ca) used till after this theorem. To these laws, which have been investigated by Sir William Hamilton In the second definition is explained what is meant by a line and by Hermann Grassmann, the former has given special names. which “ touches" a circle. Such a line is now generally called a He calls the laws expressed in tangent to the circle. The introduction of this name allows us to (1) and (3) the commutative law for addition; state many of Euclid's propositions in a much ter form. For the same reason we shall call a straight line joining two points (2) and the associative laws for addition; on the circumference of a ci-le a “chord. (6) the distributive law. Definitions 4 and 5 may Le replaced with a slight generalization 23. Having proved that these six laws hold, we can at once by the following:--. prove every one of the above propositions in their algebraical form, length of the perpendicular drawn from the point to the line. Definition.-By the distance of a point from a line is meant the The first is proved geometrically, it being one of the fundamental laws. The next two propositions are only special cases of the first. has a centre. Prop: 1 requires to find it when the circle is given, $ 27. From the definition of a circle it follows that every circie of the others we shall prove one, viz. the fourth : i.e. when its circumference is drawn. (a+b)*= (a+b) (a+b) = (a+b)a+(a+b)6 by (6) To solve this problem a chord is drawn (that is, any two points in But (a+b)a=0a+ba by (6), the circumference are joined), and through the point where this is -aatab by (s); bisected a perpendicular to it is erected. Euclid then proves, first, and (a+b) by (6) that no point of this perpendicular can be the centre, hence that the Therefore (a+b):= ca+ab+(ab+bb) centre must lie in this line; and, secondly, that of the points on the =ca+(ab tab) +bb by (4). perpendicular one only can be the centre, viz. the one which bisects 10+2ab+bb the parts of the perpendicular bounded by the circle. In the second This gives the theorem in question. part Euclid silently assumes that the perpendicular there used does In the same manner every one of the first ten propositionis therefore is incomplete. The proof of the first part, however, is cut the circumference in two, and only in two points. The proof proved. It will be seen that the operations performed are exactly the same which bisect them, the centre will be found as the point where these exact. By drawing two non-parallel chords, and the perpendiculars as if the letters denoted numbers. perpendiculars intersect. Props. 5 and 6 may also be written thus 28. In Prop. 2 it is proved that a chord of a circle lies altogether (a+b) (a - b) - 1-6. within the circle. ab=ux, What we have called the first part of Euclid's solution of Prop. 1 two kinds of angles exists the important relation expressed as may be stated as a theorem follows: Every straight line which bisects a chord, and is al right angles to it, Prop. 20. The angle at the centre of a circle is double of the angle passes through the centre of the circle. at the circumference on the same base, that is, on the same arc. The converse to this gives Prop. 3. which may be stated thus: This is of great importance for its consequences, of which the If a straight line through the centre of a circlé bisect a chord, then two following are the principal:it is perpendicular to the chord, and if ii be perpendicular to the chord Prop. 21. The angles in the same segment of a circle are equal to it bisects il. one another; An easy consequence of this is the following theorem, which is Prop. 22. The opposite ongles of any quadrilateral figure inscribed essentially the same as Prop. 43 in a circle are together equal to two right angles. Two chords of a circle, of which neither passes through the centre, Further consequences are: cannol bisect each other. Prop. 23. On the same straight line, and on the same side of it, there These last three theorems are fundamental for the theory of the cannoi be two similar segments of circles, not coinciding with one circle. It is to be remarked that Euclid never proves that a straight another; line cannot have more than two points in common with a circum Prop. 24. Similar segments of circles on equal-straight lines are ference. equal to one another. $ 29. The next two propositions (5 and 6) might be replaced by The problem Prop. 25. A segment of a circle being given to describe a single and a simpler theorem, viz: the circle of which it is a segment, may be solved much more easily Two circles which have a common centre, and whose circumferences by aid of the construction described in relation to Prop. 1, II., have one point in common, coincide. in $ 27. Or, more in agreement with Euclid's form: $ 34: There follow four theorems connecting the angles at the Two different circles, whose circumferences have a point in common, centre, the arcs into which they divide the circumference, and the cannot have the same centre. chords subtending these arcs. They are expressed for angles, arcs That Euclid treats of two cases is characteristic of Greek mathe- and chords in equal çircles, but they hold also for angles, arcs and matics. chords in the same circle. The next two propositions (7 and 8) again belong together. They The theorems are: may be combined thus: Prop. 26. In equal circles equal angles stand on equal ercs, whether If from a point in a plane of a circle, which is not the centre, straight they be at the centres or circumferences; lines be drawn to the differeni points of the circumference, then of all Prop. 27. (converse to Prop. 26). In equal circles the angles which these lines one is the shortest, and one the longest, and these lie both in stand on equal arts are equal to one another, whether they be at the that straight line which joins the given point to the centre. Of centres or the circumferences; remaining lines each is equal to one and only one other, and these Prop. 28. In equal circles equal straight lines (equal chords) cul equal lines lie on opposite sides of the shortest or longest, and make off equal arcs, the greater equal to the greater, and the less equal to equal angles with them. the less; Euclid distinguishes the two cases where the given point lies within or without the circle, omitting the case where it lies in the circum- subtended by equal straight lines. Prop: 39 (converse to Prop. 28). In equal circles equal ores are ference. $35. Other important consequences of Props. 20-22 are:From the last proposition it follows that is from a point more Prop. 31: In a circle the angle in a semicircle is a right angle; than two equal straight lines can be drawn to the circumference, but the angle in a segment greater than a semicircle is less than a righi this point must be the centre. This is Prop. 9. angle; and the angle in a segment less than a semicircle is greater than As a consequence of this we get a right angle; If the circumferences of the two circles have three points in common Prop. 32. 'If a straight line touch a circle, and from the point of they coincide. contact a straight line be drawn cutting the circle, the angles which For in this case the two circles have a common centre, because this line makes with the line touching the circle shall be equal to the from the centre of the one three equal lines can be drawn to points angles which are in the alternate segments of the circle. on the circumference of the other. But two circles which have a $36. Propositions 30, 33, 34, contain problems which are solved common centre, and whose circumferences have a point in common, by aid of the propositions preceding them: coincide. (Compare above statement of Props. 5 and 6.) Prop. 30. To bisect a given, arc, ihal is, lo divide il into two equal This theorem may also be stated thus: parts; Through three points only one circumference may be drawn; or, Prop. 3'. On a given straight line to describe a segment of a circle Three points determine a circle. containing an angle equal to a given rectilineal angle; Euclid does not give the theorem in this form. He proves, how Prop. 34. From a given circle to cut off a segment containing an ever, that the two circes cannot cut another in more than two points ongle equal to a given rectilineal angle, (Prop. 10), and that two circles cannol louch one another in more points 37. If we draw chords through a point A within a circle, they than one (Prov. 13). will each be divided by A into two segments. Between these seg, § 30. Propositions in and 12 assert that if two circles touch, then ments the law holds that the rectangle contained by them has the the point of contact lies on the line joining their centres. This gives same area on whatever chord through A the segments are taken. two propositions, because the circles may touch either internally The value of this rectangle changes, of course, with the position or externally. of A. $ 31. Propositions 14 and 15 relate to the length of chords. The A similar theorem holds if the point A be taken without the circle. first says that equal chords are equidistant from the centre, and that on every straight line through A, which cuts the circle in two points chords which are equidistant from the centre are equal; B and C, we have two segments AB and AC, and the rectangles Whilst Prop. 15 compares unequal chords, viz. Of all chords the contained by them are again equal to one another, and equal to the diameter is the greatest, and of other chords that is the greater which square on a tangent drawn from A to the circle. is nearer to the centre; and conversely, the greater chord is nearer lo The first of these theorems gives Prop: 35, and the second Prop. the centre. 36, with its corollary, whilst Prop. 37, the last of Book III., gives $32. In Prop. 16 the tangent to a circle is for the first time in the converse to Prop. 36. The first two theorems may be combined troduced. The proposition is meant to show that the straight line in one: at the end point of the diameter and at right angles to it is a tangent. If through a point A in the plane of a circle a straight line be drawn The proposition itself does not state this. It runs thus: calling the circle in B and C, ihen the rectangle AB.AC has a constant Prop. 16. The straight line drawn at right angles to the diameter value so long as the point A be fixed; and if from A a tangent AD can of a circle, from the extremity of it, falls without the circle; and no be drawn to the circle, touching at D, then the above rectangle equals the straight line can be drawn from the extremity, between thai straight square on AD. line and the circumfen zce, so as not to cut the circle. Prop. 37 may be stated thus: Corollary.--The straight line at right angles to a diameter drawn If from a point A without a circle a line be drawn culling the circle through the end point of it touches the circle. in B and C, and another line lo a point D on the circle, and AB.AC= The statement of the proposition and its whole treatment show AD?, then the line AD touches the circle al D. the difficulties which the tangents presented to Euclid. It is not difficult to prove also the converse to the general proProp, 17 solves the problem through a given point, either in the position as above stated. This proposition and its converse may be circumference or without it, to draw a tangent to a given circle. expressed as follows: Closely connected with Prop. 16 are Props. 18 and 19, which T} four points ABCD be taken on the circumference of a circle, and state (Prop. 18), that the line joining the centre of a circle to the point if the lines AB, CD, produced if necessary, meel al E, then of contact of a tangent is perpendicular to the tangent; and conversely:(Prop. 19), that the straight line through the point of contact EA.EB = EC.ED; of, and perpendicular lo, a tangeni to a circle passes through the centre and conversely, if this relation holds then the four points lie on a circle, of the circle. that is, the circle drown through three of them passes through the 33. The rest of the book relates to angles connected with a fourth. circle, viz. angles which have the vertex either at the centre or That a circle may always be drawn through three points, provided on the circumference, and which are called respectively angles that they do not lie in a straight line, is proved only later on in at the centre and angles at the circumference. ' Between these Book JV Book IV. two right angles, as all three angles together equal two right angles Thus we have to construct an isosceles triangle, having the angle at $ 38. The fourth book contains only problems, all relating to the vertex equal to half an angle at the base. This is solved in the construction of triangles and polygons inscribed in and circum. Prop. 10, by aid of the problem in Prop. 11 of' the second book. II scribed about circles, and of circles inscribed in or circumscribed we make the sides of this triangle equal to the radius of the given about triangles and polygons. They are nearly all given for their circle, then the base will be the side of the regular decagon inscribed own sake, and not for future use in the construction of figures, as in the circle. This side being known the decagon can be constructed, are most of those in the former books. In seven definitions at the and if the vertices are joined alternately, leaving out half their beginning of the book it is explained what is understood by figures number, we obtain the regular pentagon. (Prop. 11.) inscribed in or described about other figures, with special reference Euclid does not proceed thus. He wants the pentagon before to the case where one figure is a circle. Instead, however, of saying the decagon. This, however, does not change the real nature of that one figure is described about another, it is now generally said his solution, nor does his solution become simpler by not mentioning that the one figure is circumscribed about the other. We may then the decagon. state the definitions 3 or 4 thus: Once the regular pentagon is inscribed, it is easy to circumscribe Definition.-A polygon is said to be inscribed in a circle, and the another by drawing tangents at the vertices of the inscribed pentagon. circle is said to be circumscribed about the polygon, if the vertices This is shown in Prop. 12. of the polygon lie in the circumference of the circle. Props. 13 and 14 teach how a circle may be inscribed in or cirAnd definitions 5 and 6 thus: cumscribed about any given regular pentagon. Definition.- A polygon is said to be circumscribed about a circle, 44. The regular hexagon is more easily constructed, as shown and a circle is said to be inscribed in a polygon, if the sides of the in Prop. 15. The result is that the side of the regular hexagon polygon are tangents to the circle. inscribed in a circle is equal to the radius of the circle. $ 39. The first problem is merely constructive. It requires to For this polygon the other three problems mentioned are not draw in a given circle a chord equal to a given straight line, which solved. is not greater than the diameter of the circle. The problem is not 45. The book closes with Prop. 16. To inscribe a regular a determinate one, inasmuch as the chord may be drawn from any quindecagon in a given circle. If we inscribe a regular pentagon point in the circumference. This may be said of almost all problems and a regular hexagon in the circle, having one vertex in common, in this book, especially of the next two. They are: then the arc from the common vertex to the next vertex of the Prop. 2. In a given circle to inscribe a triangle cquiangular lo a pentagon is fth of the circumference, and to the next vertex of the given iriangle: hexagon is Ith of the circumference. The difference between these Prop. 3. About a given circle lo circumscribe a triangle equiangular arcs is, therefore, 1-1 - 4th of the circumference. The latter may. a given gle. therefore, be divided into thirty, and hence also in fifteen equal parts, $ 40. Of somewhat greater interest are the next problems, where and the regular quindecagon be described. the triangles are given and the circles to be found. $ 46. We conclude with a few theorems about regular polygons Prop. 4. To inscribe a circle in a given triangle. which are not given by Euclid. The result is that the problem has always a solution, viz. the The stroighe lines perpendicular lo and bisecting the sides of any centre of the circle is the point where the bisectors of two of the regular polygon meet in a point. The straight lines bisecting the angles interior angles of the triangle meet. The solution shows, though in the regular polygon meel in the same point. This point is the centre Euclid does not state this, that the problem has but one solution; of the circles circumscribed about and inscribed in the regular polygon. and also, We can bisect any given arc (Prop. 30. III.). Hence we can divide The three bisectors of Ike interior angles of any triangle meel in a a circumference into an equal parts as soon as it has been divided point, and this is the centre of the circle inscribed in the triangle. into n equal parts, or as soon as a regular polygon of n sides has been The solutions of most of the other problems contain also theorems. constructed. Henceof these we shall state those which are of special interest; Euclid Il e regular polygon of n sides has been constructed, then a regular does not state any one of them. polygon of an șides, of 4n, of 'n sides, &c., may also be construcied. $41. Prop. 5. To circumscribe a circle about a given triangle. Euclid shows how to construct regular polygons of 3. 4. 5 and is The one solution which always exists contains the following: sides. It follows that we can construct regular polygons of The three straight lines which bisect the sides of a triangle of right 3, 6, 12, 24... sides engles meel in a point, and this point is the centre of the circle circum 4. 8, 16, scribed about the triangle. 5, 10, 20, 40.... Euclid adds in a corollary the following property 15, 30, 60, 120..... The centre of the circle circumscribed about a triangle lies within, on a side of, or without the triangle, according as the triangle is The construction of any new regular polygon not included in one acute-angled, right-angled or obtuse-angled. of these series will give rise to a new series. Till the beginning of the $ 42. Whilst it is always possible to draw a circle which is inscribed 19th century nothing was added to the knowledge of regular polygons in or circumscribed about a given triangle, this is not the case with as given by Euclid. Then Gauss, in his celebrated Arithmetic, quadrilaterals or polygons of more sides of thosc for which this proved that every regular polygon of 2" + 1 sides may be constructed If this number 2*+i be prime, and that no others except those is possible the regular polygons, i.e. polygons which have all their with 2 (2"+1) sides can be constructed by elementary methods. sides and angles equal, are the most interesting. In cach of them a circle may be inscribed, and another may be circumscribed about it. This shows that regular polygons of 7.9, 13 sides cannot thus be Euclid does not use the word regular, but he describes the polygons constructed, but that a regular polygon of 17 sides is possible; for in question as equiongular and equilateral. We shall use the name 17 = 2*+1. The next polygon is one of 257 sides. The construction regular polygon. The regular triangle is equilateral, the regular becomes already rather complicated for i7 sides. quadrilateral is the square. Euclid considers the regular polygons of 4, 5, 6 and 15 sides. Book V.. For each of the first three he solves the problems-(1) to inscribe $47. The fifth book of the Elements is not exclusively geometrical. such a polygon in a given circle; (2) to circumscribe it about a It contains the theory of ratios and proportion of quantities in given circle; (3) to inscribe a circle in, and (4) to circumscribe a general. The treatment, as here given. is admirable, and in every circle about, such a polygon. respect superior to the algebraical method by which Euclid's theory. For the regular triangle the problems are not repeated, because is now generally replaced. We shall treat the subject in order to more general problems have been solved. show why the usual algebraical treatment of proportion is not really Props. 6. 7. 8 and 9 solve these problems for the square. sound. We begin by quoting those definitions at the beginning of The general problem of inscribing in a given circle a regular Book V. which are most important. These definitions have given polygon of n sides depends upon the problem of dividing the cir. rise to much discussion. cumference of a circle into n equal parts, or what comes to the same The only definitions which are essential for the fifth book are thing, of drawing from the centre of the circle r radii such that the Đefs. 1, 2, 4, 5, 6 and 7. Of the remainder 3. 8 and 9 are more angles between consecutive radii are equal, that is, to divide the than useless, and probably not Euclid's, but additions of later editors. space about the centre into equal angles. Thus, if it is required of whom Theon of Alexandria was the most prominent. Dess. 10 to inscribe a square in a circle, we have to draw four lines fram the and r1 belong rather to the sixth book, whilst all the others are centre, making the four angles equal. This is done by drawing merely nominal. The really important ones are 4, 5, 6 and 7. two diameters at right angles to one another. The ends of these $ 48. To define a magnitude is not attempted by Fuclid. The diameters are the vertices of the required square. If, on the other fast two definitions state what is meant by a “part." that is, a hand, tangents be drawn at these ends, we obtain a square circum- submultiple or mcasure, and by a " multiple" of a given magniscribed about the circle. tude. The meaning of Def. 4 is that two given quantities can have 43. To construct a regular pentagon, we find it convenient first a ratio to one another only in case that they are comparable as to to construct a regular decagon. This requires to divide the space their magnitude, that is, if they are of the same kind. about the centre into ten equal angles. Each will be rth of a right Def. 3. which is probably due to Thcon, professes to define a ratio, angle, or {th of two right angles. If we suppose the decagon con but is as meaningless as it is uncalled for, for all that is wanted is structed, and if we join the centre to the end of one side, we get an given in Dels. 5 and 7: isosceles triangle, where the angle at the centre equals fth of two In Def. 5 it is explained what is meant by saying that two magsight angles; hence each of the angles at the base will be Iths of nitudes have the same ratio to one another as two other magnitudes. 32:.. |